UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO POSGRADO EN CIENCIAS FÍSICAS INSTITUTO DE CIENCIAS NUCLEARES FUNDAMENTOS TEÓRICO-MATEMÁTICOS Y APLICACIONES DE LA NO-CONMUTATIVIDAD A LA FÍSICA TESIS QUE PARA OPTAR POR EL GRADO DE: DOCTOR EN CIENCIAS (FÍSICA) PRESENTA: LUIS ROMÁN JUÁREZ SANDOVAL TUTOR DOCTOR MARCOS ROSENBAUM PITLUCK INSTITUTO DE CIENCIAS NUCLEARES MIEMBROS DEL COMITÉ TUTOR DOCTOR JOSÉ DAVID VERGARA OLIVER INSTITUTO DE CIENCIAS NUCLEARES DOCTOR ANTONMARIA MINZONI ALESSIO IIMAS MÉXICO, D. F., AGOSTO 2014 UNAM – Dirección General de Bibliotecas Tesis Digitales Restricciones de uso DERECHOS RESERVADOS © PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL Todo el material contenido en esta tesis esta protegido por la Ley Federal del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). El uso de imágenes, fragmentos de videos, y demás material que sea objeto de protección de los derechos de autor, será exclusivamente para fines educativos e informativos y deberá citar la fuente donde la obtuvo mencionando el autor o autores. Cualquier uso distinto como el lucro, reproducción, edición o modificación, será perseguido y sancionado por el respectivo titular de los Derechos de Autor. Dedicado cariñosamente a mis padres Blanca Sandoval Barragán Moctezuma y Jesús Juárez Flores, quienes siempre reconocieron mí curiosidad por la ciencia y avivaron esa llama en cada oportunidad que se presentó. Mi más profundo anhelo es honrar el amor y el esfuerzo que ustedes incondicionalmente imprimieron durante mi formación. Resumen Se conduce un estudio formal sobre No-conmutatividad, vista como la incorporación de conmuta- dores no triviales de observables de posición mecánico-cuánticos. Las herramientas y estructuras matemáticas utilizadas se enfocan en los formalismos de cuantización por deformación y elemen- tos de geometŕıa no-conmutativa. El análisis de la Mecánica Cuántica No-conmutativa conduce a resultados importantes sobre modificaciones en la interpretación probabiĺıstica a partir de una función de cuasiprobabilidad y las ecuaciones de valores-⋆ correspondientes. Se muestra como el análisis de operadores de Heisenberg permite identificar un mecanismo dinámico para el origen del producto-⋆ de Teoŕıa de Campos No-conmutativa. Recurriendo a esquemas de cuantización axiomáticos como el de Stratonovich-Weyl y el de Berezin y al uso de estados coherentes no-conmutativos, se establece la equivalencia entre las realizaciones holomorfas del producto-⋆ de Mecánica Cuántica No-conmutativa. Dentro de los métodos de cuantización estándar se elabora un programa de cuantización canónica para generar conmutadores no triviales de operadores de posición, partiendo de acciones clásicas con constricciones. Aśı mismo se deriva la integral de trayectoria asociada a cada una de las construcciones previas y se realiza un análisis comparativo. Invocando nociones de la Geometŕıa No-conmutativa de Connes, se presenta una estructura matemática novedosa para introducir el concepto de no-conmutatividad, recurriendo a elementos de álgebras C∗ y considerando una representación torcida del grupo topológico discreto de trasla- ciones en R 3, como álgebra C∗ elemental del triple espectral. Esta formulación es implementada en el estudio del colapso cuántico de una cosmoloǵıa anisotrópica de Bianchi I. A partir de la dinámica efectiva se muestra, asintótica y numéricamente, como la no-conmutatividad induce un comportamiento oscilatorio del volúmen de la cosmoloǵıa, y que la constricción Hamiltoniana implica la ausencia de singularidades en las variables dinámicas en el régimen no-conmutativo. iii Abstract A formal study of Noncommutativity is conducted, seen as the introduction of nontrivial com- mutators between the quantum observables of position. The tools and mathematical structures used herein deal with the formalisms of deformation quantization and elements of noncommuta- tive geometry. The analysis of Noncommutative Quantum Mechanics leads to important results regarding the modification of the probabilistic interpretation that stems from the quasiprobabil- ity function and the corresponding ⋆-value equations. It is shown how the analysis of Heisenberg operators allows to identify a dynamical mechanism for the origin of the ⋆-product in Noncom- mutative Field Theory. After recurring to axiomatic quantization schemes such as Stratonovich-Weyl and Berezin’s and the use of noncommutative coherent states, an equivalence can be established among the holomorphic realizations of the ⋆-product associated to Noncommutative Quantum Mechanics. Within the standard methods of quantization a canonical quantization programme is elaborated to generate nontrivial commutators of position operators, from classical actions with constraints. By the same token the path integral is obtained for each one of the previous developments followed by a comparative analysis. By invoking notions from Connes’ Noncommutative Geometry, a novel mathematical struc- ture is presented in order to introduce the concept of noncommutativity, taking elements from C∗-algebras and making use of a twisted representation of the discrete topological group of trans- lations in R 3, as the elementary C∗-algebra of the spectral triple. This formulation is then used in the study of the quantum collapse of an anistropic Bianchi I cosmology. From the effective dynamics it is shown, asymptotically and numerically, how the noncommutativity induces an oscillatory behavior of the volume of the cosmology, and that the Hamiltonian constraint implies the absence of singularities in the dynamical variables in the noncommutative regime. v Agradecimientos ¡En buen tiempo vinimos a vivir, hemos venido en tiempo primaveral! ¡Instante brev́ısimo, oh amigos! ¡Aún aśı tan breve, que se viva! Nezahualcoyotzin Mi eterno agradecimiento a mi mentor, el Dr. Marcos Rosenbaum, sin cuyo apoyo este trabajo no habŕıa sido posible. La paciencia que mostró conmigo es equiparable a la que un padre muestra con un hijo y tanto aśı es el afecto y admiración que le tengo. Ha sido un honor y un placer trabajar todo este tiempo en compañ́ıa de su brillante y tan inspirador intelecto, al cual sólo sueño alcanzar. Sus enseñanzas, consejos, largas y memorables conversaciones, aśı como llamadas de atención han trascendido a tantos niveles que seŕıa injusto no mencionar su impacto en la vida de las personas que me rodean y quienes también se lo agradecen. Sağolun a la gentil Sra. Rosenbaum por su abrumadora calidad humana, por recibirme siempre en su hermosa casa aún si fuese de forma intempestiva y por los deliciosos platillos. También hago extensivas estas palabras de gratitud a la bella familia formada por ambos, su hija Tamara y esposo y a la preciosa nieta, quienes en cada ocasión me han hecho sentir como en casa. A los Dres. David Vergara y Antonmaria Minzoni les agradezco profundamente su cont́ınuo apoyo y gúıa durante estos años de trabajo. Estoy en deuda con el Dr. Vergara por todas sus observaciones y recomendaciones durante la redacción de tesis y por la labor titánica de haber leido diligentemente las n + 1 versiones plagadas de errores, espero honestamente haber renormalizado al menos algunos de tantos. Muchas gracias a los miembros del Jurado por transmitirme todas sus inquietudes y valiosos comentarios sobre la Tésis, permitiendo mejorar significativamente la calidad en el resultado final de este manuscrito. vii ix Quiero también mencionar al Dr. Hernando Quevedo, quien no sólo es un cient́ıfico ejemplar sino también un asombroso ser humano, por sus finas atenciones para conmigo y de las cuales me considero desmerecedor, aśı como por sus sabias palabras de aliento en los momentos precisos. Un muy sincero agradecimiento a Trini Ramı́rez por su siempre agradable compañ́ıa, por compartir su cultura excepcional sobre temas tan diversos conmigo, por tolerar mis conversa- ciones y diatribas intrascendentes y principalmente por su amistad incondicional. A mis hermanos mosqueteros y amigos entrañables David, Pablo y León, con quienes la vida dentro y fuera de la F́ısica no ha sido más que sublime. Es dif́ıcil creer que han pasado tantos años, que el cabello comienza a cambiar de color, que páıses han ido y venido, pero que nuestra amistad conmute prodigiosamente con el Hamiltoniano y esté tan intacta como durante los primeros años y sea, a mi parecer, aún más genuina. Sepan que estoy agradecido con ustedes por una miŕıada de razones y que gran parte de la persona que hoy soy, aśı como la clase de F́ısico a la que aspiro ser, es debido al ejemplo permanente que han dejado grabado en mi memoria. Finalmente agradezco el amor infinito, el apoyo incansable y la confianza inquebrantable de mi seres queridos. A mi Mamá cuya dedicación y cariño han constituido los pilares de la familia. Los valores y principios que inculcaste en mis hermanos y en mı́ han sido la fuerza motriz y el recordatorio constante de aspirar a una vida de provecho, sin pretenciones vanales y sobre todo de no renunciar a nuestras metas. Aprovecho estas ĺıneas para expresar el gran orgullo que significa ser tu hijo, por mantener cerca y vivas nuestras ráıces y que siempre haré el mejor esfuerzo por cumplir las expectativas que tienes de mı́. Gracias Papá, por introducirme a temas de interés cient́ıfico desde mis más tempranos destellos de conciencia, en gran medida la culpa de esto es tuya. Nadie más que un poĺımata como tú sabŕıa curar una fiebre infantil mayor de 40 ◦C construyendo motores eléctricos, de no ser por que yo fúı el paciente ese pasaje me pareceŕıa propio de un libro fantástico. De t́ı aprend́ı el amor y el respeto por la profesión. Tlazohkamati por los años de trabajo al lado de ustedes dos en lugares mágicos y remotos, fué un precioso regalo que cambió definitivamente mi percepción del mundo y de mı́ mismo. A Omar y Blanca, mis adorados hermanos que me han tráıdo enorme alegŕıa de la forma que sólo ustedes saben hacerlo les agradezco una vida colmada de su compañ́ıa. Son mis mejores amigos, mis cómplices y mis más estrictos cŕıticos, y han moldeado mi carácter a su forma presente. Gracias May y Simon por cuidar tan bien de estas valiosas personas en mi vida y, en el proceso, tratarme como a un hermano. Gracias a todos. Índice Índice xi 1 Introducción 1 1.1 Motivación . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Marco histórico. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Organización de la Tesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 I Conceptos preliminares 11 2 El esquema de cuantización WWGM 13 2.1 Operadores de desplazamiento . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Análisis de Fourier y equivalentes de Weyl . . . . . . . . . . . . . . . . . . . . . . 16 2.3 El producto de Groenewold-Moyal y el espacio C∞⋆ (R2n) . . . . . . . . . . . . . . 18 2.4 El principio de correspondencia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Variaciones sobre un tema de Moyal 25 3.1 Estados Coherentes Generalizados . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Correspondencia de Stratonovich–Weyl . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Śımbolos covariantes de Berezin–Weyl . . . . . . . . . . . . . . . . . . . . . . . . 32 xi xii Índice II No-conmutatividad y Geometŕıa en Mecánica Cuántica y Campos 37 4 Espacio No-conmutativo 39 4.1 Álgebra extendida de Heisenberg-Weyl hθ5 . . . . . . . . . . . . . . . . . . . . . . 40 4.2 El espacio de funciones A∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Interpretación probabiĺıstica y ecuaciones de valores-⋆ . . . . . . . . . . . . . . . 48 4.4 Operadores de Heisenberg y paréntesis de Poisson: El paso a Teoŕıa de Campos . 54 5 Representaciones alternas de la No-conmutatividad 57 5.1 Estados coherentes no-conmutativos . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Realización holomorfa del producto–∗: La equivalencia del cuantizador y el oper- ador de reflexión . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Invariancia de Reparametrización, Cuantización Canónica y No-conmutatividad . 66 5.4 Formulación de la integral de trayectoria . . . . . . . . . . . . . . . . . . . . . . . 70 5.4.1 Amplitud de transición como la traza de operadores . . . . . . . . . . . . 71 5.4.2 Acción semiclásica no-canónica . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4.3 Integral de trayectoria con estados coherentes de hθ2n+1 . . . . . . . . . . . 74 5.5 El Cálculo Espectral de Connes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 III La No-conmutatividad en el régimen Planckiano de la Cosmoloǵıa 81 6 Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I 83 6.1 Una realización de operadores mas allá del Teorema de Stone-von Neumann . . . 84 6.2 Construcción GNS y observables f́ısicos . . . . . . . . . . . . . . . . . . . . . . . 87 6.3 Cuantización del modelo cosmológico de Bianchi tipo I . . . . . . . . . . . . . . . 92 6.4 Integral de trayectoria y acción semiclásica . . . . . . . . . . . . . . . . . . . . . 95 6.5 Análisis dinámico en el régimen de fase estacionaria e interpretación de εi y µi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Índice xiii 6.6 Resultados Numéricos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Discusión, Conclusiones y Ĺıneas de Investigación Futuras 111 IV Apéndices 117 A Material complementario del formalismo WWGM 119 A.1 Propiedades algebraicas e integrales del producto ⋆~. . . . . . . . . . . . . . . . . 119 A.2 Esquema de Heisenberg en el formalismo WWGM. . . . . . . . . . . . . . . . . . 125 A.3 Valores de expectación y la función de Wigner-Szilard. . . . . . . . . . . . . . . . 127 B Invariancia de simetŕıa torcida 131 B.1 Torcedura de un álgebra de Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 B.2 Simetŕıa, deformación y torcedura de Drinfeld . . . . . . . . . . . . . . . . . . . . 131 B.3 Torcedura de Drinfeld Fθ e invariancia . . . . . . . . . . . . . . . . . . . . . . . . 133 C Cosmoloǵıa anisotrópica de Bianchi I 135 Bibliograf́ıa 139 V Art́ıculos de investigación 147 Dynamical origin of the ⋆θ- noncommutativity in field theory from quantum mechanics 149 Noncommutativity from Canonical and Noncanonical Structures 167 Canonical Quantization, Space-Time Noncommutativity and Deformed Symme- tries in Field Theory 186 Noncommutative Field Theory from Quantum Mechanical Space-Space Noncom- mutativity 202 xiv Índice Space-Time Diffeomorphisms in Noncommutative Gauge Theories 215 Lattice vortices induced by noncommutativity 236 Noncommutativity and Parametrization of Fields: The Scalar Electrodynamics Case 248 On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group 277 A Twisted C⋆ - algebra formulation of Quantum Cosmology with application to the Bianchi I model 299 Noncommutative Coherent States and Quantum Cosmology 331 Caṕıtulo 1 Introducción ”It’s beautiful isn’t it?” ... ... All throughout geometry, all throughout physics, the same idea shows up in a thousand different guises. How do you carry something from here to there, and keep it the same? You move it step by step, keeping it parallel in the only way that makes sense. You climb Schild’s ladder.” –GREG EGAN, Schild’s Ladder, 2003. 1.1 Motivación En el contexto de la filosof́ıa reduccionista de la F́ısica, ha habido gran éxito en construir teoŕıas de unificación que describen una vasta cantidad de fenómenos con el menor número de concep- tos. La primera Teoŕıa de unificación fue la Mecánica Clásica de Isaac Newton, contenida en sus Principia de finales del siglo XVII (1687), que dio una explicación mecanicista a todos los fenómenos conocidos hasta principios del siglo XIX. La segunda gran teoŕıa de unificación surgió en la segunda mitad del siglo XIX (1864) con la descripcióin del electromagnetismo mediante las ecuaciones de James C. Maxwell y, que a su vez, corresponde a la primera teoŕıa de campo, pro- porcionando a la interacción electromagnética de una velocidad finita de propagación de señales (la velocidad de la luz). Durante las primeras décadas del siglo XX nuevos fenómenos en la f́ısica fueron descubiertos, a saber: a) Las interacciones nucleares fuertes, asociadas con el núcleo atómico; b) Las inter- acciones nucleares débiles, responsables de los procesos de decaimiento-β. Sin embargo no fue sino hasta la segunda mitad del siglo XX que se logró interpretar el ”zoológico” de part́ıculas subatómicas, conocidas hasta entonces, en términos del modelo de Quarks. Igualmente la in- teraccióin débil, modelada como una teoŕıa de Yang-Mills con grupo de simetŕıa SU(2), pudo 1 2 Caṕıtulo 1. Introducción ser exitosamente unificada con la teoŕıa electromagnética, con grupo de simetŕıa U(1), en una teoŕıa Electrodébil. El producto de estos desarrollos es el Modelo Estándar de la F́ısica de part́ıculas, basado en la descripción de la materia por medio de una teoŕıa cuántica de campos invariante bajo el grupo de simetŕıa SU(3)×SU(2)×U(1). Esta teoŕıa es uno de los pilares de la F́ısica moderna y también es un ejemplo de lo que por definición debe ser capaz una teoŕıa de unificación. De igual forma, los trabajos de Relatividad Especial y General de Albert Einstein de prin- cipios del siglo XX (1905 y ∼1915 respectivamente), proporcionaron a la F́ısica de una teoŕıa unificadora, mediante una descripción elegante en el lenguaje de la geometŕıa, para fenómenos del mundo macroscópico, la Gravitación y el Espacio-Tiempo. Este formalismo ha permitido describir y predecir fenómenos astrof́ısicos con gran precisión, el cual en el ĺımite de ”bajas velocidades” recobra el modelo construido por Newton varios siglos atrás. Los preceptos de Relatividad Especial son perfectamente compatibles con el formalismo de la Teoŕıa Cuántica de Campos, siendo esto particularmente cierto en la descripción matemática de las interacciones fuerte y electrodébil mencionadas antes. Sin embargo, dada la naturaleza geométrica de Relatividad General como el modelo matemático de la Gravitación, la unificación de ésta última con las demás interacciones fundamentales no ha sido posible hasta ahora. Aunque a escalas atómicas los efectos del campo gravitacional resultan despreciables por varios órdenes de magnitud en los cálculos de espectros de enerǵıas, se cree que al incrementar la escala de enerǵıa (ó disminuir la escala de distancia) a ordenes tales como los que tuvieron lugar en los primeros instantes del universo después del Big Bang, todas las interacciones fun- damentales juegan un rol esencial y los efectos de cada una son lo suficientemente grandes como para no poder ser descartados; aśı es que una formulación cuántica del campo gravitacional y por lo tanto de la geometŕıa del espacio-tiempo, será necesaria en la descripción de dichos fenómenos a esas escalas. Es bien sabido que uno de los problemas fundamentales en las teoŕıas de campo son las divergencias que surgen en los cálculos de diagramas de Feynman en los desarrollos perturba- tivos. Los métodos de renormalización y las teoŕıas efectivas corrigen una gran cantidad de esos problemas, proporcionando a los f́ısicos de cantidades finitas acordes con las observadas en los aceleradores de part́ıculas. Sin embargo para el caso del campo gravitacional no ha sido posible construir una teoŕıa cuántica renormalizable y por ende un modelo cuántico de la gravitación. Una posibilidad para lograr esto, aunque a la fecha nigún intento ha sido próspero, pudiera ser el agregar cortes a las integrales de espacio fase de manera similar a lo que se hace en los 1.1. Motivación 3 métodos de renormalización, pero donde la forma natural de introducir este corte en la teoŕıa sea en función de la longitud de Planck ℓp. 1 Un argumento [1] conducente a establecer la existencia de una escala de longitud fundamen- tal del orden de la longitud de Planck proviene de implementar, simultáneamente, el Principio de Incertidumbre y el Principio de Equivalencia cuando se quiere resolver con cada vez mayor precisión la posición de una part́ıcula. Para observar un sistema de escalas cuánticas se utiliza algún microscopio que, esencialmente, es un dispositivo que lanza part́ıculas energéticas contra un objetivo. Estas part́ıculas deben poseer una longitud de onda de de Broglie inferior al tamaño caracteŕıstico ∆x del sistema que se observa, de forma que puedan producir patrones de difracción con información relevante del sistema. Por el simple efecto de la interacción (colisión) con las part́ıculas, el sistema recibe un intercambio de momento lineal ∆p. Del principio de incertidumbre de Heisenberg se tiene que ∆x∆p ∼ ~/2, que implica un cambio en la enerǵıa del sistema del orden ∼ ~c 2∆x y un correspondiente cambio de masa relativista ma = 1 2 ~ c∆x . 2 Ahora bien, por el principio de equivalencia mg ≡ ma = 1 2 ~ c∆x , (1.1) donde mg es la masa gravitacional asociada al sistema. Se puede considerar por simplicidad que esta masa produce un campo gravitacional de simetŕıa esférica descrito por la métrica de Schwarzschild ds2 = (c2 − 2Gmg r )dt2 − (1− 2Gmg rc2 )−1dr2 − r2dΩ2. (1.2) Entonces, cuanto mayor sea la resolución del microscopio mayor será el valor de mg, hasta que el horizonte de Schwarzschild y la precisión en las observaciones sean comparables, i.e. r ∼ ∆x, en cuyo caso ocurre r = 2Gmg c2 = G~ c3∆x ⇒ (∆x)2 ∼ G~ c3 = ℓ2p. (1.3) Desde un punto de vista conceptual y teórico éste argumento heuŕıstico apunta a la longitud de Planck como el ĺımite inferior para la precisión en cualquier medición de la posición. Puesto que la presencia del horizonte impediŕıa extraer información de la región en su interior, entonces distancias menores a ℓp no pareceŕıan poseer algún sentido operacional. Consecuentemente, los principios fundamentales de Mecánica Cuántica y Relatividad Gen- 1La longitud de Planck es una unidad de distancia construida dimensionalmente a partir de las constantes fundamentales G, c, ~ de manera que ℓp = √ G~ c3 = 1.6162× 10−35m. 2De las ecuaciones básicas de la Relatividad Especial la enerǵıa cinética asociada a una part́ıcula con momento p está dada por E = mac 2 = √ m2 0c 4 + p2c2, que en el ĺımite de altas enerǵıas permite despreciar la contribución de la masa en reposo m0, dejando únicamente el término dominante E = mac 2 ≈ p c. 4 Caṕıtulo 1. Introducción eral sugieren que seguir considerando al espacio-tiempo como una variedad diferencial, para describir fenómenos a escalas de ℓp, se torna cuestionable. Esto apunta a adoptar una nueva estructura matemática que actúe como paradigma de una geometŕıa que conduzca de un cambio del concepto de espacio a uno de álgebras de funciones en donde, generalmente, no hay análogo alguno de espacio subyacente para las teoŕıas f́ısicas. 1.2 Marco histórico. Diversas ĺıneas de investigación que buscan conciliar los dos grandes pilares de la F́ısica han abierto las puertas a nuevas áreas de estudio dentro de la F́ısica Teórica y las Matemáticas, los enfoques son variados y algunos radicalmente contrastantes. El origen de una de estas ĺıneas puede trazarse hasta Werner Heisenberg quien, durante su intercambio de correspondencia con Rudolf Peierls en 1930 [2], propuso usar una estructura no-conmutativa para las coordenadas espaciales como alternativa a una longitud de corte, con la finalidad de corregir las singulari- dades provenientes de la auto interacción del electrón en la Electrodinámica Cuántica.3 La motivación de Heisenberg era distinta a la del gedankenexperiment descrito antes, sin embargo es posible ilustrar el interés que existe actualmente en la F́ısica Teórica por la no- conmutativad del espacio. Considerando, sin pérdida de generalidad, una no-conmutatividad entre los operadores de posición cuánticos X̂, Ŷ de un espacio bidimensional, de forma que su conmutador satisfaga [X̂, Ŷ ] = Θ̂, (1.4) donde Θ̂ es un operador (anti-hermitiano) cuyo valor de expectación cumple |〈Θ̂〉| ≃ ℓ2p. Entonces se sigue [4] un principio de incertidumbre ∆x∆y ≥ 1 2 |〈Θ̂〉| ≃ ℓ2p 2 , (1.5) que en el lenguaje de la Mecánica Cuántica expresa la imposibilidad de localizar una part́ıcula dentro de una región del espacio de area inferior a ℓ2p/2 ó, de forma equivalente, una ”dis- cretización” del espacio en celdillas (en el sentido de elementos representativos para puntos indistinguibles entre śı) con un área de igual o mayor tamaño. Aśı, mediante ésta sencilla ex- tensión a la Mecánica Cuántica ordinaria, es posible incorporar formalmente un ĺımite para la precisión en cualquier medición de la posición que dé cuenta de las implicaciones de la expresión (1.3). Consecuentemente, si la mejor descripción que puede obtenerse del universo a escalas mi- 3Arthur March exploró de forma independiente esta posibilidad entre 1936 y 1954 aunque sus publicaciones recibieron poca o nula atención. Para un análisis comparativo sobre el trabajo de Heisenberg y March en el tema ver, e.g., [3]. 1.2. Marco histórico. 5 croscópicas es de naturaleza cuántica, como sucede en el caso de las part́ıculas fundamentales, entonces la suposición más simple que puede hacerse es aquella en que los principios cuánticos dictaminan que rumbo tomar en la construcción de teoŕıas que modelen la estructura del espacio- tiempo a escalas de ℓp. 4 Partiendo de espacios de Hilbert y álgebras de operadores, donde el ĺımite clásico de dichas teoŕıas se identifique con las variedades diferenciales descritas por la Relatividad Especial y General. En 1947 Hartland S. Snyder estableció, en su publicación “Quantized Space-Time” [6], los lineamientos teóricos y una amplia discusión filosófica que serviŕıan como punto de partida para el posterior estudio de los espacios no-conmutativos. La parte medular de su exposición fué el uso de operadores cuánticos para definir las coordenadas de un espacio-tiempo 4-dimensional, construidos a partir de coordenadas proyectivas. Esto le permitió incorporar conmutadores no nulos entre los operadores de espacio-tiempo similares a la expresión (1.4) y, a su vez, contar con una teoŕıa invariante de Lorentz, dado que el modelo (proyectivo) del cual part́ıa era un espacio de de Sitter. Sólo un par de años después José E. Moyal publicó un importante trabajo [7], luego de una extensa y fruct́ıfera discusión con P.A.M. Dirac, en donde reunió resultados de J. von Neumann, H. Weyl, E. Wigner y H. Groenewold (ver [8, 9, 10]). En dicho trabajo se presenta una novedosa reformulación de la Mecánica Cuántica como una teoŕıa estad́ıstica definida mediante funciones C∞(R2n) de un espacio-fase clásico con coordenadas (qi, pi), completamente equivalente a los esquemas de Heisenberg ó Schrödinger ampliamente aceptados y sólidamente establecidos para entonces.5 La importancia de los trabajos de Weyl, Wigner, Groenewold y Moyal para la F́ısica con- temporánea es indiscutible debido a su implementación en la F́ısica Experimental y Aplicada. Se ha usado ampliamente en F́ısica Nuclear en el estudio de procesos de dispersión relevantes para el desarrollo de reactores nucleares; provee algoritmos para la solución de problemas inver- sos asociados al decaimiento de isótopos radioactivos o núcleos atómicos inmersos en campos magnéticos intensos, bases para el diseño de instrumentos avanzados y precisos de imageneoloǵıa 4Actualmente no existe evidencia experimental directa que sustente esta afirmación y aún habrá que esperar algún tiempo para ello. Considerando que experimentos como el LHC, que trabaja en los órdenes de enerǵıas de 1013eV , solo logra resolver distancias no menores de 10−19m. En una comparación más clara, la diferencia de escalas entre estos órdenes de longitud y ℓp es la misma a la que existe entre las escalas mesoscópicas y las de un núcleo atómico. Esto, por supuesto, no descarta que sea posible observar efectos de gravedad cuántica indirectamente a escalas de longitud mayores, como sugieren los estudios de radiación de fondo recientemente reportados por la colaboración BICEP2 [5]. 5Aunque en general dio visto bueno a la versión final del trabajo de Moyal, Dirac mantuvo algunas dudas sobre la relevancia ó la utilidad de esta nueva presentación de la Mecánica Cuántica ya que, en contraste con los esquemas previos, hab́ıa un mayor nivel de complejidad en la teoŕıa debido a la aparición de una cuasiprobabilidad. Además, dado que en el caso clásico las funciones de distribución son siempre positivas definidas, Dirac afirmaba que para una teoŕıa donde esto no ocurŕıa no hab́ıa motivo de hacer una conexión con un esquema clásico; ver [11]. 6 Caṕıtulo 1. Introducción médica (v.gr. Resonancia Nuclear Magnética y Tomograf́ıa por emisión de Positrones); permite estudiar paquetes de fotones propagándose en fibra óptica y analizar pulsos laser ultracortos en Óptica Cuántica. También se ha aplicado en Qúımica Cuántica, Óptica Clásica, el análisis de señales en Sismoloǵıa, Bioloǵıa, Ingenieŕıa Eléctrica, etc. Sin embargo, probablemente la mayor aportación del formalismo deWeyl-Wigner-Groenewold- Moyal (que en adelante se denotará por el acrónimo WWGM) sea el lenguaje matemático desar- rollado en dichas investigaciones, el cual dió origen al celebrado producto de Groenewold-Moyal, que estableció las bases y también el primer ejemplo para lo que actualmente se conoce como una Deformación con producto-⋆ (estrella). Dicho a grosso modo un producto-⋆ reemplaza el producto punto (o de yuxtaposición) de un álgebra de funciones A, por un álgebra de funciones equipada con producto no-conmutativo A⋆. 6 La principal premisa en el formalismo WWGM es la existencia de una biyección entre op- eradores lineales que actúan sobre un espacio de Hilbert apropiado y las funciones de A⋆. Di- chos operadores pueden corresponder a los observables de alguna teoŕıa cuántica, por ejemplo, posición, momento lineal, momento angular, etc. Por lo tanto, una vez que se establece ésta biyección, se cuenta con la cuantización de algún sistema clásico. En particular el formalismo WWGM identifica operadores del álgebra Heisenberg–Weyl, que es el álgebra fundamental de conmutadores de Mecánica Cuántica, con las funciones del espacio-fase C∞(R2n) equipado con el producto-⋆ de Groenewold-Moyal. Tanto el formalismo WWGM como generalizaciones posteriores proveen ventajas concep- tuales que no se encuentran en otros métodos de cuantización, principalmente por que per- miten manejar una teoŕıa cuántica desde la perspectiva de una teoŕıa equivalente de funciones clásicas. Consecuentemente la mayoŕıa de las herramientas usadas para funciones, ya sea de tipo geométrico ó algebraico, se extienden de forma más natural. Esta particularidad de la teoŕıa permite, por ejemplo, obtener correcciones semiclásicas a las propiedades f́ısicas (como valores de expectación) de algún sistema mecánico-cuántico directamente de expansiones en potencias de ~ como series asintóticas de distribuciones de cuasiprobabilidad en reǵımenes de altas enerǵıas ó dispersivos, ver e.g. [12]. El caso análogo de bajas enerǵıas requiere, sin embargo, tomar ciertas precauciones en la forma como se efectúan dichas expansiones [13]. Dado que el estudio de escalas Planckianas corresponde naturalmente al primer caso, es inmediato que la aplicación de este tipo de formulación está exento de dichas consideraciones, constituyendo aśı una ruta genuina de cuantización para tales escalas. Por lo anterior, los productos-⋆ han sido considerados más recientemente (ver, e.g., [14]), como una plataforma lógica para estudiar modelos de teoŕıas de campos no-conmutativos con- 6Como conjuntos se cumple A ⊂ A⋆ 1.3. Organización de la Tesis 7 tinuando, en cierta forma, con la búsqueda original de Heisenberg por un método para renor- malizar teoŕıas manifiestamente divergentes, cf. [15]. En este trabajo también se utilizarán los productos-⋆ como punto de partida para estudiar propiedades fundamentales de Mecánica Cuántica y Campos en espacios no-conmutativos caracterizados por los conmutadores (1.4), aunque con principal énfasis en las consecuencias teóricas y las interpretaciones f́ısicas asociadas. En una dirección alterna, pero igualmente inspirada en las estructuras matemáticas de álgebras de operadores y espacios de Hilbert que permean la Mecánica Cuántica, se desar- rolló la teoŕıa de Álgebras C∗ que dió luz a los Teoremas de Gel’fand-Naimark [16]. Donde la consecuencia más importante de dichos teoremas en el contexto de este trabajo es la posibilidad de establecer una equivalencia entre un álgebra C∗ conmutativa y una variedad diferencial M. Lo cual evoca preguntar a que tipo de espacio equivale un álgebra C∗ no-conmutativa y que ha dado origen al estudio de la Geometŕıa No-conmutativa, desarrollada fundamentalmente por A. Connes [17]. Este cambio de paradigma propone abandonar definitivamente las nociones de cálculo diferencial e integral y de variedades tipo Hausdorff, reemplazandolos por conceptos puramente algebraicos y espectrales. En las secciones finales de éste trabajo se aborda una formulación que persigue este tipo de conceptos. 1.3 Organización de la Tesis En este trabajo se presentan resultados Teóricos y aplicaciones de la No-conmutatividad del Espacio, entendida como la incorporación de conmutadores del tipo (1.4) entre los observables mecánico-cuánticos de posición, al considerarla dentro de un contexto más serio de álgebras de operadores y las interpretaciones de observables que de ah́ı emanan, recordando la motivación subyacente de modelar la imposibilidad de realizar mediciones en la posición de un sistema cuántico con precisión infinita, en el régimen donde los principios de Relatividad General y de Mecánica Cuántica se consideran al mismo nivel. Las construcciones elaboradas en estas páginas son producto del trabajo de investigación conducido durante mi participación en el grupo de in- vestigación dirigido por el Dr. Marcos Rosenbaum, lo cual permitió generar nueve (9) art́ıculos de investigación publicados en revistas arbitradas e indizadas, con un décimo manuscrito en proceso de aceptación para su publicación, siendo autor principal en tres (3) de dichos trabajos. Estos materiales se han anexado a este trabajo de Tesis en su parte final, principalmente para su fácil localización, debido a las cont́ınuas referencias a resultados que ah́ı aparecen y que se mencionan a lo largo de éste trabajo. El formato de presentación seguido divide la Tesis en tres partes: La Primera Parte está enfocada a proporcionar las herramientas matemáticas básicas y sentar los principios teóricos utilizados en las siguientes dos partes de la Tesis. Para este fin se preparó el Caṕıtulo 2 que sirve 8 Caṕıtulo 1. Introducción a manera de repaso del método de cuantización de WWGM, mencionado previamente, aunque con variantes en el estilo adoptado para la exposición del material respecto de la forma en que usualmente se presenta en la literatura. Esto con el objetivo de hacer contacto con conceptos de álgebras de operadores de forma más natural, y que muestren con mayor claridad las motiva- ciones de caṕıtulos posteriores. También se proporciona material que acompaña a este caṕıtulo en un apéndice, donde se reproducen varias expresiones que permiten apreciar el poder teórico de éste método alterno (y autónomo) de formular la Mecánica Cuántica. En el Caṕıtulo 3 se abordan generalizaciones del método de cuantización WWGM, a saber el esquema axiomático de Stratonovich-Weyl en torno a un cuantizador y la formulación de Berezin, esta última con énfasis particular en la representación en términos del operador de reflexión. El análisis de ambas construcciones desde una perspectiva de bases supercompletas, de estados coherentes del espacio de Hilbert, proporciona un panorama más amplio del concepto de cuantización. En la Segunda Parte se implementan los métodos de cuantización discutidos en los caṕıtulos previos para el caso de la Mecánica Cuántica No-conmutativa caracterizada por un álgebra ex- tendida de Heisenberg Weyl. El Caṕıtulo 4 comprende la extensión del método de cuantización WWGM al régimen no-conmutativo, en el sentido de (1.4), proporcionando formas análogas de las expresiones usuales obtenidas de Mecánica Cuántica, lo que permite hacer comparaciones de- talladas de ambas teoŕıas y extraer diversos resultados originales. Un ejemplo notable es la inter- pretación probabiĺıstica de la teoŕıa, que establece la existencia de diversas funciones de Wigner generadas a partir de una misma función de Weyl en el caso no-conmutativo. Otro resultado importante es la identificación de un mecanismo dinámico que da origen a la no-conmutatividad de teoŕıas de campos, como consecuencia directa de la evolución de operadores en el esquema de Heisenberg. En el Caṕıtulo 5 se exploran varias representaciones de No-conmutatividad, comen- zando con la construcción de una base de estados coherentes no-conmutativos para obtener una representación holomorfa del producto-⋆ el cual posee notorias ventajas matemáticas sobre otras formulaciones. Aśı mismo se aborda la no-conmutatividad desde el esquema de la Cuantización Canónica y de la Integral de Trayectoria. Como última sección se proporciona también una śıntesis del formalismo de la Geometŕıa No-conmutativa como preámbulo para la parte final de la Tesis. Finalmente en la Tercera Parte, elaborando sobre resultados previos, se presentan argu- mentos que apuntan a la necesidad de recurrir a conceptos menos anclados a las nociones de variedades para migrar a formulaciones matemáticas en términos de álgebras de funciones como se presenta en el Caṕıtulo 6. En dicho caṕıtulo se propone una representación de operadores de Mecánica Cuántica que se aparta del Teorema de Stone-von Neumann y consecuentemente no re- curre a realizaciones diferenciales de operadores cuánticos, que están inherentemente asociados a los conceptos de variedades diferenciales. Usando técnicas de Álgebras C∗ se establece un homo- morfismo entre un álgebra de operadores acotados que actúan en el espacio de Hilbert y el álgebra del Grupo extendido de Heisenberg-Weyl. Se muestra que los observables cuánticos correspon- 1.3. Organización de la Tesis 9 dientes preservan algunas propiedades de aquellos del álgebra extendida de Heisenberg-Weyl. Esta elección de observables se utiliza posteriormente para analizar una Cosmoloǵıa Cuántica anisotrópica de tipo Bianchi I en el contexto de la integral de trayectoria, seguido del análisis de fase estacionaria y resultados numéricos. Parte I Conceptos preliminares 11 Caṕıtulo 2 El esquema de cuantización WWGM Varias de las construcciones desarrolladas en este trabajo recurren al formalismo WWGM como caso arquet́ıpico y, por lo que, el propósito de éste caṕıtulo es exhibir los principios teóricos y las interpretaciónes f́ısicas detrás de este método de cuantización, que ha dado origen a toda una rama de las matemáticas conocida como ”Cuantización por Deformación”. Con la finalidad de contextualizar éste con caṕıtulos posteriores, se optará por un lenguaje contemporáneo y una exposición que haga mayor contacto con nociones cuánticas. Por ello el orden en que varias expresiones aparecen no necesariamente coincide con la forma en que cronológicamente surgieron ó con el interés original en torno a las mismas. Aunque, como se trató de enfatizar en la Introducción, en el estudio de fenómenos f́ısicos que ocurren a escalas microscópicas es, probablemente, más sensato tomar una formulación cuántica como punto de partida para migrar después a conceptos o interpretaciones clásicas. No obstante, se tratará de preservar el esṕıritu del formalismo WWGM como el valioso método de cuantización que, por mérito propio, ocupa ya un lugar incuestionable en la F́ısica.1 2.1 Operadores de desplazamiento En [7] se considera una Mecánica Cuántica unidimensional, i.e., definida en el espacio (extendido) de Hilbert H = L2(R). Sin embargo, es conveniente hacer la generalización de las expresiones involucradas al caso H = L2(Rn). Esto resulta inmediato de tomar el producto directo de n copias del álgebra de Heisenberg-Weyl h3, en donde cada copia es generada por el operador identidad Î, un operador de posición Q̂ y un operador de momento lineal P̂ . El resultado es el álgebra h2n+1 con reglas de conmutación [Q̂i, P̂j ] = i~δij Î, i, j = 1, ..., n, (2.1) 1Para una presentación que reúne la literatura escencial sobre el formalismo WWGM ver [18]. 13 14 Caṕıtulo 2. El esquema de cuantización WWGM y el resto de los conmutadores iguales a cero.2 En la notación de bra-kets [19] una base completa y ortogonal para el álgebra h2n+1 está dada por eigen-estados de los operadores de posición: Q̂i|~q〉 = qi|~q〉, Î = ∫ Rn dq1...dqn |~q〉〈~q|, 〈~q|~r〉 = δ(q1 − r1)...δ(qi − ri)...δ(qn − rn) = δn(~q − ~r), (2.2) donde ~q, ~r constituyen vectores de R n y qi, ri sus componentes respectivas en la dirección i. El Teorema de Stone-von Neumann (ver e.g. [20]) garantiza una única representación uni- taria irreducible del álgebra (2.1), mediante grupos uniparamétricos Ui(λ), Vi(σ), i = 1, ..., n, débilmente cont́ınuos para λ, σ ∈ R, 3 que satisfacen el álgebra de trenzamiento (ó de Weyl) Ui(λ)Uj(λ) = Uj(λ)Ui(λ), Vi(σ)Vj(σ) = Vj(σ)Vi(σ), Ui(λ)Vj(σ) = e−i~λσδ i jVj(σ)Ui(λ), (2.3) estableciendo una equivalencia (unitaria) con los grupos eiλQ̂ i y eiσP̂i asociados a (2.1). Esto implica que los generadores infinitesimales de (2.3) satisfacen el álgebra (2.1) y, por lo tanto, los generadores de h2n+1 son identificados con campos vectoriales que admiten la realización diferencial usual. De acuerdo a la teoŕıa general (ver, e.g., [21]) el grupo de Lie asociado con h2n+1 puede obtenerse por la exponenciación de elementos arbitrarios X̂ ∈ h2n+1. En particular los oper- adores D̂(~x, ~y) := e i ~ (xiP̂i+yiQ̂ i), ~x, ~y ∈ R n, son elementos de grupo y, como se mostrará, son los objetos fundamentales para la construcción del formalismo WWGM.4 La ley de multiplicación de estos operadores, llamados operadores de desplazamiento, ya hab́ıa sido estudiada en [22]. Tal propiedad es consecuencia directa del teorema de Baker-Campbell-Hausdorff [23], aplicado al producto eÂeB̂ cuando el conmutador [Â, B̂] pertenece al centro del álgebra de Lie: eÂeB̂ = e 1 2 [Â,B̂]e(Â+B̂), (2.4) de forma que al sustituir  = i ~ (xiP̂i + yiQ̂ i) y B̂ = i ~ (x′jP̂j + y′jQ̂ j) en el conmutador [Â, B̂] se 2Los sub́ındices y supeŕındices tendrán la connotación tensorial usual, es decir, etiquetarán cantidades covari- antes y contravariantes respectivamente, incluida la convención de suma entre ellos, a menos que se indique lo contario. 3En el sentido de la topoloǵıa débil de operadores, para una representación de espacio de Hilbert, los elementos de matriz de Ui(λ), Vi(σ) son cont́ınuos en λ, σ ∈ R. 4La constante ~ de Planck ocurre en la exponenciación de forma que todo el argumento de la exponencial sea adimensional. 2.1. Operadores de desplazamiento 15 obtiene [ i ~ (xiP̂i + yiQ̂ i), i ~ (x′jP̂j + y′jQ̂ j) ] = − 1 ~2 { xix′j [P̂i, P̂j ] + xiy′j [P̂i, Q̂ j ] + yix ′j [Q̂i, P̂j ] + yiy ′ j [Q̂ i, Q̂j ] } , (2.5) y usando el álgebra (2.1) para resolver los conmutadores individuales simplifica la expresión en [ i ~ (xiP̂i + yiQ̂ i), i ~ (x′jP̂j + y′jQ̂ j) ] = i ~ (xiy′i − yix′i)Î. (2.6) Por lo tanto, según (2.4), el producto de dos operadores de desplazamiento está dado por e i ~ (xiP̂i+yiQ̂ i)e i ~ (x′j P̂j+y′jQ̂ j) = e i 2~ (xiy′i−yix′i)e i ~ [(xj+x′j)P̂j+(yj+y′j)Q̂ j ], (2.7) omitiendo el operador Î por obvias razones. La siguiente propiedad importante es el valor de Tr[e i ~ (xiP̂i+yiQ̂ i)] y para su cálculo se utiliza la base completa (2.2), i.e., Tr[e i ~ (xiP̂i+yiQ̂ i)] = ∫ Rn dq1...dqn 〈q1, ..., qn|e i ~ (xiP̂i+yiQ̂ i)|q1, ..., qn〉, (2.8) que, usando (2.4) y la realización Q̂i|~q〉 = qi|~q〉, puede escribirse como ∫ Rn dq1...dqn 〈q1, ..., qn|e i ~ (xiP̂i+yiQ̂ i)|q1, ..., qn〉 = ∫ Rn dq1...dqn e− i 2~ xiyi〈q1, ..., qn|e i ~ xiP̂ie i ~ yiQ̂ i |q1, ..., qn〉 = e− i 2~ xiyi ∫ Rn dq1...dqn e i ~ yiq i〈q1, ..., qn|e i ~ xiP̂i |q1, ..., qn〉. (2.9) El término que aún se encuentra entre bra-kets también puede simplificarse tomando en cuenta la realización diferencial de P̂i en la base seleccionada. Para dos estados arbitrarios |φ〉, 〈ψ| el álgebra (2.1) conduce a 〈~q|P̂i|φ〉 = −i~∂i〈~q|φ〉, (2.10) 〈ψ|P̂i|~q〉 = i~∂i〈ψ|~q〉, (2.11) por lo que evidentemente 〈~q|e i ~ xiP̂i |φ〉 = ex i∂i〈~q|φ〉 = 〈~q + ~x|φ〉, (2.12) 〈ψ|e i ~ xiP̂i |~q〉 = e−x i∂i〈ψ|~q〉 = 〈ψ|~q − ~x〉, (2.13) donde se ha usado la propiedad del operador de traslación ex i∂i sobre cada función de onda. 16 Caṕıtulo 2. El esquema de cuantización WWGM Entonces (2.12) implica que el término 〈~q|e i ~ xiP̂i |~q 〉 se puede escribir como 〈~q|e i ~ xiP̂i |~q 〉 = 〈~q + ~x|~q〉 = δn(~q + ~x− ~q) = δn(~x), (2.14) como lo confirma el cálculo análogo utilizando (2.13). Substituyendo ahora (2.14) en (2.9), ha- ciendo las integraciones correspondientes e imponiendo f(x)δ(x) = f(0)δ(x) conduce finalmente a Tr[e i ~ (xiP̂i+yiQ̂ i)] = (2π~)nδn(~x)δn(~y). (2.15) Una identidad más útil proviene de utilizar este resultado para calcular la traza en ambos lados de (2.7) Tr[e i ~ (xiP̂i+yiQ̂ i)e i ~ (x′j P̂j+y′jQ̂ j)] = e i 2~ (xiy′i−yix′i)Tr[e i ~ {(xj+x′j)P̂j+(yj+y′j)Q̂ j}] = (2π~)nδ(n)(~x+ ~x′)δn(~y + ~y′). (2.16) 2.2 Análisis de Fourier y equivalentes de Weyl La expresión (2.16) define una base completa para operadores de tipo Hilbert-Schmidt, i.e. oper- adores  ∈ End(L2(Rn)), que son funciones arbitrarias de los generadores de h2n+1 y satisfacen Tr[†] < ∞, lo que implica la existencia de un análogo en operadores de la transformada de Fourier.5 Lo anterior puede parafrasearse diciendo que todo operador de Mecánica Cuántica, asociada al álgebra (2.1), admite una descomposición uńıvoca en términos de operadores de desplaza- miento de acuerdo a la expresión  = ∫ R2n dn~xdn~y α(~x, ~y)e i ~ (xiP̂i+yiQ̂ i), (2.17) donde α(~x, ~y) constituye el “espectro” de Fourier del operador Â, al cual en la literatura se le denomina el śımbolo del operador. De manera similar a lo que ocurre con la transformada de Fourier usual, la validez de (2.17) depende de la existencia de una expresión inversa que permita determinar la función α(~x, ~y). Es directo mostrar que esta condición se cumple, multiplicando primero ambos lados de (2.17) por 5Para una desmostración completa del teorema asociado ver, e.g., [24]. 2.2. Análisis de Fourier y equivalentes de Weyl 17 e− i ~ (xj P̂j+yjQ̂ j), tomando trazas y usando (2.16): Tr[Âe− i ~ (xj P̂j+yjQ̂ j)] = ∫ R2n dn~x′dn~y′ α(~x′, ~y′ )Tr[e i ~ (x′iP̂i+y′iQ̂ i)e− i ~ (xj P̂j+yjQ̂ j)] = (2π~)n ∫ R2n dn~x′dn~y′ α(~x′, ~y′ )δn(~x′ − ~x)δn(~y′ − ~y) = (2π~)nα(~x, ~y ).  (2.18) Sin embargo, también puede pensarse en la función α(~x, ~y) como el espectro de Fourier genuino de una función clásica de espacio-fase. Es decir que, para α(~x, ~y) dada por la ecuación (2.18), hay una función AW ∈ C∞(R2n) definida por la transformada integral AW (~q, ~p) = ∫ R2n dn~xdn~y α(~x, ~y)e i ~ (xipi+yiq i), (2.19) conocida como el equivalente (ó śımbolo) de Weyl del operador Â, y cuyo sub́ındice indica que la función se obtuvo siguiendo esta serie de transformaciones. Por lo tanto las ecuaciones (2.17), (2.18) y (2.19) establecen el functor biyectivo W : End(L2(Rn)) −−−⇀↽ − C∞(R2n)  7−−−→ AW , (2.20) donde, debido a la no-conmutatividad (2.1) y a queW es biyectivo, el equivalente de Weyl de un producto de operadores arbitrarios  y B̂ corresponderá a una función (AB)W que, en general, puede diferir del producto AWBW . Tomando la integral de espacio-fase de (2.19) surge una propiedad importante de AW , ∫ R2n dn~qdn~p AW (~q, ~p) = (2π~)2nα(0, 0), (2.21) la función α(0, 0) se sustituye evaluando (2.18) en ~x = ~y = 0: ∫ R2n dn~qdn~p AW (~q, ~p) = (2π~)nTr[Â]. (2.22) Ésta expresión permite reemplazar el cálculo de trazas de operadores cuánticos por integrales en espacio-fase de equivalentes de Weyl. Conjugando (2.22) se obtiene un nuevo resultado ∫ R2n dn~qdn~p A∗W (~q, ~p) = (2π~)nTr[†], (2.23) 18 Caṕıtulo 2. El esquema de cuantización WWGM que sugiere la igualdad (A†)W (~q, ~p) = A∗W (~q, ~p), (2.24) como se confirma conjugando hermı́ticamente (2.17) seguido del cambio de variables (~x, ~y) → (−~x,−~y) y llevando a cabo el resto de las transformaciones. Entonces, si  es un operador hermitiano, la expresión análoga para † =  corresponde a A∗W (~q, ~p) = AW (~q, ~p), (2.25) es decir que un observable de Mecánica Cuántica es identificado con una función real de espacio- fase, lo cual reproduce la interpretación usual de un observable clásico.6 2.3 El producto de Groenewold-Moyal y el espacio C∞⋆ (R2n) Toda teoŕıa f́ısica autónoma permite construir magnitudes nuevas partiendo de cantidades definidas a priori dentro la misma. Este simple principio se traduce al formalismo WWGM en la necesidad de contar con una regla de composición, que relacione equivalentes de Weyl de operadores individuales con equivalentes de Weyl de productos de operadores. A continuación se verá que, para establecer anaĺıticamente la relación de (AB)W con los śımbolos AW y BW , se debe modificar la ley de multiplicación de funciones de C∞(R2n). Partiendo de la ecuación (2.17) para el caso de un producto de operadores (ÂB̂) = ∫ R2n dn~xdn~y γ(~x, ~y)e i ~ (xiP̂i+yiQ̂ i), (2.26) con γ(~x, ~y) definida por γ(~x, ~y) = ( 1 2π~ )n Tr[ÂB̂e− i ~ (xj P̂j+yjQ̂ j)], (2.27) los operadores Â, B̂ en la traza se substituyen usando nuevamente (2.17) γ(~x, ~y) = ( 1 2π~ )n ∫ R4n dn~x′dn~y′dn ~x′′dn ~y′′ { α(~x′, ~y′)β( ~x′′, ~y′′) × Tr[e i ~ (x′iP̂i+y′iQ̂ i)e i ~ (x′′iP̂i+y′′i Q̂ i)e− i ~ (xiP̂i+yiQ̂ i)] } . (2.28) El valor de Tr[e i ~ (x′iP̂i+y′iQ̂ i)e i ~ (x′′iP̂i+y′′i Q̂ i)e− i ~ (xiP̂i+yiQ̂ i)] se obtiene del uso repetido de (2.7) 6En generalizaciones posteriores del formalismo WWGM como la cuantización de Stratonovich-Weyl §3.2, las ecuaciones (2.22) y (2.24) son promovidas a postulados de la teoŕıa denominados estandarización y realidad respectivamente. 2.3. El producto de Groenewold-Moyal y el espacio C∞⋆ (R2n) 19 seguido de (2.16): Tr[e i ~ (x′iP̂i+y′iQ̂ i)e i ~ (x′′iP̂i+y′′i Q̂ i)e− i ~ (xiP̂i+yiQ̂ i)] = (2π~)ne i 2~ (x′iy′′i −y′ix′′i)δn(~x′ + ~x′′ − ~x)δn(~y′ + ~y′′ − ~y). (2.29) Aśı entonces, recurriendo a (2.19) y substituyendo (2.29) en (2.28), el śımbolo de Weyl (AB)W asociado al producto ÂB̂ puede escribirse como (AB)W (~q, ~p) = ∫ R2n dn~xdn~y γ(~x, ~y)e i ~ (xipi+yiq i) = ∫ R6n dn~xdn~ydn~x′dn~y′dn ~x′′dn ~y′′ { α(~x′, ~y′)β( ~x′′, ~y′′) × e i ~ (xipi+yiq i)e i 2~ (x′iy′′i −y′ix′′i)δn(~x′ + ~x′′ − ~x)δn(~y′ + ~y′′ − ~y) } , (2.30) e integrando sobre ~x y ~y resulta (AB)W (~q, ~p) = ∫ R4n dn~x′dn~y′dn ~x′′dn ~y′′ { α(~x′, ~y′)β( ~x′′, ~y′′) × e i ~ [(x′i+x′′i)pi+(y′i+y′′i )q i]e i 2~ (x′iy′′i −y′ix′′i) } . (2.31) Las funciones α(~x′, ~y′), β( ~x′′, ~y′′) pueden expresarse como transformadas de Fourier invir- tiendo (2.19), lo que permite incorporar los śımbolos de Weyl AW , BW expĺıcitamente en el desarrollo, i.e., α(~x′, ~y′) = ( 1 2π~ )2n ∫ R2n dn~q′dn~p′AW (~q′, ~p′)e− i ~ (x′ip′i+y′iq ′i), (2.32) β( ~x′′, ~y′′) = ( 1 2π~ )2n ∫ R2n dn ~q′′dn ~p′′BW (~q′′, ~p′′)e− i ~ (x′′ip′′i +y′′i q ′′i). (2.33) Al substituir estas expresiones en (2.31), integrando en ~x′′, ~y′′ y después en ~q′′, ~p′′ y renom- brando variables conduce a (AB)W (~q, ~p) = ( 1 2π~ )2n ∫ R4n dn~xdn~ydn~q′dn~p′ { e i ~ [xi(pi−p′i)+yi(q i−q′i)] ×AW (~q′, ~p′)BW (~q + ~x 2 , ~p− ~y 2 ) } . (2.34) Además, por las propiedades de los operadores de traslación, se tiene e i ~ [xi(pi−p′i)+yi(q i−q′i)]BW (~q + ~x 2 , ~p− ~y 2 ) = e i ~ [xi(pi−p′i)+yi(q i−q′i)]e xi 2 ∂ ∂qi e − yi 2 ∂ ∂piBW (~q, ~p) = e i ~ [xi(pi−p′i)+yi(q i−q′i)]e − i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi e i~ 2 ←− ∂ ∂qi −→ ∂ ∂piBW (~q, ~p), (2.35) 20 Caṕıtulo 2. El esquema de cuantización WWGM donde las flechas en los operadores diferenciales ←− ∂ ∂pi , −→ ∂ ∂qi , etc., distinguen si la acción de la derivación ocurre en funciones que se encuentran a la izquierda ó a la derecha respectivamente. Por lo tanto (2.35) permite extraer términos en (2.34) que ya no dependen de las variables de integración, a saber (AB)W (~q, ~p) = ( 1 2π~ )2n [ ∫ R4n dn~xdn~ydn~q′dn~p′AW (~q′, ~p′)e i ~ [xi(pi−p′i)+yi(q i−q′i)] ] × e i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) BW (~q, ~p), (2.36) y como las integraciones restantes son triviales se obtiene el resultado final (AB)W (~q, ~p) = AW (~q, ~p)e i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) BW (~q, ~p). (2.37) La expresión anterior permite definir al producto-⋆ de Groenewold-Moyal, mencionado breve- mente en la Introducción, como el operador bidiferencial ⋆~ := e i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) , (2.38) en el cual se codifica todo el formalismo WWGM.7 De acuerdo a (2.37) el mapeo W en (2.20) es ahora el isomorfismo (como functor) W : End(L2(Rn)) −−−⇀↽ − C∞⋆ (R2n) ÂB̂ 7−−−→ AW ⋆~ BW , (2.39) donde C∞⋆ (R2n) representa el álgebra de funciones de espacio-fase equipada con el producto ⋆~. 8 Las propiedades fundamentales de la estructura de C∞⋆ (R2n) surgen de analizar la imagen de los generadores de h2n+1 bajo W. Por ejemplo, en el caso de Q̂i, usando la ecuación (2.18) en conjunto con (2.2) y (2.12) se tiene ( 1 2π~ )n Tr[Q̂ie− i ~ (xj P̂j+yjQ̂ j)] = i~δn(~x) ∂δn(~y) ∂yi , (2.40) de forma que, al sustituirlo en (2.19) como un espectro de Fourier, permite calcular el equivalente de Weyl (Qi)W : (Qi)W (~q, ~p) = i~ ∫ R2n dn~xdn~y e i ~ (xipi+yiq i)δn(~x) ∂δn(~y) ∂yi = qi. (2.41) 7Varias propiedades de ⋆~ son deducidas en el Apéndice A. 8La notación matemática usual [25, 26] para un álgebra de tales caracteŕısticas es C∞(R2n)[[i~/2]]. Que simboliza un espacio que contiene a las funciones de C∞(R2n) y, además, todas las series formales de potencias en el parámetro i~/2. 2.3. El producto de Groenewold-Moyal y el espacio C∞⋆ (R2n) 21 Un desarrollo casi idéntico con P̂i da (Pi)W (~q, ~p) = pi y trivialmente IW = 1. Consecuente- mente, como era de esperarse, los generadores del álgebra son identificados con las coordenadas canónicas de espacio-fase. Utilizando estos resultados junto con (2.37) se calculan los equivalentes de Weyl de productos básicos (QiQj)W = qi ⋆~ q j = ( qi + i~ 2 ~∂ ∂pi ) qj = qiqj , (PiPj)W = pi ⋆~ pj = ( pi − i~ 2 ~∂ ∂qi ) pj = pipj , (2.42) (QiPj)W = qi ⋆~ pj = ( qi + i~ 2 ~∂ ∂pi ) pj = qipj + i~ 2 δij , que, en términos de un conmutador de funciones de C∞⋆ (R2n) definido como [f, g]⋆~ := f ⋆~ g − g ⋆~ f , se expresan ahora de manera compacta [qi, qj ]⋆~ = 0, [pi, pj ]⋆~ = 0, [qi, pj ]⋆~ = i~δij . (2.43) Comparando las relaciones (2.43) con (2.1) se confirma el isomorfismo (2.39) y se pone en evidencia la naturaleza no-conmutativa del espacio-fase C∞⋆ (R2n). Aśı mismo, con ayuda del producto ⋆~, es posible obtener más información de la expresión (2.22), ya que entonces la traza de un producto ÂB̂ se calcula según (2π~)nTr[ÂB̂] = ∫ R2n dn~qdn~p (AB)W (~q, ~p), (2.44) que, en virtud de (2.37), se lee (2π~)nTr[ÂB̂] = ∫ R2n dn~qdn~p AW (~q, ~p) ⋆~ BW (~q, ~p). (2.45) La versión correspondiente para B̂ es (2π~)nTr[B̂Â] = ∫ R2n dn~qdn~p BW (~q, ~p) ⋆~ AW (~q, ~p). (2.46) Entonces, como Tr[ÂB̂] = Tr[B̂Â], se concluye que ∫ R2n dn~qdn~p AW (~q, ~p) ⋆~ BW (~q, ~p) = ∫ R2n dn~qdn~p BW (~q, ~p) ⋆~ AW (~q, ~p), (2.47) que es el análogo integral de la propiedad ćıclica de la traza de un producto de operadores.9 9Esta es la definición de un producto-⋆ fuertemente cerrado [27]. En §A.1 se proporciona una demostración alterna de esta propiedad realizando integraciones por partes que, además, permiten sobresimplificar la ecuación 22 Caṕıtulo 2. El esquema de cuantización WWGM Esto completa el diccionario básico del formalismo WWGM, en el cual se traducen las estruc- turas matemáticas de un espacio de operadores lineales sobre L2(Rn) en aquellas de un espacio de funciones equipado con el producto–⋆~. Otras propiedades no menos importantes del formal- ismo WWGM como, por ejemplo, la descripción del esquema de Heisenberg y la interpretación probabiĺıstica en términos de la función de Wigner-Szilard se reservan para el Apéndice A. 2.4 El principio de correspondencia Para finalizar este caṕıtulo se describirá una caracteŕıstica del producto ⋆~ que amerita especial atención. Notando primero que el argumento en la exponencial (2.38) visto como un operador bidiferencial independiente, actuando sobre dos funciones arbitrarias de espacio-fase f, g, es, hasta un factor i~/2, el paréntesis de Poisson clásico: f (←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) g = ∂f ∂qi ∂g ∂pi − ∂f ∂pi ∂g ∂qi = {f, g}. (2.48) Este resultado no es casual y tiene profundas implicaciones en la forma como se conceptualiza el proceso de cuantización. Para ilustrar esto se analizan los ĺımites lim ~→0 f ⋆~ g, lim ~→0 1 i~ [f, g]⋆~ , (2.49) por inspección de (2.38) se obtiene el primer ĺımite lim ~→0 f ⋆~ g = fg, (2.50) para el segundo ĺımite es necesario desarrollar un poco más las exponenciales formales lim ~→0 1 i~ [f, g]⋆~ = lim ~→0 1 i~ [ ∞∑ n=0 (i~)n 2nn! f (←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi )n g − ∞∑ n=0 (i~)n 2nn! g (←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi )n f ] , (2.51) dado que hay cancelación de los primeros términos, por tratarse de productos de yuxtaposición, y que el proceso de ĺımite anula los términos de orden n ≥ 2 se obtiene lim ~→0 1 i~ [f, g]⋆~ = 1 2 {f, g} − 1 2 {g, f} = {f, g}, (2.52) (2.45) en la integral del producto ordinario de funciones. 2.4. El principio de correspondencia 23 luego de utilizar (2.48) para reescribir los términos restantes. Para la F́ısica el ĺımite ~→ 0 equivale a despreciar fluctuaciones cuánticas en algún sistema suficientemente macroscópico como para poder ser descrito por leyes clásicas, lo que es conocido como el principio de correspondencia.10 Por lo tanto los ĺımites (2.50) y (2.52) son consistentes con esta interpretación, ya que conducen a cantidades clásicas como el producto conmutativo de funciones y el paréntesis de Poisson respectivamente. Sin embargo, hay más información que se extrae de estas expresiones, puesto que establecen la forma precisa en que la Mecánica Clásica emerge a partir de estructuras cuánticas y, además, que la Mecánica Cuántica corrige a la Mecánica Clásica agregando términos en potencias a todos los órdenes del paréntesis de Poisson escalado por la constante de Planck. En resumen, el formalismo WWGM proporciona una teoŕıa de funciones en espacio-fase R 2n que modela la Mecánica Cuántica en R n, reemplazando los espacios de operadores por un álgebra no-conmutativa de funciones con un producto deformado y en donde, como es claro de los desarrollos previos, no hay referencias a espacios vectoriales (de Hilbert) cuyos elementos representan estados de probabilidad y que, eventualmente (ver §A.3), son sustituidos por fun- ciones de cuasiprobabilidad de espacio-fase. La generalización de éste concepto es la principal motivación dentro de la Cuantización por Deformación [26, 28, 29]. Como comentario final de esta sección y pauta de algunas secciones posteriores nótese que, a nivel de funciones clásicas y del principio de correspondencia, la expresión (2.48) admite, como se sabe, estructuras más generales (no-canónicas) de Poisson entre las variables de espacio–fase que, gracias al teorema de Darboux (ver, e.g., [30]), pueden siempre escribirse (localmente) en términos de una estructura canónica como (2.48). Al identificar (qi, pi) → za, a = 1, ..., 2n y considerando el potencial simpléctico Ω = θa(z)dz a, se obtiene la 2-forma simpléctica ω = dΩ = ωabdz a ∧ dzb, (2.53) donde ωab = ∂aθb− ∂bθa. Si ωab es una matriz no degenerada, entonces la estructura de Poisson entre las variables de espacio–fase za se define por medio de la matriz inversa ωab como {za, zb} := ωab. (2.54) Ahora bien, en vista del material presentado a lo largo de éste caṕıtulo, es leǵıtimo consid- erar el isomorfismo (2.20) en la dirección opuesta y ponderar el tipo de álgebras de operadores y productos-⋆ que originan los paréntesis no-canónicos (2.54) en el ĺımite del principio de corre- spondencia. En términos del método de cuantización canónica de Dirac [31, 32] es posible implementar 10Dirac proporciona una interesante discusión al respecto, a lo que denominó la analoǵıa clásica, ver [19]. 24 Caṕıtulo 2. El esquema de cuantización WWGM la prescripción de la analoǵıa clásica {za, zb} 7→ [Ẑa, Ẑb] = i~ωab, (2.55) de donde se observa que, además de la presencia de conmutadores entre operadores de posición y de momento, también incorpora conmutadores no triviales entre operadores de posición y entre operadores de momento. Por ejemplo, si para el caso n = 2 se tiene ωab =      0 θ 1 0 −θ 0 0 1 −1 0 0 0 0 −1 0 0      , ωab =      0 0 −1 0 0 0 0 −1 1 0 0 θ 0 1 −θ 0      , (2.56) con θ constante. Entonces en (2.55) destaca el conmutador [Q̂1, Q̂2] = i~θ, el cual tiene precisa- mente la forma del conmutador (1.4) discutido en la Introducción. Esta simple generalización del paréntesis de Poisson abre la posibilidad del estudio de diversas propiedades de teoŕıas cuánticas no-conmutativas en la etapa de pre-cuantización (ó del ĺımite semiclásico). Tal perspectiva fué abordada en detalle por nuestro grupo de investigación en ”Noncommutativity from Canonical and Noncanonical Structures, Marcos Rosenbaum, J. David Vergara and L. Román Juárez, Con- temporary Math. 462, pp. 10367-10382 (2008)” (Ref. [33]), ver Cap.5, §5.3 para una discusión más detallada. Caṕıtulo 3 Variaciones sobre un tema de Moyal El formalismoWWGM inició la revolución en métodos de cuantización basados en la deformación del producto de funciones clásicas. Un análisis completo de las diversas reglas de ordenamiento en Mecánica Cuántica (v.gr. normal, estándar, etc.) en el contexto de sus equivalentes en espacio-fase puede hallarse en [24]. Alĺı se mostró que todo ordenamiento pertenece a una familia de isomorfismos (consecuentemente de productos-⋆), conectados continuamente con el mapeo (2.39) al cual corresponde el ordenamiento simétrico (o de Weyl). Los fundamentos matemáticos para la deformación de grupos y álgebras de Lie fueron es- tablecidos por M. Gerstenhaber en [25]. En esta dirección Flato et al. desarrollaron los tra- bajos seminales [26], donde identificaron las obstrucciones para la construcción de productos-⋆ en variedades de Poisson arbitrarias. Fedosov proporcionó una solución parcial para dichas obstrucciones con su celebrado programa de cuantización [34]. Finalmente M. Kontsevich de- mostró en [35] que toda variedad de Poisson (en particular cualquier espacio-fase) de dimensión finita admite una cuantización por deformación, proporcionando a la vez su fórmula epónima del producto-⋆ universal.1 Si bien el producto de Kontsevich constituye el pináculo de la cuantización por deformación, su complejidad inherente, tanto conceptual como computacional, lo mantiene al margen de convertirse en una herramienta accesible para el estudio de problemas f́ısicos. A la par de esta corriente de investigación se encuentran construcciones alternas de productos- ⋆, análogas al formalismo WWGM, que guardan una relación más profunda con las álgebras de operadores. Tales formulaciones son particularmente atractivas desde la perspectiva del presente trabajo. En este tenor se cita primero el esquema axiomático iniciado por Stratonovich en [38] y desarrollado más ampliamente en investigaciones posteriores, e.g., [39, 40, 41]. El segundo caso 1D. Sternheimer preparó una revisión extensa sobre las investigaciones en cuantización por deformación en [36], desde sus oŕıgenes en el formalismo WWGM hasta las aportaciones de Kontsevich de finales del siglo XX y las conexiones con la teoŕıa de grupos cuánticos, la Geometŕıa No-conmutativa de Connes, los teoremas de ı́ndice de Atiyah, etc. Para una versión extendida con mayor énfasis en la fórmula de Kontsevich ver [37]. 25 26 Caṕıtulo 3. Variaciones sobre un tema de Moyal de interés corresponde a la cuantización de Berezin [42] y, en especial, su versión en términos de operadores de involución, e.g. [43]. Un logro importante obtenido por los autores de [40] fué identificar el lenguaje de los estados coherentes como el puente que conecta ambos formalismos anteriores, lo que permitió incorporar todo el potencial de dicho edificio matemático en el estudio de los productos-⋆ para obtener resultados novedosos y, a veces, insospechados. Por completez y dada su implementación en algunas secciones posteriores, se presentan a continuación los principales lineamientos teóricos de estas construcciones. 3.1 Estados Coherentes Generalizados La definición de estado coherente es tan amplia como el número de aplicaciones que puede dársele (ver e.g., [44, 45, 46]). En particular la definición de Perelomov de estado coherente generalizado [47, 48] es adecuada para el estudio de sistemas f́ısicos con algún grupo de simetŕıa subyacente, como sucede en el caso de los los sistemas elementales. Definición 1. [Estado Coherente] Un sistema elemental (clásico) [40] corresponde a una var- iedad simpléctica X que es homogénea bajo la acción de un grupo de Lie G y, consecuentemente, isomorfa a las órbitas generadas por un subgrupo maximal (subgrupo de isotroṕıa) H ⊂ G, es decir, X ≃ G/H. Śı, además, G es un grupo semisimple que actúa en un espacio de Hilbert H v́ıa una representación unitaria irreducible T (g) y H es compacto, entonces los estados co- herentes de H etiquetados por puntos de X corresponden a los estados obtenidos por la acción transitiva de G sobre el subespacio H0 ⊂ H que es H-invariante. Lo anterior implica elegir primero un estado normalizado |ϕ0〉 ∈ H0, llamado el estado base, el cual por la H-invariancia cumple T (h)|ϕ0〉 = eiα(h)|ϕ0〉, h ∈ H, (3.1) donde α(h) es una función real de h. Ahora, dado que G puede descomponerse en términos de clases laterales, i.e. G = {gxh| gx ∈ G/H, h ∈ H}, la acción arbitraria de G sobre |ϕ0〉 es T (g)|ϕ0〉 = T (gx)T (h)|ϕ0〉 = eiα(h)T (gx)|ϕ0〉, ∀g ∈ G, (3.2) donde gx es el elemento de G/H que se identifica con el punto x ∈ X. De forma que el estado T (gx)|ϕ0〉 que aparece en la expresión anterior es, según la definición dada previamente, el estado coherente generalizado asociado al punto x: |x〉 := T (gx)|ϕ0〉, ∀gx ∈ G/H, (3.3) 3.2. Correspondencia de Stratonovich–Weyl 27 que, por construcción, es también un estado normalizado de H. La propiedad fundamental de los estados coherentes generalizados es que conforman una base supercompleta de H, es decir que existe una resolución de la identidad: ∫ X dµ(x) |x〉〈x| = ÎH, (3.4) con dµ(x) la medida de Riemann invariante en X y donde, en general, la función de transición de dos estados coherentes arbitrarios cumple |〈x′|x〉| ≤ 1. (3.5) La función cont́ınua y acotada K(x′, x) := 〈x′|x〉 proporciona un ejemplo de un kernel reproducente [49], el cual satisface K(x′, x) = ∫ X dµ(y)K(x′, y)K(y, x), ∀x′, x ∈ X, (3.6) que podŕıa parecer trivial simplemente de usar (3.4). Sin embargo, contrario a lo que sucede con las funciones δ de Dirac, fijando x′ = x y notando que K(x, y) = K∗(y, x) es claro entonces que Ky := K(y, ·) ∈ H. Al evaluar un estado arbitrario en la base coherente, i.e. 〈x|ψ〉 = ψ(x), se tiene ψ(x) = ∫ X dµ(y)K(x, y)ψ(y), (3.7) que expresa la naturaleza no local de K(x, y), ya que la integral se debe evaluar expĺıcitamente en todo el espacio. Desde una perspectiva semiclásica los estados coherentes son ideales por su proximidad con estados clásicos [44] ya que saturan las relaciones de incertidumbre, además de evitar la pérdida de información en el paso de un problema cuántico a uno clásico, propiedades muy conocidas en el caso de los estados coherentes del grupo de Heisenberg-Weyl [50, 51]. 3.2 Correspondencia de Stratonovich–Weyl Es posible condensar el formalismo de Stratonovich–Weyl, en su versión más moderna (ver [39, 40, 41]), en términos de una serie de postulados para un operador pseudodiferencial Q̂ conocido como el cuantizador. Estas condiciones se basan fundamentalmente en las expresiones que condujeron a establecer el isomorfismo (2.39) en el caṕıtulo anterior y que involucraban espacios–fase euclideos. Aśı, la formulación discutida a continuación busca generalizar diversas nociones del caso plano a espacios–fase curvos. 28 Caṕıtulo 3. Variaciones sobre un tema de Moyal Asumiendo que, bajo ciertos criterios, es posible determinar el espacio–fase X de algún sistema cuántico descrito por operadores que actúan en un espacio de Hilbert H,2 entonces el cuantizador Q̂ de Stratonovich-Weyl produce el isomorfismo (como functor) Q̂ : End(H) −−−⇀↽ − C∞⋆ (X) ÂB̂ 7−−−→ FA ⋆ FB , (3.8) donde, en general, FA ⋆ FB no representa el producto-⋆ de Groenewold-Moyal, sino el producto no-conmutativo espećıfico inducido por el isomorfismo. La forma expĺıcita en que un operador arbitrario  se relaciona con su equivalente clásico de espacio-fase FA constituye la primera propiedad para Q̂ conocida como linealidad : FA(x) := Tr[ÂQ̂(x)], ∀x ∈ X, (3.9) acompañada de una inversión que denota la biyectividad de (3.8):  = ∫ X dµ(x)FA(x)Q̂(x), (3.10) es decir, decuantizar y cuantizar es posible v́ıa el mismo kernel Q̂. Si G corresponde al grupo de simetŕıa del sistema cuántico y T (g) es una representación unitaria irreducible, entonces la existencia de un cuantizador S–W bonafide, responsable del isomorfirmo (3.8), proviene de las condiciones: (i) Realidad Q̂†(x) = Q̂(x), ∀x ∈ X, (3.11a) (ii) Estandarización ∫ X dµ(x)Q̂(x) = ÎH, (3.11b) (iii) Tracialidad Tr[Q̂(x)Q̂(x′)] = δ(x− x′), (3.11c) (iv) Covariancia T (g)Q̂(x)T (g−1) = Q̂(gx), ∀g ∈ G. (3.11d) 2Por ejemplo, en el caso de Mecánica Cuántica en H = L2(Rn) el espacio–fase asociado es X = R 2n. Una demostración de ello puede hacerse en un contexto dinámico cf. (A.40). 3.2. Correspondencia de Stratonovich–Weyl 29 Es interesante notar que la condición de estandarización es un tanto redundante, como se puede mostrar tomando la integral en X de la condición de covariancia: ∫ X dµ(x)T (g)Q̂(x)T (g−1) = ∫ X dµ(x)Q̂(gx), (3.12) entonces, utilizando la propiedad de invariancia dµ(g−1x) = dµ(x) se tiene T (g) [∫ X dµ(x)Q̂(x) ] T (g−1) = ∫ X dµ(x)Q̂(x), (3.13) donde T (g) y T (g−1) se extrajeron de la integral por no tener dependencia en la variable x. Aśı, la expresión anterior implica que el operador ∫ X dµ(x)Q̂(x) conmuta con T (g) y, de acuerdo al lema de representación de Schur [52], debe ocurrir ∫ X dµ(x)Q̂(x) = k ÎH, (3.14) para alguna constante real k. De manera que basta normalizar adecuadamente al cuantizador, tomando en cuenta el valor de k, para recuperar la condición de estandarización.3 Finalmente el producto-⋆ asociado a Q̂ queda definido impĺıcitamente por la expresión FA(x) ⋆ FB(x) := FAB(x) = Tr[ÂB̂Q̂(x)], (3.15) que, con el uso repetido de (3.10), admite también una forma alterna en términos de equivalentes clásicos independientes dada por FA(x) ⋆ FB(x) = ∫ X×X dµ(x′)dµ(x′′)L(x, x′, x′′)FA(x ′)FB(x ′′), (3.16) donde la función de tres puntos L(x, x′, x′′), conocida como el trikernel, corresponde al valor de Tr[Q̂(x)Q̂(x′)Q̂(x′′)].4 Entonces, según el formalismo de Stratonovich-Weyl, el problema de obtener el producto-⋆ de un álgebra A⋆ equivale a hallar un método para construir cuantizadores. En general esto es una labor más bien artesanal que se lleva a cabo ya sea partiendo de primeros principios [39, 53] ó fijando algún ansatz para Q̂ [54, 55, 56, 57, 58]. Sin embargo, como mostraron Várilly et al. 3Sin embargo en la literatura se acostumbra enlistar las cuatro condiciones (3.11) preservando la es- tandarización. Esto se debe a que la estandarización sigue siendo necesaria para la definición de Q̂ incluso cuando se relaja la condición de covariancia. 4El producto (2.37) es, por construcción, un caso particular de (3.16), donde el cuantizador correspondiente se infiere de las expresiones (2.17-2.19), i.e.: Q̂(~q, ~p) = ( 1 2π~ )2n ∫ R2n dn~xdn~y e i ~ (xipi+yiq i)e− i ~ (xiP̂i+yiR̂ i), el cual, a su vez, satisface los postulados (3.11). Esto se confirma notando primero que (3.11a) y (3.11b) se cumplen trivialmente, mientras que las propiedades (3.11c) y (3.11d) se obtienen de usar (2.16) y (2.7) respectivamente. 30 Caṕıtulo 3. Variaciones sobre un tema de Moyal en el caso de X compacto, los estados coherentes de H son ideales para probar la existencia de cuantizadores. Cuando X corresponde al espacio–fase de un sistema elemental (§3.1) el algoritmo [40, 41] que conduce al cuantizador recurre a las propiedades de los estados coherentes (3.3), junto con una base completa y ortogonal de funciones de L2(X,µ), para generalizar la base de Fourier de operadores (2.16). En tal caso el rol de los operadores de desplazamiento es realizado por los operadores D̂Ξ := ∫ X dµ(x)YΞ(x)|x〉〈x|, (3.17) donde Ξ es un ı́ndice compuesto y las funciones YΞ cumplen ∑ Ξ YΞ(x)Y ∗ Ξ (x ′) = δ(x− x′), ∫ X dµ(x)Y ∗Ξ (x)YΞ′(x) = δΞΞ′ . (3.18) La elección natural para YΞ son las funciones armónicas, es decir, la familia de soluciones al problema de eigenvalores para el operador de Laplace–de Rham–Beltrami ∆̂ = δd + dδ (donde δ es el operador codiferencial) asociado a X, que en una base coordenada se escribe [21] ∆̂XYΞ(x) = −|g|−1/2∂α(|g|1/2gαβ∂βYΞ(x)) = λ(Ξ)YΞ(x), (3.19) con g el determinante del tensor métrico gαβ y λ(Ξ) el eigenvalor. La función de transición de dos estados coherentes admite la expansión |〈x|x′〉|2 = ∑ Ξ ξΞYΞ(x)Y ∗ Ξ (x ′), (3.20) donde ξΞ son coeficientes reales [41].5 Es posible mostrar, con ayuda de la expresión anterior y (3.18), que los operadores D̂Ξ conforman una base de Fourier de operadores, i.e. Tr[D̂ΞD̂ † Ξ′ ] = ξΞδΞΞ′ , (3.21) con lo que, de forma similar a (2.17), un operador arbitrario  ∈ End(H) puede expresarse como la combinación lineal  = ∑ Ξ AΞD̂Ξ, (3.22) donde AΞ = ξ−1Ξ Tr[ÂD̂†Ξ]. 5En el caso compacto este resultado es consecuencia directa del teorema de Peter–Weyl y, además, los coefi- cientes ξΞ están cercanamente relacionados con los coeficientes de Clebsch–Gordan de la representación T (g) cf. [40]. 3.2. Correspondencia de Stratonovich–Weyl 31 Finalmente el cuantizador de Stratonovich-Weyl que satisface (3.11) se define expĺıcitamente como Q̂(x) := ∑ Ξ ξ −1/2 Ξ Y ∗Ξ (x)D̂Ξ = ∑ Ξ ξ −1/2 Ξ YΞ(x)D̂ † Ξ. (3.23) Al susbstituir esta definición en (3.9) se halla la expresión para FA análoga a la transfor- mación (2.19): FA(x) = ∑ Ξ ξ 1/2 Ξ AΞYΞ(x), (3.24) que, ignorando por un momento el origen de los coeficientes, corresponde justamente a la trans- formada (inversa) de Fourier en X. El cuantizador (3.23) puede aplicarse indistintamente a sistemas elementales compactos ó planos mas no en casos no-euclideos no-compactos.6 La razón de esto, como se identificó du- rante la investigación desarrollada en la preparación de “On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group, L. Román Juárez and Marcos Rosenbaum, J. Phys. Math. 2, pp. 29-50 (2010)” [59], se debe a que al substituir  = ÎH en (3.22) se obtiene una forma alterna para la resolución de la identidad ÎH = ∑ Ξ ξ−1Ξ Tr[D̂†Ξ]D̂Ξ, (3.25) la cual está bien definida siempre que Tr[D̂†Ξ] exista, que equivale, según (3.17), a la existencia (incluso como distribución) de la integral ∫ X dµ(x)Y ∗Ξ (x). (3.26) Sin embargo aún para el caso no-euclideo no-compacto más simple asociado al espacio de de–Sitter bidimensional H2 = SO(2, 1)/SO(2) (también conocido como plano de Lobatchevskii ó disco de Poincaré), la integral previa no converge para ningún valor de Ξ. En dicho caso las soluciones al problema de eigenvalores (3.19) son las funciones horoesféricas [48, 60]: Φλ θ (τ, ϕ) = [cosh τ − sinh τ cos(θ − ϕ)]−1/2+iλ, λ ∈ (0,∞), τ ∈ [0,∞), θ, ϕ ∈ [0, 2π], (3.27) donde el ı́ndice compuesto Ξ es el par (λ, θ). Como la medida de Riemann en H 2 es dµ(τ, ϕ) = dτdϕ sinh τ , el conteo de potencias en el régimen asintótico muestra que la integral (3.26) diverge como eτ/2 para todo valor de (λ, θ).7 Consecuentemente los operadores D̂Ξ pierden su carácter como base de Fourier e igual- 6Aunque los autores de [41] no adviertieron esta situación. 7El mismo argumento puede generalizarse a cualquier espacio de de–Sitter SO(p, q)/SO(p, q − 1) ó anti–de Sitter SO(p, q)/SO(p− 1, q) con p ≥ q ≥ 1. Utilizando las funciones armónicas de estos espacios [61] es fácil ver que la divergencia de (3.26) es del orden de e(p+q−2)τ/2 en ambos casos. 32 Caṕıtulo 3. Variaciones sobre un tema de Moyal mente, puesto que la integral en X de (3.23) involucra la misma divergencia, la condición de estandarización (3.11b) ya no se satisface. Esta ausencia de una base de Fourier y de una definición consistente para el cuantizador constituyen un obstáculo dentro de la cuantización por deformación que es posible sortear con la construcción descrita a continuación. 3.3 Śımbolos covariantes de Berezin–Weyl De las diversas (en principio infinitas) reglas de asociación que pueden establecerse entre un álgebra de operadores y un álgebra de funciones, sólo algunas poseen interpretaciones f́ısicas trascendentes. El programa de cuantización de Berezin [42] genera un tipo especial de asociación que invoca funciones de alguna realización holomorfa para el grupo de simetŕıa subyacente, las cuales constituyen generalizaciones de los śımbolos de Wick [62].8 Un prerrequisito en esta formulación para el espacio–fase asociado a la teoŕıa cuántica es que sea una variedad simpléctica simétrica, es decir, que en cada punto de X exista una simetŕıa de reflexión. Con ello se entiende una isometŕıa σx 6= id que revierte las geodésicas que pasan por x, tal que σ2x = id, ∀x ∈ X, σx(x) = x. (3.28) Ahora, siguiendo con la construcción original de Berezin, es necesaria también una base supercompleta de H. Recordando que, en una definición más rigurosa (ver, e.g. [61]), todo espacio simétrico es, a la vez, un espacio homogéneo, entonces los estados coherentes descritos en §3.1 son de nuevo los candidatos ideales. Partiendo de los elementos anteriores, la definición para los śımbolos de Berezin del tipo más simple se sigue inmediatamente. Si  es un operador lineal que actúa en el espacio de Hilbert H, el śımbolo covariante de  es la función de espacio–fase QA(x) := 〈x|Â|x〉, (3.29) mientras que el śımbolo contravariante proviene de la expresión impĺıcita  = ∫ X dµ(x)PA(x)|x〉〈x|. (3.30) Ambas definiciones producen mapeos biyectivos entre End(H) y C∞(X) lo cual se verifica con argumentos de continuación anaĺıtica cf. [42, 48]. El eṕıteto “covariante” ó “contravariante” para cada tipo de śımbolo se debe, evidentemente, a su regla de transformación bajo la representación 8En el caso del álgebra de Heisenberg-Weyl esto refiere simplemente a los ordenamientos normal y antinormal de operadores de creación y destrucción. 3.3. Śımbolos covariantes de Berezin–Weyl 33 T (g).9 Por otro lado, la relación entre ambos śımbolos resulta de insertar (3.30) en (3.29): QA(x) = ∫ X dµ(x′)PA(x ′)|〈x|x′〉|2, (3.31) con la cual es posible evidenciar la dualidad para los śımbolos de Berezin bajo la traza de operadores. Efectivamente, en el producto de dos operadores arbitrarios  y B̂, el uso simultáneo de (3.30) conduce a ÂB̂ = ∫ X×X dµ(x)dµ(x′)PA(x)PB(x ′)〈x|x′〉|x〉〈x′|, (3.32) de donde al tomar la traza y utilizar (3.31) se obtiene Tr[ÂB̂] = ∫ X dµ(x)PA(x)QB(x) = ∫ X dµ(x)QA(x)PB(x), (3.33) que equivale a las expresiones (2.45), (2.46) y (A.23), pero donde el cálculo de la traza como una integral de espacio–fase requiere de dos śımbolos distintos en lugar de un único śımbolo de Weyl. Además de los śımbolos PA, QA, Berezin definió otra pareja de śımbolos de espacio-fase referidos como śımbolos covariantes de Weyl que, bajo circustancias adecuadas, reproducen los equivalentes de Weyl (2.19) del esquema WWGM y los equivalentes de espacio–fase (3.9) del formalismo de Stratonovich–Weyl, de forma que es posible considerarlos como generalizaciones de ambos casos. Los śımbolos de Weyl se obtienen v́ıa el automorfismo involutivo σx que, por ser un elemento del grupo de simetŕıa G del cual X es espacio homogéneo, tiene asociado un operador unitario Û(x) 6= ÎH bajo la representación T (g) el cual satisface: Û(x)2 = ÎH, ∀x ∈ X, Û(x)|x〉 = |x〉, (3.34) de donde es inmediata la hermiticidad Û(x)† = Û(x). (3.35) El śımbolo covariante de Weyl se define entonces como la función de espacio–fase wA(x) := Tr[ÂÛ(x)], (3.36) 9La notación Q y P fué introducida en los trabajos de Husimi [63] y Glauber [50] respectivamente, en el estudio de distribuciones de cuasiprobabilidad para el caso del álgebra de Heisenberg–Weyl. 34 Caṕıtulo 3. Variaciones sobre un tema de Moyal que, nuevamente, implementa la definición genérica de función clásica en términos de una traza como en casos previos. Consecuentemente, recurriendo al mismo argumento de dualidad de (3.33), el śımbolo con- travariante de Weyl corresponde a la función clásica w̃ para la cual se cumple Tr[ÂB̂] = ∫ X dµ(x)wA(x)w̃B(x) = ∫ X dµ(x)w̃A(x)wB(x), (3.37) donde  y B̂ son operadores arbitrarios. De forma que el śımbolo contravariante de Weyl de cualquier operador  es necesariamente la función w̃A definida impĺıcitamente por  = ∫ X dµ(x)w̃A(x)Û(x). (3.38) Debido a que es posible establecer relaciones biyectivas entre los śımbolos covariantes de Berezin y los de Weyl (ver [42]), entonces estos últimos también producen mapeos biyectivos entre End(H) y C∞(X). Para una representación unitaria irreducible T (g), el operador de reflexión satisface natu- ralmente casi todos los axiomas (3.11) de un cuantizador. La condición de realidad se cumple idénticamente por (3.35), mientras que la condición de covariancia se obtiene de notar que el operador de reflexión Û(x′) que deja invariante el estado coherente |x′〉, etiquetado por el punto x′ ≡ g′x y asociado con el elemento g′gx = gx′h ′, es igual al operador T (g′)Û(x)T (g′−1). En efecto, de la definición (3.3) se tiene |x′〉 = T (gx′)|ϕ0〉 = e−iα(h ′)T (g′gx)|ϕ0〉 = e−iα(h ′)T (g′)|x〉, (3.39) que junto con (3.34) conduce a T (g′)Û(x)T (g′−1)|x′〉 = e−iα(h ′)T (g′)|x〉 = |x′〉, (3.40) y, además, como Û(x)2 = ÎH, entonces [T (g′)Û(x)T (g′−1)]2 = ÎH, (3.41) que implican, necesariamente,10 T (g′)Û(x)T (g′−1) = Û(g′x).  (3.42) La condición de estandarización es consecuencia de la condición de covariancia anterior, 10Suponiendo que existe otro punto x̃ 6= x para el cual también se cumple la expresión (3.40) se obtiene la contradicción Û(x̃) = ÎH. 3.3. Śımbolos covariantes de Berezin–Weyl 35 como se mostró en la serie de ecuaciones (3.12-3.14). Por lo tanto es posible afirmar que el operador de reflexión Û(x) garantiza la existencia de al menos un cuantizador Q̂(x), śı y sólo si satisface la condición de tracialidad. Parte II No-conmutatividad y Geometŕıa en Mecánica Cuántica y Campos 37 Caṕıtulo 4 Espacio No-conmutativo Una parte del interés contemporáneo por el estudio de la no-conmutatividad del espacio-tiempo se debe al trabajo realizado dentro de la teoŕıa de cuerdas, donde, en ciertos casos ĺımite [64], se demostró la presencia de conmutadores del tipo (1.4). No obstante, la riqueza teórica obtenida con la introducción ab initio de un parámetro no nulo en los conmutadores de operadores de posición constituye una rama de estudio completamente independiente y con mérito propio. En este sentido cabe mencionar que la mayoŕıa de las investigaciones sobre no-conmutatividad en F́ısica Teórica se desarrollan en torno a la teoŕıa de campos, principalmente por su atractivo como un mecanismo para introducir cortes ultravioletas en integrales divergentes, ver e.g. [14, 65, 15] para un repaso amplio de la literatura en este contexto. Por otro lado, la no-conmutatividad extendida en el régimen mecánico cuántico, visto como el minisuperespacio de teoŕıa de campos en los ĺımites de campo libre ó de acoplamiento mı́nimo, es considerado como un contexto razonable de estudio para avanzar en la formulación no-perturbativa y libre de singularidades de la teoŕıa de campos, conducente a la Gran Unificación de las cuatro interacciones de la Naturaleza. En este caṕıtulo se analizarán las principales consecuencias teóricas de la no–conmutatividad en el contexto de una teoŕıa de deformación cuántica. Partiendo de primeros principios y uti- lizando la generalización para el esquema de cuantización WWGM (descrito en el Cap.2) corre- spondiente al álgebra extendida de Heisenberg-Weyl, que incorpora paréntesis no-nulos de los op- eradores de posición. Esto constituye una parte del trabajo de investigación conducido por nue- stro grupo y que apareció publicado en: ”Dynamical origin of the ⋆θ-noncommutativity in field theory from quantum mechanics, Marcos Rosenbaum, J. David Vergara and L. Román Juárez, Phys. Lett. A 354, pp. 389-395 (2006)” y ”On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group, L. Román Juárez and Marcos Rosenbaum, J. Phys. Math. 2, pp. 29-50 (2010)” (Refs. [66] y [59] de éste trabajo). 39 40 Caṕıtulo 4. Espacio No-conmutativo 4.1 Álgebra extendida de Heisenberg-Weyl hθ5 Se entenderá como el álgebra extendida de Heisenberg-Weyl hθ2n+1 al álgebra de Lie generada por el operador identidad Î, n operadores de posición Q̂i y n operadores de momento P̂i que satisfacen los conmutadores: [Q̂i, Q̂j ] = iθij Î, [Q̂i, P̂j ] = i~δij Î, [P̂i, P̂j ] = 0, i, j = 1, ..., n, (4.1) donde θij corresponde a una matriz constante, real y anti-simétrica. Como se mencionó en la Introducción, la incorporación de θij en el álgebra (4.1) representa un nuevo principio de incertidumbre como propiedad inherente del álgebra de generadores del grupo de Heisenberg-Weyl. Ello modifica la concepción operacional usual del espacio coordenado a nivel microscópico como variedad diferencial, haciendo de las álgebras de funciones con productos no- conmutativos del tipo de los de secciones previas los objetos centrales de estudio. Claramente el primer conmutador (4.1) no parece invariante de rotaciones o compatible con ellas, dado que los ı́ndices de las Q’s transforman bajo el grupo de rotaciones, en tanto que las θij se han tomado como escalares. Sin embargo, como se muestra en [67], desde la perspectiva del álgebra de funciones A⋆ con un producto–⋆ asociada a esta álgebra de operadores (discutida en detalle en §4.2), esto no necesariamente implica tener que renunciar a las nociones de simetŕıa usuales de la F́ısica. En virtud de que una deformación del álgebra universal envolvente de Hopf U(P ) del álgebra de Galileo (o Poincaré) P , por medio de una torcedura de Drinfeld [68] del coproducto, permite preservar la invariancia (torcida) de Galileo (Poincaré) aunque la invariancia bajo el grupo de rotaciones (Lorentz) es violada. Efectivamente, una transformación de simetŕıa sobre un álgebra de funciones A es compatible con el producto m : A ⊗ A −→ A del álgebra, lo que implica que el generador de la simetŕıa X ∈ P satisface la regla de Leibniz (derivación) cuando actua sobre un producto de funciones de A X ⊲m(f ⊗ g) = m[(X ⊲ f)⊗ g] +m[f ⊗ (X ⊲ g)] = m(∆(X) ⊲ f ⊗ g), (4.2) donde ∆(X) = X ⊗ I + I ⊗ X es la imagen de X bajo el coproducto ∆ del álgebra universal envolvente de Hopf U(P ). La torcedura de Drinfeld garantiza [69] que, para una deformación A⋆ del álgebra A, el generador de la simetŕıa X continúa actuando como derivación bajo el producto-⋆ al deformar simultaneamente el coproducto ∆ 7→ ∆⋆, de manera que la acción X ⊲ f ⋆ g = X ⊲m⋆(f ⊗ g) es obtenida por X ⊲m⋆(f ⊗ g) = m⋆(∆⋆(X) ⊲ f ⊗ g), (4.3) que es covariante a (4.2). Esto permite demostrar que los equivalentes en A⋆ de objetos como el primer conmutador en (4.1) transforman apropiadamente bajo los generadores X ∈ P y, por 4.1. Álgebra extendida de Heisenberg-Weyl hθ5 41 lo tanto, continúan siendo invariantes bajo una simetŕıa de Galileo torcida (la demostración detallada se deja para el Apéndice B). A causa de los conmutadores (4.1) la base (2.2) de Mecánica Cuántica debe reemplazarse por una base simultánea de los generadores del álgebra. Para tal fin existen diversas realizaciones multiplicativas que pueden utilizarse dependiendo de la forma de θij . En el caso más general, donde todas las componentes no diagonales de θij son distintas de cero, las bases admisibles cor- responden a los kets |p1, ..., pn〉, |q1, p2, ..., pn〉, ..., |p1, ..., pn−1, qn〉 ó cualquier otra combinación de los observables que constituya un conjunto completo de operadores conmutantes. Una real- ización que suele privilegiarse por conducir a la base usual de Mecánica Cuántica proviene de utilizar el llamado corrimiento de Bopp: X̂i := Q̂i + θij 2~ P̂j , P̂i ≡ P̂i, (4.4) de forma que con estas definiciones el conmutador para distintas X’s es [X̂i, X̂j ] = [ Q̂i + θik 2~ P̂k, Q̂ j + θjl 2~ P̂l ] = iθij + i 2 θjlδil − i 2 θikδjk = 0, (4.5) mientras que el resto de los conmutadores con P ’s permanecen inalterados. La transformación (no canónica) (4.4) permite inmediatamente construir una base completa de eigen-estados de los operadores X̂i, sin embargo la representación asociada del espacio de Hilbert corresponde a una falsa no-conmutatividad (ver e.g. [70]) y, por lo tanto, es cues- tionable que, en general, conduzca a los mismos resultados que aquellos de una representación no-conmutativa genuina. Un ejemplo de esto ocurre a nivel de Teoŕıa Cuántica de Campos donde (4.4) involucra corrimientos no locales que introducen un número infinito de derivaciones. Ello da origen al fenómeno conocido como mezcla UV/IR, el cual es un problema nada trivial en el estudio de teoŕıas renormalizables no-conmutativas (ver e.g. [15]). En vista de lo anterior es necesario partir de una representación genuinamente no-conmutativa del espacio de Hilbert para evidenciar las consecuencias directas del álgebra (4.1), optando por alguna de las bases mixtas mencionadas y donde, por simplicidad algebráica, se estudiará el caso bidimensional hθ5 dado que los resultados pueden extenderse a mayores dimensiones. Partiendo de la base multiplicativa de eigenkets |q1, p2〉,1 para la cual trivialmente se cumple Q̂1|q1, p2〉 = q1|q1, p2〉, P̂2|q1, p2〉 = p2|q1, p2〉, (4.6) el resto de la realización del álgebra puede obtenerse siguiendo procedimientos estándar (cf. [71]), emanados del Teorema de Stone-von Neumann mencionado en §2.1, mediante los subgrupos uniparamétricos Ŝ(γ) = eiγQ̂2 , T̂ (λ) = eiλP̂1 , débilmente cont́ınuos en los parámetros γ, λ ∈ R. 1Por tratarse de un sistema eucĺıdeo de baja dimensionalidad resulta conveniente usar sólo sub́ındices para denotar posición y momento. 42 Caṕıtulo 4. Espacio No-conmutativo Evaluando primero la acción de los conmutadores [Q̂1, Ŝ(γ)] = −θγŜ(γ) y [P̂2, Ŝ(γ)] = ~γŜ(γ), 2 sobre la base ortogonal Q̂1Ŝ(γ)|q1, p2〉 = (q1 − θγ)Ŝ(γ)|q1, p2〉, P̂2Ŝ(γ)|q1, p2〉 = (p2 + ~γ)Ŝ(γ)|q1, p2〉, (4.7) implican, por (4.6), que Ŝ(γ)|q1, p2〉 = |q1 − θγ, p2 + ~γ〉. (4.8) Entonces, para un valor infinitesimal de γ, la expresion previa permite extraer del desarrollo a primer orden de 〈q1, p2|Ŝ(γ)|q′1, p′2〉 la diferenciación 〈q1, p2|Q̂2|q′1, p′2〉 = (−iθ∂q1 + i~∂p2)〈q1, p2|q′1, p′2〉, (4.9) que induce la realización del operador Q̂2 en la base seleccionada, i.e. Q̂2 = −iθ∂q1 + i~∂p2 . (4.10) De manera similar, la acción no-nula del conmutador de T̂ (λ) sobre la base es Q̂1T̂ (λ)|q1, p2〉 = (q1 − ~λ)T̂ (λ)|q1, p2〉, (4.11) o equivalentemente T̂ (λ)|q1, p2〉 = |q1 − ~λ, p2〉, (4.12) que, para λ infinitesimal, en la función de transición 〈q1, p2|T̂ (λ)|q′1, p′2〉 conduce a la realización usual de P̂1 P̂1 = −i~∂q1 . (4.13) Cálculos similares pueden efectuarse para obtener la realización del álgebra en la otra base mixta |q2, p1〉 y la base de momento |p1, p2〉.3 Por otro lado el cambio de base |q1, p2〉 → |q2, p1〉 conduce a un resultado importante que es la función de transición 〈q1, p2|q2, p1〉. Esta puede derivarse [72] observando de (4.10) y (4.13) que 〈q1, p2|Q̂2|q2, p1〉 = q2〈q1, p2|q2, p1〉 = i(~∂p2 − θ∂q1)〈q1, p2|q2, p1〉, (4.14) y 〈q1, p2|P̂1|q2, p1〉 = p1〈q1, p2|q2, p1〉 = −i~∂q1〈q1, p2|q2, p1〉, (4.15) 2Debido a que en el caso bidimensional la no-conmutatividad proviene únicamente de la componente θ12 = −θ21, el uso de ı́ndices es redundante y por ello [Q̂1, Q̂2] = iθ conduce al lado derecho del primer conmutador. 3Para |p1, p2〉 la realización del álgebra se obtiene en un sólo paso utilizando (4.8) y notando que la función de transición 〈p1, p2|q ′ 1, p ′ 2〉 = 1√ 2π~ e− i ~ (q′ 1 p1+ θ 2~ p1p2)δ(p2−p′2) genera el cambio de base. Entonces Q̂1 = i~∂p1 − θ 2~ p2 y Q̂2 = i~∂p2 + θ 2~ p1 corresponden a la realización diferencial del álgebra en esta base. 4.2. El espacio de funciones A∗ 43 de manera que al combinar ambas expresiones se tiene ( q2 − θ ~ p1 ) 〈q1, p2|q2, p1〉 = i~∂p2〈q1, p2|q2, p1〉, (4.16) la cual se resuelve simultáneamente con (4.15) para obtener la solución normalizada 〈q1, p2|q2, p1〉 = 1 2π~ e i ~ (q1p1+ θ ~ p1p2−q2p2). (4.17) La expresión anterior contrasta con la forma usual de las ondas planas de Mecánica Cuántica, incorporando ahora un término no-conmutativo de momento puro que combina contribuciones de direcciones perpendiculares. 4.2 El espacio de funciones A∗ La existencia de una base completa para operadores de tipo Hilbert-Schmidt del álgebra (4.1), en términos de operadores de desplazamiento, permite obtener el equivalente no-conmutativo del formalismo WWGM. Partiendo del álgebra hθ5 y en analoǵıa con §2.1, los operadores de desplazamiento del grupo de Lie Hθ 5 son las la exponenciaciones D̂(~x, ~y) = eχ̂(~x,~y), (4.18) donde el elemento arbitrario χ̂(~x, ~y) ∈ hθ5 corresponde a χ̂(~x, ~y) = i ~ 2∑ i=1 (xiP̂i + yiQ̂i); ~x ∈ R 2, ~y ∈ R 2. (4.19) De acuerdo a la realización multiplicativa del álgebra (4.6) y a la acción de los subgrupos uniparamétricos (4.8) y (4.12) sobre la base mixta |q1, p2〉, es posible obtener la acción de D̂(~x, ~y) con ayuda del teorema BCH (2.4) dado que todos los conmutadores forman parte del centro del álgebra: D̂(~x, ~y)|q1, p2〉 = e i ~ (q1y1+p2x2)e− i 2~ (x1y1+ θ ~ y1y2−x2y2)|q1 − x1 − θy2 ~ , p2 + y2〉, (4.20) esto confirma la acción transitiva del operador de desplazamiento sobre la base. Al proyectar la expresión anterior sobre el bra 〈q1, p2| e integrar en q1 y p2 se obtiene entonces la traza del operador de desplazamiento: Tr[D̂(~x, ~y)] = ∫ R2 dq1dp2e i ~ (q1y1+p2x2)e− i 2~ (x1y1+ θ ~ y1y2−x2y2)δ ( x1 + θy2 ~ ) δ(y2) = (2π~)2δ2(~x)δ2(~y), (4.21) 44 Caṕıtulo 4. Espacio No-conmutativo que equivale a la expresión (2.15) del caso de Mecánica Cuántica usual. Un cálculo cási idéntico al realizado para llegar a la expresión (2.7), pero en donde ahora hay una contribución del conmutador entre Q̂1 y Q̂2 en el equivalente de (2.5), conduce a la ley de multiplicación de los operadores de desplazamiento D̂(~x, ~y)D̂(~x′, ~y′) = e i 2~ [~x·~y′− θ ~ (y1y′2−y′1y2)−~x′·~y]D̂(~x+ ~x′, ~y + ~y′), (4.22) de forma que incluso en el caso del álgebra extendida de Heisenberg-Weyl hθ5, las ecuaciones (4.21) y (4.22) implican que los operadores de desplazamiento forman una base ortogonal para operadores de tipo Hilbert-Schmidt, i.e.: Tr[D̂(~x, ~y)D̂(~x′, ~y′)] = e i 2~ [~x·~y′− θ ~ (y1y′2−y′1y2)−~x′·~y]Tr[D̂(~x+ ~x′, ~y + ~y′)] = (2π~)2δ2(~x+ ~x′)δ2(~y + ~y′). (4.23) Lo anterior garantiza, según la discusión en §2.2, el functor biyectivo Wθ : End(H) −−−⇀↽ − C∞(R4)  7−−−→ AW , (4.24) donde el sub́ındice θ denota que la regla de asociación depende ahora de la no-conmutatividad del espacio. Los mapeos que conducen de un álgebra a la otra asemejan a aquellos del formalismo WWGM. Tanto en lo que refiere a la descomposición uńıvoca de un operador arbitrario  = ∫ R4 d2~xd2~y α(~x, ~y)D̂(~x, ~y), (4.25) aśı como a la función de Weyl correspondiente en espacio-fase AW (~q, ~p) = ∫ R4 d2~xd2~y α(~x, ~y)e i ~ (~x·~p+~y·~q), (4.26) donde el śımbolo α(~x, ~y ) se obtiene nuevamente mediante la traza α(~x, ~y ) = (2π~)−2Tr[ÂD̂†(~x, ~y)] = (2π~)−2Tr[ÂD̂(−~x,−~y)]. (4.27) Una expresión que resulta útil es la del equivalente de Weyl en términos de la base mixta |q1, p2〉. Sustituyendo (4.27) en (4.26) y utilizando (4.20) conduce a AW (~q, ~p) = (2π~)−2 ∫ R6 d2~xd2~ydq′1dp ′ 2 { e i ~ (~x·~p+~y·~q)e− i ~ (q′1y1+p′2x2)e− i 2~ (x1y1+ θ ~ y1y2−x2y2) × 〈 q′1, p ′ 2 ∣ ∣ ∣ ∣ ∣ ∣q′1 + x1 + θy2 ~ , p′2 − y2 〉} , (4.28) e integrando en x2 y y1, seguido de la integración sobre q′1 y p′2 se simplifica en la transformada 4.2. El espacio de funciones A∗ 45 de Fourier de elementos antidiagonales de matriz AW (~q, ~p) = ∫ R2 dx1dy2 e i ~ (q2y2+p1x1) 〈 q1 − x1 2 − θy2 2~ , p2 + y2 2 ∣ ∣ ∣ ∣ ∣ ∣q1 + x1 2 + θy2 2~ , p2 − y2 2 〉 = ∫ R2 dξdη e i ~ [ (q2− θ ~ p1)η+p1ξ ]〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 . (4.29) Entonces, de acuerdo al formalismo WWGM descrito en el Caṕıtulo 2 y en particular de los resultados de §2.3, cabe formular la siguiente pregunta: ¿La biyección Wθ en (4.24) produce también un isomorfismo (como functor), semejante a (2.39), entre End(H) = {Ô(χ̂)|χ̂ ∈ hθ5}, donde Ô(χ̂) son ”funciones” de elementos de hθ5 que actúan sobre H, y el álgebra de funciones de espacio fase C∞(R4) con algún producto deformado? La respuesta a esta pregunta es afirmativa y conduce al espacio de funciones A∗, para A = C∞(R4) con producto de deformación como la composición ∗ = ⋆~ ◦ ⋆θ dado por4 ⋆~ ◦⋆θ := e i~ 2 (←−∇q · −→∇p− ←−∇p· −→∇q ) e iθ 2 ( ←− ∂ q1 −→ ∂ q2− ←− ∂ q2 −→ ∂ q1 ) = e i 2 (~ ↔ Π+θ ↔ Λ), (4.30) donde, como en §A.2, ↔ Π = ←−∇q · −→∇p − ←−∇p · −→∇q es el operador de Poisson y ↔ Λ es el operador bidiferencial no-conmutativo ↔ Λ := ←− ∂ q1 −→ ∂ q2 − ←− ∂ q2 −→ ∂ q1 . Para mostrar la contención anterior se puede recurrir a la expresión (4.29) para un producto de operadores  y B̂ (AB)W (~q, ~p) = ∫ R2 dξdη e i ~ [ (q2− θ ~ p1)η+p1ξ ]〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ÂB̂ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 , (4.31) sustituyendo el desarrollo de los operadores Â, B̂ en términos de operadores de desplazamiento (4.25) y usando (4.22) se tiene (AB)W (~q, ~p) = ∫ R10 dξdηd2~x′d2~y′d2 ~x′′d2 ~y′′ { e i ~ [ (q2− θ ~ p1)η+p1ξ ] α(~x′, ~y′)β( ~x′′, ~y′′) × 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣D̂(~x′, ~y′)D̂( ~x′′, ~y′′) ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉} = ∫ R10 dξdηd2~x′d2~y′d2 ~x′′d2 ~y′′ { e i ~ [ (q2− θ ~ p1)η+p1ξ ] e i 2~ [~x′· ~y′′− θ ~ (y′1y ′′ 2−y′′1 y′2)− ~x′′·~y′] ×α(~x′, ~y′)β( ~x′′, ~y′′) 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣D̂(~x′ + ~x′′, ~y′ + ~y′′) ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉} . (4.32) 4La elección de notación es puramente mnemotécnica, como recordatorio que ∗ tiene una ”punta” más que ⋆ y consecuentemente un producto-⋆ más. Esto puede no coincidir con la notación usada en las referencias. 46 Caṕıtulo 4. Espacio No-conmutativo Notando ahora de (4.20) que 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣D̂(~x, ~y) ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 = e i ~ (q1y1+p2x2)δ ( ξ − x1 − θ ~ y2 ) δ(η − y2), (4.33) entonces (4.32) se reduce a (AB)W (~q, ~p) = ∫ R8 d2~x′d2~y′d2 ~x′′d2 ~y′′ { e i ~ [ ~q·(~y′+ ~y′′)+~p·(~x′+ ~x′′) ] α(~x′, ~y′)β( ~x′′, ~y′′) × e i 2~ [~x′· ~y′′− θ ~ (y′1y ′′ 2−y′′1 y′2)− ~x′′·~y′] } . (4.34) Invirtiendo (4.26) para utilizar las expresiones de α(~x′, ~y′) y β( ~x′′, ~y′′) en términos de fun- ciones de Weyl individuales y siguiendo pasos idénticos a los que se tomaron para llegar de (2.31) a (2.34) permiten escribir la expresión (4.34) en la forma convolutiva (AB)W (~q, ~p) = ( 1 2π~ )4 ∫ R8 d2~xd2~yd2~q′d2~p′ { e i ~ [ (~q−~q′)·~y+(~p−~p′)·~x ] AW (~q′, ~p′) ×BW (q1 + x1 2 + θy2 2~ , q2 + x2 2 − θy1 2~ , ~p− ~y 2 ) } , (4.35) y en analoǵıa directa con (2.35) se tiene e i ~ [ (~q−~q′)·~y+(~p−~p′)·~x ] BW (q1 + x1 2 + θy2 2~ , q2 + x2 2 − θy1 2~ , ~p− ~y 2 ) = e i ~ [ (~q−~q′)·~y+(~p−~p′)·~x ] e 1 2 ~x·∇qe− 1 2 ~y·∇pe θ 2~ y2∂q1e− θ 2~ y1∂q2BW (~q, ~p) = e i ~ [ (~q−~q′)·~y+(~p−~p′)·~x ] e i~ 2 (←−∇q · −→∇p− ←−∇p· −→∇q ) e iθ 2 ( ←− ∂ q1 −→ ∂ q2− ←− ∂ q2 −→ ∂ q1 )BW (~q, ~p), (4.36) que conduce finalmente al producto deformado de funciones de espacio-fase asociado al álgebra extendida de Heisenberg-Weyl hθ5 (AB)W (~q, ~p) = AW (~q, ~p)e i~ 2 (←−∇q · −→∇p− ←−∇p· −→∇q ) e iθ 2 ( ←− ∂ q1 −→ ∂ q2− ←− ∂ q2 −→ ∂ q1 )BW (~q, ~p) := AW (~q, ~p) ⋆~ ◦ ⋆θ BW (~q, ~p). (4.37) Por lo tanto, (4.24) junto con la expresión anterior establecen un isomorfismo entre las álgebras End(H) = {Ô(χ̂)|χ̂ ∈ hθ5} y A⋆~◦⋆θ = C∞⋆~◦⋆θ(R4) de acuerdo a Wθ : End(H) −−−⇀↽ − C∞⋆~◦⋆θ(R 4) ÂB̂ 7−−−→ AW ⋆~ ◦ ⋆θ BW , (4.38) lo cual concluye la demostración.  Similarmente a la discusión de §2.3, la estructura de C∞⋆~◦⋆θ(R4) y las propiedades fundamen- tales del producto (4.30) se pueden establecer de estudiar la imagen de los generadores de hθ5, y productos elementales de los mismos, bajo Wθ. Esto es inmediato utilizando la representación 4.2. El espacio de funciones A∗ 47 (4.29), de donde trivialmente (Q1)W = q1 y (P2)W = p2 y para el los equivalentes (Q2)W y (P1)W basta insertar una resolución de la identidad Î = ∫ R2 dq′2dp ′ 1|q′2, p′1〉〈q′2, p′1|, (4.39) de forma que, por ejemplo (Q2)W = ∫ R4 dq′2dp ′ 1dξdη { e i ~ [ (q2− θ ~ p1)η+p1ξ ]〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣Q̂2 ∣ ∣ ∣q′2, p ′ 1 〉 × 〈 q′2, p ′ 1 ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉} =(2π~)−2 ∫ R4 dq′2dp ′ 1dξdη e i ~ [( q2−q′2− θ ~ (p1−p′1) ) η+(p1−p′1)ξ ] q′2 =q2, (4.40) donde se utilizó la expresión (4.17) para el cambio de base. Un cálculo idéntico conduce a (P1)W = p1 y, por lo tanto, los equivalentes de Weyl de los generadores del álgebra de Heisenberg- Weyl no-conmutativa son nuevamente identificados con las coordenadas canónicas de C∞⋆~◦⋆θ(R4). Introduciendo ahora el conmutador de funciones en C∞⋆~◦⋆θ(R4) como [f, g]⋆~◦⋆θ := f ⋆~ ◦ ⋆θ g − g ⋆~ ◦ ⋆θ f, (4.41) es sencillo ver que los equivalentes de Weyl de productos elementales de los generadores implican [q1, q2]⋆~◦⋆θ = iθ, [qi, pj ]⋆~◦⋆θ = i~δij , [p1, p2]⋆~◦⋆θ = 0, (4.42) que confirma el ismoformismo entre C∞⋆~◦⋆θ(R4) y el álgebra hθ5. El resto de las propiedades algebraicas e integrales de (4.30) coinciden, mutatis mutandis, con las del producto (2.38) estudiadas en §A.1. Esto se debe, naturalmente, a la forma del nuevo bidiferencial no-conmutativo en el producto de deformación y a su antisimetŕıa bajo el intercambio q1 ↔ q2. Dichas propiedades se utilizarán ampliamente en cálculos posteriores. Las consideraciones previas permiten, por lo tanto, generalizar los resultados al álgebra hθ2n+1 definida en (4.1). De forma que bajo el functor Wθ los elementos de End(H) sreán identificados con funciones de C∞⋆~◦⋆θ(R2n) con producto deformado ⋆~ ◦⋆θ = e i 2 (~ ↔ Π+θij ↔ Λij), (4.43) donde ↔ Λij = ←− ∂ qi −→ ∂ qj , que implica los conmutadores en C∞⋆~◦⋆θ(R2n): [qi, qj ]⋆~◦⋆θ = iθij , [qi, pj ]⋆~◦⋆θ = i~δij , [pi, pj ]⋆~◦⋆θ = 0. (4.44) 48 Caṕıtulo 4. Espacio No-conmutativo Nótese también que el álgebra A∗ = C∞⋆~◦⋆θ(R2n) contiene a la subálgebra no-conmutativa Aθ := C∞⋆θ (Rn), que corresponde a la deformación del álgebra conmutativa C∞(Rn), de funciones en coordenadas qi, con producto-⋆θ. 4.3 Interpretación probabiĺıstica y ecuaciones de valores-⋆ En §A.3 se proporciona una amplia discusión de como, dentro del formalismo WWGM, se con- ceptualizan las nociones mecánico-cuánticas del álgebra usual de Heisenberg-Weyl en el contexto de una teoŕıa f́sica de valores de expectación en espacio-fase. Esto mismo puede efectuarse ahora para el álgebra extendida de Heisenberg-Weyl hθ5 en el espacio C∞⋆~◦⋆θ(R4), partiendo del valor de expectación para un operador en términos de la matriz de densidad de von Neumann como 〈Â〉 = Tr[ρ̂Â]. (4.45) Argumentos idénticos a los que condujeron a las expresiones (2.22), (2.45) y (A.23) permiten concluir de (4.25-4.27) y (4.37) que, para el caso del álgebra extendida de Heisenberg-Weyl, la traza de un producto de operadores  y B̂ puede evaluarse como Tr[ÂB̂] =(2π~)−2 ∫ R4 d2~qd2~p AW (~q, ~p) ⋆~ ◦ ⋆θ BW (~q, ~p) =(2π~)−2 ∫ R4 d2~qd2~p AW (~q, ~p)BW (~q, ~p), (4.46) que para el valor de expectación (4.45) implica 〈Â〉 = (2π~)−2 ∫ R4 d2~qd2~p ρW (~q, ~p)AW (~q, ~p). (4.47) Es importante enfatizar que, pese a la gran semejanza de la expresión anterior con (A.46), los equivalentes de Weyl dentro de la integral de espacio-fase no corresponden, en general, con los equivalentes de Weyl usuales a causa de la no-conmutatividad. Esto es evidente en la repre- sentación integral (4.29) donde el equivalente de Weyl mecánico-cuántico se recupera únicamente en el ĺımite θ → 0. Dicha observación tiene importantes repercusiones para el equivalente de Weyl ρW (~q, ~p) de la matriz de densidad y, por lo tanto, en la interpretación probabiĺıstica de la teoŕıa no-conmutativa. Notando que la expresión (4.29) admite la forma alterna AW (~q, ~p) = ∫ R2 dξdη e i ~ [ (q2− θ ~ p1)η+p1ξ ]〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 = e− θ ~ p1∂q2 ∫ R2 dξdη e i ~ (q2η+p1ξ) 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 , (4.48) 4.3. Interpretación probabiĺıstica y ecuaciones de valores-⋆ 49 y como la presencia expĺıcita de la no-conmutatividad se encuentra ahora en el operador de traslación fuera de la integral, el cual se reduce a la identidad para θ = 0, entonces esta expresión podŕıa interpretarse como la corrección no-conmutativa del equivalente de Weyl usual calculado en la base mixta. Sin embargo, esto es únicamente una mnemotecnia ya que la representación no-conmutativa aún se encuentra codificada dentro de la base mixta. La utilidad de (4.48) es clara al aplicarla a la matriz de densidad de un ensamble puro, i.e. ρ̂ = |ψ〉〈ψ|, en donde ρW (~q, ~p) = e− θ ~ p1∂q2 ∫ R2 dξdη e i ~ (q2η+p1ξ) 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ψ 〉〈 ψ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 , (4.49) que al comparar con las expresiones de §A.3 permite escribir al equivalente deWeyl no-conmutativo de la matriz de densidad como ρW = (2π~)2e− θ ~ p1∂q2ρw+ , (4.50) donde la función de Wigner-Szilard ρw+(~q, ~p) en la base mixta |q1, p2〉 se define (Rosenbaum, Vergara y Juárez) [66] como ρw+(~q, ~p) := (2π~)−2 ∫ R2 dξdη e i ~ (q2η+p1ξ) 〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ψ 〉〈 ψ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 . (4.51) La identidad (4.50) se aleja claramente de la relación usual de Mecánica Cuántica (A.53) para el equivalente de Weyl y la función de Wigner-Szilard. Consecuentemente varios de los resultados usuales deben modificarse en el caso no-conmutativo si es que se desean mantener las mismas interpretaciones f́ısicas de Mecánica Cuántica. En efecto, al sustituir (4.50) en (4.47) se tiene que 〈Â〉 = ∫ R4 d2~qd2~p ( e− θ ~ p1∂q2ρw+ ) AW = ∫ R4 d2~qd2~p ρw+ ( e θ ~ p1∂q2AW ) , (4.52) de forma que la integral de la segunda igualdad constituye ahora la versión no-conmutativa del valor de expectación de un equivalente Weyl AW en espacio-fase, con densidad de cuasiprobabil- idad ρw+ . Es decir que para preservar a la funcion de Wigner-Szilard como la cuasiprobabilidad de espacio-fase, el cálculo de valores de expectación implica hacer la traslación de todo equiv- alente de Weyl, obtenido según (4.29), por q2 → q2 + θ ~ p1 dentro de la integración. Por otro lado es posible reinterpretar a la función (2π~)−2ρW en (4.47) como la cuasiprobabilidad de espacio-fase, manteniendo aśı intactos a los equivalentes de Weyl. Tanto (2π~)−2ρW como ρw+ son densidades de cuasiprobabilidad, como lo muestra un sencillo cálculo de integración sobre todo el espacio-fase, sin embargo los marginales de (2π~)−2ρW en los subespacios q1 − p2 y q2 − p1, i.e. las integraciones en (q2, p1) y (q1, p2) respectivamente, 50 Caṕıtulo 4. Espacio No-conmutativo implican las densidades de probabilidad 1 (2π~)2 ∫ R2 dq2dp1 ρW = 〈q1, p2|ρ̂|q1, p2〉, 1 (2π~)2 ∫ R2 dq1dp2 ρW = 〈q2, p1|ρ̂|q2, p1〉, (4.53) mientras que para ρw+ se tiene ∫ R2 dq2dp1 ρw+ = 〈q1, p2|ρ̂|q1, p2〉, ∫ R2 dq1dp2 ρw+ = 〈q2 + (θ/~)p1, p1|ρ̂|q2 + (θ/~)p1, p1〉, (4.54) que es una forma alterna para (4.50). Es decir que, traducir la información proveniente de la función de Wigner-Szilard al equivalente de Weyl de la matriz de densidad ó viceversa involucra trasladar la coordenada q2 de los elementos de matriz por ± θ ~ p1. Se debe señalar que las expresiones anteriores son ciertas únicamente en el caso no-conmutativo, y no implican que la transición entre entidades conmutativas como las de §A.3 y no-conmutativas sea mediante un cambio de variables ad-hoc como sugiere el corrimiento de Bopp (4.4) (imple- mentado aśı por varios autores en diversas investigaciones e.g. [73, 74, 75]), ya que ambas densidades de cuasiprobabilidad se han calculado en una base no-conmutativa genuina. Para estudiar con mayor cuidado esta posible fuente de confusión y la manera de esclarecerla es pru- dente considerar el caso especial del problema de eigen-valores del Hamiltoniano para estados estacionarios Ĥ|ψ〉 = E|ψ〉, (4.55) de donde se sigue inmeditamente que la matriz de densidad de ensamble puro satisface la iden- tidad de operadores Ĥρ̂ = ρ̂Ĥ = Eρ̂. (4.56) Utilizando el isomorfismo (4.38) en la igualdad anterior implica que el equivalente de Weyl no-conmutativo de la matriz de densidad satisface entonces la ecuación de valores-⋆ HW ⋆~ ◦ ⋆θ ρW = ρW ⋆~ ◦ ⋆θ HW = EρW , (4.57) que, análogamente a (A.62), es la condición necesaria y suficiente para garantizar que el valor de expectación de la función Hamiltoniana HW en espacio-fase coincida con la enerǵıa del sistema 〈HW 〉 = (2π~)−2 ∫ R4 d2~qd2~p HW (~q, ~p) ⋆~ ◦ ⋆θ ρW (~q, ~p) = E. (4.58) Es posible obtener ahora una expresión similar a (4.57) para la función de Wigner-Szilard 4.3. Interpretación probabiĺıstica y ecuaciones de valores-⋆ 51 ρw+ , notando que la propiedad5 e θ ~ p1∂q2 [ AW (q1, q2, p1, p2) ⋆~ ◦ ⋆θ BW (q1, q2, p1, p2) ] = AW ( q1, q2 + θ ~ p1, p1, p2 ) ⋆~ BW ( q1, q2 + θ ~ p1, p1, p2 ) = [e θ ~ p1∂q2AW (~q, ~p)] ⋆~ [e θ ~ p1∂q2BW (~q, ~p)], (4.59) se cumple para dos equivalentes de Weyl arbitrarios AW y BW bajo el producto (4.37), lo cual implica entonces para (4.57) que e θ ~ p1∂q2 [ HW (~q, ~p) ⋆~ ◦ ⋆θ ρW (~q, ~p) ] = [e θ ~ p1∂q2HW (~q, ~p)] ⋆~ [e θ ~ p1∂q2ρW (~q, ~p)] = E[e θ ~ p1∂q2ρW (~q, ~p)], (4.60) y, finalmente, haciendo uso de (4.50) se obtiene cf. (Rosenbaum, Vergara y Juárez) [66] [e θ ~ p1∂q2HW (~q, ~p)] ⋆~ ρw+(~q, ~p) = Eρw+(~q, ~p), (4.61) cuya integración en espacio-fase, según (4.52), es precisamente 〈Ĥ〉 = ∫ R4 d2~qd2~p ( e θ ~ p1∂q2HW ) ρw+ = E. (4.62) Al expresar ahora la representación (4.49) en la otra base mixta |q2, p1〉, utilizando las fun- ciones de transición (4.17), luego de un cálculo directo se encuentra ρW = (2π~)2e θ ~ p2∂q1ρw− , (4.63) donde la función de Wigner ρw− , en la base mixta |q2, p1〉, se define como ρw−(~q, ~p) := (2π~)−2 ∫ R2 dξdη e i ~ (q1η+p2ξ) 〈 q2 − ξ 2 , p1 + η 2 ∣ ∣ ∣ψ 〉〈 ψ ∣ ∣ ∣q2 + ξ 2 , p1 − η 2 〉 , (4.64) que junto con (4.50) permite establecer la relación entre las dos funciones de Wigner ρw+ y ρw− como e− θ ~ p1∂q2ρw+ = e θ ~ p2∂q1ρw− = (2π~)−2ρW , (4.65) que es posible verificar haciendo el cambio de base directamente en (4.64). Debido a que, de forma paralela a la propiedad (4.59), también se tiene e− θ ~ p2∂q1 [ AW (~q, ~p) ⋆~ ◦ ⋆θ BW (~q, ~p) ] = [e− θ ~ p2∂q1AW (~q, ~p)] ⋆~ [e − θ ~ p2∂q1BW (~q, ~p)], (4.66) 5Teniendo cuidado que la acción izquierda y derecha de los bidiferenciales sea exclusivamente sobre los argu- mentos q2 y p1 de ambas funciones, de lo contrario este resultado no es inmediato. 52 Caṕıtulo 4. Espacio No-conmutativo consecuentemente, la función ρw− satisface la ecuación de valores-⋆ [e− θ ~ p2∂q1HW (~q, ~p)] ⋆~ ρw−(~q, ~p) = Eρw−(~q, ~p). (4.67) Tanto (4.61) como (4.67) están expresadas en términos del producto usual de Mecánica Cuántica, donde la no-conmutatividad aparece como un corrimiento del equivalente de Weyl del Hamiltoniano HW , sin olvidar que éste último se obtiene mediante la representación no- conmutativa de los observables de posición. Ahora se puede conjeturar la existencia de una tercera función de Wigner, combinando estos dos resultados, la cual satisface la ecuación de valores-⋆ con mismo eigenvalor E [e θ ~ (p1∂q2−p2∂q1 )HW (~q, ~p)] ⋆~ ρw0(~q, ~p) = Eρw0(~q, ~p), (4.68) con la definición ρw0 := (2π~)−2e θ ~ (p1∂q2−p2∂q1 )ρW , (4.69) que no se obtiene de una realización en base mixta en el sentido de las expresiones (4.51) y (4.64) donde el parámetro θ está ausente,6 sin embargo es una función leǵıtima de Wigner para el mismo valor de enerǵıa E. Es notable que, además de (4.57), en el régimen no-conmutativo se obtengan más de una ecuación de valores-⋆ para funciones de Wigner, como consecuencia de la propiedades distribu- tivas (4.59) y (4.66),7 contrastando con el caso usual de Mecánica Cuántica en que sólo existe una. Este interesante (y también inesperado) resultado de Mecánica Cuántica No-conmutativa no ha sido reportado en algún otro lado y se presenta aqúı por primera vez. Generalizando este resultado para hθ2n+1 (con todas las θij diferentes de cero) conduce a la existencia de 2n(n−1)− 1 funciones de Wigner ρs que satisfacen las ecuaciones de valores-⋆ con forma [ exp ( 1 ~ ∑ i,j∈s θijpi∂qj ) HW (~q, ~p) ] ⋆~ ρs(~q, ~p) = Eρs(~q, ~p), (4.70) donde s es un subconjunto de ı́ndices s ⊂ {1, ..., n}.8 El efecto de las ecuaciones anteriores sobre los estados de enerǵıa es el de una degeneración 6Es posible reconstruir la base que la genera, aunque esto no proporciona información nueva. 7Nótese también que e θ 2~ (p2∂q1 −p1∂q2 )[AW ⋆~ ◦ ⋆θ BW ] 6= [e θ 2~ (p2∂q1 −p1∂q2 )AW ] ⋆~ [e θ 2~ (p2∂q1 −p1∂q2 )BW ], y, por lo tanto, un cambio de variables del tipo (4.4) no es viable para recuperar una ecuación de valores-⋆. 8El cálculo combinatorio muestra que para θij no singular en n dimensiones el número total de términos posibles ∑ i,j∈s θ ijpi∂qj corresponde a n(n−1) ∑ m=1 ( n(n− 1) m ) = n(n−1) ∑ m=0 ( n(n− 1) m ) − 1 = 2n(n−1) − 1, (4.71) en caso que θij sea singular su rango reemplaza a n en la expresión anterior. 4.3. Interpretación probabiĺıstica y ecuaciones de valores-⋆ 53 con multiplicidad igual a 2n(n−1) − 1, además de las posibles degeneraciones por la forma par- ticular de HW como, por ejemplo, en forma de niveles de Landau para el oscilador harmónico que surgen por cada uno de los corrimientos θijpi∂qj , cf.[76]. Nótese que para θ → 0 las expresiones (4.65) y (4.69) proyectan efectivamente a una sola función de Wigner que corresponde a la del caso conmutativo, lo que permite recuperar los resultados usuales de §A.3. Además, en el caso no-conmutativo las funciones ρw+ , ρw− y ρw0 de ensamble puro continúan siendo funciones genuinas de Wigner-Szilard (en el sentido de (A.66)), ya que la propiedad de proyector de la matriz de densidad ρ̂2 = ρ̂ y el isomorfismo (4.38) implican para el equivalente de Weyl ρW ⋆~ ◦ ⋆θ ρW = ρW , (4.72) que, recurriendo nuevamente a las propiedades (4.59) y (4.66) y usando (4.50), (4.63) y (4.69), conducen finalmente a (2π~)2ρw+ ⋆~ ρw+ = ρw+ , (2π~)2ρw− ⋆~ ρw− = ρw− , (2π~)2ρw0 ⋆~ ρw0 = ρw0 . (4.73) Las soluciones a las ecuaciones diferenciales (4.57), (4.61) y (4.67) son equivalentes entre śı en virtud de (4.65) y (4.69). Estas formas de escribir la ecuación de valores propios del Hamiltoniano en espacio-fase garantizan, como se mostró, que el valor promedio de la función Hamiltoniana coincida con la enerǵıa del sistema. Sin embargo, regresando a la expresión más débil (4.58) y utilizando las propiedades integrales de los productos-⋆, se pueden obtener casos especiales de ecuaciones cuyas soluciones continúen satisfaciendo el valor de expectación, pero no necesariamente posean la interpretación probabiĺıstica apropiada. Un ejemplo muy sugerente proviene de considerar un Hamiltoniano mecánico, cuyo equivalente de Weyl toma la forma HW (q1, q2, p1, p2) = 1 2m (p21 + p22) + VW (q1, q2), (4.74) donde la función VW es el equivalente de Weyl del potencial cuántico. Entonces, usando las propiedades integrales de los bidiferenciales en (4.58), el valor de ex- pectación 〈HW 〉 admite la forma integral 〈HW 〉 = (2π~)−2 ∫ R4 d2~qd2~p ( 1 2m (p21 + p22) ⋆~ ρW (~q, ~p) + VW (q1, q2) ⋆θ ρW (~q, ~p) ) = E, (4.75) de manera que basta con que ρW satisfaga la ecuación diferencial reducida 1 2m (p21 + p22) ⋆~ ρW (~q, ~p) + VW (q1, q2) ⋆θ ρW (~q, ~p) = EρW (~q, ~p), (4.76) para recuperar el valor promedio. Aśı, el término del potencial VW ⋆θ ρW no involucra más derivaciones de ρW con respecto a los momenta, lo cual invita a postular una ecuación de 54 Caṕıtulo 4. Espacio No-conmutativo Schrödinger no-conmutativa ”equivalente” de la forma 1 2m (P̂ 2 1 + P̂ 2 2 )ψ(q1, q2) + V (q1, q2) ⋆θ ψ(q1, q2) = Eψ(q1, q2). (4.77) Desde una perspectiva de ecuaciones diferenciales es válido estudiar las soluciones de (4.77) y las interpretaciones de |ψ(q1, q2)|2 (ver [73, 77]), aunque, como puede verse de las suposiciones hechas para llegar a tal expresión, desde la perspectiva de una teoŕıa formal de cuantización por deformación hay poca justificación de que sea un resultado universal dentro de la teoŕıa cuántica del álgebra (4.1). 4.4 Operadores de Heisenberg y paréntesis de Poisson: El paso a Teoŕıa de Campos Habiendo estudiado en detalle equivalentes de Weyl de operadores independientes del tiempo (de Schrödinger) es natural considerar ahora el caso de operadores de Heisenberg que, como se sabe, son obtenidos de la acción unitaria del Hamiltoniano sobre operadores de Schrödinger ÂH(t) := e i ~ tĤÂe− i ~ tĤ , (4.78) que, por tratarse puramente de una expresión de operadores que actúan en el espacio de Hilbert H, puede traducirse inmediatamente a funciones de Weyl bajo el isomorfismo (4.38) como AH W (t) = e i ~ tHW ∗ ∗AW ∗ e − i ~ tHW ∗ , (4.79) donde, como se definió en el párrafo previo a (4.30), ∗ = ⋆~ ◦ ⋆θ y las funciones e ± i ~ tHW ∗ son las series formales e ± i ~ tHW ∗ = ∞∑ n=0 (±it)n ~nn! (HW )n∗ , (4.80) con (HW )n∗ = HW ∗ . . . ∗HW ︸ ︷︷ ︸ n . Se puede mostrar fácilmente que (4.79) es correcta siguiendo argumentos similares a los que conducen a (A.31). Cuando AW no depende expĺıcitamente de t, la derivación formal de (4.79) conduce a la ecuación de evolución para equivalentes de Weyl en términos del conmutador (4.41) d dt AH W (t) = i ~ ( HW ∗AH W (t)−AH W (t) ∗HW ) = i ~ [HW , A H W (t)]∗, (4.81) ó dada la anti-hermiticidad del operador (4.30) e intercambiando el orden del segundo término 4.4. Operadores de Heisenberg y paréntesis de Poisson: El paso a Teoŕıa de Campos 55 se tiene d dt AH W (t) = −2 ~ HW sen[ 1 2 (~ ↔ Π + θ ↔ Λ)]AW := HW ↔ Mθ A H W (t), (4.82) donde, en analoǵıa con el paréntesis de Moyal usual (A.42), el operador ↔ Mθ := −2 ~ sen[12(~ ↔ Π+θ ↔ Λ)] corresponde al paréntesis no-conmutativo. En el contexto dinámico de funciones de espacio-fase dependientes del tiempo, la expresión (4.82) (o equivalentemente (4.81)) induce una estructura de Poisson {·, ·}∗ : C∞⋆~◦⋆θ(R4)⊗ C∞⋆~◦⋆θ(R4) −→ C∞⋆~◦⋆θ(R4) {f, g}∗ := g ↔ Mθ f = i ~ [g, f ]∗, (4.83) tal que la evolución de un equivalente de Weyl de un operador de Heisenberg adquiere la forma Hamiltoniana d dt AH W (t) = {AH W , HW }∗. (4.84) Partiendo de la expresión anterior y de (4.79) se puede entonces mostrar que la evolución de todo equivalente de Weyl de un operador de Heisenberg depende únicamente del paréntesis {AW , HW }∗, donde, como siempre, AW es el equivalente de Weyl del operador de Schrödinger. Efectivamente d dt AH W (t) = i ~ (HW ∗AH W −AH W ∗HW ) = i ~ e i ~ tHW ∗ ∗ (HW ∗AW −AW ∗HW ) ∗ e− i ~ tHW = e i ~ tHW ∗ ∗ {AW , HW }∗ ∗ e − i ~ tHW ∗ .  (4.85) Lo anterior significa que determinar la evolución de un equivalente de Weyl AW (~q, ~p) bajo la función Hamiltoniana HW (~q, ~p) se reduce a evaluar paréntesis elementales de qi y pi. 9 Esto muestra que, en una formulación Hamiltoniana de la Mecánica Cuántica No-conmutativa (ó en el contexto de observables dentro de una deformación de Gerstenhaber, ver [26]), las variables qi, pi, que aparecieron originalmente sólo como parámetros, son formalmente las variables dinámicas de la teoŕıa (Rosenbaum, Vergara y Juárez) [66] con estructura canónica naturalmente inducida 9Esto es siempre cierto, ya que para el producto de yuxtaposición de una función arbitraria f(~q, ~p) con una coordenada qi se tiene qif = 1 2 (qi ∗ f + f ∗ qi) y, como la estructura canónica es una derivación bajo el producto-∗ {f, g ∗ h}∗ = {f, g}∗ ∗ h+ g ∗ {f, h}∗, entonces {qi, qjf}∗ = {qi, qj}∗f + qj{qi, f}∗, {qi, qj ∗ f}∗ = {qi, qj}∗ ∗ f + qj ∗ {qi, f}∗, siguiendo el mismo argumento para las demás combinaciones posibles de q’s y p’s. 56 Caṕıtulo 4. Espacio No-conmutativo por (4.42), i.e. {q1, q2}∗ = − i ~ [q1, q2]∗ = θ ~ , {qi, pj}∗ = − i ~ [qi, pj ] = δij , {p1, p2}∗ = 0. (4.86) Similarmente, en virtud de (4.44), para mayores dimensiones se obtienen los paréntesis {qi, qj}∗ = θij ~ , {qi, pj}∗ = δij , {pi, pj}∗ = 0, (4.87) que pueden verse ahora como la estructura de espacio–fase asociada a la subálgebra de configu- ración Aθ, generada por qi’s y sus productos qi ⋆θ q j = qie i 2 θkl ↔ Λklqj , (4.88) definiendo el paréntesis de Poisson {qi, qj} como {qi, qj} := − i ~ [q1, q2]∗, (4.89) y los demás paréntesis con la forma usual, como se ve de (4.87). Esto implica que en una formulación de campos no-conmutativos definidos en Aθ, entendido esto como un módulo sobre el anillo de funciones no-conmutativas de Aθ, los campos mismos φµν...αβ...(~q) heredaran el producto-⋆θ. Caṕıtulo 5 Representaciones alternas de la No-conmutatividad Debido a que el espacio-fase de equivalentes de Weyl para operadores de Mecánica Cuántica No-conmutativa con álgebra (4.1) es, propiamente, el espacio eucĺıdeo R 2n, como se mostró desde el contexto dinámico en §4.4, esto permite formular la Mecánica Cuántica No-conmutativa dentro de los esquemas de cuantización más generales discutidos en el Cap.3. Tanto por que es posible implementar el algoritmo (3.23) para construir el cuantizador como porque existe el operador de reflexión (3.34), por tratarse de un espacio simétrico. La estrategia es construir una base supercompleta de estados coherentes no-conmutativos como se presentó en el art́ıculo de investigación ”On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group, L. Román Juárez and Marcos Rosenbaum, J. Phys. Math. 2, pp. 29-50 (2010)” (Ref. [59]). Esto permitirá hacer un análisis comparativo y mostrar un resultado importante sobre las realizaciones holomorfas de la no-conmutatividad en espacio plano. En §5.3 se estudia la no-conmutatividad desde la perspectiva de la cuantización canónica de Dirac [32], introduciendo estucturas simplécticas generalizadas, como se mencionó al final de §2.4, partiendo de conceptos de invariancia bajo reparametrización y la cuantización de teoŕıas con constricciones, de acuerdo al programa diseñado en ”Noncommutativity from Canonical and Noncanonical Structures, Marcos Rosenbaum, J. David Vergara and L. Román Juárez, Contemporary Math. 462, pp. 10367-10382 (2008)” (Ref. [33]). El formalismo de la integral de trayectoria en Mecánica Cuántica No-conmutativa para los diversos esquemas de cuantización presentados es descrito en §5.4. Finalmente en §5.5 se propor- ciona una śıntesis del atractivo formalismo matemático de Connes inspirado en las álgebras de operadores mecánico-cuánticos (como las presentadas), que busca implementar una formulación algebráica aún más sofisticada extendiéndola a los conceptos usuales de variedades diferenciales, partiendo del Teorema de Gel’fand-Naimark [16] en su iteración para álgebras no-conmutativas, para substituirlos por geometŕıas menos convencionales carentes de puntos o vecindades. 57 58 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad 5.1 Estados coherentes no-conmutativos Diversos sistemas supercompletos de estados coherentes se han construido para teoŕıas no- conmutativas (e.g. [78, 79, 80, 81]), donde es importante señalar que las representaciones en estos trabajos no son equivalentes entre śı y consecuentemente los resultados obtenidos por su implementación pueden variar. Como se mencionó en el preámbulo de éste caṕıtulo, los desar- rollos a continuación descritos se especializarán al uso de los estados coherentes construidos en (Juárez, Rosenbaum) [59], siguiendo la definición de estado coherente generalizado (3.3). Un sistema de estados coherentes no-conmutativos será aquel conformado por los estados de la Def.1, donde el grupo de simetŕıa correspondiente es el grupo de Heisenberg-Weyl extendido Hθ 2n+1 con álgebra de Lie hθ2n+1 de conmutadores (4.1). Definiendo primero una base alterna de operadores (no hermitianos) para el álgebra (4.1) como Âi := 1√ 2~ (Q̂i + 1 2~ n∑ j=1 θijP̂j + iP̂i), †i := 1√ 2~ (Q̂i + 1 2~ n∑ j=1 θijP̂j − iP̂i), (5.1) de manera que los observables Q̂i y P̂i en términos de esta base son Q̂i = √ ~ 2 [ Âi + †i + i 2~ n∑ j=1 θij(Âj − †j) ] , P̂i = −i √ ~ 2 (Âi − †i ). (5.2) lo cual, como puede verse fácilmente, permite escribir a los operadores de desplazamiento (4.18) en la forma alterna D̂(~α) = exp [ n∑ i=1 (αi † i − α∗i Âi) ] , (5.3) donde αi, α ∗ i ∈ C.1 Por lo tanto un elemento t́ıpico (exponenciación) g ∈ Hθ 2n+1 admite la descomposición en clases laterales g = g(c, ~α) = eiĉID̂(~α), (5.4) con c ∈ R. De la expresión anterior es evidente que el operador identidad Î genera el subgrupo maximal compacto U(1) el cual, de acuerdo a la discusión de §3.1, conduce al espacio homogéneo C n ≃ 1No debe confundirse la notación para la conjugación compleja z ∈ C ⇔ z∗ ∈ C con el producto-∗ definido en (4.30). 5.1. Estados coherentes no-conmutativos 59 R 2n ≃ Hθ 2n+1/U(1). Implementando ahora la definición de estado coherente generalizado (3.3), permite concluir que el sistema de estados coherentes no-conmutativos, etiquetados por puntos de C n, se obtiene de la acción transitiva de los operadores de desplazamiento sobre cualquier estado normalizado |ϕ0〉 ∈ H |~α〉 = D̂(~α)|ϕ0〉 = exp [ n∑ i=1 (αi † i − α∗i Âi) ] |ϕ0〉, (5.5) ya que todos los estados de H son U(1)-invariantes. Para elegir convenientemente el estado fiducial |ϕ0〉 nótese que, como consecuencia directa de los conmutadores (4.1), los operadores (5.1) satisfacen el álgebra n-dimensional de operadores de creación y destrucción bosónicos [Âi,  † j ] == δij , [Âi, Âj ] = [†i ,  † j ] = 0. (5.6) donde cada par Âi,  † i genera un espacio de Hilbert Hi en la base de Fock {|mi〉} tal que |mi〉 = (A†i ) mi √ mi! |0〉, 〈mi|ni〉 = δmn, mi, ni ∈ N, Âi|mi〉 = √ mi|mi − 1〉, †i |mi〉 = √ mi + 1|mi + 1〉, (5.7) y, por lo tanto, el espacio de Hilbert completoH = ⊗n i=1Hi es generado por la base {|m1, ...,mn〉}. Entonces, fijando |ϕ0〉 = |0〉,2 y con ayuda del teorema BCH (2.4), los estados coherentes (5.5) toman la forma |~α〉 = |α1, ..., αn〉 = n⊗ i=1 e− 1 2 |αi|2 ∞∑ mi=0 (αi) mi √ mi! |mi〉, (5.8) que son, precisamente, los estados coherentes usuales de Glauber (cf. [50]) en n-dimensiones correspondientes a los eigenestados del problema de valores propios Âi|~α〉 = αi|~α〉, (5.9) para Âi definido como en (5.1). La resolución de la unidad (3.4) está dada por ÎH = ∫ Cn dµ(~α, ~α∗)|~α〉〈~α|, dµ(~α, ~α∗) = 1 (2πi)n n∏ i=1 dαi ∧ dα∗i = 1 πn n∏ i=1 d2αi, (5.10) 2Nótese que cualquier otro estado |m1, ...,mn〉 podŕıa haberse utilizado para dicho fin, dado que todos ellos son estados normalizados de H. 60 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad y la función de transición de dos estados coherentes define, como se sabe, el kernel reproducente K(~α, ~β) K(~α, ~β) = 〈~α|~β〉 = e(− 1 2 |~α|2− 1 2 |~β|2+~α∗·~β), K(~α,~γ) = ∫ Cn dµ(~β, ~β∗)K(~α, ~β)K(~β,~γ), (5.11) donde ~α∗ · ~β denota el producto escalar de los vectores ~α∗ y ~β. Finalmente la acción de un operador de desplazamiento D̂(~α) sobre un estado coherente arbitrario |~β〉 es consecuencia de la ley de multiplicación de dos operadores de desplazamiento, i.e. D̂(~α)D̂(~β) = e 1 2 (~α· ~β∗− ~α∗·~β)D̂(~α+ ~β), (5.12) y, por lo tanto, D̂(~α)|~β〉 = D̂(~α)D̂(~β)|0〉 = e 1 2 (~α· ~β∗− ~α∗·~β)|~α+ ~β〉, (5.13) que muestra en forma clara por que denominar operador de desplazamiento a D̂(~α) es tan apropiado. El que los estados (5.8) resuelvan la ecuación de valores propios (5.9) conduce a la propiedad central que justificará su implementación en el contexto de Mecánica Cuántica No-conmutativa en secciones posteriores. Utilizando (5.2), entonces, es directo ver que los valores de expectación de estado coherente de los observables Q̂i y P̂i corresponden a las expresiones qi : = 〈~α|Q̂i|~α〉 = √ ~ 2 [ αi + α∗i + i 2~ n∑ j=1 θij(αj − α∗j ) ] , pi : = 〈~α|P̂i|~α〉 = −i √ ~ 2 (αi − α∗i ), (5.14) lo cual confirma la naturaleza semiclásica de los estados coherentes, como estados etiquetados por variables de espacio-fase que describen el centro de estados Gaussianos. Las expresiones (5.14) permiten utilizar variables de posición y momento simultáneas, por tratarse de valores de expectación, contrario al uso de eigenvalores del álgebra (4.1) donde, como se vió en el Cap.4, esto no es posible. Dicha propiedad es conocida ya en la Mecánica Cuántica ordinaria, sin embargo en el contexto de la No-conmutatividad del espacio cobra aún mayor importancia, ya que una descripción en términos de variables de configuración suele proporcionar mayor intuición sobre el comportamiento de un sistema que, por ejemplo, una descripción en variables mixtas del tipo (4.6). Ésta particularidad fué explotada en el estudio de un modelo de Cosmoloǵıa Cuántica No-conmutativa en ”Noncommutative Coherent States and Quantum Cosmology, Román Juárez and David Mart́ınez, arXiv:1403.2849 sometido a publicación.” (Ref. [82]). 5.2. Realización holomorfa del producto–∗: La equivalencia del cuantizador y el operador de reflexión 61 5.2 Realización holomorfa del producto–∗: La equivalencia del cuantizador y el operador de reflexión Habiendo construido el sistema de estados coherentes no-conmutativos asociados al álgebra (4.1), es posible implementar ahora el algoritmo presentado en §3.2 para obtener el cuantizador de Stratonovich-Weyl (3.23) en términos de variables holomorfas y por ende el producto-⋆ asociado. De acuerdo al método general es necesario contar con las funciones ortogonales Yζ(~α, ~α∗) de Cn, que resuelven el problema de valores propios n∑ i=1 ∂2 ∂αi∂α∗i Yζ(~α, ~α∗) = −|ζ|2Yζ(~α, ~α∗), (5.15) cuyas soluciones normalizadas, bajo la medida dµ(~α, ~α∗), son las exponenciales Yζ(~α, ~α∗) = π−n/2e(~α· ~ζ∗− ~α∗·~ζ), ~ζ ∈ C n. (5.16) Los coeficientes ξΞ que aparecen en la fórmula (3.23) se obtienen, de acuerdo a (3.21), del cálculo de Tr[D̂(~α)D̂†(~β)] que, por las expresiones (5.11-5.13) y la ortogonalidad de las funciones (5.16), es Tr[D̂(~α)D̂†(~β)] = ∫ Cn dµ(~ρ, ~ρ∗)〈~ρ|D̂(~α)D̂†(~β)|~ρ〉 = πnδ(n)(~α− ~β), (5.17) de donde se concluye que ξΞ = πn. Consecuentemente, la fórmula (3.23) del cuantizador asociado al álgebra (4.1) corresponde al kernel pseudodiferencial Q̂(~α) = π−n/2 ∫ Cn d2n~ζ Yζ(~α, ~α∗)D̂(~ζ) = ∫ Cn dµ(~ζ, ~ζ∗) e( ~α∗·~ζ−~α· ~ζ∗)D̂(~ζ), (5.18) que, como se puede verificar rápidamente, satisface todos los postulados (3.11). Por lo tanto, según la expresión impĺıcita (3.16) para el producto deformado en términos del trikernel, se puede obtener una representación en variables holomorfas del producto-⋆ calculando primero la función de tres puntos L(~α, ~β,~γ) = Tr[Q̂(~α)Q̂(~β)Q̂(~γ)], (5.19) que, por las propiedades de los operadores de desplazamiento junto con la traza (5.17) y la ortogonalidad de las funciones (5.16), es igual a L(~α, ~β,~γ) = 4e4iℑ[~α· ~γ∗+~β· ~α∗+~γ· ~β∗], (5.20) 62 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad donde ℑ(z) = 1 2i(z − z∗). Sustituyendo el resultado anterior en (3.16) conduce al producto-⋆ de equivalentes de Stratonovich- Weyl FA, FB ∈ C∞⋆ (Cn), para operadores Â, B̂ de Mecánica Cuántica No-conmutativa bajo el isomorfismo (3.8) FA(~α, ~α∗) ⋆ FB(~α, ~α∗) = 4 ∫ C2n dµ(~β, ~β∗)dµ(~γ, ~γ∗) [ e4iℑ[~α· ~γ∗+~β· ~α∗+~γ· ~β∗] ×FA(~β, ~β∗)FB(~γ, ~γ∗) ] , (5.21) que, con el cambio de variables ~γ = ~ρ+ ~α, admite la forma convolutiva FA(~α, ~α∗) ⋆ FB(~α, ~α∗) = 4 ∫ C2n dµ(~β, ~β∗)dµ(~ρ, ~ρ∗) [ e4iℑ[~α· ~ρ∗+~ρ· ~β∗] ×FA(~β, ~β∗)FB(~α+ ~ρ, ~α∗ + ~ρ∗) ] . (5.22) Empleando técnicas similares a las de las identidades (2.35) y (4.36) permite escribir e4iℑ[~α· ~ρ∗+~ρ· ~β∗]FB(~α+ ~ρ, ~α∗ + ~ρ∗) = e4iℑ[~α· ~ρ∗+~ρ· ~β∗]e 1 2 (←−∇α· −→∇α∗− ←−∇α∗ · −→∇α ) FB(~α, ~α∗), (5.23) con ←−∇α · −→∇α∗ − ←−∇α∗ · −→∇α = n∑ i=1 ( ←− ∂ αi −→ ∂ α∗i −←−∂ α∗i −→ ∂ αi), (5.24) que al substituir en (5.22) y realizando las integraciones restantes conduce al resultado final FA(~α, ~α∗) ⋆c FB(~α, ~α∗) = FA(~α, ~α∗)e 1 2 (←−∇α· −→∇α∗− ←−∇α∗ · −→∇α ) FB(~α, ~α∗). (5.25) donde el producto-⋆c, 3 inducido por el cuantizador (5.18), es el operador de variables holomorfas ⋆c := e 1 2 (←−∇α· −→∇α∗− ←−∇α∗ · −→∇α ) . (5.26) El aspecto más importante del desarrollo anterior y del producto-⋆c (5.25), como se destacó en (Juárez y Rosenbaum) [59], es la clara ausencia de la no-conmutatividad θij . Por lo tanto, todas las expresiones son indistinguibles de aquellas obtenidas con una formulación holomorfa de Mecánica Cuántica usual. Esta propiedad conduce a una notoria ventaja matemática so- bre formulaciones que invocan variables f́ısicas, ya que entonces los resultados de una teoŕıa conmutativa pueden exportarse directamente al contexto no-conmutativo y recuperar las inter- pretaciones f́ısicas más tarde usando expresiones del tipo (5.14). Efectivamente, como es fácil 3La notación escogida ⋆c sirve como mnemotécnia, indicando que es un producto en variables complejas. 5.2. Realización holomorfa del producto–∗: La equivalencia del cuantizador y el operador de reflexión 63 mostrar, para el caso en consideración se tiene ⋆c = ∗ = ⋆~ ◦ ⋆θ, (5.27) donde ⋆~ ◦ ⋆θ es el producto-∗ de R 2n definido en (4.43). La expresión (5.27) permite concluir que las definiciones (5.14) y las variables f́ısicas (qi, pi) utilizadas ampliamente en el Cap.4 corresponden al mismo conjunto de variables dinámicas. Esto proporciona una mejor interpretación ya que si bien no corresponden a valores propios (si- multáneos) de los operadores de posición y de momento, si representan la localización promedio en espacio-fase de estados coherentes no-conmutativos que resuelven Âi|~α〉 = αi|~α〉, para Âi definido en (5.1). Esta interpretación es ideal para contextos semiclásicos y particularmente en métodos asintóticos de integrales de trayectoria cf. (Juárez y Mart́ınez) [82]. Finalmente, en el marco de cuantización de Berezin-Weyl de §3.3, con ayuda de los estados coherentes (5.8), es posible estudiar el tipo de isomorfismo que el operador de reflexión (3.34) induce en los operadores de Mecánica Cuántica No-conmutativa y establecer la relación con construcciones previas. Para ello se obtendrá la representación del operador de reflexión en términos de estados coherentes, postulando primero el operador Û(0) := ∫ Cn dµ(~β, ~β∗) | − ~β〉〈~β|, (5.28) el cual como se puede ver, por las propiedades del kernel reproducente (5.11), actúa sobre cualquier estado coherente |~α〉 según Û(0)|~α〉 = ∫ Cn dµ(~β, ~β∗) | − ~β〉〈~β|~α〉 = | − ~α〉, (5.29) entonces, en particular Û(0)|0〉 = |0〉, Û(0)2 = Î, (5.30) de lo cual se concluye que Û(0) es el operador de reflexión alrededor del oŕıgen. Notando ahora que, por las propiedades (5.30), la transformación unitaria D̂(~ζ)Û(0)D̂†(~ζ) satisface (D̂(~ζ)Û(0)D̂†(~ζ))2 = Î, D̂(~ζ)Û(0)D̂†(~ζ)|~ζ〉 = |~ζ〉, (5.31) y, entonces Û(~ζ) := D̂(~ζ)Û(0)D̂†(~ζ), (5.32) es necesariamente, cf. (3.42), el operador de reflexión alrededor ~ζ ∈ C n. Efectivamente, la acción de Û(~ζ) sobre un estado coherente resulta de usar (5.13) y (5.29), de donde se tiene Û(~ζ)|~α〉 = e( ~ζ∗·~α−~ζ· ~α∗)|2~ζ − ~α〉, (5.33) 64 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad que escribiendo el argumento del estado coherente como 2(~ζ − ~α) + ~α confirma que se trata de la reflexión plana de ~α alrededor de ~ζ. Para completar la demostración, y por consistencia con (5.31), la acción repetida sobre (5.33) muestra que Û(~ζ) es una involución Û(~ζ)2|~α〉 = e( ~ζ∗·~α−~ζ· ~α∗)e[ ~ζ∗·(2~ζ−~α)−~ζ·(2 ~ζ∗− ~α∗)]|2~ζ − (2~ζ − ~α)〉 = |~α〉.  (5.34) Una representación integral de Û(~ζ) se obtiene de substituir (5.28) en (5.32) Û(~ζ) = ∫ Cn dµ(~β, ~β∗)D̂(~ζ) | − ~β〉〈~β|D̂†(~ζ), (5.35) que, evaluando la acción de los operadores de desplazamiento sobre los estados coherentes con (5.13), conduce a Û(~ζ) = ∫ Cn dµ(~β, ~β∗) e( ~ζ∗·~β−~ζ· ~β∗)|~ζ − ~β〉〈~ζ + ~β|, (5.36) ó, equivalentemente, con el cambio de variables ~ρ = ~ζ + ~β Û(~ζ) = ∫ Cn dµ(~ρ, ~ρ∗) e( ~ζ∗·~ρ−~ζ· ~ρ∗)|2~ζ − ~ρ〉〈~ρ|. (5.37) Es posible mostrar, usando expresiones previas y el desarrollo expĺıcito de estados coherentes (5.8), que la siguiente identidad es válida e( ~ζ∗·~ρ−~ζ· ~ρ∗)|2~ζ − ~ρ〉 = 1 2n ∫ Cn dµ(~λ, ~λ∗)e( ~λ∗·~ζ−~λ· ~ζ∗)e 1 2 ( ~ρ∗·~λ−~ρ· ~λ∗)|~λ+ ~ρ〉, (5.38) en virtud que el caso n-dimensional es simplemente el producto de n copias del caso unidimen- sional. Entonces, partiendo de la integral 1 2 ∫ C dµ(~λ, ~λ∗) e(λ ∗ζ−λζ∗)e 1 2 (ρ∗λ−ρλ∗)|λ+ ρ〉 = 1 2 e(ζ ∗ρ−ζρ∗) ∫ C dµ(~γ, ~γ∗) e(γ ∗ζ−γζ∗)e 1 2 (ρ∗γ−ργ∗)|γ〉 = 1 2 e(ζ ∗ρ−ζρ∗) ∫ C dµ(~γ, ~γ∗) e[γ ∗(ζ− 1 2 ρ)−γ(ζ∗− 1 2 ρ∗)]e− 1 2 |γ|2 ∞∑ m=0 γm√ m! |m〉, (5.39) y notando que para un estado coherente |σ〉 la identidad 〈m|σ〉 = ∫ C dµ(~γ, ~γ∗)〈m|γ〉〈γ|σ〉, (5.40) es equivalente a σm = ∫ C dµ(~γ, ~γ∗)e(−|γ| 2+γ∗σ)γm, (5.41) 5.2. Realización holomorfa del producto–∗: La equivalencia del cuantizador y el operador de reflexión 65 al escribir (5.39) en esta forma, mediante el reemplazo γ → √ 2γ, permite concluir 1 2 ∫ C dµ(~λ, ~λ∗) e(λ ∗ζ−λζ∗)e 1 2 (ρ∗λ−ρλ∗)|λ+ ρ〉 = e(ζ ∗ρ−ζρ∗) ∫ C dµ(~γ, ~γ∗) e[−|γ| 2+ √ 2γ∗(ζ− 1 2 ρ)]e− √ 2γ(ζ∗− 1 2 ρ∗) ∞∑ m=0 ( √ 2γ)m√ m! |m〉 = e(ζ ∗ρ−ζρ∗)e−2|ζ− 1 2 ρ| ∞∑ m=0 (2ζ − ρ)m√ m! |m〉 = e(ζ ∗ρ−ζρ∗)|2ζ − ρ〉.  (5.42) Ahora bien, dado que el término e 1 2 ( ~ρ∗·~λ−~ρ· ~λ∗)|~λ+ ~ρ〉 en la integral del lado derecho de (5.38) es precisamente la acción D̂(~λ)|~ρ〉, entonces se tiene e( ~ζ∗·~ρ−~ζ· ~ρ∗)|2~ζ − ~ρ〉 = 1 2n ∫ Cn dµ(~λ, ~λ∗)e( ~λ∗·~ζ−~λ· ~ζ∗)D̂(~λ)|~ρ〉, (5.43) que al substituir en (5.37) implica Û(~ζ) = 1 2n ∫ Cn dµ(~λ, ~λ∗)e( ~λ∗·~ζ−~λ· ~ζ∗)D̂(~λ), (5.44) donde la integral en ~ρ desaparece por ser trivialmente la resolución de la identidad. Comparando (5.44) con (5.18) se obtiene el resultado Q̂(~α) = 2nÛ(~α), (5.45) que permite concluir que los isomorfismos, y por lo tanto los productos–⋆c, inducidos por el operador de reflexión Û(~α) y el cuantizador Q̂(~α), entre el álgebra End(H) de operadores de Mecánica Cuántica No-conmutativa y el espacio de funciones holomorfas C∞(Cn), son equiva- lentes (hasta un factor de proporcionalidad 2n) y, que a su vez, en virtud de (5.14) y (5.27), son equivalentes a la realización de variables f́ısicas. Una versión del teorema anterior, utilizando elementos del kernel de Bergman (ver [83]), se obtuvo en ”On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group, L. Román Juárez and Marcos Rosenbaum, J. Phys. Math. 2, pp. 29-50 (2010)” (Ref. [59]). 66 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad 5.3 Invariancia de Reparametrización, Cuantización Canónica y No-conmutatividad Al final de §2.4 se esbozó un programa de cuantización que permite obtener conmutadores no triviales de operadores de posición cuánticos (y también de momento), basado en el método de cuantización de Dirac y la analoǵıa clásica [31, 32]. En un contexto más interesante, ésta idea permite considerar una No-conmutatividad completa del Espacio-Tiempo, ya que hasta ahora la variable temporal ha permanecido fuera de dicha posibilidad, usando el concepto de invariancia de parametrización [84, 85]. De ésta manera el tiempo es tomado como una variable dinámica más del sistema y permite introducir fácilmente una estructura no-canónica en dicho espacio-fase extendido. Entonces, de acuerdo al método general establecido en ”Noncommutativity from Canonical and Noncanonical Structures, Marcos Rosenbaum, J. David Vergara and L. Román Juárez, Contemporary Math. 462, pp. 10367-10382 (2008)” (Ref. [33]), el punto inicial es considerar sistemas invariantes de reparametrización. Esto significa que si un sistema no es naturalmente invariante bajo reparametrizaciones, entonces los parámetros originales deben promoverse al nivel de variables dinámicas. Tomando el modelo de una part́ıcula en un espacio de configuración n-dimensional y un potencial arbitrario, como ejemplo elemental de lo anterior (ver e.g. [86]), se tiene la acción Lagrangiana S = t2∫ t1 dt ( 1 2 m ( dqi dt )2 − V ( qi, t ) ) , (5.46) con i = 1, ..., n. Como el tiempo es el parámetro de esta teoŕıa, entonces, al promover t = q0(τ), extendiendo aśı el espacio-fase e introduciendo un nuevo parámetro τ , la acción (5.46) se escribe como S = τ2∫ τ1 dτ ( 1 2 m ( dqi dτ )2( dτ dt ) − V ( qi, t ) ( dt dτ )) , (5.47) al substituir q̇i = dqi dτ y q̇0 = dt dτ se tiene S = τ2∫ τ1 dτ ( 1 2 m (q̇i)2 q̇0 − V (qi, q0)q̇0 ) . (5.48) En la formulación Hamiltoniana, las variables canónicas del espacio-fase extendido son ahora (q0, p0, q i, pi), con los momentos definidos como usualmente pi = ∂L ∂q̇i y p0 = ∂L ∂q̇0 , entonces (5.48) toma la forma S = τ2∫ τ1 dτ ( p0q̇ 0 + piq̇ i − λϕ ) , (5.49) 5.3. Invariancia de Reparametrización, Cuantización Canónica y No-conmutatividad 67 donde ϕ = p0 +H ≈ 0 es una constricción (primaria) de primera clase, con la función Hamilto- niana original H = ∑ i p2i 2m + V . La constricción ϕ está asociada estrechamente con la simetŕıa de reparametrización y aparece ahora en la acción con un multiplicador de Lagrange λ. Esto permite mantener el número original de grados de libertad de la teoŕıa sin inconsistencias. Las ecuaciones de Hamilton resultantes de (5.49) implican, en particular, que q̇0 = λ. De forma que fijar una parametrización (tiempo) determina el multiplicador λ y viceversa. Por esta razón la acción resulta invariante bajo cualquier reparametrización (monotónica) τ̃ = τ̃(τ).4 Esto significa que ϕ corresponde al generador infinitesimal de una simetŕıa de norma (de reparametrización), que no modifica los resultados f́ısicos de la teoŕıa. La acción (5.49) también proporciona un ejemplo de una teoŕıa puramente constreñida (aunque artificialmente por el proceso de parametrización), donde el Hamiltoniano canónico Hc = λϕ está compuesto únicamente de la constricción. La Relatividad General es el caso por excelencia de una teoŕıa f́ısica con dicha propiedad y, como se verá en el Caṕıtulo 6, una Cosmoloǵıa homogénea, vista como su mini-superespacio, es descrita por una acción que guarda una gran similitud con la part́ıcula parametrizada por lo que, además de ser ilustrativo, éste caso de una part́ıcula permite establecer una conexión con resultados posteriores. Dentro del método de Cuantización Canónica de Dirac, en presencia de constricciones de primera clase, la condición suplementaria que asegura sin ambigüedad la evolución de los estados cuánticos (f́ısicos) sobre la superficie de constricción es ϕ̂|ψ〉F = 0, (5.51) donde ϕ̂ es el operador cuántico (hermitiano) correspondiente a la constricción clásica. Esto implica que el estado |ψ〉F es invariante bajo la acción unitaria de la constricción, i.e. eiσϕ̂|ψ〉F = |ψ〉F , σ ∈ R, (5.52) y que representa la invariancia de norma del caso cuántico, mientras que (5.51) es la forma análoga para ϕ = p0 +H ≈ 0. En una base completa |q0, qi〉, para los (ahora) operadores de Espacio-Tiempo Q̂0, Q̂i, la condición (5.51) conduce a ϕ̂ |ψ〉F = 0⇒ ( −i~ ∂ ∂t − ~ 2 2m ∇2 + V (qi, t) ) ψF (q i, t) = 0, (5.53) 4Reemplazar τ̃ = τ̃(τ) modifica la acción (5.49) de acuerdo a S = τ̃2 ∫ τ̃1 dτ̃ ( p0(q 0)′ + pi(q i)′ − λ̃ϕ ) , (5.50) donde f ′ = df/dτ̃ , para cualquier función de espacio-fase f , con λ̃ = (q0)′. 68 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad donde se hizo la identificación t = q0. Lo que permite recuperar la ecuación de Schrödinger como resultado de imponer la invariancia de reparametrización clásica a nivel cuántico. En este contexto de cuantización de teoŕıas con constricciones es posible implementar el mecanismo mencionado al principio de esta sección, para generar estructuras de Poisson arbi- trarias del tipo (2.54) (Rosenbaum, Vergara y Juárez) [33]. Partiendo de una acción invariante de reparametrización como (5.49) y doblando el número original de variables de configuración con las coordenadas simplécticas za = ( q0, qi, p0, pi ) , a = 0, ..., 2n + 1. Entonces, se mostrará que la acción general de primer orden en este espacio-fase extendido S = τ2∫ τ1 dτ (Aa(z)ż a − λϕ(z)) , (5.54) donde Aa(z) es un potencial vectorial, permite obtener paréntesis de Poisson arbitrarios. Esto como consecuencia de las constricciones presentes por haber doblado el número original de grados de libertad de la teoŕıa. Igual que antes, la función Hamiltoniana canónica es Hc = λϕ(z), (5.55) y al calcular los momenta pa, conjugados a z a, se obtienen las constricciones primarias χa = pa −Aa (z) ≈ 0. (5.56) Las constricciones χa se incorporan al Hamiltoniano total de la manera usual, v́ıa multiplicadores µa: HT = λϕ+ µaχa, (5.57) que, de demandar la conservación de las constricciones en la evolución, conduce a las condiciones de consistencia: χ̇a = {pa −Aa (z) , HT } = −λ ∂ϕ ∂za + µbωab ≈ 0, (5.58) donde ωab := ∂aAb − ∂bAa = {χa, χb}. (5.59) La notación ωab se ha usado aqúı de manera sugerente, para indicar que la estructura de Poisson arbitraria (2.54) se obtendrá de esta matriz antisimétrica. Asumiendo entonces que ωab es invertible, de forma que sea posible despejar todos los multplicadores de Lagrange µa en (5.58), se sigue de (5.59) que las constricciones χa son de segunda clase.5 Esto obliga a reemplazar el paréntesis de Poisson de funciones de espacio–fase A,B por su proyección en la 5De lo contrario necesariamente alguna de las constricciones χa es de primera clase (cf. [32, 87]), en cuyo caso el número de variables dinámicas independientes de la teoŕıa generalizada no coincidiŕıa con el número original. 5.3. Invariancia de Reparametrización, Cuantización Canónica y No-conmutatividad 69 hipersuperficie de constricciones, es decir por el paréntesis de Dirac {A,B}∗ = {A,B} − {A,χa}ωab {χb, B} , (5.60) donde ωab, es la matriz inversa de ωab. El cálculo de los paréntesis de Dirac entre las variables de configuración za, del espacio-fase extendido, se sigue inmediatamente { za, zb }∗ = ωab, (5.61) que corresponden precisamente a los paréntesis (2.54). Esto muestra como el mecanismo para generar estructuras de Poisson arbitrarias, para teoŕıas con constricciones e invariantes de reparametrización, recurre al uso de constricciones de segunda clase. Ahora es posible cuantizar la teoŕıa clásica promoviendo los paréntesis de Dirac (5.60) a los conmutadores [Ẑa, Ẑb] = i~ωab, (5.62) que, además de la no-conmutatividad del Espacio discutida ya ampliamente, tamb́ıen incorporan una no-conmutatividad de Espacio-Tiempo si ω0i 6= 0 con i = 1, ..., n. Un ejemplo interesante es considerar (5.61) para el caso de una dimensión temporal q0 = t y una dimensión espacial q1 = x, con estructura simpléctica ωab =      0 θ 1 0 −θ 0 0 1 −1 0 0 0 0 −1 0 0      , ωab =      0 0 −1 0 0 0 0 −1 1 0 0 θ 0 1 −θ 0      , (5.63) que es muy semejante a (2.56), con la diferencia que entonces los ı́ndices de ωab eran sólo espaciales y ahora son de espacio-tiempo. Cuantizando, de acuerdo a (5.62), se obtienen los conmutadores [t̂, x̂] = i~θ, [x̂, p̂x] = i~, [t̂, p̂t] = i~, [p̂t, p̂x] = 0, (5.64) con los estados f́ısicos |ψ〉F obtenidos de la condición suplementaria ϕ̂|ψ〉F = (p̂t + Ĥ(t̂, x̂, p̂x))|ψ〉F = 0. (5.65) Recordando la discusión de §4.1, es posible ahora analizar la teoŕıa cuántica eligiendo una realización para los conmutadores (5.64), que puede ser en cualquiera de las bases admisibles {|x, pt〉}, {|t, px〉} y {|pt, px〉}. Tomando, por ejemplo, la base mixta {|t, px〉}, la realización 70 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad diferencial se obtiene en forma análoga a (4.10) y (4.12): x̂ψ(t, px) = i~ (∂px − θ∂t)ψ(t, px), p̂tψ(t, px) = −i~∂tψ(t, px). (5.66) Entonces, si Ĥ(t̂, x̂, p̂x) corresponde a un Hamiltoniano mecánico de la forma Ĥ(t̂, x̂, p̂x) = p̂2x 2m + V̂ (x̂, t̂), (5.67) la realización de (5.65) en la base mixta seleccionada conduce a la ecuación diferencial ( −i~∂t + p2x 2m + V (t, i~(∂px − θ∂t)) ) ψ(t, px) = 0. (5.68) En la expresión anterior destaca el término de derivada temporal dentro del potencial, in- ducido por la no-conmutatividad. Incluso si no hubiese una dependencia temporal expĺıcita en Ĥ, la realización en la base mixta {|t, px〉} agrega contribuciones de derivadas temporales. Esto implica que, la expresión (5.68) conduce a una ecuación de tipo Schrödinger únicamente para potenciales de primer orden en x. Consecuentemente, en casos más generales, la función de onda ψ(t, px) puede no admitir la interpretación usual de amplitud de probabilidad. Es importante notar que para una matriz ωab dada, las ecuaciones (5.59) no tienen una única solución aunque todas ellas estén relacionadas por transformaciónes canónicas. Esta propiedad es relevante en la formulación de la integral de trayectoria de esta teoŕıa clásica, como se ve en la siguiente sección. Para una discusión más amplia que toma en consideración diversas soluciones y la relación entre unas y otras ver (Rosenbaum, Vergara y Juárez) [33]. 5.4 Formulación de la integral de trayectoria Además de los métodos de cuantización ya discutidos (WWGM, Schrödinger-Heisenberg-Dirac) se encuentra también la integral de trayectoria, introducida por R. Feynman [88]. Como se sabe, el uso de esta construcción funcional proporciona una ruta alterna de cuantización y es particularmente prominente en Teoŕıa de Campos y aproximaciones semiclásicas de sistemas cuánticos, entre varios otros temas relevantes en F́ısica.6 Dado que las técnicas emanadas de esta formulación serán aplicadas en secciones posteriores, aqúı se muestran los resultados de la construcción de la integral de trayectoria, para el álgebra extendida de Heisenberg-Weyl (4.1), desde tres perspectivas diferentes: 1) Vı́a el formalismo WWGM, elaborado en el Cap.4, mediante el cálculo de una traza como se mostró en (Rosenbaum, Vergara y Juárez) [91]. 2) En el contexto de cuantización canónica y estructuras de Poisson no-canónicas de la sección anterior (Rosenbaum, Vergara y Juárez) [33] y 3) Usando los estados coherentes no-conmutativos de §5.2 6Una exposición moderna sobre integrales de trayectoria puede hallarse en [89, 90] 5.4. Formulación de la integral de trayectoria 71 y el śımbolo covariante de Berezin (ó representación-Q de Husimi) (3.29) (Júarez y Mart́ınez) [82]. 5.4.1 Amplitud de transición como la traza de operadores Ya que es posible escribir una amplitud de transición de base mixta como 〈q′′1(t2), p′′2(t2)|q′1(t1), p′2(t1)〉 = 〈q′′1 , p′′2|e− i ~ (t2−t1)Ĥ |q′1, p′2〉 = Tr[ˆ̺e− i ~ (t2−t1)Ĥ ] (5.69) donde ˆ̺ := |q′1, p′2〉〈q′′1 , p′′2|, (5.70) entonces, las expresiones (4.46) y (4.80) permiten calcular (5.69) como 〈q′′1(t2), p′′2(t2)|q′1(t1), p′2(t1)〉 = (2π~)−2 ∫ R4 d2~qd2~p ̺W (~q, ~p)e − i ~ (t2−t1)HW ∗ . (5.71) El equivalente de Weyl ̺W (~q, ~p) en la expresión anterior se obtiene de acuerdo a la expresión (4.29) ̺W (~q, ~p) = ∫ R2 dξdη e i ~ [ (q2− θ ~ p1)η+p1ξ ]〈 q1 − ξ 2 , p2 + η 2 ∣ ∣ ∣ ˆ̺ ∣ ∣ ∣q1 + ξ 2 , p2 − η 2 〉 (5.72) que, por la forma expĺıcita de ˆ̺, tiene el valor ̺W (~q, ~p) = 4δ(q′′1 + q′1 − 2q1)δ(p ′′ 2 + p′2 − 2p2)e 2i ~ [(q1−q′1)p1−(p2−p′2)(q2− θ ~ p1)], (5.73) lo cual permite hacer las integraciones sobre q1 y p2 en (5.71) para llegar a 〈q′′1(t2), p′′2(t2)|q′1(t1), p′2(t1)〉 = (2π~)−2 ∫ R2 dq2dp1 { e i ~ [(q′′1−q′1)p1−(p′′2−p′2)(q2− θ ~ p1)] × (e − i ~ (t2−t1)HW ∗ ) ( q′′1 + q′1 2 , q2, p1, p′′2 + p′2 2 )} , (5.74) donde (e − i ~ (t2−t1)HW ∗ ) ( q′′1 + q′1 2 , q2, p1, p′′2 + p′2 2 ) , (5.75) significa calcular primero e − i ~ (t2−t1)HW ∗ en variables de espacio-fase no primadas y después reem- plazar las dependencias en q1 y p2 por los promedios q′′1+q′1 2 , p′′2+p′2 2 . Si bien la expresión (5.74) es una forma exacta de evaluar el propagador, los productos-∗ a todos los órdenes en el término (5.75) dificultan la evaluación. Consecuentemente en esta etapa del cálculo es preferible recurrir al método usual de dividir el intervalo temporal t2−t1 = t en N fragmentos δt, tales que t = δtN , para aproximar el valor del propagador y luego tomar el ĺımite al cont́ınuo N →∞, δt→ 0 para 72 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad recuperar el valor exacto. La partición del intervalo t conduce entonces a la expresión 〈q′′1 , p′′2|e− i ~ tĤ |q′1, p′2〉 = 〈q′′1 , p′′2| e− i ~ δtĤ × ...× e− i ~ δtĤ ︸ ︷︷ ︸ N |q′1, p′2〉 = ∫ N−1∏ i=0 dqi1dp i 2〈qi+1 1 , pi+1 2 |e− i ~ δtĤ |qi1, pi2〉, (5.76) habiendo insertado N−1 resoluciones de identidad Î = ∫ dqi1dp i 2|qi1, pi2〉〈qi1, pi2|, donde |qN1 , pN2 〉 = |q′′1 , p′′2〉 y |q01, p02〉 = |q′1, p′2〉. Sustituyendo la expresión (5.74) dentro de (5.76) resulta en ∫ N−1∏ i=0 dqi1dp i 2〈qi+1 1 , pi+1 2 |e− i ~ δtĤ |qi1, pi2〉 = ∫ N−1∏ i=0 dqi1dp i 2 ∫ N∏ j=0 dqj2dp j 1 (2π~)2 { e i ~ [(qi+1 1 −qi1)p j 1−(p i+1 2 −pi2)(q j 2− θ ~ pj1)] × (e − i ~ δtHW ∗ ) ( qi+1 1 + qi1 2 , qj2, p j 1, pi+1 2 + pi2 2 )} , (5.77) y notando que para δt→ 0 es posible aproximar (e − i ~ δtHW ∗ ) ( qi+1 1 + qi1 2 , qj2, p j 1, pi+1 2 + pi2 2 ) ≈ 1− i ~ δtHW (q̃i1, q j 2, p j 1, p̃ i 2) ≈ e− i ~ δtHW (q̃i1,q j 2,p j 1,p̃ i 2), (5.78) donde q̃i1 y p̃i2 son puntos intermedios en los intervalos (qi1, q i+1 1 ) y (pi2, p i+1 2 ), que al insertarlo en (5.77) implica 〈q′′1 , p′′2|e− i ~ tĤ |q′1, p′2〉 ≈ ∫ N−1∏ i=0 dqi1dp i 2 ∫ N∏ j=0 dqj2dp j 1 (2π~)2 { e i ~ [(qi+1 1 −qi1)p j 1−(p i+1 2 −pi2)(q j 2− θ ~ pj1)] × e− i ~ δtHW (q̃i1,q j 2,p j 1,p̃ i 2) } . (5.79) Finalmente, escribiendo la fase e i ~ [(qi+1 1 −qi1)p j 1−(p i+1 2 −pi2)(q j 2− θ ~ pj1)] = exp [ i ~ δt ( qi+1 1 − qi1 δt pj1 − pi+1 2 − pi2 δt (qj2 − θ ~ pj1) )] ≈ exp [ i ~ δt(q̇i1p j 1 − ṗi2(qj2 − θ ~ pj1)) ] , (5.80) con qi+1 1 − qi1 ≈ q̇i1δt, p i+1 2 − pi2 ≈ ṗi2δt. Entonces, ahora es claro que en el ĺımite N → ∞ 5.4. Formulación de la integral de trayectoria 73 las expresiones anteriores se vuelven exactas y (5.79) se convierte en la igualdad (Rosenbaum, Vergara y Juárez) [91] 〈q′′1 , p′′2|e− i ~ tĤ |q′1, p′2〉 = ∫ Dµ(~q, ~p)e i ~ ∫ t2 t1 dt[q̇1p1−ṗ2(q2− θ ~ p1)−HW (q1,q2,p1,p2)], (5.81) en donde la presencia expĺıcita del parámetro θ contrasta con la integral de trayectoria usual de Mecánica Cuántica. La acción semiclásica no-conmutativa se lee directamente de la expresión anterior, i.e. S = ∫ t2 t1 dt[q̇1p1 − ṗ2q2 + θ ~ ṗ2p1 −HW (q1, q2, p1, p2)], (5.82) que, debido al equivalente de Weyl HW , puede también introducir correcciones a varios órdenes en el parámetro de no-conmutatividad para cierto tipo de Hamiltonianos.7 El origen del nuevo término cinemático no-conmutativo θ ~ ṗ2p1 en la acción puede rastrearse hasta la función de transición (4.17), que implementa una elección de observables de posición manifiestamente no- conmutativos, por lo que no corresponde a un simple artefacto de cambio de variables.8 5.4.2 Acción semiclásica no-canónica Debido a su relación con una Mecánica Cuántica No-conmutativa en el ĺımite del principio de correspondencia, la acción no canónica (5.54) puede ser promovida a una acción semiclásica. De forma que la integral de trayectoria correspondiente (cf. (Rosenbaum, Vergara y Juárez) [33]), que toma en cuenta la constricción ϕ, es ∫ Dµ(za)δ(G)δ(ϕ){ϕ,G}∗e i ~ ∫ τ2 τ1 dτ(Aa(z)ża−λϕ), (5.83) donde G = G(za), es una condición de norma que depende únicamente de variables de espacio- fase y que garantiza que el paréntesis de Dirac {ϕ,G}∗ sea no degenerado (ver, e.g, [86, 93]).9 Se puede ver entonces que la integral de trayectoria (5.83) puede identificarse con la amplitud (5.81) cuando n = 2, fijando la norma usual q0(τ) = τ y resolviendo (5.59) con soluciones A1 = p1, A2 = 0, A3 = 0, A4 = θ ~ p1 − q2, (5.84) lo que conduce a la acción semiclásica S = ∫ t2 t1 dt[q̇1p1 − ṗ2q2 + θ ~ ṗ2p1 −Hc(q1, q2, p1, p2)], (5.85) donde Hc(q1, q2, p1, p2) es el Hamiltoniano clásico. Por lo que, evidentemente, si HW = Hc en 7Una derivación alterna de éste resultado puede encontrarse en [72]. 8Recientemente esta corrección cinemática no-conmutativa ha sido asociada con teoŕıas de tipo Chern-Simons [92]. 9Las funciones δ ejercen las constricciones fuertemente en la acción. 74 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad la expresión (5.81), es posible recuperar la misma amplitud de transición asociada con la acción semiclásica (5.85). Es importante notar que las soluciones (5.84) permiten fijar variacionalmente los extremos de la integral de trayectoria en las variables (q1, p2), 10 que es la razón de poder identificar el resultado con una función de transición 〈q′′1(t2), p′′2(t2)|q′1(t1), p′2(t1)〉. Es de esperar que soluciones diferentes a las ecuaciones (5.59) conduzcan a formulaciones no equivalentes de la integral de trayectoria en el régimen no-conmutativo, excepto por aquellas que estén relacionadas por transformaciones canónicas, como se mostró en (Rosenbaum, Vergara y Juárez) [33]. 5.4.3 Integral de trayectoria con estados coherentes de hθ2n+1 La construcción de la integral de trayectoria con estados coherentes (ver e.g. [94, 95]) alude a una representación manifiestamente semiclásica en virtud de las ecuaciones (5.14). Para obtener una expresión para la amplitud de transición de estados coherentes 〈~α′′(t2)|~α′(t1)〉 en forma de integral de trayectoria se puede seguir el procedimiento usual, utilizado parcialmente en (5.76), de dividir el intervalo t2 − t1 = δtN = t, insertando N resoluciones de la identidad ÎH = ∫ Cn dµ(~α, ~α∗)|~α〉〈~α|: 〈~α′′|e− i ~ tĤ |~α′〉 = 〈~α′′| e− i ~ δtĤ × ...× e− i ~ δtĤ ︸ ︷︷ ︸ N |~α′〉 = ∫ N−1∏ k=0 dµ(~αk, ~α ∗ k)〈~αk+1|e− i ~ δtĤ |~αk〉, (5.86) donde ~αN = ~α′′ y ~α0 = ~α′. La expresión (5.86) se encuentra automáticamente en todo el espacio-fase y es compar- ativamente más simple que (5.77), gracias a que los estados coherentes permiten especificar simultáneamente todas las coordenadas de espacio-fase (como valores de expectación).11 Para obtener expĺıcitamente la integral de trayectoria basta escribir el operador Hamiltoniano Ĥ en forma normal Ĥ = n∏ i=1 ∑ ri,si ξri,si( † i ) ri(Âi) si , (5.87) con coeficientes ξri,si ∈ C.12 Entonces la amplitud de transición k-ésima en (5.86) puede aprox- 10La parte cinemática en (5.85) depende estrictamente de derivadas en estas variables. 11Mientras que con el uso de bases de eigenvalores simultáneos siempre es necesario realizar una integración más en el espacio dual. 12Debido a las expresiones (5.1) y (5.6) esto siempre es posible para Hamiltonianos polinomiales ó que admiten una serie formal en potencias de Q̂i’s y P̂i’s. 5.4. Formulación de la integral de trayectoria 75 imarse fácilmente como 〈~αk+1|e− i ~ δtĤ |~αk〉 ≈ 〈~αk+1|~αk〉 − i ~ δt〈~αk+1|Ĥ|~αk〉 = 〈~αk+1|~αk〉[1− i ~ δtH(~α∗k+1, ~αk)] ≈ 〈~αk+1|~αk〉e− i ~ δtH(~α∗k+1,~αk), (5.88) en donde H(~α∗k+1, ~αk) := 〈~αk+1|Ĥ|~αk〉 〈~αk+1|~αk〉 = n∏ i=1 ∑ ri,si ξri,si(α ∗ {k+1}i) ri(α{k}i) si , (5.89) y la notación α{k}i indica la componente i-ésima del vector ~αk. Consecuentemente, insertando (5.88) en (5.86) implica 〈~α′′|e− i ~ tĤ |~α′〉 ≈ ∫ N−1∏ k=0 dµ(~αk, ~α ∗ k)〈~αk+1|~αk〉e− i ~ δtH(~α∗k+1,~αk), (5.90) y usando la función de transición de estados coherentes (5.11) se tiene 〈~αk+1|~αk〉 = exp [ −~α ∗ k+1 · ~αk+1 2 − ~α∗k · ~αk 2 + ~α∗k+1 · ~αk ] = exp [ δt ( ~αk 2 · (~α ∗ k+1 − ~α∗k) δt − ~α∗k+1 2 · (~αk+1 − ~αk) δt )] , (5.91) que al substituir en (5.90) resulta en 〈~α′′|e− i ~ tĤ |~α′〉 ≈ ∫ N−1∏ k=0 dµ(~αk, ~α ∗ k) { e− i ~ δtH(~α∗k+1,~αk) × exp [ δt ( ~αk 2 · (~α ∗ k+1 − ~α∗k) δt − ~α∗k+1 2 · (~αk+1 − ~αk) δt )]} , (5.92) cuyo ĺımite cont́ınuo, cuando N →∞, conduce a la expresión exacta 〈~α′′|e− i ~ tĤ |~α′〉 = ∫ ~α′′ ~α′ Dµ(~α, ~α∗)e ∫ t2 t1 dt[ 1 2 (~α(t)·̇~α∗(t)−~α∗(t)·̇~α(t))− i ~ H(~α(t),~α∗(t))], (5.93) donde H(~α, ~α∗) = n∏ i=1 ∑ ri,si ξri,si(α ∗ i ) ri(αi) si . (5.94) Esta representación holomorfa de la integral de trayectoria confirma las caracteŕısticas de formulaciones en términos estados coherentes (5.8), como se discutió en §5.2, en cuanto a que no hay una presencia expĺıcita de la no-conmutatividad. Para recuperar una integral de trayectoria de variables f́ısicas es necesario hacer uso de las expresiones (5.14) para reemplazar las coorde- nadas holomorfas por coordenadas de espacio-fase. En este contexto, el término cinemático de 76 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad la acción semiclásica en (5.93) es particularmente interesante ya que 1 2 (~α(t) ·~̇α∗(t)− ~α∗(t) ·~̇α(t)) = i 2~ n∑ i=1 ( piq̇i − ṗiqi + n∑ j=1 θij ~ piṗj ) , (5.95) con lo cual la acción semiclásica no-conmutativa inducida por (5.93) es S = ∫ t2 t1 dt [1 2 n∑ i=1 ( piq̇i − ṗiqi + n∑ j=1 θij ~ piṗj ) −HQ(~q, ~p) ] . (5.96) Por su obvia similitud, es inevitable tratar de establecer una relación entre (5.82), (5.85) y (5.96). Mientras que las primeras dos acciones semiclásicas pueden identificarse bajo ciertas con- sideraciones, la última pertenece a un tipo de acción diferente que, en general, no es equivalente a las primeras dos. No sólo la dependencia del término cinemático en derivadas de las variables f́ısicas permite fijar completamente todas las coordenadas de espacio-fase en los extremos,13 lo que confirma que es una acción semiclásica apropiada para la amplitud de transición (5.86). Sino que, además, por el proceso de ĺımite, la función (5.89) converge a la función Hamiltoniana HQ(~q, ~p) := H(~α, ~α∗) = 〈~α|Ĥ|~α〉, (5.97) que corresponde precisamente al śımbolo covariante de Weyl (3.29) ó Q-śımbolo de Husimi del operador Hamiltoniano. Esto implica que la acción semiclásica puede incluir términos que la hagan diferir de manera importante del equivalente de Weyl en (5.82) y, por lo tanto, las trayectorias clásicas pueden no estar relacionadas en absoluto. Aunque debe enfatizarse que tanto (5.82) como (5.96) deben conducir exactamente a la misma amplitud de probabilidad entre dos estados cuánticos 〈ψF (t2)|ψI(t1)〉 para el mismo Hamiltoniano Ĥ. Una ventaja adicional de ésta representación de integral de trayectoria es en su imple- mentación en escenarios semiclásicos, por ejemplo en un análisis de fase estacionaria (ó tipo WKB). Principalmente por que la utilización de estados coherentes garantiza que las trayec- torias clásicas (reales) describan la evolución del centro de paquetes Gaussianos, que proveen de una mejor descripción de la evolución global de los estados cuánticos. Aunado a esto, se encuentra el hecho que el Q-śımbolo de Husimi es una suavización Gaussiana del śımbolo de Weyl, por lo que sugiere ser el candidato ideal para obtener la mejor aproximación en este caso [97]. 13Esto causa que las trayectorias clásicas que van de un extremo al otro puedan pasar por el plano complejo, ver [96]. 5.5. El Cálculo Espectral de Connes 77 5.5 El Cálculo Espectral de Connes Aunque existe una diferencia de una veintena de ordenes de magnitud entre las enerǵıas de las interacciones electrodébiles y fuertes propias del Modelo Estándar y las enerǵıas a ordenes de la longitud de Planck, la presentación que se ha seguido hasta este punto, no habiendo evidencia experimental de lo contrario, asume que los principios fundamentales de la Mecánica Cuántica prevalecen aún cuando los Principios de Incertidumbre y de Equivalencia son conmensurables, modificando aśı el álgebra fundamental de Mecánica Cuántica de forma mı́nima con los conmu- tadores (4.1). Sin embargo, como se verá mas adelante, a escalas del orden de la longitud de Planck, esto puede no ser del todo válido al desechar eventualmente cualquier representación diferencial de los observables f́ısicos y con ello las nociones de variedades diferenciables que de cierta forma subyacen. De alĺı surge la idea básica de una Geometŕıa No-conmutativa (ver [17, 98, 99]), intercambiando variedades por álgebras sin dejar remanente alguno, en general, del concepto de espacio. La idea fundamental del Cálculo Espectral de Connes [17] parte de incorporar de inicio la Mecánica Cuántica en un nuevo paradigma de Geometŕıa No-conmutativa que reemplaza los cálculos diferencial e integral clásicos de acuerdo con el siguiente esquema: CLÁSICO CUÁNTICO Variable Compleja Operador en H Variable Real Operador Auto-adjunto en H Infinitesimal Operador Compacto en H Infinitesimal de orden α Operador Compacto en H cuyos valores caracteristicos µn satisfacen µn = O(n−α), n→∞ Diferencial de variable real o compleja da = [F, a] = Fa− aF Integral de infinitesimal de orden 1 Traza de Dixmier Las transiciones implicadas por la primera, segunda y quinta entrada a la tabla son enter- amente similares a las requeridas para pasar de la Mecánica Clásica a la Cuántica usual. En cuanto a la cuarta entrada de la tabla, nótese que en este marco la condición ∀ ǫ > 0, ∃ un subespacio E de dimensión finita ⊂ H : ‖TE⊥‖ < ǫ, que caracteriza a los operadores compactos T ∈ K(H) y que puede ser considerada en cierto sentido como un concepto de pequeñez de modo que estos operadores juegan el papel de in- finitesimales. El tamaño del infinitesimal T es gobernado por la rapidez de decaimiento de la secuencia {µn(T )} conforme n → ∞, en donde µn son los eigenvalores de |T | = √ T ⋆T . Consecuentemente los in- 78 Caṕıtulo 5. Representaciones alternas de la No-conmutatividad finitesimales de orden α ∈ R + los bi-ideales cuyos elementos satisfacen la condición ∃ C <∞ : µn(T ) ≤ Cn−α, ∀n ≥ 1. Ahora bien, la quinta entrada del esquema es la noción teórica-operacional del diferencial da = [F, a], (5.98) en donde a ∈ A (un álgebra involutiva de operadores en el espacio de Hilbert). Dado que el lado izquierdo de esta ecuación debe interpretarse como un infinitesimal, es necesario especificar primero las propiedades necesarias de la representación de A en el par (H, F ) de modo que [F, a] ∈ K, ∀a ∈ A. Esta representación se conoce como un modulo de Fredholm. Finalmente, en vista de los varios formalismos de cuantización descritos en caṕıtulos y sec- ciones anteriores, es posible entender la motivación para contar con una definición apropiada de la traza de operadores. En dichos casos es inmediato que ésta se encuentra invariantemente aso- ciada con una integral de śımbolos de operadores en un espacio de funciones, cf. (2.45), (3.33), (3.37) y (4.46). De forma que se puede considerar a la traza como la generalización natural de la integración para espacios no-conmutativos donde un cálculo integral ya no está disponible. En este sentido se quiere tener una ”integral” que ignore infinitesimales de orden > 1. Sin embargo, en general, un infinitesimal de orden 1 no está en el dominio de la traza ordinaria (ya que esta diverge como lnN) y, además, no se anula para infinitesimales de orden mayor. Para resolver estos dos problemas, Connes introdujo en su formalismo la traza de Dixmier [100], la cual consiste de un procedimiento invariante de escala designado precisamente para extraer el coeficiente de la divergencia (ver e.g. [98]). Su expresión está dada por: Trω(T ) = lim ω 1 lnN N−1∑ n=0 µn(T ), ∀T ≥ 0, T ∈ L(1,∞), (5.99) donde L(1,∞) es el ideal de operadores compactos que son infinitesimales de orden 1. Aqúı limω es un ĺımite homoteticamente invariante (Dixmier ha demostrado la existencia de un número infinito de ellos). De hecho, resulta que para muchos problemas de interés en la F́ısica en donde T es pseudod- iferencial y mensurable, como es el caso de teoŕıas de norma y la gravitación, la traza de Dixmier no depende del proceso ĺımite ω, de modo que el valor resultante es una integral apropiada para T en el nuevo cálculo. Mas aún, en estos casos la traza de Dixmier coincide con el residuo de Wodzicki [101]. Para completar esta breve descripción de los ingredientes que conforman la Geometŕıa Es- pectral de Connes y completar la algebraización de la Geometŕıa, es necesario adicionar a los elementos mencionados el operador auto-adjunto y no-acotado D ∈ H tal que a) El resolvente (D − λ)−1, λ 6∈ R, de D es compacto, 5.5. El Cálculo Espectral de Connes 79 b) Los conmutadores [D, a] son acotados ∀a ∈ A, y D es un operador de Dirac, el cual hace las veces de la conexión geométrica. De esta manera resulta más económico considerar como elementos básicos de la Geometŕıa Espectral el triple (A,H, D). En los capitulos siguientes haremos uso de los elementos de este formalismo necesarios para investigar la cuantización de las Cosmoloǵıas Homogéneas y los ejemplos alĺı espećıficamente considerados. Sin embargo es importante remarcar aqúı el que esta nueva noción de espacio geométrico trata en igualdad de términos el continuum como lo discreto, lo cual permite inter- pretar geométricamente el Modelo Standard como una teoŕıa puramente de norma y evidenciar el que la F́ısica de Particulas es un develado de la estructura fina del espacio-tiempo. Parte III La No-conmutatividad en el régimen Planckiano de la Cosmoloǵıa 81 Caṕıtulo 6 Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I Debe ser evidente, de las diversas representaciones de espacios de Hilbert introducidas en los capitulos previos, que una consideración seria de la no-conmutatividad, como la imposibilidad f́ısica y conceptual de realizar mediciones de distancias a escalas menores que la longitud de Planck, debe reflejarse en los conceptos geométricos básicos implicando un cambio de una for- mulación matemática en términos de ”espacios”, que en general no existen ”concretamente”, a una en términos de álgebras de funciones. Esto se manifiesta con mucho mayor claridad en el Cálculo Cuántico de Connes, descrito brevemente al final del caṕıtulo anterior, en donde la Geometŕıa Espectral está formulada en términos del triple (A,H, D). El propósito principal de este Caṕıtulo es el de proveer lo que pudiese considerarse como una formulación autoconsistente de la Cosmoloǵıa Cuántica que pueda conducir a concepciones y direcciones adicionles hacia una Gravitación Cuántica a escalas en que las implicaciones del Prin- cipio de Incertidumbre de la Mecánica Cuántica y el Principio de Equivalencia de la Gravitación se hacen conmensurables. Las construcciones teóricas y resultados aqúı presentados constituyen el material de investigación generado en la preparación del art́ıculo recientemente publicado ”Twisted C⋆-algebra formulation of Quantum Cosmology with application to the Bianchi I model, Marcos Rosenbaum, J. David Vergara, Román Juárez and A.A. Minzoni, Phys. Rev. D 89, 085038 (2014)” Ref. [102]. Ahora bien, dado que la Cosmoloǵıa Cuántica puede verse como un mini-súperespacio de la Gravedad Cuántica en donde la mayor parte de los grados de libertad han sido ”congelados” y, aunque no existe a priori razón para suponer que las conclusiones derivadas de la primera 83 84 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I puedan directamente transladarse a la segunda, es de esperarse que ciertas formulaciones de la Cosmoloǵıa Cuántica puedan contribúır más que otras como un marco de trabajo inicial para investigar lo procesos cuánticos que involucren distancias del orden de la longitud de Planck en donde las manifestaciones de la no-conmutatividad deben de ocurrir. La formulación adoptada tiene como base motivacional la observación teórica que las can- tidades f́ısicamente significativas deben ser independientes de norma, por lo que los conceptos de potenciales de norma, es decir conexiones, deben ser incorporados en la formulación de las Densidades de Acción para la descripción de nuestra percepción de la Naturaleza. Esto ha conducido naturalmente a un formalismo de Haces Fibrados para describir las fuerzas básicas de la F́ısica aśı como las Matemáticas para tratar la Teoŕıa de Norma y los Principios Varia- cionales de la Teoŕıa de Campos. Ahora bien, un haz fibrado P (M,F, τ) consiste de un espacio topológico P , una base M , una fibra t́ıpica F y una suryección continua τ : P → M , en donde en la F́ısica semiclásica M es el cont́ınuo de espacio-tiempo con Topoloǵıa Hausdorff. Más aún, se puede mostrar que un haz vectorial sobre M puede describirse puramente en términos de conceptos propios de un álgebra C⋆ conmutativa C(M).1 Pero, de acuerdo con los Teoremas de Gel’fand-Naimark [16, 104], existe una equivalencia completa entre la categoŕıa de espacios de Hausdorff (localmente) compactos y sus mapeos (propios y) continuos con la categoŕıa de álgebras C⋆ conmutativas, (no necesariamente) unitarias y sus ⋆-homomorfismos. De manera que cualquier álgebra C⋆ conmutativa puede realizarse como un álgebra C⋆ de funciones complejas valuadas sobre un espacio de Hausdorff (localmente) compacto. Esto, por otro lado, implica que el ámbito ”dual” para una topoloǵıa no-conmutativa es el álgebra C⋆ no-conmutativa [17]. Consecuentemente, partiendo de un álgebra C⋆ no-conmutativa A como un ingrediente de inicio y el análogo de los haces vectoriales - los módulos proyectivos de tipo finito sobre A - hace posible la formulación de una teoŕıa completa de Conexiones que culmina en la definición de una Acción de Yang-Mills y la consiguiente aplicación de la Geometŕıa No-conmutativa a la Teoŕıa de Campos. Por todo ello el contexto de álgebras C⋆ es una estrategia particularmente buena de la Geometŕıa No-conmutativa para la formulación de la Cuantización de Cosmoloǵıas Cuánticas Homogeneas e investigación de la Gravedad Cuántica. 6.1 Una realización de operadores mas allá del Teorema de Stone-von Neumann Volviendo al álgebra extendida de Heisenberg-Weyl hθ2n+1 en (4.1) y recordando que por el Teorema de Stone-von Neumann, mencionado al principio de éste trabajo en §2.1, la realización diferencial de los observables de posición y momento de Mecánica Cuántica No-conmutativa (4.10) y (4.13), es consecuencia directa de la equivalencia unitaria entre el álgebra C⋆ de grupos uniparamétricos Ûi(λ), V̂i(γ), débilmente cont́ınuos, que satisfacen un álgebra de trenzamiento 1Una introducción concisa a la teoŕıa de álgebras C∗ puede hallarse en [103]. 6.1. Una realización de operadores mas allá del Teorema de Stone-von Neumann 85 y los elementos del grupo de Lie Hθ 2n+1, que actúan en el espacio de Hilbert H. Por otro lado, como se muestra en §B.3, el grupo de isometŕıas de la subálgebra no-conmutativa Aθ ⊂ A∗, asociada con los operadores de posición, es el grupo de Galileo con un coproducto deformado por los generadores de traslaciones eucĺıdeas. Notando ahora de los operadores de desplazamiento (4.22) que la acción natural del subconjunto de Hθ 2n+1 generado únicamente por los operadores de posición (i.e., ~x = ~x′ = 0) sobre śı mismo es v́ıa D̂(~y)D̂(~y′) = e− i 2~2 θijyiy ′ jD̂(~y + ~y′), (6.1) entonces esta expresión establece un mapeo del grupo abeliano GT de traslaciones en R n a los automorfismos de Hθ 2n+1. En el lenguaje de la teoŕıa de representaciones la ecuación (6.1) define una representación σ-proyectiva (ó torcida) del grupo GT (ver, e.g., [105, 106, 107]), donde σ(~y, ~y′) := e− i 2~2 θijyiy ′ j es un 2-cociclo ó multiplicador de Schur con valores en el ćırculo unitario en C y satisface las propiedades σ(~y, ~y′)σ(~y + ~y′, ~y′′) = σ(~y′, ~y′′)σ(~y, ~y′ + ~y′′), σ(~y, 0) = σ(0, ~y) = 1. (6.2) A continuación se mostrará cómo estos elementos básicos permiten construir una realización del álgebra C⋆ de Weyl del tipo (6.1) con grupos uniparamétricos Ûi(λ), V̂i(γ), en el contexto de la geometŕıa no-conmutativa, que se aparta de las consecuencias del Teorema de Stone-von Neu- mann y donde las realizaciones diferenciales de los observables son eventualmente reemplazadas por entidades espectrales. Para éste propósito y como punto de inicio del análisis se substituirá el grupo GT por el grupo topológico discreto de traslaciones en R 3,2 identificado como el espa- cio vectorial T3, asociado con el espacio af́ın con topoloǵıa discreta y descomposición en clases laterales T3 = ∞∑ j1,j2,j3=−∞ (µiji)êi, ji ∈ Z, (6.3) donde êi son traslaciones básicas en R 3 (vectores unitarios), los vectores x(l) = ∑3 i=1(µij(l)i)êi ∈ T3 son elementos de R 3 como grupo y el conjunto Γ : {µij(l)i} representa una celda tridi- mensional. Para contextualizar con las ĺıneas previas y hacer más precisas las construcciones subsecuentes son necesarias las definiciones formales Definición 2. Una representación σ(x1,x2)-proyectiva Û de G en un espacio (no nulo) de Hilbert H es un mapeo del grupo G al grupo U(H) de unitarios en H tal que U(x1)U(x2) = σ(x1,x2)U(x1 + x2). (6.4) 2Se ha escogido el espacio tridimensional dada la aplicación de esta construcción a una cosmoloǵıa en secciones posteriores. 86 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I Fijando en particular U(H) ∋ σθ(x1,x2) := σ(x1,x2) = e−iπx T 1 Rx2 = e−iπθ·(x1×x2), (6.5) donde θ = ∑3 i=1 θiêi, es el dual de Hodge de la matriz de no-conmutatividad tal que θi = ǫijkθ jk, y R es la matriz antisimétrica R =    0 θ3 −θ2 −θ3 0 θ1 θ2 −θ1 0    . (6.6) Definición 3. Una realización regular proyectiva del álgebra (6.4) y (6.5) en l2(G) puede definirse como 〈x|Ûi|ξ〉 := e−2πiεixi〈x− 1 2 εiêi × θ|ξ〉 = e−2πiεixiξ(x− 1 2 εiêi × θ); ξ(x) ∈ H. (6.7) Identificando a ξ con la función correspondiente en T3 de valor uno en x y cero en cualquier otro caso, i.e. si δx ∈ l2(T3) (la función delta de Kroenecker en x) es dicha función, es inmediato que Uiδx := e−2πiεixiδ( 1 2 εiêi×θ+x), (6.8) y Ûi|x〉 = e−2πiεixi |x+ 1 2 εiêi × θ〉. (6.9) Entonces, la acción del operador unitario Ûi traslada al vector x en la dirección perpendicular a êi por 1 2εiθ. Aplicando sucesivamente (6.7) implica ÛiÛj = e−iπεiεjθ·(êi×êj)Ûi+j , (6.10) que al intercambiar ı́ndices y sustituir el resultado en (6.10) conduce a ÛiÛj = e−2iπεiεjθ·(êi×êj)ÛjÛi. (6.11) Debido a que el vector θ tiene unidades de L2, las cantidades εi deben tener unidades L−1 y las entidades εiêi × θ son vectores básicos en direcciones perpendiculares a los vectores êi y determinan los vértices fundamentales de una ret́ıcula. El álgebra anterior puede extenderse incorporando ahora los operadores unitarios V̂l := V̂ (µlêl) tales que V̂l|x〉 = |x+ µlêl〉, (6.12) 6.2. Construcción GNS y observables f́ısicos 87 de forma que V̂l también actúa en los kets |x〉 ∈ H como una traslación del vector x en la dirección êl por µl. De (6.12) se obtiene que V̂iV̂l = V̂lV̂i, (6.13) y componiendo su acción con Ûi, dada por (6.9), se tiene ÛiV̂l = e−2πiεiµl(êi·êl)V̂lÛi = e−2πiεiµlδil V̂lÛi. (6.14) Las ecuaciones (6.10), (6.11), (6.13) y (6.14) definen un homomorfismo-∗ entre ésta álgebra C∗ de Weyl A ⊂ B(H) de operadores acotados y el álgebra C∗ del grupo extendido de Heisenberg- Weyl en el sentido de (6.1). Debe notarse que las cantidades µl y εi en las relaciones anteriores aparecen estrictamente como parámetros independientes de la acción de los subgrupos discretos y que pueden tomar cualquier valor arbitrario finito. Esto puede implicar la superposición de dos reticulaciones no- conmutativas, con vértices en µl y êl · (εiêi × θ), generadas por Ûi’s y V̂l’s. Sin embargo, en la siguiente sección se mostrará que la construcción del espacio de Hilbert sobre el cual actúan estos operadores conduce naturalmente a una condición que identifica ambas reticulaciones. Coincidentemente las expresiones (6.10) y (6.11) tienen la misma apariencia que aquellas que describen al toro no-conmutativo (cf. e.g. [108]), el cual es un ejemplo protot́ıpico de una geometŕıa no-conmutativa, aunque hay ciertas diferencias que distinguen uno del otro. La realización adoptada en las expresiones (6.7) (o (6.9)) no impone condiciones de periodicidad (cf. [109]). Por otro lado, como se verá en la siguiente sección, las ecuaciones (6.9) y (6.12) constituyen un mecanismo para obtener el espacio de Hilbert completo por medio de traslaciones sucesivas, inducidas por el término de no-conmutatividad, en un vector ćıclico. La elección de ésta realización tiene importantes repercusiones f́ısicas como se verá en las secciones finales de éste caṕıtulo. 6.2 Construcción GNS y observables f́ısicos El objetivo en esta sección es recurrir al homomorfismo-∗ y utilizar el método de Gelgand- Naimark-Segal [98],[110], para obtener las formas expĺıcitas de los elementos del espacio de Hilbert H sobre los cuales actúan los operadores en A. Para dicho fin nótese que para cualquier estado funcional φ se tiene que ∀ a ∈ A ∃ φ tal que φ(a∗a) = 1, que es siempre cierta ya que el álgebra A es unitaria. Esto impica que el ideal izquierdo I = {a ∈ A |φ(a∗a) = 0} en A está vaćıo, de forma que el espacio cociente Nφ = A/Iφ ≡ A ⇒ φ es fiel. Por lo tanto, de la construcción GNS, se tiene un espacio pre-Hilbert con un producto no degenerado definido por A×A → C, 〈a, b〉 7→ φ(a∗b), (6.15) 88 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I donde Hφ es la completitud de A en ésta norma. Notando que el homomorfismo-∗ πφ : A → B(Hφ), induce una representación (A,Hφ) del álgebra C⋆ A asociando a un elemento a ∈ A un operador πφ(a)) ∈ A ⊂ B(H) según πφ(a)b = ab, (6.16) que es un operador lineal acotado bien definido en Hφ. Efectivamente, de la definición anterior se sigue que πφ(a1)πφ(a2)(b) = a1a2b = πφ(a1a2)b, (6.17) que muestra que (6.16) es efectivamente una representación.3 Para construir los elementos del espacio de Hilbert se parte de un vector privilegiado ξφ que séa ćıclico bajo πφ, i.e. tal que {π(a)ξφ|a ∈ A} es denso en Hφ. Como A es unital se puede escoger ξφ := 〈x = 0|ξφ〉 = ξφ(0, 0, 0) = I, que claramente es ćıclico siempre que los parámetros εi y µl, generados por los operadores πφ(a) = Ûi, V̂l ∈ B(Hφ), de acuerdo a (6.9) y (6.12), y que trasladan en direcciones perpendiculares unos de otros, estén relacionados apropiadamente en cuanto a que el conjunto de elementos generados por la acción de πφ(a) en ξφ sea denso en Hφ. Esto implica que las dos reticulaciones mencionadas en la sección previa coinciden, de acuerdo a las relaciones de consistencia µ1 = n1 2 ε2θ3 µ2 = n2 2 ε1θ3 (6.18) µ3 = n3 2 ε1θ2, donde, como se mostrará en §6.5, las magnitudes ni ∈ N + y ε̄i son factores de escala de las µi’s y εi’s determinadas por la relevancia relativa de los parámetros no-conmutativos en diferentes etapas de la cosmoloǵıa cuántica que se estudiará en las secciones siguientes. De hecho, es posible considerar las µi’s y εi’s como una familia de proyecciones cont́ınuas πm,n actuando en una familia de espacios topológicos Y n tales que πm,n : Y m → Y n, n ≤ m. (6.19) De forma que la variedadM con topoloǵıa de Hausdorff (Y∞) puede recuperarse como el proceso ĺımite de las preimágenes (πm,n)−1 [111]. Aún más interesante es el hecho que esta estructura algebraica posee dos ĺımites, en el ĺimite εi → 0 es inmediato que (6.9) conduce a una realización multiplicativa y µl se desacopla de (6.18), de manera que esta realización torcida del álgebra de Weyl se reduce a aquella de [112] y las reticulaciones conmutativas generadas por el espectro primitivo de ésta álgebra corresponde efectivamente al espacio estructural de configuración con topoloǵıa T1, donde como se mostrará 3En ésta construcción el álgebra C⋆ es también un modulo A de Hilbert. 6.2. Construcción GNS y observables f́ısicos 89 más adelante, la longitud elemental de las celdas inducidas por las µl’s es de orden O(λP ). Tomando entonces el ĺımite µl → 0 se recuperará el ĺımite cont́ınuo del álgebra de Heisenberg- Weyl y con ello un espacio T2 de Hausdorff.4 Ahora bien, de las expresiones (6.18), (6.9), y (6.12) se obtiene ε2θ3 =ε3θ2 ε1θ3 =ε3θ1 (6.20) ε1θ2 =ε2θ1, que implican que el subconjunto {π(Vi)ξφ} es por si mismo denso en Hφ y, en virtud de (6.16) y (6.15) (y el Teorema GNS), se tiene que para un estado funcional φ en {Vl} ⊂ A hay una representación con un vector ćıclico distinguido ξφ ∈ Hφ con la propiedad 〈ξφ, πφ(Vl)ξφ〉 = 〈I, Vl〉 = φ(Vl). (6.21) Recordando que (6.12) implica 〈x1 = 0|V̂l|ξφ〉 = ξφ(0+ µlêl) = ξφ(µlêl), (6.22) entonces, si v́ıa el homomorfismo-∗ de álgebras se asocia al elemento Vl ∈ A el operador πφ(Vl) = V̂ (−µlêl)), combinando ahora (6.21) con (6.22) permite identificar φ(Vl) con el caracter del grupo de traslaciones discretas, tal que ξkφ(xn) = e2πi ∑3 l=1 µl(klj(n)l), j(n)l ∈ Z (6.23) donde k ∈ R 3, y µl son cantidades cuyas magnitudes determinan el tamaño de la celda fun- damental de la reticulación. Es importante notar que, dado que I no contiene elementos, la representación (Hφ, ξφ) es irreducible. Las funciones ξkφ(x) en (6.23) son una representación regular irreducible unidimensional del grupo de operadores D̄k(x), del grupo abeliano de traslaciones discretas. Esto es D̄k(xn) = ξkφ(xn) = e2πi ∑ l µl(klj(n)l), (6.24) que satisface las relaciones de ortogonalidad y completez de Poisson [114] ∫ 1/2µl −1/2µl µldkl D̄ kl(j(1)l)D kl(j(2)l) = δj(1)lj(2)l , l = 1, 2, 3 ∞∑ ji=−∞ D̄ki (ji)D ki ′ (ji) = ∞∑ mi=−∞ δ(µiki − µik′i +mi), (6.25) 4En cierto sentido las relaciones (6.18) son equivalentes a la dinámica mejorada introducida en [113], que en esta construcción aparecen directamente de la consistencia requerida para las traslaciones generadas por la no-conmutatividad. 90 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I respectivamente, luego de notar que el lado derecho en la segunda ecuación de arriba es una función periódica generalizada con peŕıodo uno [115]. En vista de que las representaciones del grupo de traslaciones (6.24) son invariantes bajo el grupo rećıproco, el dominio de las compo- nentes del vector k es −1/2µi ≤ ki ≤ 1/2µi. Además, haciendo uso de la completez del espacio de kets {|k〉}, se puede escribir D̄kl(j(n)l) = e2iπj(n)lµlkl := 〈µlj(n)l|kl〉 = 〈x(n)l|kl〉, (6.26) con 3∏ l=1 ∫ 1 2µl − 1 2µl µldkl〈x(n)l|kl〉〈kl|x(n′)l〉 =: 〈x(n)|x(n′)〉 = δx(n),x(n′) . (6.27) Ahora bien, utilizando el Teorema de dualidad de Pontryagin, el dual de un grupo discreto Abeliano es un grupo compacto Abeliano, de forma que del análisis de Fourier se tiene la descomposición (para el ı́ndice i fijo) f̂(ki) = ∞∑ j(l)i=−∞ f(j(l)i) e µij(l)i(2iπki), −1/2µi ≤ ki ≤ 1/2µi, i = 1, 2, 3, (6.28) y la expresión dual f(j(l)i) = ∫ 1/2µi −1/2µi dki f̂(ki) e −ki(2iπµij(l)i). (6.29) Fijando, en particular, x(l)i := µij(l)i se puede ver que la función e2iπx(l)iki es cont́ınua y periódica en ki y entonces el polinomio ∑N l=1 f(x(l)i) e −2iπx(l)iki es una función cuasiperiódica en el sentido de Bohr (cf. [116] [117]). Si, además, tal polinomio converge uniformemente a la serie ∑∞ l=1 f(x(l)i) e 2iπx(l)iki cuando N →∞, entonces está última es también cuasiperiódica. Notando que al introducir el grupo rećıproco del grupo discreto de traslaciones en la ret́ıcula rećıproca LR := {bR = bi/µi, bi ∈ Z}, (6.30) se tiene inmediatamente de (6.28) que f̂(ki) = f̂(ki + bi/µi), (6.31) lo que confirma la afirmación debajo de la expresión (6.25) respecto del dominio fundamental de ki. Con estos resultados se pueden obtener las definiciones de observables cuánticos. Partiendo del hecho que cuando es posible implementar el Teorema de Stone-von Neumann (cf. §2.1 y §4.1) los operadores de posición y momento son equivalentes a los generadores infinitesimales de los grupos uniparamétricos del álgebra C∗ deWeyl correspondiente, que conducen a la representación 6.2. Construcción GNS y observables f́ısicos 91 diferencial usual de Mecánica Cuántica (conmutativa ó no). Es decir como los ĺımites Q̂i ≡ lim λ→0 1 2iλ [Ui(λ)− Ui(−λ)] = lim λ→0 1 2iλ [Ui(λ)− U∗i (λ)], P̂j ≡ lim γ→0 1 2iγ [Vj(γ)− Vj(−γ)] = lim γ→0 1 2iγ [Vj(γ)− V ∗j (γ)], (6.32) que evidentemente son expresiones hermitianas. Se puede generalizar este principio al álgebra C∗ de unitarios Ûi y V̂l para definir operadores autoadjuntos que corresponderán a los observables de la teoŕıa, con la reserva que no es posible tomar el ĺımite a causa del valor finito de las εi’s y µi’s. Dicho razonamiento conduce al operador de posición r̂i := − Ûi(εi)− Û †i (εi) 2iεi , (6.33) cuya realización en la base de kets se obtiene directamente de utilizar (6.9) y (6.12) r̂i|x〉 = − 1 2iεi ( e−2iπεixi |x+ 1 2 εiêi × θ〉 − e2iπεixi |x− 1 2 εiêi × θ〉 ) , (6.34) y a un operador de momento p̂l := − V̂l(µl)− V̂ † l (µl) 2iµl , (6.35) con p̂l|x〉 = 1 2iµl (|x+ µlêl〉 − |x− µlêl〉) , (6.36) de donde es claro que las unidades de los operadores están determinadas únicamente por las dimensiones de εi’s y µi’s. Ahora se puede mostrar que estas definiciones de observables reproducen las propiedades de los conmutadores del álgebra extendida de Heisenberg-Weyl (4.1). Substituyendo (6.33) en el conmutador [r̂i, r̂l] y recurriendo a (6.9), (6.10) junto con las expresiones (6.26), (6.27) se puede calcular el elemento de matriz 〈x′|[r̂i, r̂l]|x〉 = ( 2i εiεl ) sen(πεiεlθ · (êi × êl)) 3∏ m=1 ∫ 1 2 − 1 2 dk̄m { e2πik̄·(x ′−x) × cos ( 2πεiµi[ji + ( 1 2µi )k · (êi × θ)] ) cos ( 2πεlµl[jl + ( 1 2µl )k · (êl × θ)] )} , (6.37) con k̄i := µiki. De donde se infiere que la cantidad ( 2i εiεl ) sen(πεiεlθ · (êi × êl)) cos ( 2πεiµi[ji + ( 1 2µi )k · (êi × θ)] ) × cos ( 2πεlµl[jl + ( 1 2µl )k · (êl × θ)] ) , (6.38) es el śımbolo [r̂i, r̂l]sim de la acción del conmutador sobre la representación espectral del producto 92 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I 〈x′|x〉. En el ĺımite εiεlθ · (êi× êl)≪ 1 y, dado que (6.18) y (6.20) también εiµi ≪ 1, el śımbolo anterior es ahora lim [r̂i, r̂l]sim = 2iπθ · (êi × êl). (6.39) Un cálculo análogo para el conmutador [r̂i, p̂j ] usando (6.14) conduce a [r̂i, p̂l]sim = ( 2i εiµl ) sen(πεiµlδil) cos ( 2πεiµi[ji + ( 1 2µi )k · (êi × θ)] ) cos(2πµlkl), (6.40) para el cual se tiene el ĺımite lim [r̂i, p̂l]sim = 2iπδil, (6.41) y finalmente es trivial ver de (6.13) que en general [p̂i, p̂l] = 0 y, entonces, lim [p̂i, p̂l] = 0. Por lo que los śımbolos espectrales coinciden (módulo coeficientes) con los conmutadores del álgebra extendida de Heisenberg-Weyl en el ĺımite.  Las expresiones previas permiten retomar la discusión de las ĺıneas posteriores a (6.19), por que muestran la relación entre la representación torcida del álgebra C∗ de Weyl A ⊂ B(H), generada por operadores unitarios Ûi’s y V̂l’s y el álgebra extendida de Heisenberg-Weyl, la cual formalmente existe sólo en el ĺımite cont́ınuo de las reticulaciones del grupo de traslaciones discreto T3 y su reticulación rećıproca, generalizando los conceptos de observables f́ısicos dentro de un contexto puramente espectral. Esto se puede ver ahora más claramente gracias a las realizaciones (6.34) y (6.36), ya que, en los ĺımites mencionados, la primera ecuación se reduce a la realización multiplicativa del operador de posición (eigenvalor) y la segunda conduce a la diferenciación con respecto de xl. Mientras que para valores finitos de εi’s y µl’s la base de funciones cuasiperiódicas del espacio de Hilbert (6.24) no admite la interpretación usual de puntos en el espacio de configuración. En resumen, se ha visto que la no-conmutatividad de espacio del álgebra de Heisenberg puede ser expresada por una realización del grupo asociado de Heisenberg-Weyl por un álgebra C∗ A ⊂ B(H) de operadores unitarios acotados y con unidad, que actúan en un espacio de Hilbert no separable donde una base ortonormal es el conjunto de funciones cuasiperiódicas: {ξkφ(x(l)) = D̄k(x(l)) = e2iπx(l)·k}, (6.42) dadas por los caracteres en (6.23). 6.3 Cuantización del modelo cosmológico de Bianchi tipo I Por su simplicidad, la cosmoloǵıa (anisotrópica) de Bianchi I (ver Apéndice C para un breve repaso del modelo clásico) constituye un ejemplo natural para explorar las consecuencias de formulaciones cuánticas de Cosmoloǵıa, en el intento por construir teoŕıas donde los conceptos mecánico-cuánticos y de Relatividad General se encuentren al mismo nivel, y que permita mod- 6.3. Cuantización del modelo cosmológico de Bianchi tipo I 93 elar la F́ısica que pudiese haber prevalecido en los primeros instantes del Big Bang. El objeto central de estudio en este rubro (ver, e.g. [118]) es la constricción Hamiltoniana de la teoŕıa, ya que la prescripción de cuantización de Dirac (5.51) permite distinguir los estados cuánticos (geometŕıas) admisibles por medio de la ecuación de Wheeler-DeWitt [119]. Partiendo de la acción clásica (C.10) del modelo cosmológico de Bianchi I acoplado a un campo escalar sin masa Sgrav + Sϕ =~ ∫ d4x  piȧi − N(t) 4 √ 3g ( G~ c3 )  −1 2 ( 3∑ i=1 piai )2 + 3∑ i=1 (piai 2pi)     +~ ∫ d4x ( pφφ̇− N 4 √ 3g ( G~ c3 ) p2φ 2 ) , (6.43) con constricción Hamiltoniana clásica (cf. [120], [121]) Cgrav + Cφ = N(t) 4 √ 3g ( G~ c3 )    −1 2 ( 3∑ i=1 piai )2 + 3∑ i=1 (piai 2pi)  + 1 2 p2φ   = 0, (6.44) y fijando la función de lapso tal que N(t)(4(3g))− 1 2 = ( c3 G~ ) . La cuantización de éste sistema en el contexto presente se sigue de recordar que las variables dinámicas ai(t), p i(t), heredarán (ver §4.4), como operadores de Heisenberg, la no-conmutatividad espacial. Esto permite promoverlas a los operadores autoadjuntos (6.33) y (6.35), notando inclusive que esto es dimensionalmente correcto, mientras que el operador de campo escalar φ̂ y su momento conjugado p̂φ respetan el álgebra usual de la Mecánica Cuántica. Eligiendo finalmente un ordenamiento para el operador hermitiano asociado a la constricción clásica: Ĉ = Ĉgrav + Ĉφ = 1 2  − 3∑ i 6=j p̂iâiâj p̂ j + 3∑ i p̂iâ2i p̂ i  + 1 2 p̂2φ = 0̂. (6.45) Para hacer contacto con otras formulaciones se utilizarán técnicas de integral de trayec- toria en espacio-fase, discutida en el régimen no-conmutativo en §5.4), para una teoŕıa con constricciones de primera clase como lo es (6.45). Una alternativa al método usual para manejar constricciones es introduciendo la idea de un promedio sobre el grupo (ver e.g. [122, 123]). Observando que sólo el subespacio HF := {|ψ〉 ∈ H| Ĉ|ψ〉 = 0}, conocido como el espacio de Hilbert f́ısico, es relevante. Es entonces posible obtener expresiones definidas estrictamente en HF por medio de un proyector definido por la expresión general Ê := ∫ dµ(σ)ei ∑ a σaΦ̂a , (6.46) donde dµ(σ) es una medida invariante (normalizada) en la variedad de grupo generada por el 94 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I álgebra de constricciones de primera clase Φ̂a. 5 Debido a la invariancia de la medida y al álgebra de constricciones, Ê es un operador au- toadjunto e idempotente: Ê † = Ê, Ê 2 = Ê, (6.47) de forma que (6.46) es efectivamente un proyector. Entonces el proyector define un mapeo suryectivo del espacio de Hilbert total (denominado el espacio cinemático) al espacio de Hilbert f́ısico, i.e. ÊH ≡ HF . Para el caso en consideración, en que únicamente Φ̂a = Ĉ, esto se puede evidenciar fácilmente partiendo de un estado arbitrario |ψ〉 ∈ H y actuando sobre de él con el proyector Ê = ∫ dα eiαĈ , α ∈ R, (6.48) por lo tanto el estado |ψF 〉 := Ê|ψ〉 debe pertenecer al espacio de Hilbert f́ısico, como lo confirma la acción del operador unitario generado por la constricción: eiβĈ |ψF 〉 = ∫ dα ei(α+β)Ĉ |ψ〉 = ∫ dα′ eiα ′Ĉ |ψ〉 = |ψF 〉, (6.49) de donde directamente se tiene que Ĉ|ψF 〉 = 0.  En virtud de (6.47), el proyector Ê induce un producto interno enHF para estados arbitrarios de H, según la identidad 〈ψF |ϕF 〉 = 〈ψ|ʆÊ|ϕ〉 = 〈ψ|Ê|ϕ〉. (6.50) Se puede mostrar que la amplitud de transición de estados f́ısicos es una expresión definida estrictamente en HF . Considerando primero la base completa y ortogonal |x, φ〉 := |x〉|φ〉 de H, donde |x〉 := |µ1j1, µ2j2, µ3j3〉 y |φ〉 son los eigenvectores del operador de campo escalar, tal que 〈x′, φ′|x, φ〉 = δx′,xδ(φ ′, φ), (6.51) entonces, usando nuevamente (6.47), la descomposición de la función de onda ψF (x, φ) = 〈x, φ|ψF 〉 en esta base admite la forma ψF (x, φ) = 〈x, φ|Ê|ψ〉 = ∑ x′ ∫ dφ′ 〈x, φ|Ê|x′, φ′〉ψF (x ′, φ′), (6.52) y de acuerdo a (6.50) la cantidad 〈x, φ|Ê|x′, φ′〉 es el producto interno en HF para elementos de la base. Por lo tanto (6.52) es una expresión definida estrictamente en el espacio de Hilbert 5Es importante enfatizar que las σa’s deben tener las unidades apropiadas para que el argumento en la expo- nencial sea adimensional. 6.4. Integral de trayectoria y acción semiclásica 95 f́ısico y donde el kernel KF (x, φ;x ′, φ′) := 〈x, φ|Ê|x′, φ′〉 = ∫ dα 〈x, φ|eiαĈ |x′, φ′〉, (6.53) corresponde justamente al propagador en HF . 6 Es claro de la expresión anterior que el campo escalar juega el papel de un tiempo interno que permite recuperar la noción de evolución con respecto a diferentes valores de φ. 6.4 Integral de trayectoria y acción semiclásica Una vez contando con el propagador (6.53) apropiado paraHF se puede proceder, entonces, a for- mularlo en términos de la integral de trayectoria. Prestando atención al integrando 〈xf , φf |eiαĈ |xI , φI〉 para un estado inicial y uno final es claro que mantiene una semejanza con los propagadores considerados previamente en §5.4, excepto por que el término αĈ hace las veces de un Hamiltoniano Ĥ puramente matemático con un tiempo ficticio t = 1. Descomponiendo la evolución ficticia en N + 1 segmentos λ = 1 N+1 se tiene 〈xf , φf |eiαĈ |xI , φI〉 = ∑ xN ,...,x1 ∫ dφN . . . dφ1〈xN+1, φN+1|eiλαĈ |xN , φN 〉 . . . . . . 〈x1, φ1|eiλαĈ |x0, φ0〉, (6.54) donde 〈xf , φf | ≡ 〈xN+1, φN+1| y |xI , φI〉 ≡ |x0, φ0〉. El término n-ésimo en la expresión de arriba con Ĉ como en (6.45) está dado por 〈xn+1, φn+1|eiλαĈ |xn, φn〉 = 〈φn+1|e−iλαp̂ 2 φ |φn〉〈xn+1|eiλαĈgrav |xn〉 = ( 1 2π ∫ dpne iλαp2neipn(φn+1−φn) ) 〈xn+1|eiλαĈgrav |xn〉. (6.55) Es posible aproximar el término de la constricción gravitacional cuando N ≫ 1 por 〈xn+1|eiλαĈgrav |xn〉 ≈ δxn+1,xn + iλα〈xn+1|Ĉgrav|xn〉. (6.56) Usando (6.34), (6.36), aśı como (6.9–6.12) se ve que existen 16 términos que conforman la función de transición 〈xn+1|Ĉgrav|xn〉. Un término t́ıpico es, por ejemplo, 〈x(n+1)|V̂iV̂jÛiÛj |x(n)〉 = e−iπεiεj(êi×êj)·θ e−2πi(εix(n)i+εjx(n)j) × 〈x(n+1) − µiêi − µj êj |x(n) + 1 2 (εiêi + εj êj)× θ〉. (6.57) 6Los autores de [112] denominan a esta proyección de la función de transición en HF la amplitud de extracción. 96 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I el término 〈x(n+1)−µiêi−µj êj |x(n)+ 1 2(εiêi+ εj êj)×θ〉 en esta función de transición demanda entonces que se satisfagan las condiciones êl · [(εiêi ± εj êj)× θ] 2µl ∈ Z. (6.58) las cuales se cumplen idénticamente por las relaciones (6.18) y (6.20) para todos los términos en (6.56). Esto conduce entonces a 〈x(n+1)|V̂iV̂jÛiÛj |x(n)〉 = e−iπεiεj(êi×êj)·θ e−2πi(εix(n)i+εjx(n)j) 3∏ l=1 ∫ 1 2µl − 1 2µl µldk(n)l × e−2πiµlk(n)l(j(n+1)l−j(n)l)e2πik(n)l[µjδlj+µiδli+ 1 2 êl·(εiêi+εj êj)×θ]. (6.59) Continuando este proceso para todos los términos en (6.45) se puede mostrar luego de un cálculo extenso (Rosenbaum, Vergara, Juárez y Minzoni) [102] que la expresión resultante para 〈xn+1|Ĉgrav|xn〉 es 〈xn+1|Ĉgrav|xn〉 = 3∏ l=1 ∫ µldk(n)le −2πik(n)l( x(n+1)l−x(n)l) × { 1 4 3∑ i=1 1 ε2iµ 2 i sin2 [ 2πεi ( x(n)i + 1 2 3∑ l k(n)lθli )] sin2(2πk(n)iµi) −1 2 ∑ i 0 determina si la solución expande o colapsa en esa dirección respectiva- mente. Este criterio proporciona la descripción completa del sistema debajo y por encima de la escala de corte Li, imponiendo la compatibilidad para las condiciones de frontera en φ(Pi) tales que (ai)symb(Pi) = 1 εi sen(2πµiεiQ̄i(Pi)) = 2πµiQ̄i(Pi), χi(Pi) = 1 εiµi sen(2πµiεiQ̄i(Pi))sen(2πk̄i(Pi)) = 2πQ̄i(Pi)sen(2πk̄i(Pi)), (6.111) que implementa el cambio de variables f́ısicas al ir de la región debajo de la escala de corte a la región por encima. En este sentido cualquier trayectoria gobernada por una evolución del álgebra no-conmutativa con expresiones (6.88) y (6.87), con condiciones de frontera (6.109) y (6.110) obedecerá una evolución conmutativa a orden O(M) fuera de la región Planckiana determinada por (6.105- 6.108). Estos resultados pueden explicarse notando que el sistema tiene un espacio-fase 6-dimensional, del cual una proyección admisible corresponde a la gráfica (Vsymb, V̇symb) mostrada en Fig.(6.1) (que corresponde al espacio-fase de Fig.(6.4)). Esta figura muestra una órbita monotónica de colapso seguida de un comportamiento oscilatorio que escapa eventualmente a una nueva órbita que expande. Desde una perspectiva de ecuaciones diferenciales estrictamente se puede con- siderar θij = 0 y εi, µj 6= 0, entonces las Ri son claramente constantes de movimiento y las ecuaciones Riχi = ( Ri εiµi ) sen(2πεiµiQ̄i)sen(2πk̄i) = const. (6.112) proporcionan una familia de invariantes del sistema. En esta formulación el universo evoluciona en forma cuasi-periódica. Cuando θij 6= 0 los toros invariantes son sujetos a una perturbación Hamiltoniana. 106 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I 500 1000 1500 2000 2500 3000 3500 VHΦL -40000 -20000 20000 V   @ΦD Figura 6.1: Gráfica de espacio-fase de volúmen donde se observa la transición de una órbita abierta (rama inferior) a órbitas periódicas que conectan diversos toros invariantes terminando en una órbita que expande (rama superior). En la siguiente sección se presentan algunos ejemplos numéricos de cosmoloǵıas admisibles según las ecuaciones de movimiento (6.91), (6.93) y (6.96) y también usando el criterio de corte, donde se ha fijado Li = 30λp como escala de corte para continuar soluciones no-conmutativas hacia soluciones conmutativas. 6.6 Resultados Numéricos Las siguientes gráficas corresponden a simulaciones numéricas que ejemplifican los escenarios posibles, mostrando las diferencias contrastantes del caso no-conmutativo con respecto a solu- ciones del caso clásico. Considerando primero las soluciones fuertemente no-conmutativas para (6.97), que ocurren cuando el término no-conmutativo ∑3 j 6=i θij µiµj ˙̄kj en (6.87) actúa como una fuerza restitutiva (sin masa) que es conmensurable con el primer término de dicha expresión a todo momento. Esto corresponde a valores de εi tal que εiµi es de orden O(λP ). La Fig.6.2 y Fig.6.3 son ejemplos de tal regimen, obtenidos para valores numéricos de εi = 0.8(λp) −1 y εi = 0.4(λp) −1 respectivamente. Las soluciones están confinadas a escalas de volumen de Planck ya que ninguna puede acceder a la escala de corte Li. Pese a que su comportamiento es similar, el sistema de Fig.6.3 evoluciona con mayor riqueza que el de Fig.6.2 con mı́nimos y máximos globales ahora con diferentes ordenes de magnitud. Las oscilaciones heterogéneas son en ambos casos el resultado del término de fuerza no-conmutativa 6.6. Resultados Numéricos 107 -3 -2 -1 1 2 3 Φ 0.5 1.0 1.5 VHΦL Figura 6.2: Cuando εi = 0.8(λP ) −1, las soluciones de Volumen (para condiciones iniciales de los radios espec- trales de orden λp) muestran un comportamiento oscilatorio con máximos y mı́nimos dentro del mismo orden de magnitud y el sistema queda confinado a escalas de volumen Planckiano. -15 -10 -5 5 10 15 Φ 2 4 6 8 10 12 14 VHΦL Figura 6.3: Solución con ε = 0.4(λP ) −1 . Cuando ǫi toma valores más chicos el sistema puede acceder a volúmenes mayores y la interferencia constructiva de los radios espectrales permite la formación de máximos varios con órdenes de magnitud mayores que los mı́nimos. Para valores ǫi < 1/Li estos máximos alcanzan la escala de corte donde las soluciones son gobernadas por el régimen conmutativo. mencionada antes, actuando como un forzamiento, lo que modula las frecuencias de las soluciones de los radios espectrales del universo. Esto significa que la no-conmutatividad es el agente que obliga al universo a escapar de las escalas Planckianas. Entonces, considerando la evolución del sistema cuando éste se aproxima a la escala de corte por debajo, i.e. cerca de L̄i = 30 entonces, el primer término a la derecha de (6.87) se convierte en πQ̄i cos(2πk̄i)R1 con Ri de acuerdo a (6.89) y α = β = γ = 0, donde las χi’s son ahora constantes de movimiento. Para ejemplificar estas soluciones se presenta el rebote (bounce) de la Fig.6.4. En este escenario la trayectoria que colapsa (punteada) ingresa al régimen no- conmutativo por la izquierda, lo que conduce a la evolución no-conmutativa (ĺınea) debajo de la 108 Caṕıtulo 6. Cosmoloǵıa Cuántica en la representación torcida del álgebra C⋆ de Weyl: El modelo de Bianchi I escala de corte, donde se observan algunas oscilaciones no-conmutativas, hasta que los efectos del término de fuerza no-conmutativa conducen al sistema a una fase de expansión que logra alcanzar la escala de corte para continuar por una expansión cont́ınua. La Fig.6.5 proporciona mayor información sobre las interacciones subyacentes entre los radios espectrales (ai)symb cuya interferencia constructiva ó destructiva conduce al comportamiento del volúmen dentro de la región no-conmutativa. -6 -5 -4 -2 -1 0 Φ 10000 20000 30000 40000 V HΦL Figura 6.4: Solución de colapso y expansión para εi = 0.031(λP ) −1. La evolución no-conmutativa (ĺınea), compatible con las condiciones de frontera de la solución no-conmutativa (punto) que ingresa por la izquierda de la figura, permanece en la región interna por un periodo de tiempo finito antes que la interferencia constructiva de los radios espectrales lleve al sistema de nuevo a la región conmutativa hacia una expansión que se aleja de la escala de corte. Ha1Lsymb Ha2Lsymb Ha3Lsymb -6 -4 -2 0 2 Φ 5 10 15 20 25 30 HaiLsymb Figura 6.5: Radios espectrales (a1)symb, (a2)symb, (a3)symb para ε = 0.031(λP ) −1. Se muestra la interferencia constructiva y destructiva dentro del régimen no-conmutativo que conduce a la evolución del volúmen en (Fig.6.4). Para finalizar, de la evolución de las χi’s en Fig.6.6 se debe enfatizar que las regiones adiabáticas que promueven la evolución hacia regiones clásicas son cada vez más dominantes 6.6. Resultados Numéricos 109 para escalas de corte mayores, y es precisamente en la regiones adiabáticas en los extremos de la Fig.6.6 que el sistema evoluciona para φ ≷ φ(Pi) sobre esos valores constantes de χi. Χ1 Χ2 Χ3 -6 -5 -4 -3 -2 -1 0 1 Φ 0.5 1.0 1.5 2.0 2.5 3.0 Χi Figura 6.6: Gráfica de χ1, χ2, χ3, asociadas al volúmen de Fig.6.4 donde las regiones simultáneas valores constantes de χi a la izquierda y derecha de la figura conducen a la evolución asintótica del volúmen léjos de la escala de corte. Caṕıtulo 7 Discusión, Conclusiones y Ĺıneas de Investigación Futuras Partiendo de conceptos fundamentales de la F́ısica contemporánea se presentaron argumentos en favor del estudio de conmutadores no triviales de observables de posición mecánico-cuánticos. Esto como un primer paso en el intento por investigar las implicaciones de una formulación cuántica de la gravitación a escalas Planckianas, donde se espera que los efectos de la misma sean conmensurables a los de las demás interacciones fundamentales y cuya eventual descripción teórica conduzca a una unificación última de la F́ısica. Las herramientas y estructuras matemáticas utilizadas para lo anterior versaron principal- mente (aunque no exclusivamente) en torno a los formalismos de cuantización por deformación y a conceptos de geometŕıa no-conmutativa. Haciendo constante hincapié en las álgebras de oper- adores de Mecánica Cuántica y, fundamentalmente, en el papel central de los operadores unitar- ios y acotados correspondientes que actúan en espacios de Hilbert, aśı como sus representaciones. Esto es evidente ya en los Caṕıtulos 2 y 3, preliminares al análisis de la no-conmutatividad, donde se resaltaron las propiedades elementales de los operadores de desplazamiento (ó sus análogos) en diversas formulaciones de la cuantización por deformación, ya sea partiendo de primeros principios, como en el formalismo WWGM ó axiomáticamente como en la cuantización de Stratonovich-Weyl ó de Berezin y que, usualmente, son elementos del grupo de Lie asociado a la teoŕıa cuántica. En el contexto de la deformación de álgebras son estas propiedades, y prin- cipalmente la existencia de una traza bien definida, las que permiten establecer un isomorfismo entre la categoŕıa de endomorfismos (operadores lineales) de un espacio de Hilbert y la categoŕıa de funciones infinitamente diferenciables en un espacio-fase con producto deformado. Siendo esto cierto para el caso de los sistemas elementales (en el sentido de la Definición 1) compactos y planos, como se mostró al final de §3.3, mientras que para casos no-eucĺıdeos no compactos es necesario introducir otro tipo de operadores unitarios y acotados para dicho fin, como sucede por ejemplo en la construcción de Berezin con el operador de reflexión. 111 112 Caṕıtulo 7. Discusión, Conclusiones y Ĺıneas de Investigación Futuras Aśı bien, en el Caṕıtulo 4 se presentó dentro de la cuantización por deformación, con com- pleto detalle matemático, la construcción de la Mecánica Cuántica no-conmutativa, entendida como la extensión de los principios generales de Mecánica Cuántica para el caso de observ- ables de posición no-conmutantes, asumiendo que dichos principios son válidos aún a longitudes de Planck y que la noción de simetŕıa debe ser reemplazada por la de una simetŕıa torcida. Probando primero que los operadores de desplazamiento, definidos de la misma forma que aque- llos del caso conmutativo, permiten contar con una base completa de Fourier para operadores, aún en el caso no-conmutativo (como se mostró en §4.2), y que son, consecuentemente, los ob- jetos básicos para el análisis. Lo anterior permitió obtener expresiones rigurosas, usando bases mixtas de eigenestados de operadores de posición genuinamente no-conmutativos y de momento, para el isomorfismo entre el álgebra extendida de Heisenberg-Weyl y el álgebra de funciones de espacio-fase equipada con un producto-∗ de deformación. Al derivar dicho producto-∗ de forma expĺıcita se encontró que consiste de la composición del producto-⋆~, asociado a la Mecánica Cuántica usual, con un producto-⋆θ definido estrictamente en el subespacio de configuración. Este último coincide, idénticamente, con la expresión para el producto de funciones introducido (generalmente en forma heuŕıstica) en la Teoŕıa de Campos No-conmutativa, que emana de al- gunos modelos fenomenológicos de aproximaciones a bajas enerǵıas de la Teoŕıa de Cuerdas en presencia de campos magnéticos intensos. Notando, sin embargo, que en el sentido estricto de la Mecánica Cuántica son sólo los valores de expectación los que tienen un significado f́ısico, esto se traduce, en el formalismo WWGM, al hecho que el cálculo de la traza de un observable cuántico con la matriz de densidad de von Neumann implica una integral en espacio-fase de equivalentes de Weyl con la función de Wigner- Szilard. Entonces, la interpretación probabiĺıstica de espacio-fase está directamente relacionada con la definición de una densidad de cuasiprobabilidad apropiada. Como se mostró en §4.3, en el caso del álgebra extendida de Heisenberg-Weyl, es posible retener una definición para la función de Wigner-Szilard en términos de bases mixtas que asemeje la definición usual del caso conmutativo. Aunque, como también se mostró, esta definición no es única ya que las propiedades del producto de deformación en el sector no-conmutativo garantizan la existencia de más de una función de cuasiprobabilidad admisible. Analizando la modificación a la ecuación de valores-⋆, como la condición más fuerte para el valor de expectación, se estableció la relación entre las posibles funciones de cuasiprobabilidad, confirmando que en el ĺımite conmutativo θ → 0 todas ellas son proyectadas a una única función de Wigner-Szilard. Esta multiplicidad de funciones de cuasiprobabilidad implica entonces una posible degeneración en los estados de enerǵıa E en el régimen no-conmutativo. La interpretación e implementación teórica de este resultado se ha dejado abierta para trabajo futuro. Posteri- ormente, habiendo prestado atención a las propiedades del producto-∗ y de las funciones de cuasiprobabilidad bajo integrales de espacio-fase, se proporcionó una nota precautoria sobre la proliferación en Teoŕıas de Campos de expresiones que incorporan productos-⋆θ y a las cuales se les ha atribuido cierto carácter fundamental, pero que no necesariamente tienen una justificación leǵıtima dentro de la cuantización por deformación y que, en el mejor de los casos, constituyen 113 versiones reducidas ó simplificadas de las ecuaciones de valores-⋆ discutidas en detalle en éste trabajo. Del estudio de la evolución de equivalentes de Weyl de operadores cuánticos en §4.4, dentro del esquema de Heisenberg, se identificó un mecanismo dinámico para el origen del producto- ⋆θ en teoŕıa de campos no-conmutativa mencionado arriba. Esto se logró mostrando primero que las variables de espacio-fase corresponden a las variables dinámicas genuinas con estruc- tura de Poisson inducida naturalmente por la no-conmutatividad. El aspecto más importante fué la presencia de un paréntesis de Poisson no nulo para las variables de configuración de espacio-fase, consistente con el producto-⋆θ de la subálgebra de funciones Aθ de espacio-fase generada únicamente por coordenadas espaciales. Consecuentemente, una teoŕıa de campos en Aθ, definida como un módulo sobre el anillo de funciones Aθ, heredará también el producto-⋆θ. Se puede esperar que un estudio detallado de modelos exactamente solubles en este contexto ayuden, entonces, a obtener un mejor entendimiento de las consecuencias fenomenológicas de la no-conmutatividad del espacio en teoŕıa de campos. En el Caṕıtulo 5 se abordaron diversas representaciones de Mecánica Cuántica no-conmutativa vista no sólo desde la perspectiva de deformación de álgebras sino también dentro de formalismos de cuantización más familiares, a saber, la cuantización canónica y de integral de trayectoria. En §§5.1-5.2 se estudió la no-conmutatividad considerando formulaciones más axiomáticas de cuantización por deformación (en el sentido del Caṕıtulo 3), recurriendo a bases supercompletas de estados coherentes. Un resultado central de dicho análisis fué mostrar la equivalencia entre las realizaciones holomorfas del producto-∗ v́ıa el cuantizador de Stratonovich-Weyl aśı como con el operador de reflexión. Igualmente destacable fué evidenciar que la no-conmutatividad puede absorberse en las definiciones de operadores de creación y aniquilación (no observables) y, consecuentemente, las expresiones obtenidas son aplicables tanto al caso de Mecánica Cuántica conmutativa como no-conmutativa, sin que el parámetro de no-conmutatividad sea expĺıcito. Esto representa una ventaja matemática en el estudio de la no-conmutativad, ya que es posible trabajar con una realización holomorfa para llevar expresiones hasta su forma final que luego pueden reescribirse en términos de variables f́ısicas, recuperando aśı las interpretaciones teóricas. Por otro lado se sabe que el uso de estados coherentes tiene aplicaciones importantes en análisis semiclásicos de Mecánica Cuántica y, por lo tanto, este tipo de descripciones puede ser valiosa en contextos perturbativos. En virtud que el argumento de covariancia bajo simetŕıas torcidas v́ıa una torcedura de Drinfeld, discutido en §4.1, es válido tanto para el espacio coordenado como para el espacio- tiempo, entonces, es posible, en principio, extender resultados de la no-conmutatividad para observables de espacio-tiempo cuando las variables dinámicas de espacio-tiempo son promovidas a observables cuánticos. Esto es posible en una forma natural dentro del contexto de la cuan- tización canónica, como se mostró en §5.3 donde se presentó un programa de cuantización que recurre, como primer paso, al concepto de teoŕıas invariantes bajo reparametrización seguido de un análisis de constricciones para una acción genérica de primer orden. Dichos elementos 114 Caṕıtulo 7. Discusión, Conclusiones y Ĺıneas de Investigación Futuras conducen entonces a paréntesis de Dirac, definidos sobre la superficie de constricción, que in- corporan paréntesis no nulos entre coordenadas de espacio-tiempo (aśı como de momento) que al cuantizar recuperan conmutadores (incluyendo al tiempo) del tipo del álgebra extendida de Heisenberg-Weyl. El estudio con bases mixtas, como las introducidas en el Caṕıtulo 4, para un ejemplo elemental con un Hamiltoniano mecánico muestra que la no-conmutatividad permite recuperar una ecuación de tipo Schrödinger únicamente cuando el término del potencial dependa linealmente de las coordenadas de espacio. De lo contrario se obtienen expresiones con derivadas en el tiempo de orden mayor a uno que impiden recuperar una interpretación probabiĺıstica de la función de onda directamente. Esto hace contacto con el problema del tiempo (y unitariedad) en Mecánica Cuántica y Campos lo que sugiere estudiar formulaciones no-conmutativas (posi- blemente relativistas) que permitan lidiar con esta situación de forma consistente. Usando los resultados de §§5.1-5.3 se construyeron los diferentes tipos de integral de trayec- toria de Mecánica Cuántica no-conmutativa, en las realizaciones de bases mixtas y de estados coherentes no-conmutativos aśı como promoviendo la acción no canónica de primer orden men- cionada arriba a una acción semiclásica. El estudio comparativo de estas construcciones permitió identificar el tipo de Hamiltonianos y condiciones variacionales para los puntos extremos que hacen equivalentes unas con otras. Mientras que la integral de trayectoria en términos de es- tados coherentes no-conmutativos pertenece, en general, a una clase diferente cuya propiedad más atractiva es su relación directa con el Q-śımbolo de Husimi del Hamiltoniano cuántico, el cual constituye una suavización Gaussiana del equivalente de Weyl del Hamiltoniano. En una aproximación de fase estacionaria esto puede proporcionar una descripción global de la evolución de estados cuánticos, v́ıa las coordenadas del centro de paquetes Gaussianos que siguen trayec- torias dictadas por ecuaciones de Euler-Lagrange. Cabe remarcar que un común denominador en todas las expresiones de integral de trayectoria es la presencia de un término cinemático no-conmutativo dentro de la acción semiclásica, el cual combina contribuciones de momento en direcciones ortogonales y que cuyo origen proviene de una elección de observables de posición genuinamente no-conmutativos. Este término ha sido asociado recientemente en la literatura con modelos topológicos de tipo Chern-Simons y donde los observables corresponden a los la- zos de Wilson, lo que posiblemente sugiere una conexión más profunda con formalismos como la Gravedad Cuántica de Lazos en donde unas de las variables dinámicas son precisamente holonomı́as sobre un haz fibrado. Finalmente en el Caṕıtulo 6 se presentó una estructura matemática novedosa para introducir el concepto de no-conmutatividad, recurriendo a elementos de álgebras C∗. Esto con la final- idad de hacer contacto con nociones más cercanas a la Geometŕıa No-conmutativa de Connes discutida brevemente en §5.5, en donde el concepto de variedad diferencial es substituido por un triple espectral (A,H, D) con A representando un álgebra C∗, H un espacio de Hilbert y D un operador de Dirac. En efecto, la Geometŕıa No-conmutativa de Connes parte del teorema de Gel’fand-Naimark, mencionado en ĺıneas introductorias de dicho Caṕıtulo, en que el dual a un espacio-tiempo no-conmutativo debe corresponder a un álgebra C∗ no-conmutativa. Este criterio es también la base para la formulación empleada aqúı, al igual que la definición de operadores 115 acotados de la subálgebra B(A) ⊂ A y la construcción de un espacio de Hilbert generado por dicha álgebra. Mas aún, dado que nuestro objetivo final es el investigar éste nuevo formalismo para la descripción de la Gravedad Cuántica y su eventual unificación con las otras teoŕıas de campo de la materia, hemos implementado elementos del formalismo para el análisis de singular- idades que ocurren en la Cosmoloǵıa Clásica, partiendo del hecho de que la Cosmoloǵıa Cuántica puede considerarse como un minisuperespacio de una teoŕıa cuántica del campo gravitacional. Como punto de partida de lo arriba mencionado se consideró el hecho que, en el lenguaje de teoŕıa de representaciones, una subálgebra de Weyl de operadores de desplazamiento del grupo extendido de Heisenberg-Weyl, generada únicamente por operadores de posición no- conmutativos, constituye una representación σ-proyectiva del grupo de traslaciones eucĺıdeas. Esto permitió en §6.1 considerar alternativamente la representación torcida del grupo topológico discreto de traslaciones en R 3 como el álgebra C∗ elemental del triple espectral, seguida en §6.2 de la construcción de Gelfand-Naimark-Segal para generar el espacio de Hilbert correspondi- ente sobre el cual actúa. Lo anterior condujo entonces a una serie de condiciones entre los parámetros εi, µj introducidos espećıficamente en esas secciones por razones dimensionales y que, del análisis posterior, adquirieron una interpretación f́ısica como parámetros de escala de la teoŕıa intŕınsecamente relacionadas con la evolución de la cosmoloǵıa cuántica estudiada. Una diferencia fundamental de esta formulación con las demás discutidas en este trabajo es que, por construcción, se aparta de las consecuencias del teorema de Stone-von Neumann y, por lo tanto, los observables cuánticos de posición y momento, definidos a partir de los operadores acotados del álgebra C∗, no poseen una realización multiplicativa y diferencial (respectivamente) como es usual. Esto confirma que en una geometŕıa no-conmutativa el mismo concepto de puntos del espacio (y diferenciación), interpretados como los eigenvalores (simultáneos) de operadores de posición cuánticos, no es admisible. Aśı bien, la implementación de las nuevas definiciones de observables cuánticos permitió estudiar en §§6.3-6.4 el colapso cuántico de una cosmoloǵıa anisotrópica de Bianchi I v́ıa la inte- gral de trayectoria. Siguiendo métodos de fase estacionaria, para analizar la acción semiclásica efectiva, fué posible mostrar en §6.5 que las ecuaciones de evolución introducen una gran riqueza dinámica que es relevante a escalas Planckianas de longitud y que, posteriormente, se suprime a escalas mayores por medio de los parámetros de escala εi recuperando eventualmente una evolución que coincide con la evolución clásica. Esta nueva dinámica incorpora los efectos de no-conmutatividad a través de un término de forzamiento que acopla la evolución de direcciones perpendiculares. Se mostró asintóticamente y numéricamente que lo anterior induce un com- portamiento oscilatorio del volúmen de la cosmoloǵıa debido a una evolución no trivial de las variables de acción, las cuales en el caso clásico son constantes de movimiento. Aśı también se mostró que la constricción Hamiltoniana implica la ausencia de singularidades en las variables dinámicas en el régimen no-conmutativo lo que idealmente debeŕıa ser el aspecto caracteŕıstico de una teoŕıa no-conmutativa donde, formalmente, no existen los puntos geométricos que irre- mediablemente conducen a dichos escenarios. 116 Caṕıtulo 7. Discusión, Conclusiones y Ĺıneas de Investigación Futuras Si bien es cierto que no es del todo claro el grado de relevancia con que la Gravitación a nivel de mini-superespacio tiene que ver con formulaciones a niveles de midi ó superespacio y que existen en la literatura una variedad de propuestas más o menos diferentes para resolver el problema de las singularidades, el formalismo matemático empleado en el Caṕıtulo 6 tiene como motivación subyacente el antecedente que la Teoŕıa de Haces Fibrados es la más a propo para la descripción de las Teoŕıas de Campo a nivel clásico. En este contexto es bien sabido que el análogo geométrico a las secciones de un haz fibrado vectorial complejo y Hermı́tico es un módulo sobre un álgebra A = C(M), es decir un álgebra conmutativa C∗ de funciones complejas cont́ınuas en un espacio de HausdorffM localmente compacto. En vista de lo anterior es natural esperar que el análogo no-conmutativo a un haz fibrado vectorial sea provisto por un módulo proyectivo finito sobre un álgebra no-conmutativa A [98]. Como una propuesta alterna a las ya aparecidas en la literatura para abordar el problema de las singularidades clásicas en Cosmoloǵıa dentro del contexto de la no-conmutatividad, sin usar formalmente toda la maquinaria matemática desarrollada en el último caṕıtulo de este trabajo, está la posibilidad de recurrir a los estados coherentes no-conmutativos construidos en §§5.1-5.2. El razonamiento para esto se basa en las caracteŕısticas de este tipo de sistemas supercompletos del espacio de Hilbert, descritas con anterioridad, que permiten dar salida a la ausencia manifi- esta de un conjunto completo de observables de configuración simultáneos. Siendo de particular interés la propiedad que facilita un conjunto simultáneo de variables dinámicas de configuración en la forma de valores de expectación. En un trabajo recientemente enviado a publicación, se ha implementado este concepto aśı como diversas técnicas de estados coherentes no-conmutativos presentadas a lo largo de éste trabajo y su utilización en la integral de trayectoria. Ello ha permitido obtener resultados que recuperan parte del comportamiento oscilatorio del volúmen de la cosmoloǵıa de Bianchi I en escalas Planckianas, evidenciados en la formulación de álgebra C∗ torcida, incluyendo también aspectos del celebrado comportamiento de Gran Rebote (Big Bounce). En investigaciones futuras será interesante estudiar la extensión de la Geometŕıa No-con- mutativa a modelos Cosmológicos Inhomogéneos (con grados de libertad espaciales), aśı como buscar una formulación del operador de Dirac del triple espectral para una acción basada en una apropiado módulo proyectivo finito sobre un álgebra no-conmutativa A, permitiendo aśı incorporar en esa acción el hermoso resultado de Connes sobre el origen no-conmutativo de los Campos de Norma y el concomitante cambio fundamental en la topoloǵıa del espacio-tiempo que ello implica. Parte IV Apéndices 117 Apéndice A Material complementario del formalismo WWGM En este apéndice se presentan diversos resultados que, junto con los elementos del Caṕıtulo 2, sirven para establecer una perspectiva general del formalismo WWGM como método de cuantización autónomo. A.1 Propiedades algebraicas e integrales del producto ⋆~. El producto ⋆~ está definido por el operador bidiferencial (2.38) y puede, en principio, parecer un objeto matemático inexpugnable. Esto lo justifica en parte el desarrollo expĺıcito de un producto de funciones arbitrarias f, g, i.e. f ⋆~ g = ∞∑ l=0 (i~)l 2ll! f (←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi )l g = ∞∑ l=0 (i~)l 2ll! l∑ k=0 ( l k ) (−1)kf ( ←− ∂ ∂qir −→ ∂ ∂pir )l−k( ←− ∂ ∂pis −→ ∂ ∂qis )k g = ∞∑ l=0 (i~)l 2ll! l∑ k=0 ( l k ) (−1)k l−k∏ r=1 k∏ s=1 ( ∂ ∂qir ∂ ∂pis f )( ∂ ∂pir ∂ ∂qis g ) , (A.1) sin olvidar que los ı́ndices ir, is involucran sumas de n términos. Por lo tanto, aunque f ⋆~ g es simple de evaluar cuando f y g son polinomios de orden bajo, rápidamente produce términos muy engorrosos en el caso general. Sin embargo las propiedades siguientes resultan muy útiles en la manipulación de expresiones que involucran a ⋆~. 119 120 Apéndice A. Material complementario del formalismo WWGM Traslaciones. A continuación se verá como la acción, ya sea izquierda ó derecha, de ⋆~ sobre alguna función f(~q, ~p) codifica toda la información del producto (A.1) en un operador diferencial más claro. Para una función de una variable f(x) y un operador diferencial arbitrario D, la serie formal ∞∑ n=0 ←− Dn n! ∂nf(x) ∂xn , (A.2) representa un operador diferencial de orden infinito que actúa sobre funciones que multiplican por la izquierda. Esta expresión se identifica con la serie de Taylor formal del operador diferencial f(x+ ←− D), cuya forma funcional es la misma que la de f(x), haciendo el reemplazo x→ x+ ←− D . Alternativamente, visto como un operador diferencial que actúa sobre f(x), la serie (A.2) también se escribe como ∞∑ n=0 1 n! (←− D −→ ∂x )n f(x) = e ←− D −→ ∂xf(x). (A.3) lo que permite establecer la igualdad e ←− D −→ ∂xf(x) = f(x+ ←− D), (A.4) que heuŕısticamente puede tomarse como la “traslación” de f(x) por ←− D . Usando argumentos similares también se tiene f(x)e ←− ∂x −→ D = f(x+ −→ D). (A.5) Consecuentemente para (2.38) se infieren las “traslaciones” f(qi, pi)⋆~ = f ( qi + i~ 2 −→ ∂ ∂pi , pi − i~ 2 −→ ∂ ∂qi ) , ⋆~f(q i, pi) = f ( qi − i~ 2 ←− ∂ ∂pi , pi + i~ 2 ←− ∂ ∂qi ) . (A.6) Es notable que ⋆~ tenga una forma cerrada y posea la propiedad anterior. Estas caracterśticas facilitan la obtención de otras propiedades que, en general, solo pueden definirse hasta algún orden en la serie formal de un producto-⋆ y, además, requieren del uso de estructuras matemáticas más sofisticadas como los espacios de cohomoloǵıas (ver [25, 26]). Asociatividad. La asociatividad es la condición necesaria que cualquier producto-⋆ debe cumplir y para de- mostrar que ⋆~ es asociativo se empleará la propiedad (A.6). Por simplicidad de cálculo se A.1. Propiedades algebraicas e integrales del producto ⋆~. 121 restringirán las coordenadas a R 2, sin que esto afecte la generalidad de los resultados. Partiendo de la definición de asociatividad [f(q, p) ⋆~ g(q, p)] ⋆~ h(q, p) = f(q, p) ⋆~ [g(q, p) ⋆~ h(q, p)] , (A.7) se utiliza la primera identidad de (A.6) para los términos dentro de los corchetes [ f ( q + i~ 2 −→ ∂p, p− i~ 2 −→ ∂q ) g(q, p) ] ⋆~ h(q, p) = f(q, p) ⋆~ [ g ( q + i~ 2 −→ ∂p, p− i~ 2 −→ ∂q ) h(q, p) ] . (A.8) Antes de continuar expandiendo la expresión anterior es conveniente utilizar una notación auxiliar sólo para éste cálculo, de manera que puedan distinguirse correctamente las acciones de los diversos operadores diferenciales independientemente de las concatenaciones de productos. Por ejemplo, el lado izquierdo de (A.8) se reescribe como [ f ( q + i~ 2 −→ ∂pg , p− i~ 2 −→ ∂qg ) g(q, p) ] ⋆~ h(q, p), (A.9) donde el sub-sub́ındice en las diferenciaciones ∂qg ,∂pg implica que la acción es únicamente sobre los parámetros de la función g. Tomando esto en consideración para realizar los productos restantes de (A.8) se tiene f ( q + i~ 2 (−→ ∂pg + −→ ∂ph ) , p− i~ 2 (−→ ∂qg + −→ ∂qh )) g ( q + i~ 2 −→ ∂ph , p− i~ 2 −→ ∂qh ) h(q, p) = f ( q + i~ 2 −−→ ∂pgh , p− i~ 2 −−→ ∂qgh ) g ( q + i~ 2 −→ ∂ph , p− i~ 2 −→ ∂qh ) h(q, p). (A.10) Notando que el término g ( q + i~ 2 −→ ∂ph , p− i~ 2 −→ ∂qh ) h(q, p) es el mismo en ambos lados de la igual- dad, entonces basta mostrar que para el producto g · h (obviando las dependencias de las fun- ciones) se cumple f ( q + i~ 2 (−→ ∂pg + −→ ∂ph ) , p− i~ 2 (−→ ∂qg + −→ ∂qh )) g · h = f ( q + i~ 2 −−→ ∂pgh , p− i~ 2 −−→ ∂qgh ) g · h, (A.11) donde −−→ ∂pgh , −−→ ∂qgh actúan como diferenciaciones usuales sobre el producto g · h. 122 Apéndice A. Material complementario del formalismo WWGM En términos de series de Taylor formales (A.11) es ∞∑ n,m=0 (i~)n 2nn! (−i~)m 2mm! (∂nq ∂ m p f) [( ∂pg + ∂ph )n ( ∂qg + ∂qh )m g · h ] = ∞∑ n,m=0 (i~)n 2nn! (−i~)m 2mm! (∂nq ∂ m p f)∂ n p ∂ m q (g · h), (A.12) entonces, para que la igualdad se satisfaga término a término, debe ocurrir [( ∂pg + ∂ph )n ( ∂qg + ∂qh )m g · h ] = ∂np ∂ m q (g · h). (A.13) Desarrollando los binomios en el lado izquierdo y diferenciando las funciones de acuerdo a la acción espećıfica de cada operador se tiene [( ∂pg + ∂ph )n ( ∂qg + ∂qh )m g · h ] = n∑ k=0 m∑ l=0 ( n k )( m l ) (∂n−kp ∂m−lq g)(∂kp∂ l qh), (A.14) que es justamente el desarrollo de ∂np ∂ m q (g · h) de acuerdo a la regla de Leibniz, confirmando (A.13) y consecuentemente la asociatividad.  Conjugación. Si φ y ψ son dos funciones que toman valores en C, la conjugación compleja del producto φ ⋆~ ψ corresponde a (φ ⋆~ ψ) ∗ = φ∗ e − i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) ψ∗ = ψ∗ e i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) φ∗ = ψ∗ ⋆~ φ ∗. (A.15) Por lo tanto el espacio C∞⋆ (R2n) equipado con la conjugación en C constituye un álgebra-C∗ no-conmutativa . Integración. En el formalismo WWGM existen diversas expresiones que involucran la integración en espacio- fase de productos-⋆ de magnitudes f́ısicas. En general puede esperarse que cualquier función que represente algúna cantidad observable corresponda a una función de soporte compacto. Si ∂ representa genéricamente una derivada parcial en cualquiera de las coordenadas (qi, pi) de R2n, entonces, para dos funciones f, g de soporte compacto y m ∈ N, la integración por partes A.1. Propiedades algebraicas e integrales del producto ⋆~. 123 toma la forma simple ∫ R2n dn~qdn~p (∂mf(~q, ~p))g(~q, ~p) = (−1)m ∫ R2n dn~qdn~p f(~q, ~p)(∂mg(~q, ~p)), (A.16) ya que el término de frontera se anula. Escribiendo ⋆~ como el producto de operadores bidiferenciales ⋆~ = e i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi e − i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi , (A.17) el primer operador exponencial de esta expresión satisface ∫ R2n dn~qdn~p f(~q, ~p)e i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)e i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi g(~q, ~p), (A.18) donde se utilizó (A.16) para invertir la acción (las flechas) de las parciales de qi y pi, por lo que el signo del argumento se preserva. Haciendo lo análogo con el segundo exponencial se tiene ∫ R2n dn~qdn~p f(~q, ~p)e − i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)e − i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi g(~q, ~p), (A.19) y reuniendo estos dos resultados conduce a ∫ R2n dn~qdn~p f(~q, ~p)e i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi e − i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)e i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi e − i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi g(~q, ~p) = ∫ R2n dn~qdn~p g(~q, ~p)e i~ 2 ←− ∂ ∂qi −→ ∂ ∂pi e − i~ 2 ←− ∂ ∂pi −→ ∂ ∂qi f(~q, ~p) = ∫ R2n dn~qdn~p g(~q, ~p) ⋆~ f(~q, ~p), (A.20) donde en el penúltimo paso hubo un reordenamiento de términos para evidenciar el producto ⋆~. Por lo tanto se recupera la expresión (2.47): ∫ R2n dn~qdn~p f(~q, ~p) ⋆~ g(~q, ~p) = ∫ R2n dn~qdn~p g(~q, ~p) ⋆~ f(~q, ~p). (A.21) Por otra parte también puede invertirse la acción de una sola derivada parcial en (A.18) y (A.19) en lugar de ambas, para que el operador diferencial actúe únicamente sobre f ó g, en 124 Apéndice A. Material complementario del formalismo WWGM cuyo caso ∫ R2n dn~qdn~p f(~q, ~p) ⋆~ g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)e − i~ 2 −→ ∂ ∂qi −→ ∂ ∂pi e i~ 2 −→ ∂ ∂pi −→ ∂ ∂qi g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)e − i~ 2 ←− ∂ ∂qi ←− ∂ ∂pi e i~ 2 ←− ∂ ∂pi ←− ∂ ∂qi g(~q, ~p) = ∫ R2n dn~qdn~p f(~q, ~p)g(~q, ~p), (A.22) que, como puede verse, resulta de la antisimetŕıa del argumento bidiferencial en ⋆~ bajo el intercambio de qi por pi. Esta misma conclusión puede esperarse con cualquier otro producto-⋆ de caracteŕısticas similares. Una aplicación importante del resultado anterior puede hacerse para la expresión (2.45): (2π~)nTr[ÂB̂] = ∫ R2n dn~qdn~p AW (~q, ~p)BW (~q, ~p). (A.23) Fórmula de Parseval-Plancherel. Si φ̃ y ψ̃ corresponden a las transformadas de Fourier (normalizadas) de un par de funciones cuadráticamente integrables φ y ψ, entonces la ecuación (A.22) asegura que la fórmula de Parseval-Plancherel sigue siendo válida en C∞⋆ (R2n): ∫ R2n dn~qdn~p φ(~q, ~p) ⋆~ ψ ∗(~q, ~p) = ∫ R2n dn~xdn~y φ̃(~x, ~y)ψ̃∗(~x, ~y). (A.24) En particular, todos los equivalentes de Weyl de operadores del tipo Hilbert–Schmidt (Tr[†] < ∞) satisfacen ésta fórmula ya que usando (2.24) y (A.23) se tiene (2π~)nTr[†] = ∫ R2n dn~qdn~p |AW (~q, ~p)|2 <∞. (A.25) Como el śımbolo de un operador definido de acuerdo a (2.18) es también la transformada de Fourier del equivalente de Weyl, los resultados previos permiten establecer las siguientes igualdades: (2π~)nTr[ÂB̂†] = ∫ R2n dn~qdn~p AW (~q, ~p)B∗W (~q, ~p) = (2π~)2n ∫ R2n dn~xdn~y α(~x, ~y)β∗(~x, ~y). (A.26) A.2. Esquema de Heisenberg en el formalismo WWGM. 125 A.2 Esquema de Heisenberg en el formalismo WWGM. Usando los resultados de §2.3 es posible definir los equivalentes de Weyl de operadores de Heisen- berg. Sabiendo que para cualquier operador  y un Hamiltoniano Ĥ, el operador de Heisenberg Â(t) se obtiene bajo la transformación unitaria generada por el operador de evolución e i ~ tĤ , i.e. ÂH(t) = e i ~ tĤÂe− i ~ tĤ . (A.27) La ruta más corta para obtener el equivalente deWeyl de este operador es a partir del isomorfismo (2.37), entonces AH W (t) = (e i ~ tH)W ⋆~ AW ⋆~ (e − i ~ tH)W , (A.28) donde (e± i ~ tH)W representa el śımbolo de Weyl de e± i ~ tĤ . Expandiendo la exponencial formal e± i ~ tĤ se tiene e± i ~ tĤ = ∞∑ n=0 (±it)n ~nn! Ĥn, (A.29) y aplicando repetidamente (2.37) en esta expresión conduce a (e± i ~ tH)W = ∞∑ n=0 (±it)n ~nn! HW ⋆~ . . . ⋆~ HW ︸ ︷︷ ︸ n = ∞∑ n=0 (±it)n ~nn! (HW )n⋆~ , (A.30) donde, como se evidenćıa en esta serie de igualdades, el término (HW )n⋆~ representa el monomio de orden n bajo el producto ⋆~ del equivalente de Weyl de Ĥ. Consecuentemente (A.28) se reescribe como AH W (t) = e i ~ tHW ⋆~ ⋆~ AW ⋆~ e − i ~ tHW ⋆~ , (A.31) donde e ± i ~ tHW ⋆~ representa la serie formal en el lado derecho de (A.30). Debido a que AH W (t) constituye el equivalente de Weyl de un operador de Heisenberg entonces, según el isomorfismo (2.39), debe poseer propiedades similares a las de ÂH(t). Efectivamente, evaluando (A.31) en t = 0 se tiene AH W (0) = AW , (A.32) que corresponde a la expresión que conecta los esquemas de Heisenberg y Schrödinger en el formalismo WWGM. Suponiendo que ∂t = 0, entonces la ecuación de evolución para operadores de Heisenberg que proviene de derivar (A.27) con respecto a t es dÂH(t) dt = i ~ [Ĥ, ÂH(t)], (A.33) cuya versión análoga para equivalentes de Weyl resulta consecuentemente de diferenciar formal- 126 Apéndice A. Material complementario del formalismo WWGM mente (A.31) con respecto a t dAH W (t) dt = i ~ ( HW ⋆~ e i ~ tHW ⋆~ ⋆~ AW ⋆~ e − i ~ tHW ⋆~ − e i ~ tHW ⋆~ ⋆~ AW ⋆~ e − i ~ tHW ⋆~ ⋆~ HW ) , (A.34) usando (A.31) permite simplificar el lado derecho en dAH W (t) dt = i ~ ( HW ⋆~ A H W (t)−AH W (t) ⋆~ HW ) = i ~ [HW , A H W (t)]⋆~ , (A.35) que bien habŕıa podido obtenerse simplemente de calcular el equivalente de Weyl de la expresión (A.33). Sin embargo, este cálculo es útil para corroborar la consistencia del formalismo WWGM. Similarmente a lo que se hizo en §2.4, la ecuación (A.35) puede analizarse en el contexto del principio de correspondencia. Notando primero que, como consecuencia de (2.50), cualquier función AW ∈ C∞⋆~ (R2n) satisface lim ~→0 AW = A ∈ C∞(R2n), (A.36) donde A es la función de espacio-fase que resulta de igualar a cero todos los términos que contengan ~ en la forma expĺıcita de AW . Por lo tanto, al evaluar (A.35) en ~→ 0 y usando (2.52) sucede lim ~→0 dAH W (t) dt = dAH(t) dt = −{H,AH(t)} = {AH(t), H}. (A.37) Esta expresión corresponde a la ecuación de evolución en la formulación Hamiltoniana de la Mecánica Clásica cf. [124], para la Hamiltoniana H. En §§(2.2-2.4) se utilizó un conjunto de coordenadas (qi, pi) para C∞⋆~ (R2n) designadas a priori como las coordenadas de espacio-fase, basando esto puramente en el hecho que todo R 2n representa algún espacio-fase de Mecánica Clásica, aunque, estrictamente, esto es cierto siempre que las variables (qi, pi) constituyan un conjunto de variables canónicas. La expresión (A.35) permite afirmar desde un contexto dinámico que (qi, pi) forman un conjunto genuino de coordenadas de espacio fase. Partiendo de los operadores de Heisenberg (R̂i)H(t), (P̂i) H(t) cuyos equivalentes de Weyl (Ri)HW (t), (Pi) H W (t) satisfacen las ecuaciones de evolución d(Ri)HW (t) dt = i ~ [HW , (R i)HW (t)]⋆~ , d(Pi) H W (t) dt = i ~ [HW , (Pi) H W (t)]⋆~ , (A.38) A.3. Valores de expectación y la función de Wigner-Szilard. 127 evaluando en t = 0 y usando (A.32) se obtiene d(Ri)HW (t) dt |t=0 = i ~ [HW , q i]⋆~ , d(Pi) H W (t) dt |t=0 = i ~ [HW , pi]⋆~ . (A.39) Expandiendo los conmutadores en ambas expresiones y utilizando (A.6) se concluye d(Ri)HW (t) dt |t=0 = ∂HW ∂pi ≡ q̇i, d(Pi) H W (t) dt |t=0 = − ∂HW ∂qi ≡ ṗi, (A.40) que corresponden a las ecuaciones de Hamilton de un sistema con coordenadas canónicas (qi, pi) descrito por la Hamiltoniana HW . La solución a la ecuación (A.35) se obtiene en términos del celebrado paréntesis de Moyal, que surge de hacer la siguiente observación: [HW , A H W (t)]⋆~ = HW e i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) AH W (t)−HW e − i~ 2 ( ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi ) AH W (t) = 2iHW sen ( ~ 2 ↔ Π ) AH W (t), (A.41) donde ↔ Π := ←− ∂ ∂qi −→ ∂ ∂pi − ←− ∂ ∂pi −→ ∂ ∂qi . Entonces la ecuación diferencial (A.35) es ahora dAH W (t) dt = HW ↔ MAH W (t), (A.42) con el paréntesis de Moyal definido por ↔ M := −2 ~ sen ( ~ 2 ↔ Π ) , cf. [7]. Integrando formalmente la expresión anterior conduce al resultado final AH W (t) = etHW ↔ MAH W (0). (A.43) A.3 Valores de expectación y la función de Wigner-Szilard. La interpretación probabiĺıstica de la Mecánica Cuántica, a base de estados normalizados de H que satisfacen la ecuación de Schrödinger, es lo que le proporciona su estatus como una teoŕıa f́ısica de mediciones. Las únicas cantidades cuánticas medibles de algún fenómeno microscópico son los valores de expectación 〈Â〉 = Tr[ρ̂Â], (A.44) 128 Apéndice A. Material complementario del formalismo WWGM donde { ∈ End(H)|  = †} y ρ̂ constituye la matriz de densidad de von Neumann, definida para un ensamble mixto por ρ̂ := ∑ λ ℘λ|ψλ〉〈ψλ|, ℘λ ∈ [0, 1], Tr[ρ̂] = ∑ λ ℘λ = 1. (A.45) En el caso de Mecánica Cuántica ordinaria en R n, resulta evidente de (A.23) que el valor de expectación (A.44) puede expresarse también como 〈Â〉 = ( 1 2π~ )n ∫ R2n dn~qdn~p ρW (~q, ~p)AW (~q, ~p), (A.46) donde ρW es el equivalente de Weyl de ρ̂. Esta expresión evoca a primera instancia el valor promedio clásico de una función AW , para una distribución de probabilidad ρW /(2π~) n y, como se muestra a continuación, tal sospecha no es por completo errónea. La expresión para ρW se obtiene, por supuesto, de las ecuaciones (2.18) y (2.19): ρW (~q, ~p) = ( 1 2π~ )n ∫ R2n dn~xdn~y Tr[ρ̂e− i ~ (xiP̂i+yiR̂ i)]e i ~ (xipi+yiq i), (A.47) sustituyendo (A.45) en Tr[ρ̂e− i ~ (xiP̂i+yiR̂ i)] y realizando cálculos similares a aquellos de §2.1 se encuentra Tr[ρ̂e− i ~ (xiP̂i+yiR̂ i)] = ∑ λ ℘λ ∫ Rn dn~x′e− i ~ yi(x ′i+xi 2 ) 〈~x′|ψλ〉〈ψλ|~x′ + ~x〉, (A.48) que al reemplazar dentro de (A.47) e integrar sobre ~y y ~x′ produce ρW (~q, ~p) = ∑ λ ℘λ ∫ Rn dn~x e i ~ xipi 〈 ~q − ~x 2 ∣ ∣ ∣ψλ 〉〈 ψλ ∣ ∣ ∣~q + ~x 2 〉 . (A.49) Para simplificar la exposición en lo que sigue se utilizará una matriz de densidad de ensamble puro, en cuyo caso: ρW (~q, ~p) = ∫ Rn dn~x e i ~ xipiψ∗ ( ~q + ~x 2 ) ψ ( ~q − ~x 2 ) , (A.50) donde las proyecciones 〈 ~q − ~x 2 ∣ ∣ ∣ψ 〉 y 〈 ψ ∣ ∣ ∣~q + ~x 2 〉 se han escrito como las funciones de onda correspondientes. De ésta forma expĺıcita para ρW se observa que, pese a ser real ya que ρ̂ es hermı́tica, no es necesariamente positiva semidefinida como sucede con las distribuciones de probabilidad clásicas.1 1La literatura dedicada al estudio de esta propiedad es extensa, ver e.g., [18], para una revisión contemporánea del tema. A.3. Valores de expectación y la función de Wigner-Szilard. 129 En virtud de la desigualdad de Cauchy-Schwarz ∣ ∣ρW (~q, ~p) ∣ ∣2 ≤ ∫ Rn dn~x ∣ ∣ ∣ψ ( ~q + ~x 2 )∣ ∣ ∣ 2 ∫ Rn dn~y ∣ ∣ ∣ψ ( ~q − ~y 2 )∣ ∣ ∣ 2 , (A.51) y a la normalización de las funciones de onda, se tiene entonces que ρW es una función acotada de espacio-fase |ρW (~q, ~p)| ≤ 2n. (A.52) La expresión (A.50) no es más que la definición de la distribución ρw de Wigner-Szilard introducida por razones distintas en [9], salvo por el factor de proporcionalidad (2π~)n, i.e. ρw := ρW (2π~)n , (A.53) con lo cual (A.46) puede escribirse como 〈AW 〉 = ∫ R2n dn~qdn~p ρw(~q, ~p)AW (~q, ~p). (A.54) Utilizando (2.22) para evaluar la integral en espacio-fase de (A.53) se ve que ρw es una densidad: ∫ R2n dn~qdn~p ρw(~q, ~p) = Tr[ρ̂] = 1, (A.55) como se puede corroborar haciendo el cálculo expĺıcito con (A.50). Esto se relaciona directamente al hecho que las densidades de probabilidad mecánico-cuánticas en espacio de configuración y espacio de momento se obtienen de los marginales ∫ Rn dn~p ρw(~q, ~p) = ψ∗(~q)ψ(~q), (A.56) ∫ Rn dn~q ρw(~q, ~p) = ψ̃∗(~p)ψ̃(~p), (A.57) donde ψ̃(~p) = 1 (2π~) n 2 ∫ Rn dn~q e− i ~ qipiψ(~q). (A.58) Todas estas propiedades hacen de ρw una densidad de cuasiprobabilidad. Como los estados f́ısicos que conforman ρ̂ resuelven i~∂t|ψ〉 = Ĥ|ψ〉, entonces la derivada parcial de ρ̂ con respecto a t conduce a la ecuación de von Neumann i~∂tρ̂ = [Ĥ, ρ̂], (A.59) similar a la ecuación de evolución (A.33), pero con un cambio de signo ya que los estados |ψ〉 se encuentran en el esquema de Schrödinger. Consecuentemente, emulando los argumentos de la 130 Apéndice A. Material complementario del formalismo WWGM sección anterior, ρw satisface ∂tρw = 1 i~ [HW , ρw]⋆~ = ρw ↔ MHW , (A.60) conocida como la ecuación de Moyal, que es la generalización mecánico–cuántica de la ecuación de Liouville en espacio-fase. Para estados estacionarios, i.e. Ĥ|ψ〉 = E|ψ〉, la matriz de densidad de ensamble puro cumple Ĥρ̂ = ρ̂Ĥ = Eρ̂, (A.61) cuyo equivalente de Weyl proporciona la versión análoga en espacio-fase de la ecuación de valores propios del Hamiltoniano HW ⋆~ ρw = ρw ⋆~ HW = Eρw, (A.62) conocida como la ecuación de valores-⋆ (estrella) [125, 126]. Como corolario la integral en espacio-fase ∫ R2n dn~qdn~p HW (~q, ~p) ⋆~ ρw(~q, ~p) = E, (A.63) muestra que el valor esperado de HW coincide con la enerǵıa del sistema, corroborando que HW es una función Hamiltoniana. Una última propiedad importante dentro de varias aplicaciones de la función de Wigner- Szilard es la condición de estado puro, notando que en dicho caso como consecuencia de la propiedad de proyector (idempotencia) de la matriz de densidad ρ̂2 = ρ̂, (A.64) el equivalente de Weyl bajo (2.39) satisface la ecuación ρW ⋆~ ρW = ρW , (A.65) ó igualmente, usando (A.53), para la función de Wigner-Szilard (2π~)nρw ⋆~ ρw = ρw. (A.66) Entonces, sólo una función de cuasiprobabilidad que satisfaga (A.66) y (A.55) puede consid- erarse como una función de Wigner-Szilard de ensamble puro genuina, esto permite construir ansatz en casos donde resolver anaĺıticamente (A.62) no es trivial. Apéndice B Invariancia de simetŕıa torcida B.1 Torcedura de un álgebra de Hopf Dada un álgebra de Hopf (B, µ,∆, ı, ε, S),1 se define una torcedura (cf. [128]) como el elemento invertible F ∈ B ⊗ B que satisface (F ⊗ I)(∆⊗ id)F = (I⊗ F )(id⊗∆)F , (B.1) (ε⊗ id)F = I = (id⊗ ε)F . (B.2) Estas condiciones permiten utilizar la torcedura F , junto con el coproducto original ∆, para construir un nuevo coproducto ∆F v́ıa la transformación de similitud ∆F (a) = F∆(a)F−1, a ∈ B, (B.3) y nueva ant́ıpoda SF (a) = S(F(1))F(2)S(a)S(F−1(1) )F −1 (2) (en la notación de Sweedler [127]). De forma que (B, µ,∆F , ı, ε, SF ) es también un álgebra de Hopf, denominada la torcedura de (B, µ,∆, ı, ε, S) por F . B.2 Simetŕıa, deformación y torcedura de Drinfeld Para un espacio de funciones A = C∞(M) sobre alguna variedad (homogénea) y el álgebra de Lie P de simetŕıas deM, es posible generar un tipo especial de torcedura F del álgebra universal envolvente de Hopf U(P ), partiendo de una deformación mλ en el parámetro λ (producto-⋆) del 1Una clara exposición sobre esta estructura matemática puede hallarse en [127]. 131 132 Apéndice B. Invariancia de simetŕıa torcida producto m en A (veáse Lema 6.2.10 de Ref. [128]): f ⋆ g = mλ(f ⊗ g) = m(f ⊗ g) + ∞∑ n=1 λnBn(f, g), f, g ∈ A, (B.4) donde Bn : A⊗A → A son operadores (invariantes de izquierda) bidiferenciales y bilineales. Entonces, como los elementos X ∈ P son identificados con campos vectoriales en M, los operadores diferenciales D : A → A son identificados con elementos de U(P ) y los operadores bidiferencialesBn con elementos de U(P )⊗U(P ). SiDx es el operador diferencial correspondiente al elemento x ∈ U(P ) (y similarmente para elementos de U(P ) ⊗ U(P )), se tiene la acción covariante del operador Dx ⊲ m(f ⊗ g) = m(∆(x) ⊲ f ⊗ g), f, g ∈ A, (B.5) que, en particular, reproduce la regla de Leibniz para los elementos X ∈ P : X ⊲m(f ⊗ g) = m(∆(X) ⊲ f ⊗ g) = m[(X ⊲ f)⊗ g] +m[f ⊗ (X ⊲ g)], (B.6) ya que ∆(X) = X ⊗ I+ I⊗X, por tratarse de elementos primitivos de U(P ). Sean ahora F̃n ∈ U(P ) ⊗ U(P ) correspondientes a los operadores bidiferenciales Bn, tales que F̃ := I⊗ I+ ∞∑ n=1 λnF̃n, (B.7) permite reescribir el producto mλ como mλ = m ◦ F̃ . (B.8) Implementando la expresión anterior y (B.5) en la propiedad de asociatividad del producto mλ conduce a (∆⊗ id)F̃(F̃ ⊗ I) = (id⊗∆)F̃(I⊗ F̃ ), (B.9) que implica que Fλ := F̃−1 satisface la propiedad básica (B.1) para una torcedura, mientras que la condición counital (B.2) se cumple trivialmente por la forma de (B.7). La torcedura Fλ, construida de esta manera, define, entonces, la torcedura de Drinfeld [68] (U(P ), µ,∆λ, ı, ε, Sλ) del álgebra universal envolvente de Hopf U(P ) con coproducto ∆λ = Fλ∆F−1λ . Es inmediato probar ahora que los elementos x ∈ (U(P ), µ,∆λ, ı, ε, Sλ) preservan la covari- B.3. Torcedura de Drinfeld Fθ e invariancia 133 ancia al actuar sobre f ⋆ g = mλ(f ⊗ g). Efectivamente Dx ⊲ mλ(f ⊗ g) = Dx ⊲ m[F−1λ (f ⊗ g)] = m[∆(x)F−1λ ⊲ (f ⊗ g)] = m[F−1λ Fλ∆(x)F−1λ ⊲ (f ⊗ g)] = mλ[∆λ(x) ⊲ (f ⊗ g)], (B.10) donde se usó (B.5) en la segunda igualdad. Claramente la expresión anterior y (B.5) son co- variantes, con el producto y coproducto deformados reemplazando el producto y coproducto originales. Esto significa que, para el álgebra A⋆, se recupera la noción de simetŕıas generadas por elementos de P , consistente con la deformación del coproducto de U(P ), a través de una torcedura de Drinfeld que, además, deja P y U(P ) intactas. B.3 Torcedura de Drinfeld Fθ e invariancia Para el álgebra de funciones A∗ de §4.2 con estructura no-conmutativa (4.44), equivalente al álgebra extendida de Heisenberg-Weyl (4.1), y espećıficamente para la subálgebra Aθ, generada por variables qi con producto-⋆θ, se puede obtener inmediatamente la torcedura de Drinfeld correspondiente al álgebra universal envolvente de Hopf U(P ), donde P es naturalmente el grupo de Galileo. Esto se logra leyendo directamente F−1θ del producto-⋆θ: qi ⋆θ q j = qie i 2 θkl ↔ Λklqj , (B.11) donde ↔ Λkl = ←− ∂ qk −→ ∂ ql . Entonces, dado que los generadores de traslaciones espaciales Pi ∈ P son identificados con los campos vectoriales i∂qi , es trivial ver que F−1θ = e− i 2 θij(Pi⊗Pj), (B.12) y, por lo tanto, Fθ = e i 2 θij(Pi⊗Pj). (B.13) Como prueba de consistencia con los resultados de la sección previa, se puede verificar que (B.13) satisface efectivamente la condición (B.1). Usando las definiciones anteriores y el resultado de covariancia (B.10) se demuestra, fi- nalmente, que el conmutador [qi, qj ]⋆θ = iθij , equivalente al primer conmutador en (4.1), es invariante bajo la acción de los generadores de simetŕıas Pi (traslaciones) y Mij (rotaciones). 134 Apéndice B. Invariancia de simetŕıa torcida Primero, dado que Pi = i∂qi , la acción F−1θ ⊲ (qi ⊗ qj) se simplifica en F−1θ ⊲ (qi ⊗ qj) = [I⊗ I− i 2 θkl(Pk ⊗ Pl)]q i ⊗ qj = qi ⊗ qj + i 2 θkl(δik ⊗ δjl ), (B.14) y como m[∆(Pa) ⊲ (q i ⊗ qj − qj ⊗ qi)] = Pa ⊲ (q iqj − qjqi) = 0, m[∆(Mab) ⊲ (q i ⊗ qj − qj ⊗ qi)] =Mab ⊲ (q iqj − qjqi) = 0, (B.15) entonces, de acuerdo a (B.10) y usando (B.8) y (B.14), se calcula la acción Pa ⊲ [q i, qj ]⋆θ = mθ[∆θ(Pa) ⊲ (q i ⊗ qj − qj ⊗ qi)] = m[∆(Pa)F−1θ ⊲ (qi ⊗ qj − qj ⊗ qi)] = m[∆(Pa) ⊲ (q i ⊗ qj − qj ⊗ qi + i 2 θkl(δik ⊗ δjl − δ j k ⊗ δil))] = Pa ⊲ (iθ ij) = 0, (B.16) y análogamente Mab ⊲ [q i, qj ]⋆θ = mθ[∆θ(Mab) ⊲ (q i ⊗ qj − qj ⊗ qi)] = m[∆(Mab)F−1θ ⊲ (qi ⊗ qj − qj ⊗ qi)] = m[∆(Mab) ⊲ (q i ⊗ qj − qj ⊗ qi + i 2 θkl(δik ⊗ δjl − δ j k ⊗ δil))] =Mab ⊲ (iθ ij) = 0.  (B.17) Apéndice C Cosmoloǵıa anisotrópica de Bianchi I El modelo cosmológico anisotrópico de Bianchi I, el cual corresponde a un espacio eucĺıdeo, es descrito por el elemento de ĺınea ds2 = −N2(t)dt2 + gij(τ)dx idxj , gij(τ) = a2i (τ)δij , (C.1) donde N(τ) es la función de lapso y las cantidades ai(τ) caracterizan el ”tamaño” del Universo en tres direcciones tipo espacio independientes. La separación ADM [129] de la acción de Einstein-Hilbert para la métrica anterior está dada por Sgrav = c3 G ∫ dtd3x [ πij ġij − N(t) √ (3)g ( πijπij − 1 2 (πii) 2 )] = c3 G ∫ dtd3x [ πij ġij − N(t) √ (3)g ( πijπklgikgjl − 1 2 (πijgij) 2 )] , (C.2) donde (3)g = Det(gij) y π ij son los momenta conjugados a gij . 1 Al susbstituir expĺıcitamente las componentes gij en la acción y usando la definición πi := 2πijaj , es posible reescribir (C.2) en términos del par canónico (ai, π i) como Sgrav = c3 G ∫ dtd3x [ πiȧi − N(t) 4 √ (3)g ( (πi)2(ai) 2 − 1 2 (πiai) 2 )] , (C.3) con constricción Hamiltoniana Cgrav = N(t) 4 √ (3)g ( (πi)2(ai) 2 − 1 2 (πiai) 2 ) ≈ 0. (C.4) 1Por tratarse de un modelo cosmológico homogéneo, las contribuciones a la acción por los términos de super- momento se anulan trivialmente y en el caso particular del modelo de Bianchi I la contribución del escalar de 3-curvatura es cero. 135 136 Apéndice C. Cosmoloǵıa anisotrópica de Bianchi I El que la acción gravitacional (C.3) corresponda a una teoŕıa puramente constreñida (ver, e.g., §5.3) y, por lo tanto, sin evolución, es consecuencia de que la Relatividad General sea una teoŕıa manifiestamente covariante bajo difeomorfismos (reparametrizaciones), sin embargo es posible fijar el valor de la función de lapso N(t) 4 √ (3)g = 1 para recuperar una densidad Hamiltoniana. Esto permite utilizar el tiempo cosmológico t para obtener las ecuaciones de evolución: ȧi = 2δji π ja2j − (πjaj)ai = [2δji π jaj − (πjaj)]ai, π̇i = −2δij(πj)2aj + πi(πjaj) = −[2δijπjaj − (πjaj)]π i, (C.5) y como las cantidades π1a1 = χ1, π2a2 = χ1, π3a3 = χ3 son claramente constantes de movimiento, entonces las soluciones tienen la forma ai(t) = ai(t0)e (t−t0)ηi , πi(t) = πi(t0)e −(t−t0)ηi , (C.6) con ηi = (χi − χj − χk)|τ0 para i, j, k ćıclicos. Lo anterior significa que, dependiendo del signo de ηi, las soluciones (C.6) conducen a un comportamiento de expansión infinita ó colapso asintótico (singular) para t → ±∞ en cada variable de espacio-fase. Debido a la invariancia bajo reparametrización, el tiempo cosmológico t es arbitrario y no posee un significado f́ısico real, por lo que la acción (C.3) suele acoplarse mı́nimamente a la acción de un campo escalar sin masa, independiente de coordenadas espaciales, cuya evolución monotónica actúe como tiempo interno respecto del cual se midan los valores de las demás variables dinámicas. La acción total en este caso está dada por Sgrav + Sϕ = c3 G ∫ dtd3x [ πiȧi − N(τ) 4 √ (3)g ( (πi)2(ai) 2 − 1 2 (πiai) 2 )] + ~ ∫ d4x ( pϕϕ̇− 1 2 N √ 3g p2ϕ ) , (C.7) donde claramente el término de campo escalar se encuentra ya en unidades de acción, por lo que la integral es adimensional. Con la finalidad de tener ambos términos en las mismas unidades y que permitan factorizar una constricción total es necesario determinar primero las unidades de las variables dinámicas (ai, π i). Notando del elemento de ĺınea (C.1) que hay dos posibilidades para esto, donde en el primer caso (como es lo usual) [ai] = 1 mientras que en el segundo [ai] = L. Para el primer caso se encuentra [πi] = L−1 y para el segundo [πi] = L. En lo sucesivo se asumirá el segundo caso por razones propias de este trabajo. Entonces es posible definir una cantidad nueva pi := c3 G~ πi 137 con unidades de longitud inversa, de forma que (C.7) se puede escribir como Sgrav + Sϕ =~ ∫ d4x  piȧi − N(t) 4 √ 3g ( G~ c3 )  −1 2 ( 3∑ i=1 piai )2 + 3∑ i=1 (piai 2pi)     +~ ∫ d4x ( pϕϕ̇− 1 2 N √ 3g p2ϕ ) , (C.8) y finalmente con las redefiniciones pφ := ( 4c3 G~ ) 1 2 pϕ, y φ̇ := ( G~ 4c3 ) 1 2 ϕ̇, (C.9) donde tanto pφ como φ̇ son ahora adimensionales, se llega a Sgrav + Sϕ =~ ∫ d4x  piȧi − N(t) 4 √ 3g ( G~ c3 )  −1 2 ( 3∑ i=1 piai )2 + 3∑ i=1 (piai 2pi)     +~ ∫ d4x ( pφφ̇− N 4 √ 3g ( G~ c3 ) p2φ 2 ) , (C.10) Consecuentemente la constricción Hamiltoniana total clásica es: Cgrav + Cφ = N(t) 4 √ 3g ( G~ c3 )    −1 2 ( 3∑ i=1 piai )2 + 3∑ i=1 (piai 2pi)  + 1 2 p2φ   = 0. (C.11) Bibliograf́ıa [1] S. Doplicher, K. Fredenhagen, and J. Roberts Comm. Math. Phys., vol. 172, p. 187, 1995. [2] S. Lee, Sir Rudolf Peierls, Selected Private and Scientific Correspondence, vol. 1. 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Parte V Art́ıculos de investigación 147 ar X iv :h ep -t h /0 6 0 4 0 3 8 v 1 5 A p r 2 0 0 6 Dynamical origin of the ⋆θ- noncommutativity in field theory from quantum mechanics Marcos Rosenbaum, J. David Vergara and L. Román Juárez Instituto de Ciencias Nucleares, UNAM, A. Postal 70-543, México D.F. Abstract We show that introducing an extended Heisenberg algebra in the context of the Weyl-Wigner-Groenewold-Moyal formalism leads to a deformed product of the clas- sical dynamical variables that is inherited to the level of quantum field theory, and that allows us to relate the operator space noncommutativity in quantum mechan- ics to the quantum group inspired algebra deformation noncommutativity in field theory. Key words: Noncommutativity, star-products PACS: 02.40Gh, 11.10.Nx 1 Introduction Theoretical physics has provided us a fairly deep understanding of the micro- scopic structure of matter, but very little is known regarding the microscopic structure of space-time. From a methodological point of view, the use of a noncommutative struc- Preprint submitted to Elsevier Science 1 February 2008 ture for space-time coordinates had already been proposed in the early days of field theory as a failed hope at finding an effective and Lorentz invariant cutoff needed to control the ultraviolet divergences plaguing the theory. From a conceptual and theoretical point of view there is a simple heuristic argu- ment - based on Heisenberg’s Uncertainty Principle, the Einstein Equivalence Principle and the Schwarzschild metric - which shows that the Planck length seems to be a lower limit to the possible precision measurement of position, and that shorter distances do not appear to have an operational meaning [1]. Thus Quantum Mechanics and Field Theory, at dimensions of the order of the Planck length, ought to incorporate in their very structure the noncom- mutativity of space-time by replacing the concept of a space-time point by a cell of a dimension given by the Planck scale area. Under these premises the very concept of manifold as an underlying mathematical structure of physi- cal theories becomes questionable and some people are convinced that a new paradigm of geometrical space is needed. The noncommutative geometry of Connes [2], which by resorting to arbitrary and noncommutative C∗-algebras dualizes geometry and replaces its usual notions of manifolds and points by a new calculus based on operators in Hilbert space and the use of spectral analysis, epitomizes this line of thought. More recently there has been further evidence of space-time noncommutatitvity [3] coming from certain models of string theory which, although with a geometry quite different from that of noncommutative geometry is not incompatible with it, and has led to the same issue of noncommutativity of space-time at short distances. In the noncommutative quantum field theory rooted on the phenomenology of the low energy approximation of string theory in the presence of a strong magnetic background, the fields on a target space of space-time canonical co- ordinates are replaced by a C∗-algebra of functions with a deformed product 2 given by the so called Groenewold-Moyal star-product: f(x) ⋆θ g(x) = f(x)e( i 2 ←− ∂ iθij −→ ∂ j)g(x), (1) where the constant real and invertible anti-symmetric tensor θij has dimen- sions of length squared. One interpretation (see e.g. [4]) for the origin of this noncommutativity is based on postulating the replacement of the space-time argument of canonical coordinates xi of field operators by a “space-time” of Hermitian operators obeying the Heisenberg algebra [x̂i, x̂j ] = iIθij , i, j = 1, . . . , 2d (2) where I is an identity operator. Operators O(x̂), acting on a Hilbert space of delta-function normalizable functions in d-dimensions, are then defined in terms of the basic operators (2) by means of the Weyl basis g(α, x̂) = eiαix̂ i . Using now the Weyl-Moyal correspondence O(x̂) = ∫ d2dα g(α, x̂)ÕW (α), (3) where ÕW (α) is the Fourier transform of the Weyl function corresponding to O, it follows, in complete analogy to the results derived from the Weyl- Wigner-Groenewold-Moyal (WWGM) formalism of quantum mechanics (see the following section), that the Weyl function corresponding to the operator product O1O2 is given by (O1)W ⋆θ (O2)W . (4) For a review of noncommutative quantum field theory based on these criteria see, e.g., [5]. An alternative and Lorentz invariant (in the twisted symmetry sense) inter- pretation of the origin of the star-product (1) comes from considering the 3 twisted coproduct of the Hopf algebra H of the universal enveloping U(P) of the Poincaré algebra P. It can be shown (see e.g. [9]) that for a certain Drinfeld twisting of the coproduct with an invertible F ∈ U(P) ⊗ U(P) such that F12(∆⊗ id)F = F23(id⊗∆), (ǫ⊗ id)F = 1 = (id⊗ ǫ)F , (5) this coproduct induces a deformation in the product, m → mF , of the mod- ule algebra A = C∞(M) over H, such that the action of H on A preserves covariance, i.e. h ⊲ mF (a⊗ b) = m ◦ [(F−1 (1) ⊲ a)⊗ (F−1 (2) ⊲ b)] = a ⋆θ b, (6) where a, b ∈ A and h ∈ H, and we have used the Sweedler notation through- out. In particular, considering the coordinates xi as elements of A, equation (6) implies that [xi, xj ]⋆θ ≡ xi ⋆θ x j − xj ⋆θ x i = iθij . (7) Note, however, that although both of the above described representative lines of thought lead to the same algebra of operators for noncommutative quantum field theory, the origins of this noncommutativity appear to be quite different. In the later case, as has been stressed by Chaichian et al., the product (7) is inherited from the twist of the operator product of quantum fields and no noncommutativity of the coordinates was used in the derivation of (6); while in the line of thought described in [4] the assumed noncommutativity of the space-time operators forms an essential part of the ensuing arguments. How- ever, the inference that the multiplication in the algebra of fields is given by the star-product (6) is an external ingredient imported from the phenomenology of string theory. 4 Since quantum mechanics is strongly interwoven into noncommutative geom- etry, and since single particle quantum mechanics can be seen, in the free field or weak coupling limit, as a mini-superspace sector of quantum field theory where most degrees of freedom have been frozen (i.e, as a one-particle sector of field theory), it is suggestive that a further study of quantum mechanics in this noncommutative context, and in particular in the WWGM formalism based on a Heisenberg algebra extended to incorporate space noncommuta- tivity, may help to shed some additional light on the origins of the product (7) in the algebra of noncommmutative field theory. Observe, however, that in the strict sense of quantum mechanics only expec- tation values have a physical meaning. This, in the WWGM quantum for- malism, translates to the fact that the c-equivalent of a quantum operator, or to that effect of a product of operators, appears together with the Wigner quasi-distribution function inside of a phase-space integral. In the case of the standard Heisenberg algebra of usual quantum mechanics, the Wigner func- tion is the same as the Weyl equivalent of the von Neumann density matrix and the Weyl equivalent of a product of operators (given by the Groenewold- Moyal product of their respective Weyl equivalents) is indeed the c-function that would appear in the integrand multiplying the Wigner function. On the other hand, as it is shown in the next section, this is not true for the case of a quantum mechanics with an extended Heisenberg algebra. In fact, as shown in equations (27) or (28) there, either of which can be used to evaluate the expectation value of a product of operators, the Weyl equivalent of a product of operators (given by (30) with a composite ⋆-product defined by (25), (31) and (32) ) is not the one required in the integrands in order to arrive at the correct expectation values. Hence this ⋆-product does not appear as a nat- 5 ural ingredient of the quantum mechanical formalism when considering only Schrödinger operators. The purpose of this work is to show nonetheless that when considering in addi- tion Weyl equivalents of Heisenberg operators, the ⋆θ product for the algebra of what can then be identified as canonical dynamical variables, emerges nat- urally within the theory and thus allows for a further link between the points of view of quantum operator space noncommutativity, as presented in [4], and the quantum group inspired algebra deformation noncommutativity, discussed in [9]. Lastly we could expect as well that a detailed study of exactly solvable models in the frame of this extended Heisenberg algebra WWGM formalism may also be helpful to achieve a further understanding of the possible phe- nomenological consequences in space of the noncommutativity in field theory. In this context, the above observations as well as some additional ones con- tained below are also pertinent to some works that have appeared recently in the literature on what has been called noncommutative quantum mechanics. 2 Quantum Mechanics on Extended Heisenberg Algebras in the WWGM Formalism By an extended Heisenberg algebra we understand the algebra of position and momentum operators satisfying the commutation relations [R̂i, R̂j ] = iθij , (8) [P̂i, P̂j] = i~θ̄ij , (9) [R̂i, P̂j] = i~δij , (10) 6 where R̂i,P̂i i = 1, . . . , d are the components of the position and momentum quantum operators, respectively, with component eigenvalues on Rd, and θij and θ̄ij are evidently antisymmetric matrices, which in the most general case can be functions of the generators of the above algebra. For our present pur- poses and algebraic simplicity, in what follows we shall set θ̄ij = 0 and d = 2, and consider only the zeroth order constant term of the Taylor expansion of θ12 ≡ θ. (For θ constant, the formalism described below can be generalized to include more spatial dimensions in a fairly straightforward way, and it also can be extended to incorporate space-time noncommutativity by parameterizing the time and considering it as an extra variable. See for example [10]). From an intrinsically noncommutative operator point of view, the development of a formulation for the quantum mechanics based on the above extended Heisen- berg algebra of operators requires first a specification of a representation for the generators of the algebra, second a specification of the Hamiltonian which governs the time evolution of the system and last a specification of the Hilbert space on which these operators and the other observables of the theory act. As for the choice of the Hilbert space, a reasonable assumption is that it can be taken to be the same as that for the corresponding system in the usual quantum mechanics, but for a realization of the extended Heisenberg algebra, because of the noncommutativity (8), we can not use configuration space as a basis. We can use, however, for a basis either of the eigenkets |p1, p2〉, |q1, p2〉, |q2, p1〉, of the commuting pairs of observables (P̂1, P̂2), (R̂1, P̂2), or (R̂2, P̂1), respectively, or any combination of the (R,P ) such that they form a complete set of commuting observables. Having in mind generalizations to include the noncommutativity (9), we choose as the realization of our extended Heisenberg algebra the one based on |q1, p2〉. 7 The construction follows standard procedures (cf.[6]): Consider the unitary op- erator Ŝ(γ) = eγR̂2 (γ is an arbitrary parameter) and evaluate its commutators with R̂1 and P̂2. It is easy to show that Ŝ(γ)|q1, p2〉 = |q1 − θγ, p2 + ~γ〉. (11) Assuming now that γ is an infinitesimal and evaluating 〈q1, p2|Ŝ(γ)|q′1, p′2〉 to first order in γ results in 〈q1, p2|R̂2|q′1, p′2〉 = (−iθ∂q1 + i~∂p2 )〈q1, p2|q′1, p′2〉, so the realization of R̂2 in this basis is R̂2 = −iθ∂q1 + i~∂p2 . (12) Considering next the unitary operator Ŝ(λ) = eλP̂1 and following a similar procedure we get P̂1 = −i~∂q1 . (13) The representations for the remainder of the generators R̂1 and P̂2 of the algebra are obviously simply multiplicative. (Note that by making use of (11) we can readily make the change of basis |q1, p2〉 → |p1, p2〉 and derive the representations R̂1 = i~∂p1 and R̂2 = i~∂p2 + θ ~ p1 for the extended Heisenberg algebra generators in the momentum representation. In this case P̂1 and P̂2 are obviously just multiplicative. All our calculations could then be related to that basis.) For later calculations we shall be needing to evaluate the transition function 〈q1, p2|q2, p1〉. This can be derived [7] by noting that 〈q1, p2|R̂2|q2, p1〉 = q2〈q1, p2|q2, p1〉 = i(~∂p2 − θ∂q1 〈q1, p2|q2, p1〉, (14) 8 and 〈q1, p2|P̂1|q2, p1〉 = p1〈q1, p2|q2, p1〉 = −i~∂q1 〈q1, p2|q2, p1〉. (15) Combining these two expressions yields (~q2 − θp1)〈q1, p2|q2, p1〉 = i~∂p2 〈q1, p2|q2, p1〉, (16) which can be readily solved to give, after normalization, 〈q1, p2|q2, p1〉 = 1 2π~ exp[− i ~ (q2p2 − θ ~ p1p2 − q1p1)]. (17) Making use of (17) and the Baker-Campbell-Hausdorff (BCH) theorem, it is fairly direct to show that 1 (2π~)2 Tr{exp[ i ~ ((y − y′) · R̂ + (x− x′) · P̂)]} = δ(x− x′)δ(y − y′), (18) where x = (x1, x2) y = (y1, y2). Thus for our extended Heisenberg algebra also the {(2π~)−1 exp[ i ~ (y · R̂ + x · P̂)]} form a complete set of orthonormal operators. and any Schrödinger operator (which may depend explicitly on time) A(P̂, R̂, t) can be written as A(P̂, R̂, t) = ∫ ∫ dx dyα(x,y, t) exp[ i ~ (x · P̂ + y · R̂)], (19) where, by (18), the c-function α(x,y, t) is determined by α(x,y, t) = (2π~)−2Tr{A(P̂, R̂, t) exp[− i ~ (x · P̂ + y · R̂)]}. (20) The Weyl function corresponding to the quantum operator A(P̂, R̂, t) is then given by AW (p,q, t) = ∫ ∫ dx dy α(x,y, t) exp[ i ~ (x · p + y · q)] = ∫ ∫ dx1dy2e i ~ (x1p1+y2q2)〈q1 − x1 2 − θy2 2~ , p2 + y2 2 |Â|q1 + x1 2 + θy2 2~ , p2 − y2 2 〉. (21) 9 To derive the expectation value of a product of two Schrödinger operators, one writes the expectation value of the product in terms of the von Neumann density matrix ρ as 〈Â1Â2〉 = Tr[ρÂ1Â2], (22) and evaluates the trace in the above chosen basis. After a rather lengthy but fairly straightforward calculation the result obtained is 〈Â1Â2〉 = ∫ . . . ∫ dp1dp2dq1dq2 1 (2π~)2 ∫ dξdηe− i ~ (ηq2−ξp1) 〈q1 − ξ 2 , p2 − η 2 |ρ|q1 + ξ 2 , p2 + η 2 〉e 1 ~ θp1∂q2 ((A1)W ⋆~ (A2)W ), (23) where ⋆~ := exp[ i~ 2 Λ] := exp [ i~ 2 ( ←−∇q · −→∇p − ←−∇p · −→∇q) ] , (24) is the Gronewold-Moyal star-product bidifferential of the usual WWGM quan- tum mechanics formalism. If we now let ρ(Wigner) := 1 (2π~)2 ∫ dξdηe− i ~ (ηq2−ξp1)〈q1 − ξ 2 , p2 − η 2 |ρ|q1 + ξ 2 , p2 + η 2 〉 (25) denote the standard Wigner quasi-probability distribution in our chosen basis, then (23) reads as 〈Â1Â2〉 = ∫ ∫ dpdq ρ(Wigner)e 1 ~ θp1∂q2 ((A1)W ⋆~ (A2)W ). (26) Note that we could equally well have integrated the above equation by parts to get 〈Â1Â2〉 = ∫ ∫ dpdq ρW ((A1)W ⋆~ (A2)W ). (27) where the Weyl function ρW corresponding to ρ is related to ρ(Wigner) by ρW = e− 1 ~ θp1∂q2 (ρ(Wigner)), (28) in contradistinction to what happens in the usual quantum mechanics where they are the same. So in the calculation of the expectation value of the product 10 of two Schrödinger operators, the quantities that enter in the quantum me- chanics based on the extended Heisenberg algebra are either ((A1)W ⋆~(A2)W ), when averaging with ρW , or e 1 ~ θp1∂q2 ((A1)W ⋆~(A2)W ) when averaging with the usual Wigner function. However, also contrary to what happens in ordinary quantum mechanics, these quantities are not equal to the Weyl equivalent (Â1Â2)W of the product Â1Â2. To evaluate (Â1Â2)W we use (20) and (21), and following steps entirely anal- ogous to the ones treated in more detail in the following section when consid- ering Heisenberg operators, it can be shown that (Â1Â2)W = (Â1)W ⋆(Â2)W , (29) where ⋆ is defined by the composition of operator bi-differentials: ⋆ := ⋆θ ◦ ⋆~, (30) with ⋆~ as defined in (24) and ⋆θ := e iθ 2 ( ←− ∂ q1 −→ ∂ q2 − ←− ∂ q2 −→ ∂ q1 ). (31) Furthermore and similarly to what occurs in ordinary quantum mechanics, there is a stronger star-value equation related to (27). There are again how- ever important differences. Thus, given a Hamiltonian operator Ĥ and a pure energy state satisfying the eigenvalue equation Ĥ|ψ〉 = E|ψ〉, it can be shown that the star-value equation for the quantum mechanics with our extended Heisenberg algebra is H̄W ⋆~ ρ(Wigner) = E ρ(Wigner), (32) where H̄W (p,q) = e 1 ~ θp1∂q2HW (p,q). (33) 11 Because of space limitations we omit here the details of the proof of this theorem. These, together with other more detailed aspects of our previous discussion as well examples where specific implications of the quantum me- chanics here summarized are displayed and compared with other approaches, will be dealt with in a forthcoming paper to appear elsewhere. 3 Weyl Equivalent of Heisenberg Operators Let ΩH := Ω(P̂, R̂, t) := e it ~ ĤΩ(P̂, R̂, 0)e− it ~ Ĥ , (34) be the Heisenberg operator corresponding to the Schrödinger operator Ω(P̂, R̂, 0). As for Schrödinger operators the c-function αΩ(x,y, t), associated with the Weyl function (ΩH)W defined as in (21), is given by (see (20)) αΩ(x,y, t) = (2π~)−2Tr{e it ~ ĤΩ(P̂, R̂, 0)e− it ~ Ĥe− i ~ (x·P̂+y·R̂)}. (35) Differentiating (21) with respect to t and taking the Fourier transform gives immediately ∂αΩ ∂t = i(2π~)−2 ~ ∫ dq1dp2〈q1 − x1 2 − y2θ 2~ , p2 + y2 2 |[H,ΩH ]|q1 + x1 2 + y2θ 2~ , p2 − y2 2 〉 × exp[− i ~ (y1q1 + x2p2)]. (36) Consider now the quantity ∫ dq1dp2 exp[− i ~ (y1q1+x2p2)]〈q1− x1 2 −y2θ 2~ , p2+ y2 2 |HΩH|q1+ x1 2 + y2θ 2~ , p2− y2 2 〉 12 which, after making use of (19), (17), the BCH theorem and performing several fairly direct integrations, yields (2π~)−2 ∫ dq1dp2 exp[− i ~ (y1q1 + x2p2)] 〈q1 − x1 2 − y2θ 2~ , p2 + y2 2 |HΩH|q1 + x1 2 + y2θ 2~ , p2− y2 2 〉 = ∫ dx′dy′αH(x′y′)αΩ(x− x′,y − y′, t) × exp[ i 2~ (−y ′ 1y2θ ~ + y1y ′ 2θ ~ + x′2y2 − x2y ′ 2 + y1x ′ 1 − y′1x1)]. (37) Rewriting (37) in terms of HW and (ΩH)W , by making use of the Fourier inverse of the first equality in (21), and substituting the result into (36) it readily follows that ∂αΩ ∂t = i(2π~)−8 ~ ∫ . . . ∫ dp′dq′dp′′dq′′dx′dy′e− i ~ (x′·p′+y′·q′) ×[HW (p′,q′)ΩH W (p′′,q′′, t)− ΩH W (p′,q′)HW (p′′,q′′, t)] × exp[− i ~ ((x− x′) · p′′ + (y − y′) · q′′)] × exp[ i 2~ (−y ′ 1y2θ ~ + y1y ′ 2θ ~ + x′2y2 − x2y ′ 2 + y1x ′ 1 − y′1x1)] . (38) Finally, double Fourier transforming both sides of (38), rearranging terms and performing the integrals, yields ∂ΩH W ∂t = i ~ [HW (p,q)⋆ΩH W (p,q)− ΩH W (p,q)⋆HW (p,q)]. (39) Note that by interchanging the ordering of the Weyl functions in the second term inside the square brackets in (39), we alternatively have ∂ΩH W ∂t = i ~ HW [e i 2 (~Λ+θΛ′) − e− i 2 (~Λ+θΛ′)] ΩH W = −2 ~ HW sin[ 1 2 (~Λ + θΛ′)] ΩH W , (40) where Λ := ←−∇q · −→∇p − ←−∇p · −→∇q, Λ′ := ←− ∂ q1 · −→∂ q2 −←−∂ q2 · −→∂ q1 . (41) 13 Equation (40) can be formally integrated to give ΩH W (p,q, t) = exp{−2t ~ HW sin[ 1 2 (~Λ + θΛ′)]}ΩW (p,q, 0). (42) Note that (39) is in agreement with the derivation in [8] for the time evolution of the Wigner function, although the calculation there is somewhat circular from our point of view as it assumes the ⋆θ-product to be valid ab initio. 4 Noncommutative Field Theory from extended Heisenberg alge- bra Quantum Mechanics Up to this point in the WWGM formalism the q’s and p’s (the continuum of eigenvalues of R̂ and P̂) are only variables of integration . In order to be able to interpret them as canonical dynamical variables, as it is the case for ordinary WWGM quantum mechanics, let us consider the specific cases when the Heisenberg operator ΩH in Section 3 is P̂(t) or R̂(t). Making use of (21) and (42), and recalling that PW (p,q, 0) = p and RW (p,q, 0) = q, we get for this particular cases, and a mechanical Hamiltonian of the form Ĥ = P̂2 2m + V (R̂), dPH W dt |t=0 =−1 ~ H(~Λ + θΛ′)p = −∇qV, d(RH 1 )W dt |t=0 == −1 ~ H(~Λ + θΛ′)q1 = p1 2m + θ ~ ∂q2 V, d(RH 2 )W dt |t=0 == −1 ~ H(~Λ + θΛ′)q2 = p2 2m − θ ~ ∂q1 V. (43) 14 Introducing now the following fundamental Poisson brackets as part of the algebra structure of the q’s and p’s: {pi, pj} = 0, {qi, qj} = θij ~ , {qi, pj} = δij , (44) we have that (43) read d(PH i )W dt |t=0 = {pi, H} = ṗi, d(RH i )W dt |t=0 = {qi, H} = q̇i, (45) and therefore with this additional Poisson structure the q’s and p’s satisfy the Hamilton equations and can be considered formally as canonical dynamical variables in the theory. A representation for the above Poisson brackets can be constructed by defining the twisted product qi ⋆θ qj := qie i 2 ∑ lm ←− ∂ ql θlm −→ ∂ qmqj , (46) where we have generalized our arguments to Rd (with d ≥ 2), and letting {qi, qj} := − i ~ [qi, qj]⋆θ := − i ~ [qi ⋆θ qj − qj ⋆θ qi]. (47) We can consequently argue that the noncommutativiy of the extended Heisen- berg algebra in Quantum Mechanics manifests itself as a twisting in the prod- uct of the algebra of the corresponding classical canonical dynamical variables which, in accordance with [9], may be interpreted in turn as an Abelian Drin- feld twisting of the coproduct in the Hopf algebra H of the universal envelope U(G) of the Galileo symmetry algebra. If we now view the module algebra Aθ (the so called Groenewold-Moyal plane), described in the Introduction, as a certain completion of the algebra generated by the qi and describe fields as elements of Aθ, then fields will clearly inherit the ⋆θ-product. 15 As a final parenthetical remark, note from Sec 2 that in all the expressions based on the WWGM formalism containing the θ, it always appears in the form of of the quotient θ ~ . If we claim that the noncommutativity (8) in the extended Heisenberg algebra is originated from quantum gravity, then it is reasonable to assume (as already mentioned in the Introduction) that θ ∼ l2p = k~ c3 , where lp is the Planck length and k is the gravitational coupling constant. Thus θij ~ ∼ k c3 . This shows then that corrections, due to this noncommutativity, to calculations such as energy spectra and equations of motion such as (43), are indifferent to the value of ~, and that even in the limit ~ → 0 there is what may appear as a remanent of quantum gravity. Acknowledgments The authors acknowledge partial support from CONACyT projects UA7899-F (M.R.) and 47211-F(J.D.V.) and DGAPA-UNAM grant IN104503 (J.D.V.). References [1] Doplicher S., Fredenhagen, K. and Roberts, J.E., Comm. Math. Phys. 172, 187 (1995). [2] Connes A 1994 Noncommutative Geometry, Academic Press San Diego, California. [3] C. S. Chu and P. M. Ho, Nucl. Phys. B 550, 151 (1999). N. Seiberg and E. Witten, JHEP 9909, 032 (1999) [arXiv:hep-th/9908142]. [4] L. Alvarez-Gaume and S. R. Wadia, Phys. Lett. B 501, 319 (2001) [arXiv:hep-th/0006219]. 16 [5] Szabo, R.J., Phys. Rep. 378, 207-299 (2003). [6] A. Messiah 2000 Quantum Mechanics, Dover Publications . [7] C. Acatrinei, JHEP 0109:007 (2001)[arXiv:hep-th/0107078] [8] A. Eftekharzadeh and B.L. Hu, Braz.J.Phys.35,333-342 (2005) [arXiv:hep-th/0504150]. [9] M. Chaichian, P.P. P. Presnajder and A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005) [arXiv:hep-th/0409096] M. Chaichian, P.P. Kulish, K. Nishijima and A. Tureanu, Phys. Lett. B 604, 98-102 (2004) J. Wess, [arXiv:hep-th/0408080] P. Kosinski and P. Maslanka, [arXiv:hep-th/0408100] Ch. Blohmann, “Paris 2002, Physical and Mathematical aspects of symmetries” [arXiv:QA/0408080] R. Oeckl, Nucl. Phys. B 581, 559 (2000) J. Lukierski and M. Woronowicz, [arXiv:hep-th/0508083] [10] A. Pinzul and A. Stern, Phys. Lett. B 593, 279 (2004) [arXiv:hep-th/0402220]. R. Banerjee, B. Chakraborty and S. Gangopadhyay, J. Phys. A 38, 957 (2005) [arXiv:hep-th/0405178]. M. Rosenbaum, J. D. Vergara and L. R. Juarez, in preparation. 17 ar X iv :h ep -t h /0 6 1 0 1 5 0 v 2 1 2 O ct 2 0 0 6 Contemporary Mathematics Noncommutativity from Canonical and Noncanonical Structures Marcos Rosenbaum, J. David Vergara, and L. Román Juárez This paper is dedicated to J. Plebanski. Abstract. Using arbitrary symplectic structures and parametrization invari- ant actions, we develop a formalism, based on Dirac’s quantization procedure, that allows us to consider theories with both space-space as well as space-time noncommutativity. Because the formalism has as a starting point an action, the procedure admits quantizing the theory either by obtaining the quantum evolution equations or by using the path integral techniques. For both ap- proaches we only need to select a complete basis of commutative observables. We show that for certain choices of the potentials that generate a given sym- plectic structure, the phase of the quantum transition function between the admissible bases corresponds to a linear canonical transformation, by means of which the actions associated to each of these bases may be related and hence lead to equivalent quantizations. There are however other potentials that re- sult in actions which can not be related to the previous ones by canonical transformations, and for which the fixed end-points, in terms of the admissible bases, can only be realized by means of a Darboux map. In such cases the original arbitrary symplectic structure is reduced to its canonical form and therefore each of these actions results in a different quantum theory. One in- teresting feature of the formalism here discussed is that it can be introduced both at the levels of particle systems as well as of field theory. 1. Introduction In recent years, space-time noncommutativity has become the subject of in- creasing interest. In field theory stimulated by some results in low energy string theory, and in quantum mechanics because it is in the context of this formalism that space-time noncommutativity is more naturally understood in terms of space and time operators acting on a Hilbert space and also, because quantum mechanics viewed as a minisuperspace reduction of field theory, could reasonably be expected to provided further insight into how quantum mechanical noncommutativity reflects 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. Non-commutative theories, The first author was supported in part by CONACyT Grant #UA7899-F. The second author was supported in part by grants: DGAPA-IN104503, CONACyT 47211-F. The third author was partially supported by a SNI fellowship. c©2006 American Mathematical Society 1 2 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ itself in field theory. Some of the more relevant work related to the approach here considered may be found in [1], [2], [3], [4], [5], [6], [7]. An interesting idea that allows us to consider in a full setting the space-time noncommutativity in the context of particle mechanics, is to use the concept of parametrization invariance [5], [7]. In this way the time is taken as an extra canon- ical variable of the system and it is then easy to introduce a non-canonical structure in this extended phase- space. The usual way to study the parametrization invari- ance of a system is by using the Dirac method of canonical analysis. Because not all the momenta are independent due to the invariance under parametrizations, this ap- proach requires that a constraint on the system be introduced. For a parametrized particle, this constraint is at the classical level the Hamilton-Jacobi equation and at the quantum level the Schrodinger equation. So the Dirac method associates to the symmetry of parametrizations the classical or quantum evolution equations [8]. Here we want to generalize the above mentioned procedure in order to be able to consider noncommutative theories at the quantum level resulting both from canonical and non-canonical structures. The noncommutativity will then appear as a consequence of the existence of second class constraints, and the implementation of these constraints in terms of Dirac brackets. The interesting point of the procedure is that on the one hand we get the classical and quantum evolution equations for the noncommutative systems and on the other hand we also obtain a classical action that can be quantized using the path integral formalism. Furthermore, the analysis is not restricted to noncommutative theories with constant deformation parameters, since the procedure naturally incorporates arbitrary canonical potentials. Another interesting property of the method is that it can be naturally extended to field theory. Our starting point is to consider a parametrization invariant system. This means that if the system is not naturally invariant under parametrizations we pro- mote the original parameters of the theory, for example the time in the case of particle dynamics, to the level of canonical variables. The second step is to per- form the canonical analysis of this theory. One point that we must be careful with is that, since we add new variables to the system, we have to introduce constraints associated to the parametrization invariance symmetry of the theory in order that the number of degrees of freedom are preserved. The third step is to introduce an arbitrary canonical potential that allows us to realize the required noncommutativ- ity. The next step is to show that under the Dirac brackets the first class constraint (or constraints) generate the symmetry. This means that we will probably need to modify the constraints. At this point, if we have several constraints, we need to check that the algebra of these first class constraints closes. Once we finish this pro- cedure we obtain the quantum evolution equations for our system. Alternatively, we can introduce the canonical potential in the action and select an appropriate basis in order to quantize the system using the path integral formalism. For certain choices of the potentials that generate a given symplectic structure, the phase of the quantum transition function between the admissible bases corresponds to a linear canonical transformation, by means of which the actions associated to each of these bases may be related and hence lead to equivalent quantizations. We must stress that in contradistinction to the case when time plays the role of a parameter, the canonical transformation here is implemented in an extended phase space, where the time and its conjugate momentum are included. NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 3 With the purpose of examining all the above mentioned facets of the space-time noncommutativity, our presentation has been structured as follows: In Section 2 we consider the canonical formalism of parametrization invariant systems. In Section 3 we introduce an arbitrary symplectic structure in the action, and after the canonical analysis we construct the Dirac brackets associated to the theory and also obtain the action for the reduced system. In Section 4, we quantize the theory using different bases, and using both path integral methods and the quantum evolution equations. We conclude the paper with some remarks and possible extensions. 2. Parametrization invariant systems We begin here by reviewing the essentials of the canonical analysis of parametrized systems following the approach in [8]. To this end, consider the action for a particle in a N -dimensional configuration space, in an arbitrary potential: (2.1) S = t2 ∫ t1 dt ( 1 2 m ( dqi dt )2 − V ( qi, t ) ) , where i = 1, . . . , N . In this action the time t plays the role of a parameter in the theory. To study the non-commutativity of the space and time it is more convenient to consider the time as another coordinate of our theory, i.e. we extend our configuration space with one extra dimension t = q0. To do this, we parametrize the action by introducing a new parameter τ and assume that the coordinates qi(t) are scalars under this parametrization, i.e., t→ τ qi (t)→ qi (τ) (2.2) The action (2.1) takes the form (2.3) S = τ2 ∫ τ1 dτ ( 1 2 m ( dqi dτ )2( dτ dt ) − V ( qi, t ) ( dt dτ ) ) , where t = q0 now plays the role of a new coordinate in the theory. Making the identifications q̇i ≡ ( dqi dτ ) and q̇0 ≡ dt dτ , we can rewrite (2.3) in the form (2.4) S = τ2 ∫ τ1 dτ ( 1 2 m (q̇i)2 q̇0 − V (qi, q0)q̇0 ) . In Hamiltonian form the action (2.4) reads (2.5) S = τ2 ∫ τ1 dτ ( p0q̇ 0 + piq̇ i − λϕ ) , where ϕ = p0 + H ≈ 0 is the first class primary constraint associated to the symmetry under parametrizations (which needs to be included in (2.5) in order to account for the fact that by introducing a new variable in the theory, restrictions must be added to the physical evolution of the system that indicate that the N +1 new coordinates are not all independent), H is the canonical Hamiltonian of the action (2.1), and λ(τ) is a Lagrange multiplier. The action (2.5) is invariant up to 4 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ a total derivative under the transformations generated by the constraint ϕ, given by (2.6) δq0 = {q0, εϕ} , δp0 = {p0, εϕ} , δpi = {pi, εϕ} , δqi = { qi, εϕ } δλ = ε̇, where the variation of the Lagrange multiplier is imposed in such way that when varying the action it should vanish up to a boundary term. Following Dirac [11], we propose that at the quantum level the physical sates of the theory are invariant under the above transformations, i.e., (2.7) eiεϕ̂ |ψ〉P = |ψ〉P . So in infinitesimal form we get (2.8) ϕ̂ |ψ〉P = 0. We thus see that the constraint leads to a supplementary condition on the physi- cal states, and is another way to reduce the quantum theory to its physical sector without imposing a gauge condition. Now if we consider the configuration representation with basis |q0, qi〉, equation (2.8) yields, (2.9) ϕ̂ |ψ〉P = 0⇒ ( −i~ ∂ ∂t − ~2 2m ∇2 + V (qi, t) ) ψ(qi, t) = 0, where we have identified t = q0. We therefore obtain the Schrodinger equation as a result of imposing at the quantum level the classical invariance under parametriza- tions of the theory. In the following section we shall apply the same procedure to the case of arbitrary symplectic structures. 3. Non-commutativity and Dirac Brackets Let za = ( q0, qi, p0, pi ) , with a = 1, ..., 2N + 2, denote the 2N + 2 phase-space variables of a parametrized system in the Hamiltonian formulation. In this case we don’t have a second order action to begin with as in (2.1). We can however consider a general first order action, equivalent to (2.5), given by (3.1) S = τ2 ∫ τ1 dτ (Aa(z)ża − λϕ(z)) , where Aa(z) is a vector potential which we shall use to generate an arbitrary sym- plectic structure associated to the Poisson brackets in the Hamiltonian formulation. Applying the Dirac’s method for constrained systems, we have from (3.1) that the corresponding canonical Hamiltonian is given by (3.2) Hc = λϕ(z), and the canonical momenta lead to the set of primary constraints, (3.3) χa = pza −Aa (z) . Consequently, the total Hamiltonian for this theory is (3.4) HT = λϕ+ µaχa. NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 5 Moreover, from the evolution of the constraints we obtain the following consistency conditions (3.5) χ̇a = {pza −Aa (z) , HT } = −λ ∂φ ∂za + µbωab ≈ 0, where (3.6) ωab := ∂aAb − ∂bAa = {χa, χb}. This antisymmetric matrix will play the role of the symplectic structure of the the- ory. Assuming further that ωab is invertible so all the Lagrange’s multipliers µa in (3.5) can be determined, it then follows from (3.6) that the constraints χa are second class. Note that in the case where the symplectic structure is degenerate, at least one of the χa’s will be first class, but in this case the number of degrees of freedom of the generalized theory will not correspond to the degrees of freedom of the original theory. Hence in what follows we will assume that all the constraints χa are second class. Now, in order to impose these constraints as strong conditions when quantizing, we construct the associated Dirac brackets which are given by (3.7) {A,B}∗ = {A,B} − {A,χa}ωab {χb, B} , where ωab, is the inverse matrix of ωab. Computing the Dirac’s brackets of the coordinates with the above expression we obtain (3.8) { za, zb }∗ = ωab. Thus, quantizing a theory constrained by symmetries under parametrization results in the noncommutativity of the quantum operators corresponding to the phase space coordinates: (3.9) [ ẑa, ẑb ] = i~ωab. The simplest case corresponds to the usual Heisenberg algebra of ordinary Quantum Mechanics, for which the inverse matrix of the canonical symplectic structure takes the form (3.10) Jab := ωab|θ=0 = ( 0 I −I 0 ) . 4. Non-commutative Quantum Mechanics In the previous section we have considered a general procedure for quantizing a theory with an arbitrary symplectic structure. One interesting feature of this formalism is that by including time as a canonical variable allows us to consider also noncommutativity between the time and the spatial coordinates. Now, given such a symplectic structure we can quantize either by using the Dirac’s procedure where the first class constraints act as operators on the physical states, imposing supplementary conditions on them, and the Dirac brackets of the second class con- straints are replaced by commutators, or, alternatively, we can also quantize by first evaluating the generating potentials of the symplectic structure and then applying path integral methods in order to derive the Feynman propagators. It should be noted, however, that for a given symplectic structure the solution for the potentials Aa is not unique, although all the possible resulting actions and 6 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ resulting classical theories are related by canonical transformations. Furthermore, in the Dirac quantization the commutators (3.9) of the generators of the extended Heisenberg algebra define the possible complete sets of commuting observables of the theory and the correlative admissible bases (labeled by the eigen- values of these sets). For each of these admissible bases, we obtain a realization of the Heisenberg algebra and of the subsidiary condition (2.8) and, correspondingly in the path integral formalism, the Feynman propagators derived from the transition functions in each of these bases. This means that in the path integral calculation of a transition function, the only admissible actions are those for which the fixed end-points in a variational principle are the same as the dynamical variables label- ing the basis used for the evaluation of the transition function. Note finally that there are also actions originating from solutions of (3.6) for which no fixed end-points, corresponding to one of the admissible bases in the Dirac quantization exits. However, can be defined using a Darboux map. This map, involves introducing new dynamical variables in terms of linear combinations of the original ones and, consequently implies a change in the initial symplectic structure to a canonical one. Compatible, although non-equivalent, path integral and Dirac quantizations result from promoting to the rank of operators these new variables, which will satisfy the Heisenberg algebra of ordinary quantum mechanics. So in these cases the deformation of the symplectic structure at the classical level is reflected at the quantum level in a deformed Hamiltonian while the standard Heisenberg algebra of the usual quantum mechanics is preserved. To further illustrate the above observations, we next consider some examples of quantum noncommutativity schemes in the context of both the Dirac and path integral formalisms. For analytical simplicity we assume a 1+1 space-time, gener- alization to higher order dimensions is fairly straightforward. 4.1. Space-time noncommutativity. Let us consider first the case where the Dirac brackets (3.8) determine a symplectic structure of the form (4.1) ωab =     0 θ 1 0 −θ 0 0 1 −1 0 0 0 0 −1 0 0     , ωab =     0 0 −1 0 0 0 0 −1 1 0 0 θ 0 1 −θ 0     . Quantizing according to Dirac’s prescription by using (3.9) leads to the commuta- tors (4.2) [t̂, x̂] = i~θ, [x̂, p̂x] = i~, [t̂, p̂t] = i~, [p̂t, p̂x] = 0, and, using (2.8), to the supplementary condition (4.3) ϕ̂|ψ〉 = 0, where ϕ̂ is given by (4.4) ϕ̂ = p̂t +H(t̂, x̂, p̂x). It is obvious from (4.2) that for a mechanical Hamiltonian the sets of complete commuting observables in this case are {x̂, p̂t}, {t̂, p̂x} and {p̂t, p̂x}. The admissible bases in Hilbert space are then {|x, pt〉}, {|t, px〉} and {|pt, px〉}, respectively. NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 7 4.1.1. Basis |x, pt〉. For the basis {x̂, p̂t} the algebra (4.2) is realized by (4.5) t̂ψ(x, pt) = i~ (∂pt + θ∂x)ψ(x, pt), p̂xψ(x, pt) = −i~∂xψ(x, pt), while the remaining generators of the extended Heisenberg algebra are just mul- tiplicative quantities. Also projecting on (4.3) with 〈x, pt| and substituting (4.5) into (4.4), with a Hamiltonian of the form H = p2 x 2m + V (x, t), yields the subsidiary condition (4.6) ( pt − ~2 2m ∂2 x + V (x, i~(∂pt + θ∂x)) ) ψ(x, pt) = 0 on the wave function ψ(x, pt). One interesting feature of the Dirac quantization resulting from the use of this basis is that for a t independent potential, equation (4.6) becomes (4.7) ( pt − ~2 2m ∂2 x + V (x) ) ψ(x, pt) = 0. For such a time independent Hamiltonian, (4.7) may be interpreted as an eigen- value equation, with −pt the energy eigenvalues of the system and ψ(x, pt) the corresponding eigenvectors. Note that the energy spectrum of the resulting theory does not have any corrections from the noncommutativity of the space-time. A similar result was obtained by Balachandran, et al [4] by means of a very different approach. Now, in order to obtain the equivalent quantization by means of path integrals, we need to compute the transition function 〈x(τ2), pt(τ2)|x(τ1), pt(τ1)〉. For this purpose we need first to derive the appropriate action function, which according to our previous observations has to have as fixed end-points the variables x, pt. This, as implied by (3.6), requires in turn deriving the proper generating potentials Aa(z) for the symplectic structure (4.1) by solving the equations, (4.8) ∂A1 ∂pt − ∂A3 ∂t = 1, ∂A2 ∂px − ∂A4 ∂x = 1, ∂A4 ∂pt − ∂A3 ∂px = θ. It is not difficult to verify that the needed solution is (4.9) A1 = 0, A2 = px, A3 = −(t+ θpx), A4 = 0. In fact, Inserting (4.9) in the action (3.1) results in (4.10) S1 = τ2 ∫ τ1 dτ (pxẋ− θpxṗt − tṗt − λ (pt +H(t, x, px))) , which indeed has the appropriate variational fixed end-points x, pt. With (4.10) we can now compute the propagator (4.11) 〈x(τ2), pt(τ2)|x(τ1), pt(τ1)〉 = ∫ DtDptDxDpxδ(χ)δ(ϕ){ϕ, χ}∗ exp( i ~ S1), where we have introduced a canonical gauge fixing condition χ = χ(τ, t, pt, x, px). This gauge must first be a good canonical gauge in the Dirac’s sense, i.e. the Dirac bracket {ϕ, χ}∗ must be invertible and second the gauge must be consistent with 8 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ the boundary conditions. Because, we are fixing at the end points (x, pt), it is not possible to use the usual gauge t = f(τ), we will use instead the gauge condition (4.12) χ = x− f(τ) ≈ 0. The Dirac’s bracket between this gauge condition and the constraint is given by (4.13) {ϕ, χ}∗ = −px m + θ ∂V ∂t This gauge is a good canonical gauge for px 6= 0, in which case the path integral has two different branches, one corresponding to px > 0 and the other for negative px. It can also be seen that this term leads to corrections of first order in θ which are, however, proportional to the time dependence of the potential. Consequently, if we assume that the potential is time independent, this corrections cancel and we can then integrate (4.11) over t to obtain 〈x(τ2), pt(τ2)|x(τ1), pt(τ1)〉 = ∫ DxDptDpxδ(x− f)δ(ϕ)δ(ṗt) ( −px m ) × ( exp ( i ~ ∫ τ2 τ1 dτ (px(ẋ− θṗt)− λϕ(pt, px, x)) )) . (4.14) Note now that the only dependence on θ in the above expression appears multi- plying ṗt, but taking into account that this term is zero due to the delta functional in the path integral we do not get nonconmmutative corrections to the propagator. This is in agreement with our previous results derived by using the Dirac’s quanti- zation. 4.1.2. Basis |t, px〉. Let us next consider the basis {t̂, p̂x} in which the operators x̂ and p̂t are realized by (4.15) x̂ψ(t, px) = i~ (∂px − θ∂t)ψ(t, px), p̂tψ(t, px) = −i~∂tψ(t, px). In the Dirac quantization we have that a realization of the supplementary condition (2.8) in this basis results from projecting with 〈t, px| and substituting (4.15) into the first class constraint (4.4), we thus get (4.16) ( −i~∂t + p2 x 2m + V (t, i~(∂px − θ∂t)) ) ψ(t, px) = 0. Note that contrary to what we had in the case of the basis {|x, pt〉} where the supplementary condition was independent of time, here we have a time evolution equation. However, because of the time derivative in the potential in (4.16) we may lose the usual probability amplitude interpretation for ψ(t, px) for time derivatives of order higher than one, regardless of whether or not the potential has an explicit dependence on time. It is conceivable, nonetheless, that for certain forms of the potential a probabilistic interpretation may be recovered by modifying the product in the algebra of the wave functions or by redefining hermicity, in analogy to what occurs in Feshbach-Villars formulation of the Klein-Gordon equation. NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 9 It is natural to ask how is (4.16) related to (4.7) for a time independent poten- tial. For this purpose note that 〈t, px|V (x̂)|ψ〉 = V (i~(∂px − θ∂t))ψ(t, px) = ∫ dx dptV (i~(∂px − θ∂t)) 〈t, px|x, pt〉ψ(x, pt). (4.17) But (4.18) 〈t, px|x̂|x, pt〉 = x〈t, px|x, pt〉 = i~(∂px − θ∂t)〈t, px|x, pt〉. So (4.19) V (i~(∂px − θ∂t)) 〈t, px|x, pt〉 = V (x)〈t, px|x, pt〉, and using (4.20) 〈t, px|x, pt〉 = (2π~)−1e− i ~ (xpx−θptpx−tpt), (see e.g. [13] for details of a procedure used to derive a similar transition function), we get (4.21) 〈t, px|V (x̂)|ψ〉 = (2π~)−1 ∫ dx dptV (x) e− i ~ (xpx−θptpx−tpt)ψ(x, pt). Finally, substituting this result in (4.16) we get the integro-differential equation ( −i~∂t + p2 x 2m ) ψ(t, px)+ (2π~)−2 ∫ dx dpt dt ′ dp′xV (x) e− i ~ [x(px−p′ x )−θpt(px−p′ x )−(t−t′)pt]ψ(t′, p′x) = 0. (4.22) On the other hand, if ψ(x, pt) is a solution of (4.7) then (4.23) ψ(t, px) = ∫ dxdpt〈t, px|x, pt〉ψ(x, pt) = (2π~)−1 ∫ dxdpte − i ~ (xpx−θptpx−tpt)ψ(x, pt), is a solution of(4.16). Indeed acting with ( −i~∂t + p2 x 2m + V (t, i~(∂px − θ∂t)) ) on (4.23) and making use of (4.17) and (4.21) we get ( −i~∂t + p2 x 2m + V (t, i~(∂px − θ∂t)) ) ψ(t, px) = (2π~)−1 ∫ dx dpt e − i ~ (xpx−θptpx−tpt)[pt − ~2 2m ∂2 x + V (x)]ψ(x, pt). (4.24) Now, if ψ(x, pt) satisfies (4.7) the right side of (4.24) is zero, hence ψ(t, px) as given by (4.23) satisfies (4.16). Q.E.D. Let us now turn to the path integral quantization for this case and the calcula- tion of the propagator 〈t(τ2), px(τ2)|t(τ1), px(τ1)〉. The appropriate solution to the equations (4.8) for which t, px are the fixed end points of the action are (4.25) A1 = pt, A2 = 0, A3 = 0, A4 = θpt − x. 10 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ Inserting this solution into the action (3.1) we then obtain, (4.26) S2 = τ2 ∫ τ1 dτ ( ptṫ+ θptṗx − xṗx − λϕ ) , Observe that the action (4.26) and the action (4.10) are indeed related by a linear canonical transformation generated by F1 = pxx− θptpx − ptt. The propagator for the admissible basis {|t, px〉} is then (4.27) 〈t(τ2), px(τ2)|t(τ1), px(τ1)〉 = ∫ DtDptDxDpxδ(χ)δ(ϕ){ϕ, χ}∗ exp( i ~ S2), and for the boundary conditions that we are considering, the usual gauge (4.28) t = f(τ), is a good gauge condition. Assuming now that the Hamiltonian is independent of t, we can easily integrate (4.27) over the variables t and pt, using the gauge condition (4.28) and the con- straint, we get 〈f(τ2), px(τ2)|f(τ1), px(τ1)〉 = ∫ DxDpx(−1− θ∂xV ) exp ( −i ~ ∫ f2 f1 df ( (θH + x) dpx df +H ) ) , (4.29) where the parametrization in the action has been eliminated. Note that in the limit θ = 0 both (4.16) and (4.29) reduce to the usual Quan- tum Mechanics. The same is true for a free particle, as it is immediately evident from (4.16), and it is also follows for (4.29) since in this case the Hamiltonian is independent of x, so by integrating over this variable the term with θ = 0 disap- pears. 4.1.3. Basis |pt, px〉. To conclude our analysis of the Dirac and path integral quantization realized on the three admissible bases for the extended Heisenberg algebra (4.2) that we are studying in this section, consider now the representation of the operators (t̂, x̂) in |pt, px〉. For this basis we have (4.30) t̂ψ(pt, px) = (i~∂pt + aθpx)ψ(pt, px), x̂ψ(pt, px) = (i~∂px + (1 + a)θpt)ψ(pt, px). It is interesting to note that in this representation we have introduced an extra parameter a, that can translate the noncommutativity from the coordinate opera- tor to the time operator. (Observe that this characteristic is also present when we impose noncommutativity of the space so we can also translate the noncommuta- tivity parameter from one coordinate to the another). For this representation the constraint equation (4.3) takes the form (4.31) ( pt + p2 x 2m + V (i~∂pt + aθpx, i~∂px + (1 + a)θpt) ) ψ(pt, px) = 0. Note that in this case, when the potential is time independent so that (4.31) reduces to (4.32) ( pt + p2 x 2m + V (i~∂px + (1 + a)θpt) ) ψ(pt, px) = 0, NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 11 we do have noncommutative corrections except when we choose the parameter a = 0, or for the case of a free particle. For the path integral formulation in this basis, an appropriate action (having pt, px as fixed end-points) is given by (4.33) S3 = τ2 ∫ τ1 dτ (−tṗt + aθptṗx − (1− a)θpxṗt − xṗx − λϕ)) , from which we can obtain results equivalent to those derived from the analysis of the constraint equation (4.31). Contrary to the actions S1 and S2 which are unique solutions of (3.6) for their corresponding fixed end-points, there are several canonically equivalent admissible actions with fixed points pt, px. Thus, for example, S4 = ∫ τ2 τ1 dτ(−tṗt−θpxṗt−xṗx) can be obtained from S3 by substracting the total derivative of F2 = aθptpx from the integrand in S3. Other canonically equivalent actions follow from S3 and S4 by means of the generator F3 = θptpx. 4.1.4. Noncanonical related actions. Up to this point we have considered path integral quantizations based on actions which are compatible with the extended Heisenberg algebra (4.2), derived by means of the Dirac quantization procedure. There are, however, other solutions to the equations (4.8) which, although indis- tinguishable at the classical level from the ones considered so far, they are not canonically related to them, in the sense that there is no generating function for mapping canonically the actions resulting from these solutions to the ones previ- ously considered. We shall see that in these cases the transformations needed for fixing the end-points required for a path integral quantization are actually trans- formations which map the original phase-space variables with symplectic structure (4.2) to another set of variables related to the canonical symplectic structure (3.10). Classically, as it is well known from the Darboux theorem [12], this map is always possible (at least locally). To each of these Darboux maps corresponds, however, a different quantum mechanics, generated by what in some works in the literature has been called the equivalent of the Seiberg-Witten map for “noncommutative quantum mechanics”. To exhibit in more detail the above considerations, let us begin with the solu- tions: (4.34) A1 = pt, A2 = px, A3 = 0, A4 = θpt, A1 = pt, A2 = px, A3 = −θ 2 px, A4 = θ 2 pt. With the first set of equations in (4.34), the canonical action takes the form (4.35) S5 = τ2 ∫ τ1 dτ (pt(t+ θpx)• + pxẋ− λ (pt +H(t, x, px))) . We therefore see from (4.35) that from the original phase-space variables of the theory we do not have a set of fixed end-points for the action from which a quan- tization can be developed. Nonetheless a natural pair (t̃, x) can be constructed by 12 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ making the change of variables (4.36) t̃ := t+ θpx, x̃ = x, where t̃ is a new canonical variable associated to the time. In terms of this new pair of variables, the symplectic structure is reduced to (3.10), and introducing this new time in the action (4.35), results in (4.37) S5 = τ2 ∫ τ1 dτ ( pt ˙̃t+ pxẋ− λ ( pt +H(t̃− θpx, x, px) ) ) . Note that if the original Hamiltonian was time-dependent, the modified one intro- duces a new kind of interaction that is proportional to the parameter θ of noncom- mutativity and to the momenta in the spatial direction. Also note that in terms of the modified symplectic structure (3.10) the Dirac brackets (3.8) lead, upon quantization, to the commutators (4.38) [ t̃, pt ] = i~, [ t̃, x ] = 0, [x, px] = i~, [x, pt] = 0, [px, pt] = 0. From these commutators we clearly see that a new complete set of commuting observables is (ˆ̃t, x̂), which label the admissible associated basis of coordinate states { ∣ ∣t̃, x 〉 }. The Dirac’s supplementary condition in this basis is now, (4.39) ( −i~ ∂ ∂t̃ + Ĥ(t̃+ i~θ∂x, x,−i~∂x) ) ψ(x, t̃) = 0, and we note that in the case that the Hamiltonian does not depend explicitly on the time the Schrödinger equation is not modified by the noncommutativity. Now, if we consider the second set in (4.34) of solutions to (4.8) the resulting action is given by (4.40) S6 = τ2 ∫ τ1 dτ ( pt(t+ θ 2 px)• + px(x − θ 2 pt) • − λ (pt +H(t, x, px)) ) . Following the same logic as in the previous case, it is natural to introduce in this equation the new set (ť = t + θ 2px, x̌ = x − θ 2pt) of time and spatial coordinate. Here then the action (4.40) is reduced to (4.41) S6 = τ2 ∫ τ1 dτ ( pt ˙̌t+ px ˙̌x− λ ( pt +H(ť− θ 2 px, x̌+ θ 2 pt, px) ) , ) and, upon Dirac quantization, the corresponding new set of dynamical observables satisfies the following commutation relations, (4.42) [ ˆ̌t, ˆ̌x ] = 0, [ ˆ̌t, pt ] = i~, [ ˆ̌x, p̂x ] = i~, [p̂t, p̂x] = 0. Using as a complete set of commuting observables the variables (ˆ̌t, ˆ̌x), the new supplementary Dirac condition is (4.43) ( −i~ ∂ ∂ť +H(ť+ i ~θ 2 ∂x̌, x̌− i ~θ 2 ∂ť, −i~∂x̌) ) ψ(ť, x̌) = 0. For this Schrödinger equation we see that including the case when the Hamiltonian does not depend explicitly on time we do have modifications originated by the NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 13 noncommutativity. Furthermore we see that the new theory could be non-unitary, since partials with respect to ť appear to an order that depends on the kind of interaction. This type of quantization can been formulated directly by using the Moyal product: (4.44) H(ť+ i ~θ 2 ∂x̌, x̌− i ~θ 2 ∂ť, −i~∂x̌)ψ(ť, x̌) = H(ť, x̌, −i~∂x̌) ⋆θ ψ(ť, x̌), where (4.45) ⋆θ = exp [ i ~θ 2 (←− ∂ ť −→ ∂ x̌ − ←− ∂ x̌ −→ ∂ ť ) ] . So for this selection of symplectic potentials the theory is not unitary and this result is equivalent to the obtained in Ref. [15] in the context of noncommutative field theory. To quantize these two cases by means of the path integral method we make use of the basis { ∣ ∣t̃, x 〉 } and the respective actions (4.37) and (4.41) to compute the propagator (4.46) 〈t̃2, x2|t̃1, x1〉. Following the normal procedure to quantize a theory with first class constraints [8], we have only two extra points to consider. First we have to impose a gauge condition, which in this case can be the normal canonical gauge t̃ = f(τ), since in difference with the approach used in [5] and [7] we are imposing the noncommu- tativity at the level of the action, using the symplectic structure, and not at the level of the gauge condition. The second point that we need to take into account is the extra appearance in the Hamiltonian of the θpx shifted term when we have a t dependent theory, this can imply that it may not be possible to compute the path integral over the momenta. These are however the usual problems that one finds when computing path integrals with actions in terms of variables with powers larger than two. One additional point to notice is that for both types of solutions of the equations (4.8) considered in this section, the Dirac constraint is not modified, since in both cases the new time is canonical conjugated to the original pt and then the constraint generates the parametrization invariance. It is not difficult to see that this is not the case when the above analysis is extended to the more general case of symplectic structures that upon quantization result in an extended Heisenberg algebra that includes noncommutativity of the momenta. For such a generalization one would have to consider a symplectic structure of the form (4.47) ωab =     0 θ 1 0 −θ 0 0 1 −1 0 0 β 0 −1 −β 0     , ωab = 1 γ     0 β −1 0 −β 0 0 −1 1 0 0 θ 0 1 −θ 0     , where (4.48) γ = 1− βθ. 14 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ Here the quantization of the Dirac brackets would then result in the extended Heisenberg algebra (4.49) [t̂, x̂] = i~θ, [x̂, p̂x] = i~, [t̂, p̂t] = i~, [p̂t, p̂x] = i~β, and, in contradistinction to what occurred for the previously considered symplectic structure, we would only have two complete sets of commuting fundamental ob- servables: (x̂, p̂t) and (t̂, p̂x), with their respective admissible bases: {|x, pt〉} and {|t, px〉}. Except for some differences such as the ones mentioned above, the analysis of the Dirac and path integral quantizations relative to these bases, as well as others re- sulting from considering canonical transformations of their respective associated actions followed by Darboux maps, is qualitatively similar (see [14]) to what we have already done, so for the sake of brevity we shall omit the details here. Rather, and in preparation for a future investigation of how our analysis of space-time noncommutativity in the discrete realm of quantum mechanics can be extended to the continuum of relativistic field theory, we turn next our consideration to the case of a relativistic particle. 4.2. Space-time Noncommutativity for a Relativistic particle. Our starting point is the action for the free relativistic particle (4.50) S = τ2 ∫ τ1 dτ ( −m √ −ηαβẋαẋβ ) , α = 1, . . . , n. In Hamiltonian form we have (4.51) S = τ2 ∫ τ1 dτ ( pαẋ α − λ ( p2 +m2 )) , where now the first class primary constraint ϕ is given by (4.52) ϕ = p2 +m2 ≈ 0. As discussed above in Sec. 3, for an arbitrary symplectic structure the action (4.51) has the form (4.53) S = τ2 ∫ τ1 dτ (Aa(z)ża − λ (ϕ(z))) , za = (xα, pα), a = 1, . . . , 2n. Again, arising from the definition of the momenta, we have the primary constraints (4.54) χa = pza −Aa (z) . These constraints are second class and the corresponding Dirac brackets are iden- tical in form to those in the non-relativistic case, given by Eq.(3.8). Let us consider now a symplectic structure which is determined by the following Dirac brackets involving the space-time and momentum variables: (4.55) {xα, xβ}∗ = θαβ , {xα, pβ}∗ = δα β , NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 15 where θαβ is a constant antisymmetric tensor. Then, the symplectic structure takes the explicit form: (4.56) ωab = ( θαβ I −I 0 ) , ωab = ( 0 −I I θαβ ) . Note, as it was the case before, that the solutions of (4.57) ωab = ∂aAb − ∂bAa, for the generating potentialsAa, are not unique, but they are all related by canonical transformations. One possible covariant solution is (4.58) Aα = 0, An+α = −xα − θαβ 2 pβ, α = 1 . . . n. Introducing this symplectic potential in the action (4.53), we obtain (4.59) S = τ2 ∫ τ1 dτ ( −xαṗα + θαβ 2 pαṗβ − λ ( p2 +m2 ) ) . So the variables with fixed end points in the action are the momenta pα. Dirac quantization in this case results in the commutators (4.60) [x̂α, x̂β ] = i~θαβ , [x̂α, p̂β] = i~δα β , [p̂α, p̂β] = 0, and the supplementary condition (4.61) ϕ̂(x̂, p̂)ψ(p) = (pαp α +m2)ψ(p) = 0, referred to the admissible basis {|p〉}. As is to be expected, this merely states that (pαp α +m2) = 0. To compute the propagator for the theory (4.59) using path integrals, the more convenient technique is to use a non-canonical gauge and the BFV-BRST path integral procedure [8]. The full action, after introducing the gauge fixing term and ghost terms, is (4.62) S = τ2 ∫ τ1 dτ ( −(xα + θαβ 2 pβ)ṗα − λπ̇ − P ˙̄C + ĊP̄ − iP̄P − λ ( p2 +m2 ) ) . Here the boundary conditions on the ghost, the momenta conjugate to the coordi- nates pα and the Lagrange multiplier π are π(τ1) = π(τ2) = C̄(τ1) = C̄(τ2) = C(τ1) = C(τ2) = 0,(4.63) pα(τ1) = pα1, pα(τ2) = pα2.(4.64) From the path integral over the ghosts we get a multiplicative factor of (τ1 − τ2), this term is very useful since it allows to eliminate the dependence of the propagator on the parameter τ . Using the path integral over xα, we obtain delta functions that we then use to integrate over the momenta pα. As a result these integrals cancel the θαβ correction term, since this term is multiplied by ṗα. So, finally we get the usual propagator in the basis where the momenta are fixed at the end points (4.65) 〈pβ(τ2)|pα(τ1)〉 = −iηαβδ(pα2 − pα1) p2 +m2 . This result is fully consistent with the previous result (4.61). 16 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ Other admissible bases compatible with the Heisenberg algebra (4.60) are ob- tained from (4.59) by a canonical transformation generated by F = pαx α, for α fixed. These sets of admissible bases are {|xα, pβ, pγ , pλ〉; α 6= β 6= γ 6= λ}. Refered to them, the Dirac subsidiary condition results in (4.66) ϕ̂(x̂, p̂)ψ(xα, pβ, pγ , pλ) = ( −~2(∂xa)2 + (pβ)2 + (pγ)2 + (pλ)2 ) ψ(xα, pβ , pγ , pλ) = 0, where indices here are not summed over. So, even though the deformation parameter θ does not appear in these constraint equations the space-time noncommutativity is reflected in their violation of Lorentz invariance. On the other hand, canonically transforming (4.59) with F = pαx α, where now we sum over α, we get, after regrouping terms, (4.67) S = τ2 ∫ τ1 dτ ( pα(xα + θαβ 2 pβ)• − λ ( p2 +m2 ) ) . Here we see that it is natural to define as fixed end-point variables of the action the new set of coordinates given by (4.68) x̃α = xα + θαβ 2 pβ. The Dirac bracket between these new coordinates vanishes and, in consequence, so does their commutator: (4.69) [ x̃α, x̃β ] = 0, while (4.70) [x̃α, pβ ] = i~δα β . Note, however, that (4.68) is a Darboux map and not a canonical transformation of the action (4.59). Consequently this is a different Dirac quantization, related to the canonical symplectic form and not to the original one given by (4.56). The Dirac supplementary condition in this case is (4.71) ϕ̂(ˆ̃x, p̂)ψ(x̃) = ( −∂x̃α ∂x̃α +m2 ) ψ(x̃) = 0. So, quantizing the theory in this way we obtain that a relativistic particle satisfies the Klein-Gordon equation, and thus arrive at the well known result that for a free particle we do not obtain any deformation of the theory. However, if we consider that the particle lives in a given background, we will get the deformation produced by the new choice of coordinates. To further illustrate this point, consider the interaction of the relativistic particle with a constant external field. Here the constraint will be of the form (4.72) ( Πµ − 1 2 Fµνx ν )( Πµ − 1 2 Fµσxσ ) +m2 ≈ 0. Using the x̃α coordinates, which will have the same form as in (4.68), except for the substitution pβ → Πβ , the Dirac supplementary condition in the basis {|x̃α〉} NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 17 is of the form [( −i~∂µ − 1 2 Fµν ( x̃ν + i~ 1 2 θνρ∂ρ )) × ( −i~∂µ − 1 2 Fµσ ( x̃σ + i~ 1 2 θσρ∂ ρ )) +m2 ] ψ(x̃) = 0, (4.73) which indeed shows corrections containing the deformation parameter θ. 5. Concluding remarks We have seen that according to the Dirac quantization scheme for constrained systems, it is the first class constraints and the symplectic structure resulting from the Dirac brackets that uniquely define a particular quantum theory, irrespectively of the fact that there are many possible solutions for the potentials Aa correspond- ing to the same symplectic structure ω. On the other hand, if we use these solutions as the starting point for evaluating the action in the path integral formulation, then depending on the type of solutions that we propose for the equations (3.6), we could get different quantizations. We have seen moreover, that if there is a linear canon- ical transformation relating these actions, as is the case for the actions S1, S2 and S3 considered in subsections 4.1.1-4.1.3, then the corresponding quantizations are actually equivalent to each other and differ only by the fact that they are referred to the three admissible bases compatible with the extended Heisenberg algebra (4.2). Indeed, the phases of the quantum mechanical transition functions corresponding to changes between these bases (cf. e.g. Eq. (4.20)) are nothing other than the classical generating functions of the linear canonical transformations among the three actions, and the associated symplectic transformation leaving invariant their common symplectic structure ω is, for each of these three cases, the identity ele- ment of the group. Alternatively, for the type of solutions to (3.6) leading to the actions consid- ered in subsection 4.1.4, the situation is actually quite different because there is no generating function that permits to canonically transform such actions to the ones previously considered, and because at the classical level fixing the end-points of these actions involves a change of variables in extended phase-space which results in a Darboux map from the original symplectic structure to the canonical one given by (3.10). Quantizing in these cases via either the Dirac or path integral formalisms is then tantamount to applying standard quantum mechanics with a Hamiltonian modi- fied with the new variables, which are formally promoted to the rank of operators satisfying the commutation relations (4.42). But in axiomatic quantum mechanics the operators acting on vectors in Hilbert space are observables, i.e. operators functions of the basic dynamical variables of the theory, with eigenvalues given by quantities measurable by experiment. For the systems we have been considering and the construction followed in subsection 4.1.4, this would imply that the new time and coordinate variables are the observables of the theory and, since they obey the commutation relations (4.42), the new time and coordinate operators commute. Physically this would then mean that experiments could be designed to measure simultaneously the eigenvalues of these space-time operators. This, however, begs the question of what is then the true physical interpretation for the θ parameter that 18 MARCOS ROSENBAUM, J. DAVID VERGARA, AND L. ROMÁN JUÁREZ appears in the modified quantum expressions of the theory, such as the Hamilton- ian? We could try to further argue that both the old and new space-time operators are observables and that θ reflects the noncommutativity of the old observables. This, however, brings in a somewhat Bohmian flavor of hidden variables to the new quantization which is, to say the least, subject to questioning (for additional argu- ments regarding this issue see [16]). Thus, from our point of view, it would seem preferable to conclude that in the case of the quantizations discussed in subsection 4.1.4, the term “space-time noncommutativity” is a misnomer. Nonetheless, since the different quantizations here discussed lead to different (at least conceptually) experimental predictions, it is experiment then that will determine which, if any, of these theories can be closer related to reality. The same can be said regarding the different cases discussed in Section 4.2 for the relativistic particle. Of course it could also be contended that the use of the Dirac and path integral quantizations, which have been so successful in extending classical mechanics and field theory to a certain range of the quantum realm, is not justified a priori when dealing with distances of the order of the Planck length where quantum gravity becomes relevant. This could very well be so and it may involve having to drop the very concept of manifold, which underlies the mathematics of all of our present day physical constructions, in favor of new geometrical paradigms in which quantization is built in ab initio, such as the noncommutative geometry proposed by Connes [17] a few years ago. Be it as it may, we believe that the analysis presented here, the more axiomatic one presented in [13] and references within, as well as many other related works that have appeared in the literature, could provide some guidance for further work in that ultimate direction. References [1] S. Doplicher, K. Fredenhagen and J. E. Roberts, Commun. Math. Phys. 172, 187 (1995); D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli, Phys. Lett. B 533, 178 (2002). [2] Y. Liao and K. Sibold, Eur. Phys. J. C 25, 469 (2002) [arXiv:hep-th/0205269]. [3] K. Fujikawa, Phys. Rev. D 70, 085006 (2004) [arXiv:hep-th/0406128]. [4] A. P. Balachandran, T. R. Govindarajan, C. Molina and P. Teotonio-Sobrinho, JHEP 0410 (2004) 072 [arXiv:hep-th/0406125]. [5] A. Pinzul and A. Stern, Phys. Lett. B 593, 279 (2004) [arXiv:hep-th/0402220]. [6] P. Heslop and K. Sibold, Eur. Phys. J. C 41, 545 (2005) [arXiv:hep-th/0411161]. [7] R. Banerjee, B. Chakraborty and S. Gangopadhyay, J. Phys. A 38, 957 (2005) [arXiv:hep-th/0405178]. [8] M. Henneaux and C. Teitelboim, “Quantization of gauge systems,”, Princeton U.P., Princeton, New Jersey, 1992. [9] C. Acatrinei, JHEP 0109, 007 (2001) [arXiv:hep-th/0107078]. [10] C. A. Vaquera-Araujo, J. L. Lucio M, Non-Commutative Mechanics as a modification of space-time, [arXiv:math-ph/0512064]. [11] P.A.M. Dirac, “Lectures on Quantum Mechanics”, Belfer Graduate School of Science Mono- graphs Series No.2 (Yeshiva University, New York, 1964); E.C.G. Sudarshan and N. Mukunda, “Classical Dynamics: A Modern Perspective” (Wiley & Sons, New York, 1974); A. J. Hanson, T. Regge and C. Teitelboim, “Constrained Hamiltonian Systems” (Accademia Nazionale dei Lincei, Roma 1976); K. Sundermeyer, “Constrained Dynamics With Applications To Yang- Mills Theory, General Relativity, Classical Spin, Dual String Model,” Lect. Notes Phys. 169, 1 (1982). [12] V. I. Arnold, ”Mathematical Methods of Classical Mechanics” Springer, New York 1989. NONCOMMUTATIVITY FROM CANONICAL AND NONCANONICAL STRUCTURES 19 [13] M. Rosenbaum, J.D. Vergara and L.R. Juárez, “Dynamical origin of the ⋆-noncommutativity in field theory from quantum mechanics”, Phys. Lett. A, in press. [14] M. Rosenbaum, J.D. Vergara and L.R. Juárez, in preparation. [15] M. Chaichian, A. Demichev, P. Presnajder and A. Tureanu, Eur. Phys. J. C 20, 767 (2001) [arXiv:hep-th/0007156]. [16] G. D. Barbosa and N. Pinto-Neto, Phys. Rev. D 69, 065014, (2004) [17] A. Connes, “Noncommutative Geometry”, Academic Press, San Diego, Cal. (1994). Instituto de Ciencias Nucleares, UNAM, A. Postal 70-543,México D.F., México. E-mail address: mrosen@nucleares.unam.mx Instituto de Ciencias Nucleares, UNAM, A. Postal 70-543,México D.F., México. E-mail address: vergara@nucleares.unam.mx Instituto de Ciencias Nucleares, UNAM, A. Postal 70-543,México D.F., México. E-mail address: lromanjs@gmail.com ar X iv :h ep -t h /0 6 1 1 1 6 0 v 1 1 4 N o v 2 0 0 6 Canonical Quantization, Space-Time Noncommutativity and Deformed Symmetries in Field Theory Marcos Rosenbauma, J. David Vergarab and L. Roman Juarezc Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543 , México D.F., México a)mrosen@nucleares.unam.mx, b)vergara@nucleares.unam.mx, c)roman.juarez@nucleares.unam.mx ABSTRACT Within the spirit of Dirac’s canonical quantization, noncommutative spacetime field theories are intro- duced by making use of the reparametrization invariance of the action and of an arbitrary non-canonical symplectic structure. This construction implies that the constraints need to be deformed, resulting in an automatic Drinfeld twisting of the generators of the symmetries associated with the reparametrized theory. We illustrate our procedure for the case of a scalar field in 1+1- spacetime dimensions, but it can be readily generalized to arbitrary dimensions and arbitrary types of fields. 1 2 1. Introduction It has been considered as common wisdom among practitioners of noncommutative field theory that at the first quantization level, fields are elements of an algebra where multiplication is deformed by means of the Moyal ⋆-product [1]. This anzatz, which originated in a basically heuristic fashion from some results in string theory [2], is based on an analogy with the Weyl-Wigner-Groenewold-Moyal (WWGM) formalism of Quantum Mechanics. But in Quantum Mechanics time is a parameter of the theory and, in order for spacetime to have a truly noncommutativity physical meaning we need to consider both space and time as observables represented by noncommutative operators and include them as dynamical variables in an extended Heisenberg algebra. Moreover, as we have shown elsewhere [3], the ⋆-product deformation of functions of spacetime then results naturally in the WWGM formalism when considering in this extended context the algebra of the Weyl-equivalent functions corresponding to operator functions of the Heisenberg space and time operators. Other approaches for constructing a noncommutative spacetime Quantum Mechanics have been based on the idea of promoting the time parameter to the rank of a coordinate by means of a reparametrization, whereby time becomes a function t(τ) of a new parameter τ and thus becomes a coordinate on the same level as the spatial coordinates xi(τ), either by fixing the gauge degrees of freedom [4], [5], [6] or by deforming the symplectic structure of the theory [7]. An important feature of these formulations is that, because additional degrees of freedom are added to the original theory, first class constraints appear in the reparametrized theory. In order to eliminate these additional degrees of freedom one can apply gauge conditions or follow Dirac’s quantization method and operate with the constraints on the state vectors in order to obtain the physical states of the system. Now, when going on to field theory both the time and space coordinates play the role of parameters of the field, so applying commutation relations to them is, to say the least, even more unclear; as it is the relation of this procedure to the operator spacetime noncommutativity in Quantum Mechanics, particularly when we view the latter as a minisuperspace of the former and in the light of what we have just said above. In order to shed some additional insight on some of these issues, we explore in the present work how the above refereed reparametrization formalism can be extended to the case of field theory on a noncommutative space-time. However, since we are now dealing with a system with an infinite number of degrees of freedom, the basic idea here is to promote the coordinates of the space-time, that are the parameters on which the field depends, to new fields in the ensuing reparametrized theory. This idea in not new in the case of commutative spacetime. For example in [8] such a construction of a field theory was used as a model when considering the canonical quantization of gravity. Making use of the results in that work, it is possible to construct the reparametrized theory for any field theory, with as many constraints as the number of coordinate fields being added. In addition, as it occurs in the case of General Relativity, the parametrized field theory is also invariant under diffeomorphisms, so such a construction provides an ideal arena for studying these symmetries at the quantum level there. It is interesting to note that this idea was also used in the context of string theory as a means for constructing a theory which would be independent of the background [9]. Once the spacetime coordinates are promoted to the rank of fields, it does make sense to impose commutation relations among them. This can be achieved by deforming the symplectic structure in the original theory and thus arriving at a noncommutative field theory. Such a theory is already at the 3 first quantization level radically different from the usual one, because - since the coordinate fields do not commute - we can not use their eigenstates as configuration space bases to construct amplitudes of the state vector, which will then necessarily have to be either functions of both the eigenvalues of the momenta field operators as well as of some of the coordinate fields (those that commute among themselves), or only of the eigenvalues of the commuting momenta fields. Another important point that we analyze in this paper is the deformed symmetries that appear in the noncommutative theory. According to our procedure, the nature of these deformed symmetries appears automatically since, when deforming the symplectic structure the algebra of the constraints is broken and, in order to preserve it, it is necessary to deform the generators of the symmetry by means of what turns out to be a Drinfeld twist. The algorithm suggested by our procedure for this twist is quite straightforward to implement and can be readily generalized to other types of ⋆-products as well as to situations where noncommutativity involves both spacetime and momenta variables. 2. Spacetime Noncommutativity in Field Theory In a previous paper [7] noncommutative space-time quantum mechanical theories were constructed by using a reparametrization invariant action where the time parameter is elevated to the rank of a dynamical variable. Furthermore, in order to consider the noncommutativity between the space- time coordinates, an arbitrary non-canonical symplectic structure was introduced that, together with Dirac’s Hamiltonian method, leads to Dirac brackets for the space-time dynamical variables, which when quantized may be interpreted as noncommutative. As mentioned in the Introduction, we shall apply this procedure to the case of fields in order to investigate the implications of noncommutativity of spacetime as field variables on the algebra of the reparametrized fields. 2.1. Reparametrization of the scalar field. To illustrate the procedure, consider for simplicity the case of a scalar field in a D + 1−dimensional Minkowski spacetime with signature (1,−1, . . . ,−1) and with a potential V (φ). The corresponding action is then (2.1) S = ∫ dxdt ( 1 2 ηµν∂µφ∂νφ− V (φ) ) . In order to parameterize the full spacetime, let us write (2.2) t = t(τ,σ), xi = xi (τ,σ) , so that the new action in terms of the new parameters τ,σ reads (2.3) S = ∫ dτdDσ √−g ( 1 2 gµν∂µφ∂νφ− V (φ) ) , with the inverse metric gαβ given by (2.4) gαβ = ∂σα ∂xµ ∂σβ ∂xµ and g := det(gµν) where √−g = J is the Jacobian of the transformation. Also, in (2.3) we are making the identification ∂0 ≡ ∂τ and ∂i = ∂σi . The canonical momentum associated to the field φ is Pφ = J ∂τ ∂xµ ∂σα ∂xµ ∂φ ∂σα , σ0 = τ, σi ≡ σi, ,(2.5) 4 and, following [8], we define the canonical momenta associated to the spacetime coordinates as pν ≡ −J ∂τ ∂xµ T µ ν ,(2.6) where T µ ν = ∂µφ∂νφ − δµ ν (1 2∂ ρφ∂ρφ − V (φ)) is the unparametrized energy-momentum tensor of the field. In terms of this momenta the Hamiltonian action becomes (2.7) S = ∫ dτdDσ ( Pφφ̇+ pµẋ µ − λν ( pν + J ∂τ ∂xµ T µ ν )) , where we have introduced the definition of the momenta (2.6) as Hamiltonian constraints due to the fact that the right hand side of (2.6) is independent of the velocities when the energy-momentum tensor is expressed as a function of the canonical variables φ, Pφ [8]. We can write an alternate expression for the action (2.7), based on the ADM-type decomposition of spacetime Σ × R, where R is the temporal direction and Σ is a space-like hypersurface of constant τ , by introducing the vectors si with components sµ i = ∂σixµ tangent to Σ and the unit vector n̂, normal to this hypersurface, with components (2.8) nµ = ( √ g00ẋµ + g0i √ g00 ∂xµ ∂σi ) , i = 1, . . . , d. Furthermore, constructing from sµ i the orthonormal basis ûi = αj i sj , we can write the (D+1)-vector constraint Π, with components Πν ≡ pν + J ∂τ ∂xµ T µ ν , as (2.9) Π ≡ (n̂n̂ + γijûiûj) ·Π = n̂Φ0 + γijûiΦj , where (2.10) I := (n̂n̂ + γij ûiûj), is the unit dyadic, multiplication is with the Lorentzian metric, (2.11) Φ0 := n̂ ·Π = nµ(pµ + J ∂τ ∂xν T ν µ) = 1 2 √−γ ( P 2 φ + γγij∂σiφ∂σjφ ) + nµpµ + √−γ V (φ) , (2.12) Φj := sj ·Π = (∂σjxµ)(pµ + J ∂τ ∂xν T ν µ) = Pφ∂σjφ+ pµ∂σjxµ, and where γij ≡ gij is the D-metric of the Σ-hypersurface, γij is the inverse matrix to γij and γ is the determinant of γij . Inserting now (2.9) into (2.7) we can write (2.13) S = ∫ dτdDσ ( Pφφ̇+ pµẋ µ −NH⊥ −N iHi ) , after identifying the proyections (−γ)− 1 2 (λ · n̂), γjkαi kλ · ûj of the Lagrange multipliers with the lapse and shift functionsN andN i, respectively, so thatH⊥ = √−γΦ0 is the super-Hamiltonian and Hi = Φi are the super-momenta for the system. The Poisson brackets of these super- Hamiltonian and super-momenta are given by [10] {H⊥(σ, τ),H⊥(σ′, τ} = D ∑ i=1 (Hi(σ, τ) +Hi(σ ′, τ))∂σiδ(σ − σ′), {Hi(σ, τ),Hk(σ′, τ)} = (Hk(σ, τ)∂σiδ(σ − σ′) +Hi(σ ′, τ))∂σkδ(σ − σ′),(2.14) {H⊥(σ, τ),Hi(σ ′, τ)} = (H⊥(σ, τ) +H⊥(σ′, τ))∂σiδ(σ − σ′), from where we see that the constraints are first class. 5 Let us now further simplify the calculations and the basic steps leading to a noncommutative field theory by consider first our scalar field to be propagating in a flat space-time with Minkowskian coor- dinates (t, x) and signature (1,−1). In this case (2.15) gµν = g−1 ( t ′2 − x′2 −(t′ṫ− x′ẋ) −(t′ṫ− x′ẋ) ṫ2 − ẋ2 ) , and (2.16) g := det(gµν) = −(ṫx′ − ẋt′)2, where the primes denote partials with respect to σ while the dots are partials with respect to τ . Explicit expressions for the momenta canonical to t, x and φ can be derived from (2.5) and (2.6) or, even simpler, directly from (2.3), (2.15) and (2.16). They are given by: pt = − 1√−g (ṫφ′2 − t′φ′φ̇)− x′V (φ)− x′ 2g [(t′2 − x′2)φ̇2 − 2(t′ṫ− x′ẋ)φ′φ̇+ (ṫ2 − ẋ2)φ′2], px = 1√−g (ẋφ′2 − x′φ′φ̇) + t′V (φ) + t′ 2g [(t′2 − x′2)φ̇2 − 2(t′ṫ− x′ẋ)φ′φ̇+ (ṫ2 − ẋ2)φ′2],(2.17) Pφ = − 1√−g [(t′2 − x′2)φ̇− (t′ ṫ− x′ẋ)φ′]. From these expressions it can be readily verified that (2.18) ptṫ+ pxẋ+ Pφφ̇ = L = √−g ( 1 2 gµν∂µφ∂νφ− V (φ) ) . Furthermore, because we are introducing the fields t(τ, σ) and x(τ, σ) as new degrees of freedom, the theory must have constraints in the Hamiltonian formalism. Specifically, since instead of our two original phase space degrees of freedom we now have six, we thus need four relations which we can get by two primary first class constraints, and two gauge conditions. The primary constraints follow from specializing (2.11) and (2.12) to the case D = 1 and are explicitly given by (2.19) H⊥ = 1 2 ( P 2 φ + φ′2 ) + ptx ′ + pxt ′ + ( x ′2 − t′2 ) V (φ) ≈ 0, H1 = pxx ′ + ptt ′ + Pφφ ′ ≈ 0. Defining (2.20) H⊥,1[f ] := ∫ dσf(σ)H⊥,1(σ, τ), it can then be shown that {H⊥[f ],H⊥[g]} = H1[fg ′ − gf ′], {H1[f ],H1[g]} = H1[fg ′ − gf ′],(2.21) {H⊥[f ],H1[g]} = H⊥[fg′ − gf ′]. Moreover, since the test functions f and g are arbitrary, we can take the functional derivatives of (2.21) relative to them to arrive at {H⊥(σ, τ),H⊥(σ′, τ} = (H1(σ, τ) +H1(σ ′, τ))δ′(σ − σ′), {H1(σ, τ),H1(σ ′, τ)} = (H1(σ, τ) +H1(σ ′, τ))δ′(σ − σ′),(2.22) {H⊥(σ, τ),H1(σ ′, τ)} = (H⊥(σ, τ) +H⊥(σ′, τ))δ′(σ − σ′), where δ′(σ−σ′) := ∂σδ(σ−σ′), which reproduce (2.14) for the case D = 1. Note that these constraints close in the constant τ Poisson brackets according to the Virasoro algebra without a central charge 6 and they are first-class, as we already know. But first class constraints are generically associated with gauge invariance, which in this case is the invariance of the action (2.3) under two-dimensional reparametrizations, with its generators satisfying the algebra (2.22). Moreover since H = ∫ dσ(NH⊥ +N1H1) is the Hamiltonian of the theory, it clearly follows that (2.23) Ḣ⊥,1 = {H⊥,1, H} ≈ 0, so the constraints are preserved by the “time” τ evolution. Next, in order to introduce space-time noncommutativity in the Dirac quantization procedure for the above theory, we need to implement an additional general symplectic structure into our formalism. 2.2. Symplectic structure. For this purpose consider the following general first order action: (2.24) S = ∫ dτdσ ( Aa(z)ża −NH̃⊥ −N1H̃1 ) , with symplectic variables za = (t, x, φ, pt, px, Pφ). Here H̃⊥ and H̃1 are weakly zero and appropriately modified first-class constraints to be specified below. The six potentials Aa play the role of momenta canonically conjugate to the za. The action (2.24) allows us to generate an arbitrary symplectic structure associated to the Poisson brackets in the Hamiltonian formulation, but in order that it be equivalent to the action (2.13) forD = 1, we need six additional second-class primary constraints (these, together with the two first-class constraints and their corresponding two compatibility conditions, give the relations needed to eliminate ten of the twelve degrees of freedom in the za’s). The additional second-class constraints follow by noting that the canonical momenta conjugate to za are given by (2.25) πza = ∂ża ( Aa(z)ża −NH̃⊥ −N1H̃1 ) = Aa(z), and since they are independent of the velocities they lead to the constraints (2.26) χa = πza −Aa ≈ 0. Hence the action of our constrained system is now given by (2.27) S = ∫ dτdσ (Aa(z)ża −HT ) , with (2.28) HT = NH̃⊥ +N1H̃1 + µaχa. Note that from (2.26) we have (2.29) {χa, χb} = ∂Ab ∂za − ∂Aa ∂zb := ωab, so the constraints χa are indeed second-class (note that the Poisson brackets here are to be evaluated in the extended phase-space (za, πa) ). Moreover, in order that the consistency conditions (2.30) χ̇a = {χa, ∫ dσHT } = −N ∂H̃⊥ ∂za −N1 ∂H̃1 ∂za + µbωab ≈ 0, (2.31) ˙̃H⊥,1 = {H̃⊥,1, ∫ dσHT } = µa{H̃⊥,1, ∫ dσχa} ≈ 0, 7 be satisfied, we need, solving (2.30) for µa, that (2.32) µa = ωab ( N ∂H̃⊥ ∂zb +N1 ∂H̃1 ∂zb ) , and also that (2.33) ωab ( ∂H̃⊥ ∂za ∂H̃1 ∂zb ) ≈ 0, which results from inserting (2.32) into (2.31) and using the arbitrariness of the Lagrange multipliers. Introducing now the Dirac brackets (2.34) {ξ, ρ}∗ := {ξ, ρ} − {ξ, χa}ωab{χb, ρ}, it readily follows that (2.35) {H̃⊥, H̃1}∗ = ωab ( ∂H̃⊥ ∂za ∂H̃1 ∂zb ) . Hence, in order to satisfy the compatibility condition (2.31) we need to chose our modified constraints H̃⊥, H̃1 such that their Dirac bracket is weakly zero. We shall defer the proof that such a choice indeed exist for later on, and note at this point that (2.36) {χa, χb}∗ = 0. We can therefore treat the χa as strongly zero in our formalism, after replacing the Poisson brackets by the Dirac brackets. Note also that (2.34) implies (2.37) {za, zb}∗ = ωab, and by assuming further that the symplectic structure is determined by (2.38) ωab =           0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 1 0 0 0 θ 0 0 1 0 −θ 0 0 0 0 1 0 0 0           , ωab =           0 θ 0 1 0 0 −θ 0 0 0 1 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0           , we find that (2.38), incorporates spacetime noncommutativity into the formalism. In particular upon quantization, the strong equations χa = 0 need to be promoted to a relation between quantum operators: (2.39) π̂za − Âa = 0, and we have from (2.37) that at equal τ (2.40) [ t̂ (τ, σ) , x̂ (τ, σ̃) ] = iθδ (σ − σ̃) , [ t̂ (τ, σ) , p̂t (τ, σ̃) ] = iδ (σ − σ̃) , [x̂ (τ, σ) , p̂x (τ, σ̃)] = iδ (σ − σ̃) , [ φ̂ (τ, σ) , P̂φ (τ, σ̃) ] = iδ (σ − σ̃) . 8 We turn now to the derivation of the explicit form for the modified first-class constraints H̃⊥ and H̃1, by observing that the formalism requires that their algebra should now close relative to the Dirac- brackets. This can be achieved by further noting that (2.41) {t̃, x̃}∗ = 0, where (2.42) t̃ = t+ θ 2 px, x̃ = x− θ 2 pt. This selection of the t̃, x̃, variables is not unique, since there exist an infinite number of possible choices all of which are related by canonical transformations that leave invariant the symplectic structure (2.38). At the quantum level, however, only those theories which are related by linear canonical trans- formations will be equivalent. Now, taking into account that the Dirac-bracket algebra of the variables (t̃, x̃, φ, pt, px, Pφ) is the same as the Poisson algebra of (t, x, φ, pt, px, Pφ), it therefore follows that by setting H̃⊥,1(z a) = H⊥,1(z̃ a) we immediately have {H̃⊥(τ, σ), H̃⊥(τ, σ′}∗ = (H̃1(τ, σ) + H̃1(τ, σ ′))δ′(σ − σ′), {H̃1(τ, σ), H̃1(τ, σ ′)}∗ = (H̃1(τ, σ) + H̃1(τ, σ ′))δ′(σ − σ′),(2.43) {H0(τ, σ),H1(τ, σ ′)}∗ = (H̃⊥(τ, σ) + H̃⊥(τ, σ′))δ′(σ − σ′), with (2.44) H̃⊥ = 1 2 (P 2 φ + φ ′2) + pt(x− θ 2 pt) ′ + px(t+ θ 2 px)′ + ( (x− θ 2 pt) ′2 − (t+ θ 2 px) ′2 ) V (φ) ≈ 0, H̃1 = px(x− θ 2 pt) ′ + pt(t+ θ 2 px)′ + Pφφ ′ ≈ 0. When quantizing, the constraints H̃⊥,1 are promoted to the rank of operators satisfying the subsidiary conditions ˆ̃H⊥|Ψ〉 = 0, ˆ̃H1|Ψ〉 = 0. (2.45) Also for consistency we need that at the quantum level the additional condition (2.46) [ ˆ̃H⊥, ˆ̃H1]|Ψ〉 = 0, be satisfied. This implies that the commutator of the first class constraint operators has to be of the form (2.47) [ ˆ̃H⊥(τ, σ), ˆ̃H1(τ, σ ′)] = ĉ⊥(σ, σ′) ˆ̃H⊥ + ĉ1(σ, σ ′) ˆ̃H1, where, in general, the ĉ⊥,1 are functions of the field operators that need to appear to the left of the ˆ̃H⊥,1. This, in turn, involves finding the operator ordering needed to achieve this requirement in order to have an appropriate quantum theory. In the present case this does not constitute an important issue, since ordering for the super-Hamiltonian is immaterial and the difference in placing the momenta to the right or to the left of the coordinates in the super-momentum leads to a term which in the basis |t(σ), px(σ), φ(σ)〉 (see paragraph following Eq.(2.51) below) is of the form ∂σδ(σ − σ′)|σ=σ′Ψ(t(σ), px(σ), φ(x(σ), t(σ)), τ) and which, because of the antisymmetry of the delta function 9 derivative, can be put equal to zero. We therefore choose the following ordering for the ˆ̃H⊥,1: (2.48) ˆ̃H⊥ = 1 2 ( P̂φ 2 + φ̂ ′2 ) + p̂t(x̂− θ 2 p̂t) ′ + p̂x(t̂+ θ 2 p̂x)′ − ( (t̂+ θ 2 p̂x) ′2 − (x̂− θ 2 p̂t) ′2 ) V ( φ̂ ) ≈ 0, ˆ̃H1 = p̂x(x̂ − θ 2 p̂t) ′ + p̂t(t̂+ θ 2 p̂x)′ + P̂φφ̂ ′ ≈ 0. Making repeated use of the identity (2.49) f(σ′)δ′(σ − σ′) = f ′(σ)δ(σ − σ′) + f(σ)δ′(σ − σ′) in the evaluation of the commutator of these two operators, we get 2P̂φ(σ)P̂φ(σ′)δ′(σ − σ′) = (P̂ 2 φ(σ) + P̂ 2 φ(σ′))δ′(σ − σ′), 2ˆ̃x′(σ)ˆ̃x′(σ′)δ′(σ − σ′) = (ˆ̃x′2(σ) + ˆ̃x′2(σ′))δ′(σ − σ′), 2ˆ̃t′(σ)ˆ̃t′(σ′)δ′(σ − σ′) = (ˆ̃t′2(σ) + ˆ̃t′2(σ′))δ′(σ − σ′), ( p̂t(σ)ˆ̃x′(σ′) + p̂t(σ ′)ˆ̃x′(σ) ) δ′(σ − σ′) = ( p̂t(σ)ˆ̃x′(σ) + p̂t(σ ′)ˆ̃x′(σ′) ) δ′(σ − σ′), ( p̂x(σ)ˆ̃t′(σ′) + p̂x(σ′)ˆ̃t′(σ) ) δ′(σ − σ′) = ( p̂x(σ)ˆ̃t′(σ) + p̂x(σ′)ˆ̃t′(σ′) ) δ′(σ − σ′), (ˆ̃x′2(σ) + ˆ̃t′2(σ))[V (φ̂(σ)), P̂φ(σ′)]φ′(σ′) = i(ˆ̃x′2(σ′) + ˆ̃t′2(σ′))∂σV (φ̂(σ))δ(σ − σ′) = i(ˆ̃x′2(σ′) + ˆ̃t′2(σ′)) ( V (φ̂(σ′))− V (φ̂(σ)) ) δ′(σ − σ′). (2.50) From these relations it follows that (2.51) [ ˆ̃H⊥(τ, σ), ˆ̃H1(τ, σ ′)] = i ( ˆ̃H⊥(τ, σ) + ˆ̃H⊥(τ, σ′) ) δ′(σ − σ′). Hence our choice (2.48) is indeed of the form (2.47) and results in an appropriate Dirac quantization of the theory. In this parametrized quantization all the dynamics is hidden in the constraints although, because of the noncommutativity of the coordinate field operators t(τ, σ), x(τ, σ), we can not construct configuration space state functionals of the form Ψ[t(σ), x(σ), φ(σ), τ ] = 〈t(σ), x(σ), φ(σ)|Ψ(τ)〉 with the usual interpretation of a probability amplitude that the scalar field φ have a definite distribution φ(σ) on a curved spacelike hypersurface defined by t = t(σ), x = x(σ) at time τ . (Note that in the Schrödinger picture the dynamical variables do not depend on τ). We can, however, construct state amplitudes from mixed momenta and reduced configuration space eigenkets such as |t(σ), px(σ), φ(σ)〉. In this basis x̂ and p̂t are represented by x̂ = i ( δ δpx(σ) − θ δ δt(σ) ) ,(2.52) p̂t = −i δ δt(σ) ,(2.53) so that from (2.48) we get: ( θ 2 ∂ ∂σ δ2 δt(σ)δt(σ) − ∂ ∂σ δ δt(σ) δ δpx(σ) ) Ψ[t(σ), px(σ), φ(σ), τ ] = [ 1 2 ( − δ2 δφ(σ)δφ(σ) + φ′2 ) + px(t′ + θ 2 p′x)− ( (t′ + θ 2 p′x)2 + ∂2 ∂σ2 ( δ δpx(σ) − θ 2 δ δt(σ) )2 ) V (φ) ] Ψ, (2.54) and (2.55) [ px ∂ ∂σ ( δ δpx(σ) − θ 2 δ δt(σ) ) − ( t′ − θ 2 p′x ) δ δt(σ) − φ′ δ δφ ] Ψ[t(σ), px(σ), φ(σ)] = 0. 10 Thus, introducing noncommutativity by parametrizing the action in the Dirac first quantization of the scalar field scheme leads us necessarily to the above twofold infinity of coupled equations. The equations (2.54) and (2.55) are the analogous of the Wheeler-De Witt equations for our noncommutative scalar field, and they can not be reduced to a Schrödinger-like equation as in the commutative case, because here we can not solve explicitly the super-Hamiltonian and super-momentum constraints for the momenta pt and px. It is not our objective here to investigate this system any further or the issue of second quantization. We shall consider instead in the following section the deformed symmetries which result from the deformed constraints of the theory, which in turn result from the space-time noncommutativity, and derive a general anzatz for constructing these deformed symmetries for any field theory. 3. spacetime noncommutativity and deformed symmetries We have seen that the Dirac-bracket algebra (2.43) together with (2.44) provides an algorithm for constructing the deformed gauge symmetries associated with the reparametrization invariance of the action (2.3), where a symplectic structure was introduced in order to allow for the appearance of spacetime noncommutativity when applying Dirac’s procedure for canonical quantization to the original action. In fact, making use of (2.37) one can show that (3.56) {tn(τ, σ), xm(τ, σ′)}∗ = nmθtn−1(τ, σ)xm−1(τ, σ′)δ(σ − σ′). On the other hand, evaluating the Moyal product (xµ)n ⋆θ (xν)m with the bidifferential (3.57) ⋆θ := exp [ i 2 θµν ∫ dσ′′ ←− δ δxµ(τ, σ′′) −→ δ δxν(τ, σ′′) ] , and comparing with (3.56), we have that (3.58) {tn(τ, σ), xm(τ, σ′)}∗ ∼= [tn(τ, σ), xm(τ, σ′)]⋆θ := tn(τ, σ) ⋆θ x m(τ, σ′)− xm(τ, σ′) ⋆θ t n(τ, σ). More generally, for Dirac- brackets of arbitrary A(τ, σ), B(τ, σ) functionals of t(τ, σ), pt(τ, σ), x(τ, σ), px(τ, σ), φ(τ, σ) and Pφ(τ, σ) we get (3.59) {A(τ, σ), B(τ, σ′)}∗ ∼= [A(τ, σ), B(τ, σ′)]⋆θ , after identifying the momenta in the left side of the above equation with their corresponding differential operators on the right side. We thus have a morphism from the Poisson-Dirac algebra of functionals of t, x, φ, pt, px and Pφ, to the algebra of differential operators obtained from these functionals (after mak- ing the maps pt 7→ −iδ/δt, px 7→ −iδ/δx Pφ 7→ −iδ/δφ) with multiplication given by the ⋆θ-product commutator. As a parenthetical remark we find it interesting to recall here that in the process of reparametrization the space-time parameters of the original action were elevated to the rank of dynam- ical variables and, as we have shown elsewhere [3], when considering quantum mechanical deformations from the point of view of the Weyl-Wigner-Groenewold-Moyal formalism, the multiplication of elements of the algebra of functions of the space-time dynamical variables had to be modified precisely with the ⋆-operator (3.57). 11 Applying now the above described algebra morphism to (2.43) results in [ H̃⋆ ⊥(τ, σ), H̃⋆ ⊥(τ, σ′) ] ⋆θ = ( H̃⋆ 1(τ, σ) + H̃⋆ 1(τ, σ ′) ) δ′(σ − σ′), [ H̃⋆ 1(τ, σ), H̃⋆ 1(τ, σ ′) ] ⋆θ = ( H̃⋆ 1(τ, σ) + H̃⋆ 1(τ, σ ′) ) δ′(σ − σ′),(3.60) [ H̃⋆ ⊥(τ, σ), H̃⋆ 1(τ, σ ′) ] ⋆θ = ( H̃⋆ ⊥(τ, σ) + H̃⋆ ⊥(τ, σ′) ) δ′(σ − σ′). Here the notation H̃⋆ ⊥,1 stands for the differential operators (3.61) H̃⋆ ⊥,1(τ, σ) := H⊥,1(τ, σ) exp [ − i 2 θµν ∫ dσ′′ ←− δ δxµ(τ, σ′′) −→ δ δxν(τ, σ′′) ] , and their algebra multiplication µθ is given by (3.62) µθ(H̃⋆ i ⊗ H̃⋆ j ) = H̃⋆ i ⋆ H̃⋆ j , i, j =⊥, 1. Note that from (3.61) it follows that (3.63) [ H̃⋆ i (τ, σ), H̃⋆ j (τ, σ′) ] ⋆θ = [Hi(τ, σ),Hj(τ, σ ′)] e h − i 2 θµν R dσ′′ ←− δ δxµ(τ,σ′′) −→ δ δxν (τ,σ′′) i , i, j =⊥, 1 and substituting (3.61) and (3.63) into (3.60) we get [H⊥(τ, σ),H⊥(τ, σ′)] = (H1(τ, σ) +H1(τ, σ ′)) δ′(σ − σ′), [H1(τ, σ),H1(τ, σ ′)] = (H1(τ, σ) +H1(τ, σ ′)) δ′(σ − σ′),(3.64) [H⊥(τ, σ),H1(τ, σ ′)] = (H⊥(τ, σ) +H⊥(τ, σ′)) δ′(σ − σ′), which is the algebra of differential operator generators isomorphic to the non-deformed algebra (2.22). Furthermore, since by (2.19) (3.65) {φ,H⊥} = (x′2 − t′2)√−g φ̇+ (t′ṫ− x′ẋ)√−g φ′, {φ,H1} = φ′, the generators Hi of (2.22) - the Virasoro algebra V - can be viewed as derivations acting on elements φ(t(τ, σ), x(τ, σ)) of the algebra of functions A, with point multiplication µ. That is, {φ,H⊥} ∼=Ĥ⊥ ⊲ φ = ( (x′2 − t′2)√−g ∂τ + (t′ ṫ− x′ẋ)√−g ∂σ ) ⊲ φ {φ, Ĥ1} ∼=Ĥ1 ⊲ φ = ∂σ ⊲ φ, (3.66) In addition, since Ĥi ∈ V̂ is a (infinite dimensional) Lie algebra, its universal envelope U(V̂) can be given the structure of a Hopf algebra with coproduct (3.67) ∆(Ĥi) = Ĥi ⊗ 1 + 1⊗ Ĥi, i =⊥, 1 and antipode (3.68) S(Ĥi) = −Ĥi, i =⊥, 1, so A is a left module-algebra over U(V̂). In parallel, for the symplectic structure (2.38) we have the algebra V̂⋆ of derivation operators ˆ̃H⋆ i , defined in analogy to (3.61) by (3.69) ˆ̃H⋆ ⊥,1(τ, σ) := Ĥ⊥,1(τ, σ) exp [ − i 2 θµν ∫ dσ′′ ←− δ δxµ(τ, σ′′) −→ δ δxν(τ, σ′′) ] , 12 with multiplication µθ generated by (3.59), and the corresponding left module algebra Aθ over U(V̂⋆), whose elements are now functions φ(t(τ, σ), x(τ, σ)) with multiplication µθ inherited from (3.58). From (3.69) it immediately follows that (3.70) ˆ̃H⋆ i ⋆θ φ(t, x) = Ĥi ⊲ φ(t, x), so the action of elements of the twisted algebra V̂⋆ on elements of Aθ is equal to the action of the corresponding elements of the untwisted algebra on the corresponding elements of the ordinary algebra A of functions of commuting variables. Thus the morphism from V̂ to V̂⋆ by (3.71) Ĥi 7→ ˆ̃H⋆ i induces the morphism from A to Aθ by (3.72) µ(f(t, x)⊗ g(t, x)) 7→ µθ(f(t, x) ⊗ g(t, x)). Let us next consider the symmetries associated with the canonical transformation (3.73) Hτ [ξ] = ∫ dσ ( ξ0(τ, σ)H⊥(τ, σ) + ξ1(τ, σ)H1(τ, σ) ) , in order to make contact with some related results appearing in the literature. We thus have δφ = {φ,Hτ [ξ]} ∼= Ĥτ [ξ] ⊲ φ = ξ0 (x′2 − t′2)√−g φ̇+ ( ξ0 (t′ ṫ− x′ẋ)√−g + ξ1 ) φ′ δt = {t,Hτ [ξ]} = ξ0x′ + ξ1t′(3.74) δx = {x,Hτ [ξ]} = ξ0t′ + ξ1x′. On the other hand, it is evident that the Lagrangian in (2.3) is invariant under the infinitesimal general coordinate transformations τ → τ + ρ0(τ, σ) σ → σ + ρ1(τ, σ), (3.75) from where it follows that δρφ = −ρ0∂τφ− ρ1∂σφ, δρt = −ρ0∂τ t− ρ1∂σt,(3.76) δρx = −ρ0∂τx− ρ1∂σx. We can relate the generator (3.73) to the diffeomorphism (3.76) by equating the last two equations in (3.74) to the last two equations in (3.76) and solving for ξ0 and ξ1. We thus get ξ0 = (t′ẋ− x′ ṫ) (x′2 − t′2) ρ 0, ξ1 = (t′ ṫ− x′ẋ) (x′2 − t′2) ρ 0 − ρ1. (3.77) The consistency of this solution can be checked by substituting it into the first equation in (3.74) and verifying that it yields the first equation in (3.76). Consequently (3.78) δρφ = Ĥτ [ξ(ρ)] ⊲ φ ∼= {φ,Hτ [ξ(ρ)]}, with the components of ξ(ρ) given by (3.77). Hence (3.79) δρ = Ĥτ [ξ(ρ)] = −(ρ0ṫ+ ρ1t′)∂t − (ρ0ẋ+ ρ1x′)∂x = −(ρt∂t + ρx∂x), 13 where we have re-expressed the vector field δρ in terms of the spacetime basis components {ρt := (ρ0ṫ+ ρ1t′), ρx := (ρ0ẋ+ ρ1x′)}. Applying now the derivation δη := −ηt∂t−ηx∂x to (3.78) and subtracting from the result the expression with inverted order of the derivations we get [δη, δρ]φ ∼= {{φ,Hτ [ξ(ρ)]}, Hτ [ξ(η)]} − {{φ,Hτ [ξ(η)]}, Hτ [ξ(ρ)]} = {{Hτ [ξ(η)], Hτ [ξ(ρ)]}, φ} = −(−ηλ∂λρ µ + ρλ∂λη µ)∂µφ = −(η × ρ)µ∂µφ = δη×ρ φ,(3.80) after making use of the Jacobi identity. We therefore have an homomorphism between the algebra of diffeomorphisms in two-dimensions (3.81) [δη, δρ] = δη×ρ and the Poisson algebra H generated by (3.82) {Hτ [ξ(η)], Hτ [ξ(ρ)]} = Hτ [ξ(η × ρ)]. In going on to the noncommutative spacetime case, we proceed according to our previously derived algorithm, i.e. we replace the Poisson-brackets by Dirac-brackets and t→ t̃, x→ x̃. Hence we can now write (3.83) H ∋ Ĥτ [ξ(ρ)] 7→ Ĥ⋆ τ [ξ̃(ρ)] = δ⋆ ρ = ∫ dσ(ξ̃0 ˆ̃H⋆ ⊥ + ξ̃1 ˆ̃H⋆ 1) ∈ H ⋆, and (3.84) {φ,Hτ [ξ(ρ)]} ∼= δρ ⊲ φ 7→ δ⋆ ρ ⋆ φ(t(τ, σ), x(τ, σ)); φ ∈ Aθ. Note that equations (3.83) and (3.84) provide an explicit expression for the mapping δρ 7→ δ⋆ ρ, such that (3.82) becomes (3.85) [ δ⋆ ρ , δ ⋆ η ] ⋆θ = δ⋆ η×ρ, and (3.86) δ⋆ ρ ⋆ (f ⋆ g) = δρ(f ⋆ g). We can now compare some of our results with those obtained in [11]. Thus, we have that our equation (3.69) for the twisted derivations ˆ̃H⋆ i corresponds to equation (3.26) in [11], while the algebra (3.85) and the derivation δ⋆ ρ correspond to equations (5.3) and (5.4) there. Note also that since the universal envelope U(H⋆) in our formalism can be given the structure of a Hopf algebra, we can obtain an explicit expression for the coproduct by making use of the duality between product and coproduct, followed by the application of equations (3.86) and (3.61). Thus we have µθ ◦∆(δ⋆ ρ)(f ⊗ g) = δ⋆ ρ ⋆ (f ⋆ g) = δρ(f ⋆ g) = µ(δρ ⊗ 1 + 1⊗ δρ)(e i 2 θµν∂µ⊗∂νf ⊗ g) = µθ ◦ [ (δ⋆ ρ ⊗ 1 + 1⊗ δ⋆ ρ)e i 2 θµν∂µ⊗∂ν (f ⊗ g) ] = µθ ◦ [ e− i 2 θµν∂µ⊗∂ν (δρ ⊗ 1 + 1⊗ δρ)e i 2 θµν∂µ⊗∂ν ] (f ⊗ g). (3.87) This result also compares with the Leibnitz rule given by equation (5.9) in [11]. Further note that if we let F = e− i 2 θµν∂µ⊗∂ν ∈ U(H)⊗ U(H), and define f ⋆ g = µθ(f ⊗ g) := µ(F−1 ⊲ (f ⊗ g)), we then have δρ(f ⋆ g) = δρ ⊲ µ(F−1 ⊲ (f ⊗ g)) = µ[(∆δρ)F−1 ⊲ (f ⊗ g)] = µF−1[(F(∆δρ)F−1)(f ⊗ g)](3.88) = µθ[(F(∆δρ)F−1)(f ⊗ g)]. 14 Thus, the undeformed coproduct of the symmetry Hopf algebra U(H) is related to the Drinfeld twist ∆F by the inner endomorphism ∆Fδρ := (F(∆δρ)F−1) and, by (3.88), it preserves the covariance: δρ ⊲ (f · g) = µ ◦ [∆(δρ)(f ⊗ g)] = (δρ(1) ⊲ f) · (δρ(2) ⊲ g) θ→ δ⋆ ρ ⊲ (f ⋆ g) = (δ⋆ ρ(1) ⊲ f) ⋆ (δ⋆ ρ(2)) ⊲ g),(3.89) where we have used the Sweedler notation for the coproduct. Consequently, the twisting of the coproduct is tied to the deformation µ→ µθ of the product when the last one is defined by (3.90) f ⋆ g := (F−1 (1) ⊲ f)(F−1 (2) ⊲ g). A more extensive discussion of the application of some of these algebras to the construction of a deformed differential geometry for gravity theories may be found also in [11] as well as other works cited therein. If we now assume that the coefficients of the vector fields δξ are linear in the spacetime variables, then the generators δρ in (3.87) become the infinitesimal generators of the Poincaré transformations, and the coproduct defined in this equation reduces to the twisted coproduct considered by e.g. [12]. We would like to stress, however, that while all the above mentioned papers, as well as a large number of others appearing in the literature, start from equating spacetime noncommutativity with the noncommutativity of the parameters of the functions denoting classical fields, and deforming the algebra of these fields via the Moyal ⋆-product (with this anzatz originating in a basically heuristic fashion from some results in string theory), none of the algebras V̂⋆,H⋆ and Aθ in our approach are assumed a priori. On the contrary, they appear naturally, as does the spacetime noncommutativity, as a consequence of implementing Dirac’s canonical quantization formalism for constrained systems with an arbitrary symplectic structure. Note, in particular, that in our formalism the space-time variables are dynamical, as would be expected when viewing quantum mechanics as a minisuperspace of field theory, and their noncommutativity results from the quantization of their Dirac-brackets. The deformation of the module-algebra A - in which the fields originally lived - to Aθ ∋ φ, so that by (3.71) and (3.72) functions of the field multiply according to µθ is, in our formalism, again a consequence of the spacetime noncommutativity resulting from the quantization of the Dirac-brackets, and the concomitant deformation of the constraints associated with the symmetries of the field Lagrangian. Finally, it should be obvious by mere observation of the notation already introduced, how our al- gorithm can be readily extended to higher dimensional noncommutative space-times with constant parameters of noncommutativity. Thus, the commutator relations for the spacetime coordinate fields at equal times will now be given by (3.91) [xµ(τ,σ), xν(τ,σ′)] = iθµνδD(σ − σ′), where θµν = const. As in the bi-dimensional case, we can also introduce a new set of commuting coordinate fields defined by (3.92) x̃µ(σ) = xµ(σ) + θµν 2 pν(σ), from which new constraints can be constructed having the form H̃⊥ = 1 2 ( P 2 φ + γ̃γ̃ij∂σiφ∂σjφ ) + √ −γ̃ ñνpν − γ̃V (φ) ≈ 0, H̃i = Pφ∂σiφ+ pµ∂σi x̃µ. (3.93) 15 Making use the algebra morphism discussed at the beginning of this section we then arrive at the quantum algebra [ H̃⋆ ⊥(τ,σ), H̃⋆ ⊥(τ,σ′) ] ⋆θ = D ∑ i=1 ( H̃⋆ i (τ,σ) + H̃⋆ i (τ,σ ′) ) ∂σiδ(σ − σ′), [ H̃⋆ i (τ,σ), H̃⋆ j (τ,σ′) ] ⋆θ = ( H̃⋆ i (τ,σ)∂σj δ(σ − σ′) + H̃⋆ j (τ,σ ′)∂σiδ(σ − σ′) ) ,(3.94) [ H̃⋆ ⊥(τ,σ), H̃⋆ i (τ,σ′) ] ⋆θ = ( H̃⋆ ⊥(τ,σ) + H̃⋆ ⊥(τ,σ′) ) ∂σiδ(σ − σ′). With the constraints (3.93) it is possible to construct a quantum theory in the Schrödinger represen- tation analogous to (2.54) and (2.55). As in that case, however, since these constraints are no longer linear and algebraic in the momenta (they contain mixed products of the pµ’s and their derivatives), it is not possible to solve explicitly for the spacial momenta in order to construct a Schrödinger type equation. Nonetheless, it is still possible to show that the action in the reduced configuration space is in agreement with the usually proposed noncommutative field theory for a scalar field. As for the generalization to (D+1)-Minkowski spacetime of the symmetries and twisted symmetries elaborated above for the D = 1 case, the results follow through directly by replacing ξ0, ξ1 by ξ0 = − 1 g00 √−g ρ 0,(3.95) ξi = g0i g00 ρ0 − ρi.(3.96) These expressions can be inferred immediately from (3.77). 4. Concluding remarks We have shown in this paper how, by considering a parametrized field theory, it is possible to introduce spacetime noncommutativity from first principles. We have accomplished this by resorting to an extended phase-space, leading to second class constraints which, in order to remove them according to the Dirac quantization procedure, lead in turn to Dirac-brackets. The latter then result in a deformed symplectic structure for the spacetime coordinates and corresponding canonical momenta, which yield the desired noncommutativity. An important characteristic of our formulation is the automatic deformation of the symmetry gener- ators when the symplectic structure is deformed. Such a deformation being imposed by the consistency conditions on the constraints (see discussion in subsection 2.2), which have as a result that the algebra of the deformed constraints is maintained in the noncommutative case. This provides us then with a straightforward algorithm for constructing the Drinfeld twist of the Hopf algebras that one can associate with the reparametrization symmetry groups. In addition, our formalism can be readily extended to spacetimes of any dimensions and to the consideration of different possible types of deformed products, of which the Moyal product is just a particular case. Thus the formalism here described may turn out to be also useful for achieving a better understanding of twisted symmetries in Yang-Mills field theories, since in this case, in addition to the constraints associated with the reparametrization, we will also have the constraints associated with invariance under the gauge transformations Aµ(x)→ U(x)Aµ(x)U−1(x) + iU(x)∂µU −1(x), so the full set must then be analyzed in order to see how it is to be twisted when noncommutativity is introduced. 16 REFERENCES [1] R.J. Szabo, Physics Reports 378 207-299, (2003). [2] N. Seiberg and E. Witten, JHEP 09 (1999) 032. [3] M. Rosenbaum, J.D. Vergara and L.R. Juárez, Phys. Lett. A, 354, 389 (2006). [4] A. Pinzul and A. Stern, Phys. Lett. B 593, 279 (2004). [5] R. Banerjee, B. Chakraborty and S. Gangopadhyay, J. Phys. A 38, 957 (2005). [6] S. Ghosh and P. Pal, Phys.Lett. B 618, 243 (2005). [7] M. Rosenbaum, J.D. Vergara and L.R. Juárez, [arXiv:hep-th/0610150]. [8] K. Kuchar, “Canonical Quantization of Gravity”, in “Relativity, Astrophysics and Cosmology.” W. Israel (Ed.), Reidel Pub. Co., Dordrecht, Holland (1973). [9] S. R. Das and M. A. Rubin, Prog. Theor. Phys. Suppl. 86, 143 (1986). [10] P.A.M. Dirac, “Lectures on Quantum Mechanics”, Belfer Graduate School of Science Monographs Series No.2 (Yeshiva University, New York, 1964); A. J. Hanson, T. Regge and C. Teitelboim, “Con- strained Hamiltonian Systems” (Accademia Nazionale dei Lincei, Roma 1976); K. Sundermeyer, Lect. Notes Phys. 169, 1 (1982). [11] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, Class. Quant. Grav. 22, 3511 (2005). [12] M. Chaichian, P.P. P. Presnajder and A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005). M. Chaichian, P.P. Kulish, K. Nishijima and A. Tureanu, Phys. Lett. B 604, 98-102 (2004). J. Wess, [arXiv:hep-th/0408080]. P. Kosinski and P. Maslanka, [arXiv:hep-th/0408100] Ch. Blohmann, “Paris 2002, Physical and Mathematical aspects of symmetries” [arXiv:math.QA/0402199]. R. Oeckl, Nucl. Phys. B 581, 559 (2000). J. Lukierski and M. Woronowicz, Phys. Lett. B 633, 116 (2006). ar X iv :0 7 0 9 .3 4 9 9 v 1 [h ep -t h ] 2 1 S ep 2 0 0 7 ICN 2007-002 Noncommutative Field Theory from Quantum Mechanical Space-Space Noncommutativity Marcos Rosenbauma, J. David Vergarab and L. Roman Juarezc Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543 , México D.F., México a)mrosen@nucleares.unam.mx, b)vergara@nucleares.unam.mx c)roman.juarez@nucleares.unam.mx ABSTRACT We investigate the incorporation of space noncommutativity into field theory by extending to the spec- tral continuum the minisuperspace action of the quantum mechanical harmonic oscillator propagator with an enlarged Heisenberg algebra. In addition to the usual ⋆-product deformation of the algebra of field functions, we show that the parameter of noncommutativity can occur in noncommutative field theory even in the case of free fields without self-interacting potentials. 1 2 1. Introduction Particle Quantum Mechanics can be viewed in the free field or weak coupling limit as a minisuperspace sector of quantum field theory where most of the degrees of freedom have been frozen. It is thus a very convenient arena for further investigating the implications of the quantum mechanical spacetime noncommutativity in the formulation of field theories, as well as for evaluating the justification of some statements that are considered as generally accepted wisdom among practitioners of noncommutative field theory. Cf e.g. [1] -[14] and references therein for related works, albeit in a somewhat different spirit from the problem considered here, on noncommutative quantum mechanics. Further, in [15] noncommutativity was considered within the context of the Weyl-Wigner-Gröenewold-Moyal (WWGM) formalism for extended Heisenberg algebras, and its relation to the Bopp shift map (or what some authors refer to as the quantum mechanical equivalent of the Seiberg-Witten map) for expressing the algebra of extended Heisenberg operators in terms of their commutative counterparts was discussed and results were compared for problems previously studied in some of the above cited works. Moreover, the canonical, noncanonical and the possible quantum mechanical nonunitarity nature of some of these maps was additionally analyzed in [16, 17]. The results found there are conceptually relevant to our approach here, since as shown in several of the examples considered, transforming a problem in NCQM into a commutative one does not always lead to two unitarily equivalent quantum mechanical formulations. A case in point, arises when we compare some of our results and their physical implications with those obtained by the procedure followed in Ref [18]. The comparison is quite pertinent since both approaches are analogous in that they both have a quantum mechanical minisuperspace as a starting point for a construction of a field theoretical model. Indeed, the original quantum Hamiltonian in [18], modulo some irrelevant normalizations, is the same as the one considered here. On the other hand, the extended original Heisenberg algebra used in that work (Eq(2.6)) is different from ours because the authors there require to introduce (for their latter arguments) also noncommutativity of the momenta operators. Making then a linear transformation (actually a Bopp shift map) to a new set of quantum variables which satisfies the usual Heisenberg algebra results in their new Hamiltonian (2.9). The remainder of the construction in [18] follows from the above. Note, however, that the two decoupled quantum oscillators obtained in that work are not the same as ours (to see this it suffices to set B̂ = 0, in their equations (2.10) and compare them with our equation (3.60)). Thus the quantum mechanical problems implied by Eqs. (2.5) and (2.9) in [18] are not unitarily equivalent. In fact, the quantum mechanical problem that is actually considered there is that of a two dimensional anharmonic oscillator with a particular choice of frequencies containing some constant terms labeled with the symbols θ̂ and B̂, which can not truly be identified with the noncommutativity of any of the observables generated by the Heisenberg algebra (2.8) characterizing the quantum problem that at the end of the day is involved in that work. Moreover, the field constructed in [18] is a complex scalar field, which is not so in our case, and the Feynman propagator derived there and given in Eq(3.5) is quite different from ours (cf Eq(4.66)). The most important difference being that (3.5) in Ref [18] satisfies a highly non-local differential equation which violates both ordinary as well as twisted Poincaré invariance, while the symmetries of the Feynman propagator we derive in this work are in agreement with recent results on twisted NCQFT. Based on the above remarks and recalling that observables in quantum mechanics are represented by Hermitian operators acting on a Hilbert space, noncommutativity of the dynamical variables of a quantum mechanical system can be readily understood as the noncommutativity of their corresponding operators. In this way the physical argument that measurements below distances of the order of the 3 Planck length loose operational significance [19], can be mathematically described by extending the usual Heisenberg algebra of ordinary quantum mechanics to one including the noncommutativity of the operators related to the spacetime dynamical variables. Consistently in this paper we shall therefore use a quantum basis which is fully compatible with the noncommutativity of the coordinates. In particular in order to formulate space noncommutativity in Quantum Mechanics we use the extended Heisenberg algebra with generators satisfying the commutation relations [ Q̂i, Q̂j ] = iθij , [ Q̂i, P̂j ] = i~δij,(1.1) [ P̂i, P̂j ] = 0, (these could of course be generalized even more by also postulating noncommutativity of the momenta). The parameters θij of noncommutativity in (1.1) have dimensions of (length)2 and can, in general, be themselves arbitrary antisymmetric functions of the spacetime operators. However, most of the work so far appearing in the literature assumes for simplicity that these parameters are constant, and so shall we in what follows. The observables formed from the generators in (1.1) act on a Hilbert space which is assumed to be the same as the one for ordinary quantum mechanics, for any of the admissible realizations of the extended noncommutative Heisenberg algebra. Furthermore, utilizing the WWGM formalism for a quantum mechanical system, with observables obeying the above extended Heisenberg algebra, we showed in a previous paper [17] that the Weyl equivalent to a Heisenberg operator Ω(P̂, Q̂, t) satisfies the differential equation (1.2) ∂ΩW (p,q, t) ∂t = −2 ~ HW sin   1 2 (~Λ + ∑ i6=j θijΛ ′ ij)  ΩW (p,q, t), where Λ = ←−∇q · −→∇p − ←−∇p · −→∇q, Λ′ij = ←− ∂ qi −→ ∂ qj ,(1.3) and HW is the Weyl equivalent to the quantum Hamiltonian. Making use of (1.2) we further showed in [17] that in the WWGM formalism of quantum mechanics, the labeling variables q,p, can be interpreted as canonical classical dynamical variables provided their algebra A is modified with a multiplication given by the star-product: (1.4) qi ⋆θ qj := qi ( e 1 2 P k,l θkl ←− ∂ qk −→ ∂ ql ) qj . Applying these results to the simple case of a two dimensional harmonic oscillator satisfying the algebra (1.1), which is taken as the unfrozen mode, or the one particle sector of a two-component vector (or composite system) field, and using spectral analysis in order to reconstruct the corresponding quantum field, we shall show how the parameter of the quantum mechanical noncommutativity appears in the theory even for the case of a free field. This novel result, which as we shall see is a quite natural consequence of our approach, and contrasts with the usually made assumption that the presence of noncommutativity in field theory is manifested only through the deformation of the multiplication in the algebra of the fields [20], [21], [22]. 4 2. The quantum mechanics of the harmonic oscillator in noncommutative space As discussed in [17], a configuration space basis for the quantum mechanics with an extended Heisen- berg algebra generated by (1.1) is not an admissible basis, since the position operators Q̂i, Q̂j, i 6= j, do not simultaneously form part of a complete set of commuting observables. For i, j = 1, 2, then the only admissible bases for such a case are either one of the 3 sets of kets {|q1, p2〉}, {|q2, p1〉} and {|p1, p2〉}, where the labels of the kets are the eigenvalues of the possible sets of commuting observables. Let us now consider the first of these bases and use the WWGM formalism and the results in [17] in order to evaluate the transition amplitude 〈q′′1 (t2), p ′′ 2 (t2)|q′1(t1), p′2(t1)〉, for a quantum 2-dimensional harmonic oscillator with Hamiltonian (2.5) Ĥ = 1 2m ( P̂ 2 1 + P̂ 2 2 ) + mω2 2 ( Q̂2 1 + Q̂2 2 ) . ¿From the results in Sec. 2 of the above cited paper, it can be seen that this transition amplitude is given by 〈q′′1 (t2), p ′′ 2 (t2)|q′1(t1), p′2(t1)〉 = 〈q′′1 (t1), p ′′ 2(t1)|e− i ~ Ĥ(t2−t1)|q′1(t1), p′2(t1)〉 = Tr[ρe− i ~ Ĥ(t2−t1)](2.6) = ∫ dp1dp2dq1dq2ρW e θ ~ p1∂q2 e − i ~ HW (t2−t1) ⋆ , where θ := θ12, (2.7) ρ := |q′1(t1), p′2(t1)〉〈q′′1 (t1), p ′′ 2(t1)|, ρW is its corresponding Weyl function: (2.8) ρW = (2π~)−2 ∫ dξdηe− i ~ (ηq2−ξp1)〈q1 − ξ 2 , p2 − η 2 |ρ|q1 + ξ 2 , p2 + η 2 〉, andHW := 1 2m ( p2 1 + p2 2 ) + mω2 2 ( q21 + q22 ) is the Weyl function associated with the quantum Hamiltonian (2.5). Substituting (2.7) into (2.8) gives (2.9) ρW = 4 (2π~)2 δ(q′′1 + q′1 − 2q1)δ(p ′′ 2 + p′2 − 2p2) exp [ − i ~ (2p2 − 2p′2)q2 + i ~ (2q1 − 2q′1)p1 ] , which, when inserted in its turn into (2.6), yields 〈q′′1 (t2), p ′′ 2 (t2)|q′1(t1), p′2(t1)〉 = ∫ dp1dq2e − i ~ (p′′2−p′2)q2 exp{ i ~ [(q′′1 − q′1) + θ ~ (p′′2 − p′2)]p1} × (e − i ~ HW (t2−t1) ⋆ )(p1, p′2 + p′′2 2 , q′1 + q′′1 2 , q2).(2.10) Note now that for an infinitesimal transition with t1 = t, t2 = t+ δt and q′′1 − q′1 = q̇′′1 δt, p ′′ 2 − p′2 = ṗ′′2δt, (2.10) reads (2.11) 〈q′′1 (t+ δt), p′′2(t+ δt)|q′1(t), p′2(t)〉 = e i ~ [q̇′1p1−ṗ′2q2+ θ ~ ṗ′2p1]δte− i ~ Hcl(p1,p′2,q′1,q2)δt, where Hcl (= HW for the case here considered) is the classical Hamiltonian resulting from making the replacements Q̂→ q and P̂ → p in the original quantum Hamiltonian (2.5). Following Feynman’s path integral formalism, the transition over a finite time interval is then given by (2.12) 〈q′′1 (t2), p ′′ 2 (t2)|q′1(t1), p′2(t1)〉 ∼ ∫ Dq1Dp2Dp1Dq2 exp{ i ~ ∫ t2 t1 [q̇1p1 − ṗ2q2 + θ ~ ṗ2p1 −Hcl]dt}. 5 This result (for an alternate derivation see [23] and related work in [24]-[27]) provides an univocal procedure for obtaining the Feynman propagator in spacetime noncommutative quantum mechanics as well as the expression for the deformed classical action, which in our particular case is given by (2.13) S(q1, p2, q2, p1, t) = ∫ t2 t1 [q̇1p1 − ṗ2q2 + θ ~ ṗ2p1 −Hcl]dt. Let us next re-write the action (2.13) in the form (2.14) S = ∫ dt [ p1q̇1 − ṗ2(q2 − θ ~ p1)− p1 2 2m − p2 2 2m − mω2 2 q1 2 − mω2 2 q2 2 ] , which, when setting (2.15) q̃2 = q2 − θ ~ p1, results in (2.16) S = ∫ dt [ p1q̇1 − ṗ2q̃2 − p1 2 2m − p2 2 2m − mω2 2 q1 2 − mω2 2 (q̃2 + θ ~ p1) 2 ] . Fixing now p1 and q̃2 at the end points and varying with respect to these variables we get q̇1 = p1 m + m2ω2θ ~ (q̃2 + θ ~ p1),(2.17) ṗ2 = −mω2(q̃2 + θ ~ p1).(2.18) ¿From the above we derive p1 = mq̇1 + mθ ~ ṗ2(2.19) q̃2 = − 1 mω2 [ṗ2 + m2ω2θ ~ (q̇1 + θ ~ ṗ2)].(2.20) Substituting these last expressions into (2.16) shows that (2.12) may be reduced to (2.21) 〈q′′1 (t2), p ′′ 2 (t2)|q′1(t1), p′2(t1)〉 ∼ ∫ ∫ Dq1Dp2e i ~ S(q1,p2,t), with (2.22) S(q1, p2, t) = ∫ dt [ m 2 q̇21 + mθ ~ ṗ2q̇1 + ( 1 2mω2 + mθ2 2~2 ) ṗ2 2 − p2 2 2m − mω2 2 q21 ] . Note that by varying (2.22), it follows that the canonical dynamical variables q1 and p2 obey the set of second order coupled ordinary differential equations (2.23) ( q̈1 p̈2 ) = − ( m2ω4θ2 ~2 + ω2 −ω2θ ~ −m2ω4θ ~ ω2 )( q1 p2 ) , which, when diagonalized, decouple into two harmonic oscillators with frequencies given by (2.24) ω1,2 = ω [ 1 + m2ω2θ2 2~2 ± mωθ 2~ √ 4 + m2ω2θ2 ~2 ] 1 2 . Hence, the energy eigenvalues of (2.5) are (2.25) E = ~ω1(n1 + 1 2 ) + ~ω2(n2 + 1 2 ). It is pertinent to emphasize here that the change of variables at the classical level involved in Eq. (2.15) does not correspond to a Bopp shift, so it also does not follow that making such a change of 6 variables in the action (2.14) implies that we are passing from NCQM to ordinary quantum mechanics. 2.1. Hamiltonian formulation. Consider now the Lagrangian L in the action (2.22) and make the identifications (2.26) z1 := q1, z2 := p2 mω , so that both z1 and z2 have dimension of length. Furthermore, introducing the dimensionless quantity θ̃ : (2.27) θ̃ = mωθ ~ , with m,ω, being some characteristic mass and frequency, respectively, to be further specified below, we can then write (2.28) L = 1 2 [ ż2 1 − ω2z2 1 + ż2 2 − ω2z2 2 + 2θ̃ż1ż2 + θ̃2ż2 2 ] . The momenta canonical to the zi’s are π1 = ż1 + θ̃ż2, π2 = ż2 + θ̃ż1 + θ̃2ż2.(2.29) Inverting (2.29) we have (2.30) ( ż1 ż2 ) = ( 1 + θ̃2 −θ̃ −θ̃ 1 )( π1 π2 ) , from where it follows that (2.31) H = π1ż1 + π2ż2 − L = 1 2 [ (1 + θ̃2)π2 1 + π2 2 − 2θ̃π1π2 + ω2z2 1 + ω2z2 2 ] . Making use of the theory of quadrics we can diagonalize (2.31) by first solving for the eigenvalues λ1,2 of the characteristic determinant of the matrix ( 1 2 (1 + θ̃2) − θ̃ 2 − θ̃ 2 1 2 ) . We thus get (2.32) λ1,2 = 1 2 ( 1 + θ̃2 2 ± θ̃ 2 √ 4 + θ̃2 ) . Hence H = (π1, π2) ( M̃ ) (M) ( 1 2 (1 + θ̃2) − θ̃ 2 − θ̃ 2 1 2 ) ( M̃ ) (M) ( π1 π2 ) + ω2 2 ( z2 1 + z2 2 ) = (π′1, π ′ 2) ( λ1 0 0 λ2 )( π′1 π′2 ) + ω2 2 ( z′21 + z′22 ) ,(2.33) where (2.34) ( π′1 π′2 ) = (M) ( π1 π2 ) , ( z′1 z′2 ) = (M) ( z1 z2 ) , 7 and the entries of the symmetric matrix (M) = ( m11 m12 m21 m22 ) are given by (2.35) m11 = − 1 √ 1 + ω2 2 ω2 , m12 = − ω2 2 ω2 − 1 θ̃ √ 1 + ω2 2 ω2 , (2.36) m21 = 1 √ 1 + ω2 1 ω2 , m22 = ω2 1 ω2 − 1 θ̃ √ 1 + ω2 1 ω2 where, by using (2.24) and (2.27), one can readily verify that m12 = m21 as required. If we finally let (2.37) z′i = (λi) 1 2xi, π′i = (λi) − 1 2πxi , i = 1, 2, we arrive at (2.38) H = π2 x1 + π2 x2 + 1 4 ( ω1 2x2 1 + ω2 2x 2 2 ) . It should be clear from the above calculations that the transformed variables xi, πxi remain canoni- cally conjugate to each other. Thus it follows from the Hamilton equations that (2.39) πxi = 1 2 ẋi, so the Lagrangian (2.28) now reads (2.40) L = 1 4 ( ẋ2 1 + ẋ2 2 − ω2 1x 2 1 − ω2 2x 2 2 ) . Variation of this expression with respect to xi yields (2.41) ẍi + ω2 i xi = 0, which are indeed the equations of motion for two decoupled harmonic oscillators with respective fre- quencies ωi, as asserted previously. Furthermore, it can be readily verified that the point transformations (2.42) ( πx1 πx2 ) = ( m11 √ λ1 m12 √ λ1 m21 √ λ2 m22 √ λ2 )( π1 π2 ) , and (2.43) ( x1 x2 ) = ( m11√ λ1 m12√ λ2 m21√ λ1 m22√ λ2 )( z1 z2 ) , are canonical, with generating function (2.44) F2(z1, z2, πx1 , πx2) = ∑ i,j mij√ λi zjπxi . We also have that when substituting (2.43) into the Lagrangian (2.40) we recover (2.28) and that the Jacobian of each the transformations (2.42) and (2.43) is equal to 1 2 , so that Dx1Dx2 = 1 2 Dz1Dz2. Consequently, the quantum mechanics derived from the path integral with the action (2.22) is uni- tarily equivalent to the path integral formulation based on the action resulting from the diagonalized Lagrangian (2.40). 8 3. Field theoretical model Paralleling standard quantum field theory we next construct a noncommutative field theory over a (1 + 2)-Minkowski space by taking an infinite superposition of the quantum mechanical harmonic os- cillator minisuperspaces described by (2.40). Each of these oscillators consists of the pair x1(k), x2(k), labeled by the continuous parameter k and satisfying (2.41). Thus in our construction, the quantum mechanical spacial noncommutativity will reflect itself both in the deformation parameter dependence of the different frequencies of the pairs of oscillators, as well as in the twisting of the product of the algebra of the resulting fields. Consequently this simple model shows that spacetime noncommutativity can be present in field theory even in the absence of self-interaction potentials. Let us consider a field system Φi(q, t), i = 1, 2, over a (1 + 2)-Minkowski space-time, satisfying the uncoupled Klein-Gordon field equations (3.45) ( 2 + µ2 1 0 0 2 + µ2 2 )( Φ1(q1, q2, t) Φ2(q1, q2, t) ) = 0, where (3.46) Φi(q, t) = (2π)−1 ∫ dk xi(k, t) e ik.q ⋆θ , and (3.47) eik.q ⋆θ := 1 + ik.q + 1 2 (ik.q) ⋆θ (ik.q) + . . . . Note that in the above definition of the field system in terms of its Fourier transform we have used the star-exponential for describing plane waves. Our rationale for this is based on the observation made in ([15]) where, by making use of the WWGM formalism and elements of quantum group theory, we show that quantum noncommutativity of coordinate operators in the extended Heisenberg algebra leads to a deformed product of the classical dynamical variables that is inherited to the level of quantum field theory. This deformed product is the so called Moyal star-product defined in (1.4). Thus, expressing the fields as in (3.46) guarantees explicitly that they are elements of the deformed algebra Aθ with the ⋆-multiplication. Note also that in (3.45) the D’Alembertian is given by (3.48)  2 = ∂2 t − ∂̄†i ∂̄i, with the anti-hermitian derivation ∂̄i defined by [28]: (3.49) ∂̄i = θ−1 ij adqj , and ∂̄†i = −∂̄i, i = 1, 2, and where the adjoint action is realized by the twisted product commutator (3.50) [qi, qj ]⋆θ := qi ⋆θ qj − qj ⋆θ qi. Thus, the algebra (1.4) has been incorporated into (3.45) through the defining Fourier transformation equation (3.46) for the fields since these, as functions of the qi’s, they inherit the ⋆-multiplication and are therefore also elements of the twisted algebra Aθ. Now, by making use of the Baker-Campbell-Hausdorff theorem, together with the commutator (3.50) as well as of the identity [q2, q n 1 ]⋆θ = −inθq(n−1) 1 , we have that ∂̄1(e ik.q ⋆θ ) = θ−1[q2, e ik1q1eik2q2e i 2 k1k2θ]⋆θ = k1e ik.q ⋆θ ,(3.51) 9 and (recalling that ∂̄†i = −∂̄i) (3.52) ∂̄†1∂̄1(e ik.q ⋆θ ) = −k2 1e ik.q ⋆θ . Similarly (3.53) ∂̄†2∂̄2(e ik.q ⋆θ ) = −k2 2e ik.q ⋆θ . We therefore find that the field equations (3.45) read (3.54) (2 + µ2 i )Φi(q, t) = (2π)−1 ∫ dk[ẍi + (k2 + µ2 i )xi]e ik.q ⋆θ = 0, i = 1, 2. Using next the orthonormality (3.55) (2π)−2 ∫ ∫ dq1dq2e ik.q ⋆θ e−ik′.q ⋆θ = δ(k− k′), and the dispersion relation (3.56) k2 + µ2 i = k2 0 = ω2 i (k), we obtain from the right hand of (3.54): (3.57) ẍi(k, t) + ω2 i (k)xi(k, t) = 0. Observe that ωi(k), i = 1, 2, in (3.56) is given by (2.24) with ω → ω(k) and θ → θ(k) being now respectively the wave vector dependent frequency in (1.1) and the noncommutative parameter of the quantum mechanical system for each k in the spectral decomposition (3.46). Comparing (3.57) with (2.41), and observing that according to our definition (2.27) we now have θ(k) = ~θ̃ mω(k) , we choose θ(k) such that θ̃ remains a pure number independent of k. We then have that the Lagrangian (2.40) for the pair of decoupled harmonic oscillators xi(k, t) can be seen, for a fixed value of the continuum parameter k, as a minisuperspace of the full field theory characterized by the action: S = ∫ dtdq1dq2 L = 1 2 ∫ dtdq1dq2 [ Φ̇†1 ⋆θ Φ̇1 − (∂̄iΦ1) † ⋆θ ∂̄iΦ1 − µ2 1Φ † 1 ⋆θ Φ1 +Φ̇†2 ⋆θ Φ̇2 − (∂̄iΦ2) † ⋆θ ∂̄iΦ2 − µ2 2Φ † 2 ⋆θ Φ2 + 1 2 (Φ†1 ⋆θ J1(q, t) +J†1 (q, t) ⋆θ Φ1) + 1 2 (Φ†2 ⋆θ J2(q, t) + J†2(q, t) ⋆θ Φ2) ] , (3.58) after adding two arbitrary external driving sources. Note that in the above expression we have formally included the ⋆-product for the algebra of the fields, even though in fact, in the absence of field interaction potentials, these could be ignored in view of the identity (3.59) ∫ dq1dq2f(q) ⋆θ g(q) = ∫ dq1dq2f(q)g(q), which follows directly by parts integration. However, also note that the noncommutativity parameter θ̃ will still be present in the frequencies ωi(k) even in such a case, since these now read (3.60) ω1,2(k) = ω(k) [ 1 + θ̃2 2 ± θ̃ 2 √ 4 + θ̃2 ] 1 2 . 10 4. Path integral and Feynman propagator In order to derive the Feynman propagator for our theory, we use (3.46) and a similar expression for the Fourier transform F̃i of the sources Ji together with (3.55), as well as the transformations xi((k, t) = (2π)− 1 2 ∫ dk0e ik0tx̃i(k, k0), Fi((k, t) = (2π)− 1 2 ∫ dk0e ik0tF̃i(k, k0).(4.61) We thus get S = 1 2 ∫ dk0dk  (k2 0 − k2 − µ2)   ∑ i=1,2 x̃i(k, k0)x̃i(k,−k0)   + x̃1(k, k0)F̃1(k,−k0) + x̃1(k,−k0)F̃1(k, k0) + x̃2(k, k0)F̃2(k,−k0) + x̃2(k,−k0)F̃2(k, k0) ] . (4.62) Following standard procedures (see e.g. [29]), we now make the change of variables x̃1(k, k0) = Z1(k, k0) + β(k0)F̃1(k, k0) + γ(k0)F2(k, k0), x̃2(k, k0) = Z2(k, k0) + λ(k0)F̃1(k, k0) + ν(k0)F2(k, k0).(4.63) Inserting (4.63) into (4.62) and requiring that terms linear in the Zi’s cancel, allows us to fix the parameters β, γ, λ, ν as: β(k0) = (k2 0 − k2 − µ2)−1, λ(k0) = γ(k0) = 0, ν(k0) = −(k2 0 − k2 − µ2) −1 .(4.64) If we next replace (4.64) into the action resulting from (4.62) by the above procedure, we derive the following contribution to the integrand in that action from the terms quadratic in the sources: 〈Z0[J ]〉 := −1 2 ∫ . . . ∫ dq dq′ dt dt′ ( J†1 (q, t) J†2 (q, t) ) × ( D1(q− q′, t− t′) 0 0 D2(q− q′, t− t′) )( J1(q ′, t′) J2(q ′, t′) ) , (4.65) where Di(q− q′, t− t′) are the Feynman propagators: (4.66) Di(q − q′, t− t′) = (2π)−3 ∫ . . . ∫ dk dk0 ( e−i[k0(t−t′)−k.(q−q′)] k2 0 − ω2 i (k) + iǫ ) , i = 1, 2, and the ω2 i (k) are given by (3.60). Note that these propagators satisfy the Klein-Gordon equations (4.67) (  2 + µ2 i ) Di(q− q′, t− t′) = −δ(q− q′)δ(t− t′). Observe also that (4.67) is invariant under the twisted Poincaré transformations discussed in [30], since the D’Alembertian, as defined in (3.48), is invariant under these transformations and the indices i = 1, 2, are not space-time indices. In consequence of the above, the vacuum to vacuum amplitude for our theory is thus given by (4.68) W [J ] = W [0]e i ~ 〈Z0[J]〉, 11 and the classical fields Φ (0) (cl)i ≡ −i δ ln W0 δJ † i (q,t) = δZ0 δJ † i (q,t) satisfy the driven Klein-Gordon field equations (4.69) (2 + µ2 i )Φ (0) (cl)i = 1 2 Ji. 5. Second Quantization Let us promote the xi in (3.46) to the rank of operators and, similarly to that equation, let us define field canonical momenta by (5.70) Π̂i = (2π)−1 ∫ dk π̂i(k, t) e −ik.q ⋆θ , with x̂i, π̂j , satisfying now the commutation relations [x̂i(k, t), π̂j(k ′, t)] = i~δijδ(k − k′), [x̂i(k, t), x̂j(k ′, t)] = 0,(5.71) [π̂i(k, t), π̂j(k ′, t)] = 0. Assuming further that Φ̂i and Π̂i are real, we have by Hermicity that (5.72) x̂†i (k, t) = x̂i(−k, t), π̂†i (k, t) = π̂i(−k, t), and we also take ωi(k) = ωi(−k). Next let x̂i(k, t) = √ ~ 2ωi(k) ( âi(k, t) + â†i (−k, t) ) , π̂i(k, t) = i √ ~ωi(k) 2 ( â†i (k, t)− âi(−k, t) ) .(5.73) It readily follows from (5.73) and (5.71) that [âi(k, t), â † j(k ′, t)] = δijδ(k− k′), (5.74) [âi(k, t), âj(k ′, t)] = 0, [â†i (k, t), â † j(k ′, t)] = 0. So â†i and âi, are the usual Fock creation and destruction operators. Note however that while particle- antiparticle degeneracy at the dispersion relation level is preserved for a given value of the label i = 1, 2, the energies of the particles-antiparticles created (destroyed) by â†i (âi) are different and are given by ~ωi. 6. Discussion and Conclusions Spacetime noncommutativity in field theory is understood in some circles as a merely convenient way to describe a special type of interaction. Such a description consisting in mathematically deforming the product in the algebra of field functions by means of the so-called Moyal star-product. However, as we tried to stress throughout the paper, referring to the formalism under such premises as spacetime noncommutativity is, at best, a misnomer since the arguments of the fields are parameters of the theory. Speaking about noncommutativity in this context then has little physical basis, beyond the rather loose analogy of the Moyal product with the Groenewold- Moyal product occurring in the WWGM phase- space formulation of quantum mechanics. One of our contentions here has been, however, that there is more physical substance to that designation if one recalls the operational nature of observables in 12 quantum mechanics from where noncommutativity of the dynamical variables of the system is readily understood then as the noncommutatitivity of their corresponding operators. Furthermore, based on the concept that quantum mechanics can be viewed as a minisuperspace sector of field theory, where only a few degrees of freedom are unfrozen, we have used the quantum mechanics of a harmonic oscillator over an extended Heisenberg algebra, to construct a field theoretical model which inherits the space-space noncommutativity of the quantum mechanical problem. An interesting feature of our construction is that it shows that the global symmetry of the original theory (2.5) is broken by the noncommutativity. This in turn implies that if at the level of field theory the index tagging the fields denotes a composite system of scalar fields (and not the components of a vector field), then the noncommutativity can be seen as giving rise to a field doublet (or more generally an n-tuplet) of slightly different masses where classical Lorentz symmetry for each member is broken, but each one satisfies a deformed Klein-Gordon equation which is invariant under a twisted Lorentz symmetry. On the other hand, if the labeling of the fields is taken as corresponding to that of a vector field of spacetime dimensions then, because of the mass differences, both classical and twisted Lorentz invariance are broken by the noncommutativity. This symmetry breaking and mass differences resulting from the presence of noncommutativity is in some way reminiscent of the spontaneous symmetry breaking mechanism that occurs in the Standard Model, but without the appearance of a Goldstone boson. In addition, by thinking of noncommutativity of spacetime as the quantum mechanical operator algebra expressing the loss of operational meaning for localization at distances of orders smaller than the Planck length, it then follows that minisuperspaces based on noncommutative spacetimes have to be at least of two dimensions, and the fields constructed from them must necessarily contain the presence of the parameter of noncommutativity even in the absence of self-interacting potentials. An alternate way to mathematically express the physical argument that measurements below dis- tances of the order of the Planck length loose operational significance, can be accomplished, both at the quantum mechanical and field theoretical level, by using parametrization invariance of the action and following the canonical quantization approach of embedding a spatial manifold Σ in the spacetime manifold. Such an approach, whereby the embedding variables acquire a dynamical interpretation, which, in turn, gives physical sense to their noncommutativity and is achieved by the inclusion of a general symplectic structure in the formalism, has been analyzed extensively by the authors elsewhere [31]. The deformed algebra of the constraints resulting from the parametrization and general symplec- tic structure of the theory is particularly convenient for analyzing the twisting of its symmetries and for indeed thinking of a true physical spacetime noncommutativity as underlying the merely axiomatic mathematical deformation of the algebra product describing a certain type of interactions in field theory. Finally, we note that although our construction has been restricted for simplicity to two spacial dimensions and to bosonic fields, it can be generalized to allow for higher dimensional spaces in a conceptually straightforward (albeit algebraically more complicated) way, and to the case of fermionic fields by including Grassmanian variables in the construction of the spectral oscillators. Acknowledgements The authors acknowledge partial support from CONACyT project UA7899-F (M. R.), DGAPA- UNAM grant IN104503-3 (J.D.V.) and SEP-CONACyT project 47211-F (J.D.V). 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Juárez, [arXiv:hep-th/0611160]. ar X iv :0 8 0 7 .2 5 0 8 v 1 [h ep -t h ] 1 6 J u l 2 0 0 8 Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 055, 21 pages Space-Time Diffeomorphisms in Noncommutative Gauge Theories⋆ Marcos ROSENBAUM, J. David VERGARA and L. Román JUAREZ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F., México E-mail: mrosen@nucleares.unam.mx, vergara@nucleares.unam.mx, roman.juarez@nucleares.unam.mx Received April 11, 2008, in final form June 25, 2008; Published online July 16, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/055/ Abstract. In previous work [Rosenbaum M. et al., J. Phys. A: Math. Theor. 40 (2007), 10367–10382] we have shown how for canonical parametrized field theories, where space- time is placed on the same footing as the other fields in the theory, the representation of space-time diffeomorphisms provides a very convenient scheme for analyzing the induced twisted deformation of these diffeomorphisms, as a result of the space-time noncommutati- vity. However, for gauge field theories (and of course also for canonical geometrodynamics) where the Poisson brackets of the constraints explicitely depend on the embedding variables, this Poisson algebra cannot be connected directly with a representation of the complete Lie algebra of space-time diffeomorphisms, because not all the field variables turn out to have a dynamical character [Isham C.J., Kuchař K.V., Ann. Physics 164 (1985), 288–315, 316– 333]. Nonetheless, such an homomorphic mapping can be recuperated by first modifying the original action and then adding additional constraints in the formalism in order to retrieve the original theory, as shown by Kuchař and Stone for the case of the parametrized Maxwell field in [Kuchař K.V., Stone S.L., Classical Quantum Gravity 4 (1987), 319–328]. Making use of a combination of all of these ideas, we are therefore able to apply our canonical reparametrization approach in order to derive the deformed Lie algebra of the noncommu- tative space-time diffeomorphisms as well as to consider how gauge transformations act on the twisted algebras of gauge and particle fields. Thus, hopefully, adding clarification on some outstanding issues in the literature concerning the symmetries for gauge theories in noncommutative space-times. Key words: noncommutativity; diffeomorphisms; gauge theories 2000 Mathematics Subject Classification: 70S10; 70S05; 81T75 1 Introduction Within the context of quantum field theory, a considerable amount of work has been done re- cently dealing with quantum field theories in noncommutative space-times (NCQFT). One of the most relevant issues in this area is related to the symmetries under which these noncom- mutative systems are invariant. The most recent contention being that NCQFT are invariant under global “twisted symmetries” (see, e.g., [5]). This criterion has been extended to the case of the twisting of local symmetries, such as diffeomorphisms [6], and this has been used to propose some noncommutative theories of gravity [6, 7, 8]. Another possible extension of this idea is to consider the construction of noncommutative gauges theories with an arbitrary gauge group [9, 10]. Regarding this latter line of research there is, however, some level of contro- versy as to whether it is possible to construct twisted gauge symmetries [11, 12, 13]. In this ⋆This paper is a contribution to the Special Issue on Deformation Quantization. The full collection is available at http://www.emis.de/journals/SIGMA/Deformation Quantization.html 2 M. Rosenbaum, J.D. Vergara and L.R. Juarez work we address this issue from the point of view of canonically reparametrized field theories. It is known indeed that for the case of field theories with no internal symmetries, it is pos- sible to establish, within the framework of the canonical parametrization, an anti-homorphism between the Poisson algebra of the constraints on the phase space of the system and the al- gebra of space-time diffeomorphisms [2, 3]. Using this anti-homomorphism we were able in [1] to show how the deformations of the algebra of constraints, resulting from space-time non- commutativity at the level of the quantum mechanical mini-superspace, are reflected on the twisting of the algebra of the fields as well as in the Lie algebra of the twisted diffeomorphisms and in the ensuing twisting of the original symmetry group of the theory. However, as it has also been noted by Isham and Kuchař in [2, 3], for the case of gauge theories, there are some difficulties in representing space-time diffeomorphisms by an anti-homomorphic mapping into the Poisson algebra of the dynamical variables on the extended phase space of the canonically reparametrized theory, due to the fact that because of the additional internal symmetries some components of the field loose their dynamical character and appear as Lagrange multipliers in the formalism. Nonetheless, as it was exemplified in [4] for the case of the parametrized Maxwell field, such difficulties can be circumvented and the desired mapping made possible by adding some terms to the original action and some additional constraints in order to recover the original features of the theory. Making therefore use of the specific results derived by Kuchař and Stone in [4] for the parametrized Maxwell field and the re-established mapping between the space-time diffeomor- phisms and the Poisson algebra of the modified theory, together with our previous results in [14] – whereby noncommutativity in field theory, manifested as the twisting of the alge- bra of fields, has a dynamical origin in the quantum mechanical mini-superspace which, for flat Minkowski space-time, is related to an extended Weyl–Heisenberg group – and including this results into a generalized symplectic structure of the parametrized field theory [1], we show here how our approach can be extended to gauge field theories thus allowing us to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms, as well as to con- sider how the gauge transformations act on the twisted algebras of gauge and particle fields. Hopefully this approach will help shed some additional univocal light on the above mentioned controversy. The paper is organized as follows: In Section 2 we review the essential aspects of the construc- tion of canonical parametrized field theories and representations of space-time diffeomorphisms, following [2, 3, 15, 16]. In Section 3 we show how the formalism can be extended to the case of parametrized gauge field theories by making use of the ideas formulated in [4] in the context of Maxwell’s electrodynamics. Section 4 summarizes in the language of Principal Fiber Bundles (PFB) some of the basic aspects of the theory of gauge transformations which will be needed in the later part of the work. In Section 5 we combine the results of the previous sections in order to extend the formalism to the noncommutative space-time case, by deforming the symplec- tic structure of the theory to account for the noncommutativity of the space-time embedding coordinates. We thus derive a deformed algebra of constraints in terms of Dirac-brackets which functionally satisfy the same Dirac relations as those for the commutative case and can therefore be related anti-homomorphically to a Lie algebra of generators of twisted space-time diffeomorphisms. On the basis of these results we further show how, in order to preserve the consistency of the algebra of constraints, the Lie algebra of these generators of space-time diffeomorphisms and those of the gauge symmetry are in turn related. Finally by extending the algebra of twisted diffeomorphisms to its universal covering, it was given an additional Hopf structure which allowed us to relate the twisting of symmetry of the theory to the Drinfeld twist. Space-Time Diffeomorphisms in Noncommutative Gauge Theories 3 2 Space-time diffeomorphisms in parametrized gauge theories As it is well known, see e.g. [2, 3], for Poincaré invariant field theory on a flat Minkowskian background, each generator of the Poincaré Lie algebra, represented by a dynamical variable on the phase-space of the field, is mapped homomorphically into the Poisson bracket algebra of these dynamical variables. On a curved space-time background field theories are not Poincaré invariant but, by a para- metrization consisting of extending the phase-space by adjoining to it the embedding variables, they can be made invariant under arbitrary space-time diffeomorphisms [17, 18]. Hence space- time parameters are raised to the level of fields on the same footing as the original fields in the theory. Moreover, in this case it can also be shown [2] that: a) An anti-homomorphic mapping can be established from the Poisson algebra of dynamical variables on the extended phase-space and the Lie algebra £diffM of arbitrary space-time diffeomorphisms. Thus, {Hτ [ξ],Hτ [η]} = −Hτ [£ξη], where ξ, η ∈ £ diffM are two complete space-time Hamiltonian vector fields on M, Hτ [ξ] := ∫ Σ dσ ξαHα, and Hα are the constraints (supermomenta and superHamiltonian) of the theory, satisfying the Dirac vanishing Poisson bracket algebra {Hα(σ),Hβ(σ′)} ≃ 0. (2.1) b) The Poisson brackets of the canonical variables representing the £diffM correctly induce the displacements of embeddings accompanied by the evolution of the field variables, predicted by the field equations. For the prescribed pseudo-Riemannian backgroundM, equipped with coordinates Xα, repa- rametrization involves a foliation Σ×R of this space-time, where R is a temporal direction labeled by a parameter τ and Σ is a space-like hypersurface of constant τ , equipped with coordinates σa (a = 1, 2, 3), and embedded in the space-time 4-manifold by means of the mapping Xα = Xα(σa). This hypersurface is assumed to be spacelike with respect to the metric gαβ onM, with signature (−,+,+,+). Let now the embedding functionals Xα a(σ,X) := ∂Xα(σ) ∂σa and nα(σ,X), defined by gαβX α an β = 0, and gαβn αnβ = −1, (2.2) be an anholonomic basis consisting of tangent vectors to the hypersurface and unit normal, respectively. We can therefore write the constraints Hα as Hα = −H⊥nα +HaXα a, where H⊥ and Ha are the super-Hamiltonian and super-momenta constraints, respectively. Using this decomposition the Dirac relations (2.1) can be written equivalently as {H⊥(σ),H⊥(σ′)} = 3 ∑ a=1 γabHb(σ)∂σaδ(σ − σ′)− (σ ↔ σ′), {Ha(σ),Hb(σ ′)} = Hb(σ)∂σaδ(σ − σ′) +Ha(σ ′)∂σbδ(σ − σ′), {Ha(σ),H⊥(σ′)} = H⊥(σ)∂σaδ(σ − σ′), 4 M. Rosenbaum, J.D. Vergara and L.R. Juarez where γab is the inverse of the spatial metric γab(σ,X) := gαβ(X(σ))Xα aX β b. Also, as a consequence of the antihomomorphism between the Poisson algebra of the con- straints and £ diffM we can write Hτ [ξ] ❀ Ĥτ [ξ] ≡ δξ = ξα(X(τ,σ)) ∂ ∂Xα ∣ ∣ ∣ ∣ X(τ,σ) . Indeed, since [η, ρ] = £ηρ we have [δη , δρ]φ = δ£ηρφ = Ĥτ [£ηρ] ⊲ φ ∼= {φ,Hτ [£ηρ]} = Ĥτ [η] ⊲ [Ĥτ [ρ] ⊲ φ]− Ĥτ [ρ] ⊲ [Ĥτ [η] ⊲ φ] ∼= {{φ,Hτ [ρ]},Hτ [η]} − {{φ,Hτ [η]},Hτ [ρ]} = −{φ, {Hτ [η],Hτ [ρ]}} after resorting to the Jacobi identity and where φ is some field function in the theory. Making use of this antihomomorphism as well as of the dynamical origin of ⋆-noncommutati- vity in field theory from quantum mechanics exhibited in [14], we have considered in [1] the extension of the reparametrization formalism and the canonical representation of space-time diffeomorphisms to the study of field theories on noncommutative space-times. More specifically, in that paper we discussed the particular case of a Poincaré invariant scalar field immersed on a flat Minkowskian background, and showed that the deformation of the algebra of constraints due to the incorporation of a symplectic structure in the theory originated the Drinfeld twisting of that isometry. However, although the formalism developed there can be extended straight- forwardly to any field theory with no internal symmetries, for the case of parametrized gauge theories some additional complications arise, as pointed out in [3] and [4], due to the fact that the components of the gauge field perpendicular to the embedding are not dynamical but play instead the role of Lagrange multipliers which are not elements of the extended phase space and therefore can not be turned into dynamical variables by canonical transformations. To do so, and recover the anti-homomorphism between the algebra of space-time diffeomorphisms and the Poisson algebra of constraints it is necessary to impose additional Gaussian conditions. The simplest case where such a procedure can be exhibited is the parametrized electromagnetic field. This has been very clearly elaborated in [4], so we shall only review those aspects of that work needed for our presentation. 3 Parametrized Maxwell field and canonical representation of space-time diffeomorphisms Consider a source-free Maxwell field in a prescribed pseudo-Riemannian space-time represented by the action S = −1 4 ∫ d4X √−g gµνgαβFµαFνβ , (3.1) where Fµα = A[µ,α] := Aµ,α − Aα,µ. In the canonical treatment of the evolution of a field one assumes it to be defined on a space-like 3-hypersurface Σ, equipped with coordinates σ, which is embedded in the space-time manifold M by the mapping Xµ : (σ) = Xµ(σa), a = 1, 2, 3. By adjoining the embedding variables to the phase space of the field results in a parametrized field theory where the space-time coordinates have been promoted to the rank of fields. In Space-Time Diffeomorphisms in Noncommutative Gauge Theories 5 terms of the space-time coordinates σα = (τ,σ) determined by the foliation M = R × Σ, the action (3.1) becomes S = −1 4 ∫ dτd3σ √−ḡḡµν ḡαβFµαFνβ , t ∈ R, (3.2) with the inverse metric ḡαβ given by ḡαβ = ∂σα ∂Xµ ∂σβ ∂Xµ , which can be therefore seen as a function of the coordinate fields. In (3.2) ḡ := det(ḡµν) where√−ḡ = J is the Jacobian of the transformation. In order to carry out the Hamiltonian analysis of the action (3.2), we define in similar way to (2.2) the tangent vectors to Σ, Xα a and the unit normal nα = −(−ḡ00)− 1 2 ḡ0ρ ∂Xα ∂σρ . We thus arrive at S[Xµ, Pµ, Aa, π a, A⊥] = ∫ dτd3σ ( PαẊ α + πaȦa −NΦ0 −NaΦa −MG ) , (3.3) where N and Na are the lapse and shift components of the deformation vector Nα := ∂Xα/∂τ , M = NA⊥ −NaAa, and Aa := Xα aAα, A⊥ := −nβAβ are the tangent and normal projections of the gauge potential. The constraints Φ0, Φa and G in (3.3) are defined by: Φ0 = Pαn α + 1 2 γ−1/2γabπ aπb + 1 4 γ1/2γacγbdFabFcd, Φa = PαX α ,a + Fabπ b, G = πa ,a, (3.4) where γab, γ are the metric components on Σ and their determinant, respectively. These con- straints satisfy the relations: {Φ0(σ) +A⊥(σ)G(σ),Φ0(σ ′) +A⊥(σ′)G(σ′)} = [ γab(σ)Φb(σ)+γab(σ′)Φb(σ ′) ] δ,a(σ,σ ′), {Φa(σ)−Aa(σ)G(σ),Φb(σ ′)−Ab(σ ′)G(σ′)} = (Φb(σ)−Ab(σ)G(σ))δ,a(σ,σ ′) + ( Φa(σ ′)−Aa(σ ′)G(σ′) ) δ,b(σ,σ ′), {Φa(σ)−Aa(σ)G(σ),Φ0(σ ′) +A⊥(σ′)G(σ′)} = (Φ0(σ) +A⊥(σ)G(σ)) δ,a(σ,σ ′), {Φ0(σ), G(σ′)} = 0, {Φa(σ), G(σ′)} = 0. (3.5) From here we see that the Gauss constraint G is needed to achieve the closure of the algebra of the super-Hamiltonian and super-momenta constraints, Φ0, Φa, under the Poisson-brackets. However, because of the gauge invariance implied by the Gauss constraint G ≈ 0, the scalar potential A⊥ occurs in (3.5) not as a dynamical variable but as a Lagrange multiplier. The end result of this mixing of constraints and consequent foliation dependence of the space-time action in gauge theories, is that the super-Hamiltonian, nαHα = H⊥ := Φ0(σ) +A⊥(σ)G(σ), and the supermomenta, Xα aHα = Ha := Φa(σ)−Aa(σ)G(σ), constraints do not satisfy the Dirac closure relations (2.1) ({Hα(σ),Hβ(σ′)} ≃ 0), so we do not have a direct homomorphic map from the Poisson brackets algebra of constraints into the Lie algebra of space-time diffeomorphisms for such theories. Nonetheless, this difficulty can be 6 M. Rosenbaum, J.D. Vergara and L.R. Juarez circumvented by turning the scalar potential into a canonical momentum π (via the relation π = √ γA⊥) conjugate to a supplementary scalar field ψ and prescribing their dynamics by imposing the Lorentz gauge condition. The new super-Hamiltonian and super-momenta ∗H⊥ := H⊥ − √ γγabψ,aAb, ∗Ha := Ha + πψ,a, (3.6) of the modified theory satisfy the Dirac closure relations, and the mapping ξ → ∗Hτ [ξ] = ∫ Σ dσ ′ ξα(X(σ′)) ∗Hα results in the desired anti-homomorphism: {∗Hτ [ξ], ∗Hτ [ρ]} = −∗Hτ [£ξρ], (3.7) from the Lie algebra £diffM∋ ξ, ρ into the Poisson algebra of the constraints on the extended phase space Aa, π a, ψ, π, Xα, Pα of the modified electrodynamics with the space-time action: S(φ,ψ) = ∫ M d4X √−g ( −1 4 FαβFαβ + ψ,αg αβAβ ) . (3.8) Note however that in order to recover Maxwell’s electrodynamics from the dynamically minimal modified action (3.8), one needs to impose the additional primary and secondary constraints C(σ) := ψ(σ) ≈ 0, G(σ) ≈ 0 (3.9) on the phase space data. In this way, the new algebra of constraints leading to vacuum electro- dynamics from (3.8) is: {∗H⊥(σ),∗H⊥(σ′)} = γab(σ) ∗Hb(σ)δ,a(σ,σ ′)− (σ ↔ σ′), {∗Ha(σ),∗H⊥(σ′)} = ∗H⊥(σ)δ,a(σ,σ ′), {∗Ha(σ),∗Hb(σ ′)} = ∗Hb(σ)δ,a(σ,σ ′)− (aσ ↔ bσ′), {C(σ),∗H⊥(σ′)} = (γ)− 1 2 (σ)G(σ)δ(σ,σ′), {C(σ),∗Ha(σ ′)} = C,a(σ)δ(σ,σ′), {G(σ),∗H⊥(σ′)} = ( (γ) 1 2 (σ)γab(σ)C,b(σ)δ(σ,σ′) ) ,a , {G(σ),∗Ha(σ ′)} = ( G(σ)δ(σ,σ′) ) ,a . (3.10) This Poisson algebra implies that once the constraints (3.9) are imposed on the initial data they are preserved in the dynamical evolution generated by the total Hamiltonian associated with (3.8), so that if the derivations ∗Ĥτ [ξ] := δξ representing space-time diffeomorphisms start evolving a point of the extended phase space lying on the intersection of the constraint surfaces ∗H⊥(σ) ≈ 0 ≈∗ Ha(σ) and C(σ) := ψ(σ) ≈ 0 ≈ G(σ), the point will keep moving along this intersection. In summary, we have seen that for canonically parametrized field theories with gauge sym- metries in addition to space-time symmetries the Poisson algebra of the constraints does not agree with the Dirac relations and, therefore, cannot be directly interpreted as representing the Lie algebra of the generators of space-time diffeomorphisms. The reason being that because of the gauge invariance there are additional constraints in the theory which cause that not all the relevant variables are canonical variables. Following the arguments in [4] for the case of the elec- tromagnetic field, we have seen that these difficulties can be circumvented by complementing the original action (3.1) with the addition of a term, containing the scalar field ψ, that enforces the Lorentz condition, so the modified action is given by (3.8). Varying this action with respect to the gauge potential Aα gives 1 2 (|√g| 12Fαβ),β = |√g| 12 gαβψ,β, (3.11) Space-Time Diffeomorphisms in Noncommutative Gauge Theories 7 which therefore implies that the modified action introduces a source term into the Maxwell equations, so the dynamical theory resulting from (3.8) is not the same as Maxwell’s electrody- namics in vacuum. It is interesting to observe, parenthetically, that the charge source on the right of (3.11) is a real field and not a complex one as one would have expected. The dynamical character of ψ, however, is evident when differentiating this last equation with respect to Xα whereby, due to the vanishing of the left side, this field must satisfy the wave equation ψ,α ,α = 0. Consequently, in order to recover Maxwell’s electrodynamics it was required that ψ vanish or at least that it is a space-time constant. This was achieved by simply imposing additional constraints on the phase space data, given by (3.9), which (c.f. equation (4.10) in the next section) implies loosing the generator of gauge transformations. This procedure, and its generalization to the case of non-Abelian Yang–Mills fields then allows (still within the canonical group theoretical framework) to undo the projection and replace the Poisson bracket relations (3.5) by the genuine Lie algebra £diffM of space-time diffeomorphisms. Note that even though the algebra in (3.10) involves derivatives of the constraints G(σ) and C(σ), these derivatives can be removed by simply using the identity J(σ′)δ,a(σ,σ ′) = J(σ)δ,a(σ,σ ′) + J,a(σ)δ(σ,σ′), so the algebra does close, as it is to be expected from counting degrees of freedom. As a consequence the elements ∗Hτ [ξ], together with Gτ [ᾱ] := ∫ dσ ᾱ(X(σ))G(σ) and Cτ [β̄] := ∫ dσ β̄(X(σ))C(σ), form a closed algebra under the Poisson brackets. On the basis of the above discussion let us now derive explicit expressions for the generators of the Lie algebra of space-time diffeomorphisms associated with the anti-homomorphism (3.7) and investigate whether these Lie algebra can be extended with the smeared elements Gτ [ᾱ] and Cτ [β̄] and, if so what would be the interpretation of such an extension. For this purpose let us first begin by deriving the Poisson bracket of the projection Aa of the 4-vector potential field Aα on the hypersurface Σ with ∗Hτ [ξ]. Making use of (3.4) and (3.6) we get {Aa(σ), ∗Hτ [ξ]} = ∫ dσ′ {Aa(σ),−ξα(σ′)nα(σ′)∗H⊥ + ξαXα b(σ′) ∗Hb} = −ξαnαγ − 1 2 γabπ b + (ξαAα),a + ξαXα bFba = (£ξAβ)Xβ a, (3.12) after also making use of the expression πa := δL δȦa(σ) = −γ 1 2γabF⊥b, for the momentum canonical conjugate to Aa (c.f. equation (3.10) in [4]). Now, since the right side of (3.12) represents another gauge vector potential on Σ, it clearly follows that {{Aa(σ), ∗Hτ [ξ]}, ∗Hτ [η]} = (£η£ξAβ)Xβ a, and interchanging the symbols ξ, η on the left side above, substracting and using the Jacobi identity, yields {Aa(σ), {∗Hτ [ξ], ∗Hτ [η]}} = −(£[ξ,η]Aβ)Xβ a. We can therefore write the map {Aa(σ), ∗Hτ [ξ]}❀ ∗Ĥτ [ξ] ⊲ Aa(σ), 8 M. Rosenbaum, J.D. Vergara and L.R. Juarez where δξ ≡ ∗Ĥτ [ξ] := (Xβ a ◦£ξ) (3.13) is a derivation operator which when acting on a 4-vector potential Aβ it projects its Lie derivative onto the hypersurface Σ. Consider next the Poisson bracket of the scalar field ψ with ∗Hτ [ξ]. Again, from (3.4) and (3.6) we get {ψ(σ), ∗Hτ [ξ]} = [ξα(−nαγ − 1 2πa ,a +Xα aψ,a)](σ). (3.14) Similarly for the time evolution of ψ, derived from the total Hamiltonian, we obtain ψ̇ = {ψ(σ), ∫ dσ′ (N ∗H⊥(σ′) +Na ∗Ha(σ ′))} = Nγ− 1 2πa ,a +Naψ,a. (3.15) Moreover, since ψ̇ := ∂Xα ∂τ ψ,α = Nαψ,α = Nα(nαψ,⊥ +Xα aψ,a) = −Nψ,⊥ +Naψ,a, which when substituted into (3.15) implies that ψ,⊥ = −γ− 1 2πa ,a, and hence (from (3.14)) that {ψ(σ), ∗Hτ [ξ]} = (ξαψ,α)(σ) = £ξψ(σ). It clearly follows from this that {ψ(σ), {∗Hτ [ξ], ∗Hτ [η]}} = −£[ξ,η]ψ(σ) so for the action of ∗Hτ [ξ] on scalar fields we can therefore also write the morphism (3.13), ∗Hτ [ξ] ❀ δξ ≡ ∗Ĥτ [ξ] := (Xβ a ◦ £ξ), provided it is naturally understood that the surface projection Xβ a acts as an identity on scalars. It should be clear from the above analysis that these derivations δξ, as defined in (3.13), are indeed full space-time diffeomorphisms. Let us now turn to the elements G(σ) and C(σ) of the algebra of constraints (3.10). The Poisson algebra of the mapping ᾱ→ Gτ [ᾱ] = ∫ Σ dσ ′ ᾱ(X(σ′))G(σ′), with Aa is {Aa(σ), Gτ [ᾱ]} = −∂aᾱ, (3.16) and, making use of (3.12), we get {{Aa(σ), Gτ [ᾱ]}, ∗Hτ [ξ]} = −(£ξ∂βᾱ)Xβ a. (3.17) Inverting the ordering of the constraints in the above brackets we also have {{Aa(σ), ∗Hτ [ξ]}, Gτ [ᾱ]} = −{(£ξAβ)Xβ a, Gτ [ᾱ]} = −∂a(ξ c∂cᾱ). (3.18) Subtracting now (3.17) from (3.18), and making use of the Jacobi identity on the left side of the equation, results in {Aa(σ), {∗Hτ [ξ], Gτ [ᾱ]}} = ∂a(ξ ⊥ᾱ,⊥). (3.19) Note that we could equally well have gotten this result by identifying Gτ [ᾱ] with a derivation through the map Gτ [ᾱ] ❀ Ĝτ [ᾱ] := − ∫ Σ dσ′(∂bᾱ)(σ′) δ δAb(σ′) , (3.20) Space-Time Diffeomorphisms in Noncommutative Gauge Theories 9 which could be seen as resulting from integrating the smeared constraint by parts and identifying the canonical momentum πb with the functional derivative: πb ❀ π̂b := δ δAb(σ′) . Indeed, acting first on Aa with the derivation operator (3.13) gives δξ ⊲ Aa ≡ ∗Ĥτ [ξ] ⊲ Aa := (Xβ a ◦£ξ)Aa = −ξαnαγ − 1 2γabπ b + (ξαnαA⊥),a + (ξc),aAc + ξcAa,c, which, when followed by the action of (3.20) results in Ĝτ [ᾱ] ⊲ (δξ ⊲ Aa) = − ∫ Σ dσ′(∂bᾱ)(σ′) δ δAb(σ′) ((ξαnαA⊥),a + (ξc),aAc + ξcAa,c) (σ) = −∂a(ξ c∂cᾱ). (3.21) Alternating the order of the above derivations, a similar calculation gives δξ ⊲ (Ĝτ [ᾱ] ⊲ Aa) = −δξ ⊲ ∂aᾱ = £ξ(∂β ᾱ)Xβ a = −∂a(ξ γ∂γᾱ), and subtracting from this (3.21) yields [δξ, Ĝτ [ᾱ]] ⊲ Aa = −∂a(ξ ⊥∂⊥ᾱ), which could be thought to imply an algebra homomorphism when compared with (3.19). Ob- serve, however, that if we evaluate the Poisson bracket of ∗Hτ [ξ] and Gτ [ᾱ] directly from (3.4) and (3.6) we get {∗Hτ [ξ], Gτ [ᾱ]} = ∫ Σ dσ ( −ξ⊥ᾱ,⊥G+ ξ⊥(∂aᾱ)γ 1 2 γabC,b ) (σ) = −Gτ [ξ ⊥ᾱ,⊥]− Cτ [(ξ ⊥(∂aᾱ)γ 1 2 γab),b]. (3.22) This result remains compatible with (3.19) because C(σ) acts as a projector when operating on the gauge vector field Aa. But, because the right hand side of the equation contains a linear combination of the smeared constraints Gτ and Cτ , there is no way that we could implement the mapping (3.20) to get an homomorphism between the Poisson bracket (3.22) and the Lie bracket [δξ, Ĝτ [ᾱ]], as may be easily seen in fact when calculating the later with (3.13) and (3.20). Similarly, if we now consider the Poisson bracket of the map β̄→Cτ [β̄]= ∫ Σ dσ ′β̄(X(σ′))C(σ′) with ∗Hτ [ξ] we find (again making use of (3.4) and (3.6)) that {Cτ [β̄], ∗Hτ [ξ]} = ∫ Σ dσ[ξαβ̄,αC + ξ⊥β̄γ− 1 2G+ ξaβ̄C,a](σ) = Cτ [ξ αβ̄,α − (ξaβ̄),a] +Gτ [ξ⊥β̄γ− 1 2 ]. (3.23) However, if we were to assume valid the derivation operator map Cτ [β̄] ❀ Ĉτ [β̄] = ∫ Σ dσβ̄ δ δπ(σ) , it would then clearly follow that [δξ, Ĉτ [β̄]] ⊲ Aa = 0. This result immediately enters into conflict with (3.23), where such a morphism of algebras, involving Ĉτ [β̄] together with (3.20), would yield {Aa, {Cτ [β̄], ∗Hτ [ξ]}}❀ [δξ, Ĉτ [β̄]] ⊲ Aa = −∂a(ξ ⊥β̄γ− 1 2 ). Consequently, the largest Lie algebra that we can associate with the Poisson algebra (3.10) is the one of space-time diffeomorphisms, given by the homomorphism implied by (3.7) and originating from the sub-algebra of the super-Hamiltonian and super-momenta described by the first 3 equations in (3.10). We shall return to this observation later on, as it is essential for our conclusions. First we need however to relate our results derived so far with some basic aspects of gauge theory as formulated from the point of view of principal fiber bundles. 10 M. Rosenbaum, J.D. Vergara and L.R. Juarez 4 Gauge transformations Recall (c.f. e.g. [21]) that a gauge transformation of a principal fiber bundle (PFB) π : P →M, with structure Lie group G, is an automorphism f : P → P such that f(pg) = f(p)g and the induced diffeomorphism f̄ : M → M, defined by f̄(π(p)) = π(f(p)), is the identity map f̄ = 1M (i.e. π(p) = π(f(p))). Moreover, if we define f : P → P by f(p) = pζ(p), where ζ is an element of the space C(P,G) of all maps such that ζ(pg) = g−1 · ζ(p) = Adg−1ζ(p) (so G acts on itself by an adjoint action), then C(P,G) is naturally anti-isomorphic to the group of gauge transformations GA(P ). That is, for f, f ′ ∈ GA(P ) and ζ, ζ ′ ∈ C(P,G) we have that (f ◦ f ′)(p) = p(ζ ′(p)ζ(p)). From the above, it can be readily shown that f∗(σu∗X) = d dt ( Rζ(p)−1◦ζ(σu(γ(t)))f(p) ) |t=0 +Rζ(p)∗(σu∗X), where X ∈ TM or, writing ζ(p)−1◦ζ(σu(γ(t))) := etb as an element of a one-parameter subgroup of G, f∗(σu∗X) = b∗f(p) +Rζ(p)∗(σu∗X), where b∗f(p) is the fundamental vector field on f(p) corresponding to b = L−1 ζ(p)∗ζ∗(σu ∗X). (4.1) Consequently, (σ∗uf ∗ω)(X) = b +Ad(σ∗ uζ)(X)−1(σ∗uω)(X). (4.2) In the above expressions, ωf(p) is a connection 1-form at f(p) ∈ P , (f∗ω)p is its pull-back to p with the gauge map f and (σ∗uf ∗ω)π(p) is in turn its pull-back with the local section σu to a 1-form on U ⊂ M, the map γ : R → U is a curve in the base manifold with d dtγ(t)|t=0 = X, and (σ∗uζ)(X µ) is a space-time-valued element of G. Write now ζ as an element of a one-parameter subgroup of C(P,G) by means of the expo- nential map ζ = exp(−tαBTB), (4.3) where αBTB := α is an element of the gauge algebra space C(P, g), and the TB denote the basis matrices of the Lie algebra g associated with G. Replacing (4.3) into (4.1) and (4.2) we get (σ∗u(Rexp(−tαB(p)TB)) ∗ω)(X) = d ds [exp(tᾱB(X)TB) exp(−sᾱB(γ(s))TB)]|s=0 + Adexp(tᾱB(X)TB)(σ ∗ uω)(X), (4.4) where ᾱB := (σ∗uα B). The infinitesimal version of (4.4) follows directly by differentiating both sides of the above equation with respect to the parameter t and evaluating at zero. We therefore arrive at δᾱA := d dt (σ∗u(Rexp(−tαBTB)) ∗ω)|t=0 = −dᾱ− [A, ᾱ] = −Dᾱ ∈ Λ̄1(M, g), (4.5) where Λ1(M, g) denotes the space of 1-forms on M valued in the Lie algebra g. Making use of (4.5) in the expression for the Yang–Mills curvature: F := DA = dA+ 1 2 [A,A], Space-Time Diffeomorphisms in Noncommutative Gauge Theories 11 we obtain that δᾱF = [ᾱ, F ]. (4.6) In the particular case where the one-parameter group is Abelian, it immediately follows that (4.5) and (4.6) simplify to δᾱA = −idᾱ, (4.7) and δᾱF = 0. This last result merely states the well know fact that the electromagnetic field strength is gauge independent (i.e. it is independent of the choice of local trivialization). Moreover, since (4.7) implies that δᾱAµ = −i∂µᾱ, we obtain, by projecting on the sheet Σ with Xµ a, δᾱAa = −i ∂aᾱ(X(σ)). (4.8) Let us now turn to the Gauss constraint G(σ), introduced in (3.4), and to the smearing map ᾱ→ Gτ [ᾱ] = ∫ Σ dσ′ᾱ(X(σ′))G(σ′). (4.9) Comparing (3.16) with (4.8) we see that i{Aa, Gτ [ᾱ]} ∼= δᾱAa, (4.10) so the Poisson bracket of the projection Aa of the gauge 4-vector on the space-like hypersurface Σ with the Gauss constraint smeared with the scalar function ᾱ(X(σ′)) is the same as the pullback to M of the infinitesimal action of the gauge algebra of the PFB with group U(1) on the connection one-form ω (c.f. equation (4.5)) evaluated on a tangent vector to Σ. In addition, for f ∈ GA(P ), it is a simple matter to show that if ω is a connection 1-form then the pullback f∗ω is also a connection 1-form. This theorem follows immediately by noting first that the action of f∗ω on a fundamental vector yields its corresponding Lie algebra generator, and second that the requirement ωpg(Rg∗X) = Adg−1ωp(X) in the definition of a connection 1-form is directly satisfied when acting on ω with the pullback of f ◦Rg = Rg ◦ f , which in turn is equivalent the automorphism condition f(pg) = f(p)g. Let now V be a vector space on which G acts from the left. If Lg : V → V is linear, then the homomorphism G → GL(V ) by g 7→ Lg is a representation of G. In this case C(P, V ) will denote the space of all maps ζ : P → V such that ζ(pg) = g−1 ·τ(p) and the elements of C(P, V ) correspond to particle fields. In particular, C(P, V ) = Λ̄0(P, V ), where, in general, Λ̄k(P, V ) is the space of V -valued differential k-forms ϕ on P such that R∗gϕ = g−1 · ϕ, ϕ(Y1, . . . ,Yk) = 0, if any one of the Y1, . . .Yk ∈ TpP is vertical. Making now use of the exponential map (4.3) it readily follows that f∗ϕ = ζ−1 · ϕ. (4.11) 12 M. Rosenbaum, J.D. Vergara and L.R. Juarez Or, differentiating with respect to t and evaluating at t = 0, we arrive at the following infinitesimal version of (4.11): δᾱϕ̄ = ᾱBTB · ϕ̄. (4.12) Furthermore, related to our discussion in the following sections, note that from the definition of diffeomorphisms we have that Rg ◦f = f ◦Rg, thus acting with the pull-back of this equality on any element κ ∈ Λ̄k(P, V ), and recalling that the action of the differential f∗ on a fundamental field B∗ is a fundamental field, it then immediately follows that (f∗κ)(B∗) = κ(B∗) = 0. Hence f∗κ ∈ Λk(P, V ), k = 0, 1, 2 . . . , and since C(P, V ) = Λ̄0(P, V ) it also follows that the gauge group GA(P ) acts on particle fields via pull-back, so that f∗ϕ(p) = ϕ(f(p)), (4.13) i.e. if ϕ is a particle field, so is also f∗ϕ. Using the above results we can now formulate the multiplication rules for gauge and particle fields under gauge transformations, when pulled-back to the base space M. Thus, given two g-valued potential 1-forms A,A′ ∈ Λ1(M, g), their product is defined by [A,A′] := ( Aa ∧A′b ) ⊗ [Ta, Tb], while the product of two particle fields ϕ1, ϕ2 ∈ C(P, V ) is by simple point multiplication. Now, as shown previously, the action of an element f ∈ GA(P ) on a connection 1-form and on a particle field is via pull-back (c.f. equations (4.2) and (4.13)) and since the pull-back of a connection is a connection and the pull-back of a particle field is a particle field, it therefore follows that f : [A,A′] ❀ [(σ∗uf ∗ω1), (σ ∗ uf ∗ω2)], f : (σ∗uϕ1)(π(p)) · (σ∗uϕ2)(π(p)) ❀ (σ∗uf ∗ϕ1)(π(p)) · (σ∗uf∗ϕ2)(π(p)). By (4.5) and (4.12), the infinitesimal expression for the above is: δᾱ ( [A,A′](X1,X2) ) := µ [ (δᾱ ⊗ 1 + 1⊗ δᾱ) ( Aa(X1)⊗A′b(X2)−Aa(X2)⊗A′b(X1) )] ⊗ [Ta, Tb] = (δᾱA a ∧A′b −Aa ∧ δᾱA′b)(X1,X2)⊗ [Ta, Tb], (4.14) and δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p))) = δᾱ(ϕ̄1(π(p))) · ϕ̄2(π(p)) + ϕ̄1(π(p)) · δᾱ(ϕ̄2(π(p))), (4.15) respectively. This last result implies that under an infinitesimal gauge transformation the pro- duct of two particle fields transforms according to the Leibniz rule. We can therefore give this infinitesimal transformations the structure of a Hopf algebra with coproduct ∆δᾱ = δᾱ⊗1+1⊗δᾱ, so that δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p))) = µ[∆δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p)))]. From the above discussion we can derive some additional insight into the implications of the PFB point of view of gauge transformations on our previous results. We thus see that since gauge transformations are automorphisms on the fibers that project to the identity on the base space, the Gauss constrain – which we have seen here to be related to the pull-back of the infinitesimal gauge transformations, and which was shown in Section 3 to be needed in order to Space-Time Diffeomorphisms in Noncommutative Gauge Theories 13 close the algebra in (3.5) – occurs in the extended algebra (3.10) primarily as part of the super- Hamiltonian and super-momenta associated with the Lie algebra of space-time diffeomorphisms. Its independent appearance is then only as a constraint which, together with C(σ) ≃ 0, have to be implemented at the end as strong conditions in order to recover the Maxwell theory. This provides an additional natural explanation for why these two constraints can not be mapped into derivations that could lead to an enlarged Lie algebra beyond the one of the space-time diffeomorphisms. 5 Noncommutative gauge theories With these results in hand, let us now consider an approach for extending the theory of gauge fields to the noncommutative space-time case, by specifically concentrating on the vacuum Maxwell field discussed in the last two sections, and by following the procedure introduced in [1]. Recall, in particular, that – because of the anti-homomorphism that can be established between the Poisson sub-algebra of the constraints occurring in the first 3 lines of (3.10), for the modified theory in extended phase space, and the Lie algebra £diffM – we can use the latter to investigate the deformed space-time isometries of the system by requiring that this sub-algebra of constraints, modified by the noncommutativity of space-time, should continue obeying the Dirac relations, relative to the Dirac brackets resulting from admitting an arbitrary symplectic structure in the action (3.3). This, as shown in [1], was needed in turn in order to incorporate into the parametrized canonical formalism the dynamical origin of star-noncommutativity from quantum mechanics [14]. Moreover, since the constraints depend on the metric of the embedding space-time, this last step would require in general a well developed theory of quantum mechanics in curved spaces and knowledge of the commutators of the operators representing the phase space coordinates. We shall defer such more general considerations for some future presentation, and concentrate here only on the case of fields on flat Minkowski space-time and the corresponding quantum mechanics for the extended Weyl–Heisenberg group. Consequently, admitting a symplectic structure in the action (3.8) we have S[z] = ∫ d4σ ( B(z)Aż A −Nα(∗H̃α)−MG(σ)− TC(σ) ) , with the symplectic variables zA = (Xα, Aa, ψ;Pα, π a, π) and symplectic potentials B(z)A to be determined by a prescribed symplectic structure. Here M , T are the additional Lagrange multipliers needed to recover Maxwell’s electrodynamics and the tildes on the constraints needed of the formerly introduced quantities, in order that their Dirac-bracket algebra originated by the new symplectic structure is identical to their sub-algebra in (3.10). That is, we want to maintain the algebra of these constraints invariant by utilizing new twisted generators. (Observe however, that since the G(σ) and C(σ) can not form part of our Lie algebra of space-time isometries, but are strictly constraints to be implemented in order to retrieve Maxwell’s electromagnetism, their action on gauge and particle fields will be determined by the arguments given at the end of this section.) As noted in [1], the symplectic structure is defined by, ωAB := ∂BB ∂zA − ∂BA ∂zB , (5.1) from where we can readily solve for the symplectic potentials, which are defined up to a canonical transformation. The resulting second-class constraints can then be eliminated by introducing Dirac brackets, according to a scheme analogous to the one described in the above cited paper, from where the inverse of the symplectic structure is additionally defined through the Dirac- brackets for the symplectic variables zA. Hence the Dirac brackets for the symplectic variables 14 M. Rosenbaum, J.D. Vergara and L.R. Juarez are given by {zA, zB}∗ := {zA, zB} − {zA, χC} ωCD{χD, z B} = ωAB , (5.2) where χA = πzA − B(z)A ≃ 0 are the second-class constraints. More specifically, based on the premise that quantum mechanics is a minisuperspace of field theory and for a quantum mechanics on flat Minkowski space-time based on the extended Weyl–Heisenberg group, we have shown in [14] that the WWGM formalism implies that, for the phase space variables to have a dynamical character, we need to modify their algebra by twisting their product according to µ(Xα ⊗Xβ) ❀ µθ(X α ⊗Xβ) := Xα(τ,σ) ⋆θ X β(τ,σ′), (5.3) where ⋆θ := exp [ i 2 θµν ∫ dσ′′ ←− δ δXµ(τ,σ′′) −→ δ δXν(τ,σ′′) ] , (5.4) and where, since the embedding space-time variables are functionals of the foliation, we use functional derivatives. Also, since fields are in turn functions of the embedding space-time variables their multiplication in the noncommutative case is inherited from (5.3). Moreover, using this ⋆-product we can now define the commutator [Xα(τ,σ),Xβ(τ,σ′)]θ := Xα(τ,σ) ⋆θ X β(τ,σ′)−Xβ(τ,σ′) ⋆θ X α(τ,σ) = iθαβδ(σ,σ′), (5.5) and let {Xα,Xβ}∗ = [Xα(τ,σ),Xβ(τ,σ′)]⋆θ = iθαβδ(σ,σ′). On the other hand, defining the map X̃α = Xα + θαβ 2 Pβ , (5.6) it follows from (5.2) that {X̃α, X̃β}∗ = 0, (5.7) and {∗H̃α(~σ),∗ H̃β(~σ′)}∗ = 0. Thus, in parallel to (3.7), we have {∗H̃τ [ξ], ∗H̃τ [ρ]}∗ = −∗H̃τ [£ξρ]. Furthermore, making the identification Pβ = −i δ δXβ in the Darboux map (5.6) we can write X̃α ❀ ˆ̃Xα = (Xα) ⋆−1 θ := (Xα) exp [ − i 2 θµν ∫ dσ′′ ←− δ δXµ(τ,σ′′) −→ δ δXν(τ,σ′′) ] , (5.8) where the bi-differential acting from the right on the embedding coordinates Xα is the inverse of (5.4). Hence {X̃α, X̃β}∗ ∼= [ ˆ̃Xα, ˆ̃Xβ ]⋆θ = [Xα,Xβ ] ⋆−1 θ = 0, Space-Time Diffeomorphisms in Noncommutative Gauge Theories 15 since under point multiplication the embedding coordinates commute. So the map (5.8) retrie- ves (5.7). In addition, since multiplication in the algebra of the operators ˆ̃Xα is by the ⋆θ-product we can generalize the last result to {(X̃α)m, (X̃β)n}∗ ∼= [( ˆ̃Xα)m⋆ , ( ˆ̃Xβ)n⋆ ]⋆θ = [(Xα)m, (Xβ)n] ⋆−1 θ = 0. We can therefore conclude from the above that, when replacing the functional dependence on the embedding variables in the constraints in (3.10) by the “tilde” variables (5.6) and the point multiplication of fields by their ⋆-product, the functional form of their algebra is evidently preserved for the noncommutative case. That is, {∗H̃τ [ξ], ∗H̃τ [η]}∗ ∼= [∗Ĥτ [ξ], ∗Ĥτ [η]]⋆ ⋆ −1 θ , (5.9) and ∗Ĥτ [ξ] = δξ ❀ ∗Ĥτ [ξ] ⋆ −1 θ = δ⋆ ξ , (5.10) where the multiplication µθ of the algebra of generators of diffeomorphisms δ⋆ ξ ∈ £diffM is via the ⋆θ-product. Consequently, by using the example of a modified electromagnetism within the context of canonical parametrized field theory, it was shown that, by including additional constraints, Maxwell’s equations could be recovered as well as the possibility of also establishing for gauge field theories the anti-homomorphism between Dirac-brackets of the modified constraints and space-time diffeomorphisms. Furthermore using our previous results in [1] where it was shown that noncommutativity in field theory – manifested as the twisting of the algebra of fields – has a dynamical origin in the quantum mechanical mini-superspace which, for flat Minkowski space-time, is related to an extended Weyl–Heisenberg group, and including these results into the symplectic structure of the parametrized field theory then allowed us to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms, as shown by (5.9) and (5.10) above. Moreover, making use of (5.10) we can summarize the action of space-time diffeomorphisms on particle fields associated with gauge theories, and the transition of the theory to the non- commutative space-time case by means of the following functorial diagrams: ∗Hτ [ξ] ∈ V θ−−−−→ V⋆ ∋ ∗H̃τ [ξ] = ∫ d~σ(ξ̃⊥ ∗H̃⊥ + ξ̃a ∗H̃a) C   y C   y ∗Ĥτ [ξ] ∈ V̂ C(θ)−−−−→ V̂⋆ ∋ ∗Ĥτ [ξ] ⋆ −1 θ ≡ δ⋆ ξ (5.11) (where V denotes the space of constraints satisfying the algebra (3.10), V⋆ is the corresponding space of constraints for the space-time noncommutative case with the embedding coordinates mapped according to (5.6) and V̂ , V̂⋆ denote the spaces of the Lie algebra of diffeomorphisms and their corresponding twisted form, respectively); ϕ̄ ∈ A δξ−−−−→ A ∋ δξ ⊲ ϕ̄ D   y D   y ϕ̄ ∈ Aθ D(δ⋆ ξ ) −−−−→ Aθ ∋ δ⋆ ξ ⊲ ϕ̄ = δ⋆ ξ ⋆θ ϕ̄(X(τ,σ)) (5.12) (here A denotes the module algebra of particle fields ϕ̄ ∈ C(M, V ) with point multiplication µ and Aθ is its noncommutative twisting with ⋆-multiplication µθ := µ ◦ e i 2 θµν∂µ⊗∂ν ). 16 M. Rosenbaum, J.D. Vergara and L.R. Juarez It then follows from these two diagrams that {ϕ̄, ∗Ĥτ [ξ]} ∼= δξ ⊲ ϕ̄ 7→ δ⋆ ξ ⋆θ ϕ̄(X(τ,σ)) = ∗Ĥτ [ξ] ⊲ ϕ̄. (5.13) Note that the diagrams (5.11), (5.12) and equation (5.13) provide an explicit expression for the mappings δρ 7→ δ⋆ ρ , which in turn imply [ δ⋆ ρ , δ ⋆ η ] ⋆θ = δ⋆ £ρη, and δ⋆ ρ ⋆θ (ϕ̄1 ⋆θ ϕ̄2) = δρ(ϕ̄1 ⋆θ ϕ̄2), (5.14) where ϕ̄1, ϕ̄2 ∈ Aθ. Note also that the universal envelopes U(V̂) and U(V̂⋆) of the derivations δξ and twisted derivations δ⋆ ξ can be given the structure of Hopf algebras. Thus, in particular, we can obtain an explicit expression for the coproduct in U(V̂⋆) by making use of the duality between product and coproduct, followed by the application of equation (5.14). We get µθ ◦∆(δ⋆ ρ)(ϕ̄1 ⊗ ϕ̄2) = δ⋆ ρ ⋆θ (ϕ̄1 ⋆θ ϕ̄2) = δρ(ϕ̄1 ⋆θ ϕ̄2) = µ(δρ ⊗ 1 + 1⊗ δρ)(e i 2 θµν∂µ⊗∂ν ϕ̄1 ⊗ ϕ̄2) = ∑ n 1 n! ( i 2 )n θµ1ν1 · · · θµnνn [ (δ⋆ ρ ⋆θ ∂µ1...µnϕ̄1)e − i 2 θµν←−∂ µ −→ ∂ ν ⋆θ ∂ν1...νnϕ̄2 + (∂µ1...µnϕ̄1)e − i 2 θµν←−∂ µ −→ ∂ ν ⋆θ (δ⋆ ρ ⋆θ ∂ν1...νnϕ̄2) ] = µθ ◦ [ e− i 2 θµν∂µ⊗∂ν (δ⋆ ρ ⊗ 1 + 1⊗ δ⋆ ρ)e i 2 θµν∂µ⊗∂ν ] (ϕ̄1 ⊗ ϕ̄2). This result compares with the Leibniz rule given in [6]. Furthermore, if we let F = e− i 2 θµν∂µ⊗∂ν ∈ U(V̂)⊗ U(V̂), and define ϕ̄1 ⋆θ ϕ̄2 = µθ(ϕ̄1 ⊗ ϕ̄2) := µ(F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2)), we then have [22, 23]: δρ(ϕ̄1 ⋆θ ϕ̄2) = δρ ⊲ µ(F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2)) = µ[(∆δρ)F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2))] = µF−1[(F(∆δρ)F−1)((ϕ̄1 ⊗ ϕ̄2)))] = µθ[(F(∆δρ)F−1)((ϕ̄1 ⊗ ϕ̄2)))]. (5.15) Thus, the undeformed coproduct of the symmetry Hopf algebra U(V̂) is related to the Drinfeld twist ∆F by the inner endomorphism ∆Fδρ := (F(∆δρ)F−1) and, by virtue of (5.15), it preserves the covariance: δρ ⊲ ((ϕ̄1 · ϕ̄2))) = µ ◦ [∆(δρ)(ϕ̄1 ⊗ ϕ̄2))] = (δρ(1) ⊲ ϕ̄1) · (δρ(2) ⊲ ϕ̄2) θ→ δ⋆ ρ ⊲ (ϕ̄1 ⋆θ ϕ̄2) = (δ⋆ ρ(1) ⊲ ϕ̄1) ⋆θ (δ⋆ ρ(2)) ⊲ ϕ̄2), where we have used the Sweedler notation for the coproduct. Consequently, the twisting of the coproduct is tied to the deformation µ→ µθ of the product when the last one is defined by ϕ̄1 ⋆θ ϕ̄2 := (F−1 (1) ⊲ ϕ̄1)(F−1 (2) ⊲ ϕ̄2). We want to reiterate at this point that the ⋆-product, associated with the algebra Aθ, that we have been considering here is the one originated when considering in turn the flat-Minkowski space-time quantum mechanics generated by the extended Weyl–Heisenberg group H5, for the even more particular case of an extension of the Lie algebra of H5 by the commutator [Xµ,Xν ] = Space-Time Diffeomorphisms in Noncommutative Gauge Theories 17 iθµν , for the simplest case when θµν = const. In this case the generators δρ of isometries become the infinitesimal generators of the Poincaré group of transformations, and the coproduct defined in this equation reduces to the twisted coproduct considered by e.g. [24] (see also e.g. [5] and [25, 26]). Since the embedding coordinates in the canonical parametrized theory can in general be associated to a curved space-time manifold and, since the constraints and related diffeomorphisms are constructed for such spaces, it seems possible in principle that our formalism could be extended to curved space-time backgrounds with a ⋆-product determined by the Lie algebra associated with, for instance, a given homogeneous space. This would imply finding first the equivalent of the mapping (5.6) and also, of course, the realization of this map in terms of the ⋆-product, perhaps by a procedure based on the deformation quantization formalism developed by Stratonovich [27]. A fairly simple example of the above is the Darboux map given in [29], for the case of the Snyder algebra [28]. However, finding a full realization of the ⋆-product is a more difficult job. In equation (4.15) of the previous section we derived the expression for the infinitesimal gauge transformation on a product of particle fields in A. Let us now consider the effect of such a gauge transformation on the product of two particle fields in Aθ when we have space- time noncommutativity. For this purpose we first recall equation (4.13) which shows that if ϕ is a particle field, so is its gauge transformation by pull-back, i.e. ϕ ∈ C(P, V ) ⇒ ϕ′ := f∗ϕ ∈ C(P, V ). From this it follows that to a given element of C(P, V ) we can always associate another one which is the pull-back of the former, thus the twisted product of the pull-back with the section σu of any pair of particle fields can be written as ϕ̄′1 ⋆θ ϕ̄ ′ 2 = (σ∗u(f∗ϕ1)) ⋆θ (σ∗u(f∗ϕ2)). Observe however that, because of the noncommutativity that the algebra (5.5) of the embedding coordinates is required to satisfy, the pull-back to M of the gauge transformation (4.11) now should be understood as σ∗uf ∗ϕ = ζ̄−1 ⋆ (X) ⋆θ ϕ̄(X); so that ϕ̄′1 ⋆θ ϕ̄ ′ 2 = (ζ̄−1 ⋆ ⋆θ ϕ̄1) ⋆θ (ζ̄−1 ⋆ ⋆θ ϕ̄2), (5.16) where, due to the noncommutativity, equation (4.3) is replaced by ζ̄−1 ❀ ζ̄−1 ⋆ = exp⋆(tᾱ(X)) := 1 + tᾱ + t2 2 ᾱ ⋆θ ᾱ + · · · . Using the infinitesimal version of this map we have that ϕ̄′1 = ϕ̄+ ᾱ⋆θ ϕ̄, so that (5.16) becomes δᾱ : (ϕ̄1 ⋆θ ϕ̄2) := ϕ̄′1 ⋆θ ϕ̄ ′ 2 = (ᾱ(X) ⋆θ ϕ̄1(X)) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ (ᾱ(X) ⋆θ ϕ̄2(X)). (5.17) By a similar argument, since f ∈ GA(P ) also maps connections into connections, its infinitesimal action on the ⋆-product of two gauge fields (c.f. (4.14)) goes into δᾱ : ( [A,A′]⋆θ (X1,X2) ) := − [( dᾱA(X1) + 1 2 cACD[AC(X1), ᾱ D(X1)]⋆θ ) ⋆θ A ′B(X2) − ( dᾱA(X2) + 1 2 cACD[AC(X2), ᾱ D(X2)]⋆θ ) ⋆θ A ′B(X1) +AA(X1) ⋆θ ( dᾱB(X2) + 1 2 cBCD[A′C(X2), ᾱ D(X2)]⋆θ ) −AA(X2) ⋆θ ( dᾱB(X1) + 1 2 cBCD[A′C(X1), ᾱ D(X1)]⋆θ )] ⊗ [TA, TB ]. Note that we have written the last two equations for the general case of any group of gauge transformations, where ᾱ(X) = ᾱBTB , in order to underline the fact that, because of the ⋆- product in the multiplication of the fields one needs to apply the constraint that these NC gauge 18 M. Rosenbaum, J.D. Vergara and L.R. Juarez groups have to be in the fundamental or adjoint unitary representation (i.e. TA ∈ U(n)), since only in this representation the gauge group closes (c.f. e.g. [12, 19]). See however also [20] for arguments tending to circumvent this constraint). Hence, in the NC case the generators of gauge symmetry act on particle fields with the fundamental representation ϕ̄ ❀ ϕ̄′ = ζ−1 ⋆ ⋆θ ϕ̄ = exp⋆(tᾱ(X)) ⋆θ ϕ̄, (5.18) while on gauge fields the action is via the adjoint representation A(X) ❀ A′(X) = ζ−1 ⋆ ⋆θ A(X) ⋆θ ζ⋆ + ζ−1 ⋆ ⋆θ (dζ⋆)(X). (5.19) Equations (5.18) and (5.19) agree with those on which [11] is based when remarking on some of the conclusions on deformed gauge theories arrived at in [10, 9, 30, 31]. Indeed, one basic idea in this other approach of gauge twisted theories is the assumption that the gauge generators δᾱ := ᾱ(X) = ᾱB(X)TB act on particle and gauge fields with the usual point product, so instead of (5.17) they define δᾱ(ϕ̄1 ⋆θ ϕ̄2) := (δᾱϕ̄1) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ (δᾱϕ̄2). (5.20) Moreover, by assuming that the algebra of the gauge generators can be given an additional Hopf bialgebra structure, and that the derivatives of any order of the gauge and particle fields are, as noted in [11], in the same representation of the gauge algebra as the fields themselves, one could further write δᾱ(ϕ̄1 ⋆θ ϕ̄2) = (ᾱ(X)ϕ̄1) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ ᾱ(X)ϕ̄2. = µ ◦ (δᾱ ⊗ 1 + 1⊗ δᾱ) ◦ (e i 2 θµν∂µ⊗∂ν ϕ̄1 ⊗ ϕ̄2) = µθ[(∆ Fδᾱ) ◦ (ϕ̄1 ⊗ ϕ̄2)]. (5.21) Assuming a scalar particle field for simplicity and setting ϕ̄2 = ∂µϕ̄ and ϕ̄1 = ∂µϕ̄ †, it can be readily seen that one immediate consequence of the extra assumption leading to equating the last two lines in (5.21) with the first one is that the latter then yields: δᾱ(∂µϕ̄ † ⋆θ ∂µϕ̄) = 0, which implies that the kinetic terms in the Lagrangian of the particle fields are invariant by themselves, so there would be no need to introduce the gauge potentials to achieve gauge invari- ance of the theory. Consequently, since (5.21) only fully agrees with (5.17) when ᾱ is coordinate independent, there appears to be a discrepancy as a consequence of local internal symmetry between assuming the validity of (5.20) and some essential aspects of the theory of gauge inva- riance. Recall furthermore, that a Drinfeld twist (c.f. e.g. [22, 23, 32]) involves a simultaneous and covariant deformation of the product of an algebra A of functions and the coproduct of a bial- gebra H. More specifically, the algebra A is a module algebra (H-module algebra) over a Hopf bialgebra whose elements are in the universal enveloping algebra U(L) of a Lie algebra L, such that if x ∈ L then ∆(x) = x⊗ 1 + 1⊗ x, and x(ab) = x(a)b+ ax(b) ∀ a, b ∈ A, so that x acts as a derivation. On the other hand, as shown by equations (4.9) and (4.10), the infinitesimal gauge transformation of the gauge potential is given by the Poisson bracket of the smeared Gauss constraint Gτ [ᾱ] with the gauge potential; but, as it was also shown in Section 3 of this paper, the δᾱ can not be made isomorphic to a derivation operator acting as such on the gauge potentials or particle fields, contrary to the case of the smeared super-Hamiltonian and super-momenta constraints. Consequently the algebra of the infinitesimal gauge transformations can not be considered as part of the Hopf algebra of the space-time diffeomorfisms δξ, associated with Lie Space-Time Diffeomorphisms in Noncommutative Gauge Theories 19 algebra L and its universal envelope, from which a Drinfeld twist could be properly constructed. Note also that in the context of the canonical parametrized formalism, the Gauss constraint is defined on the spacelike hypersurface Σ and, again contrary to the super-Hamiltonian and super-momenta constraints, does not depend on the embedding variables. This translates in the fact that for the NC case the space-time diffeomorphisms δξ, on the one hand, and the infinitesimal gauge transformations δᾱ, on the other, act quite differently on the gauge and particle fields. This is clearly seen when comparing the actions (5.10) and (5.18) on the gauge and particle fields, as well as their actions (5.15) and (5.17) on their respective products. It thus appears from our present results as well as from those in [1] (where the noncommu- tative reparametrized scalar field was considered and its respective constraints together with their anti-homomorphic relation to space-time diffeomorphisms was explicitly established), that it might not be possible to extend the concept of a Drinfeld twist symmetry to include gauge symmetries, when considering the minimal coupling of gauge and particle fields in order to in- vestigate a full model of NC theory in the context of the canonical reparametrized theory (see e.g. [12] regarding this point). However, if one were to consider relaxing the concept of twisted symmetries and modify the definition of a deformed Leibniz rule (such as the one exhibited in (5.20)), several different twists and gauge invariants may be constructed that would lead to alternate formulations for NC gauge theories. Some new ideas in this context that might help to remove some of the inconsistencies pointed out here as well as elsewhere, are discussed in [33, 34]. This would involve, essentially, assuming different deformations of products of elements in the same algebra of space-time functions A, when considering different transformation groups. Such an assumption however, would be hard to reconcile with the point of view that the product in this algebra of functions is inherited from the deformation of the algebra of space-time coordinates and its dynamical origin in the quantum mechanical mini-superspace. As it was remarked previously the ⋆-product considered so far applies to an underlying flat Minkowski space-time, and the corresponding twisted isometries refer then to the Poincaré group. It is interesting to observe, however, that our formalism admits a natural extension of (5.4) which allows us to consider much more general symplectic structures than (5.1) that would imply noncommutativity among all the symplectic variables zA = (Xα, Aa, ψ;Pα, π a, π). Moreover, because of the appearance of the embedding metric in the canonical parametrized formalism, this could lead in turn to the possibility of extending our analysis to the case of twisted isometries on curved space backgrounds. Even within the flat Minkowski space-time case, we could have a more general symplectic structure that would lead to a different ⋆-product with bi-differentials involving some of the other fields in the theory. Consider for instance the symplectic structure resulting in the Dirac brackets: {Xα,Xβ}∗ = iθαβ, {Xα, Pβ}∗ = iδβ α, {Pα, Pβ}∗ = 0, {Aa, Ab}∗ = 0, {Aa, π b}∗ = iδb a, {πa, πb}∗ = iβab, (5.22) (and the remainder equal to zero). Here the Darboux map, that takes us from the extended algebra (5.22) to the usual Heisenberg algebra, is given by the transformations: X̃α = Xα + θαβ 2 Pβ , π̃a = πa + βab 2 Ab. (5.23) These maps are unique up to a canonical transformation on the phase-space (Xα, Pα, Aa, π a). In order to construct the deformed constraints, note that in the expressions for Φ0 and Φa in (3.4) there appear the projectors nα(σ,X) and Xα a (σ,X) as well as the 3-metric γab, all of which are functionals of the space-time embedding coordinates Xα. These quantities thus need 20 M. Rosenbaum, J.D. Vergara and L.R. Juarez to be modified according to (5.23). On the other hand, the Gauss constraint also requires to be modified in order that the Dirac bracket algebra of the new constraints be the same as the Poisson algebra of the original ones. The resulting deformed constraints are then: Φ̃0 = Pαñ α + 1 2 γ̃−1/2γ̃abπ̃ aπ̃b + 1 4 γ̃1/2γ̃acγ̃bdFabFcd, Φ̃a = PαX̃ α ,a − Fabπ̃ b, G̃ = π̃a ,a, where the tilde on top of a symbol denotes the replacement of the space-time coordinates accord- ing to (5.23). However, one point to observe is even that the constraints have been deformed by the fact that their algebra involves now Dirac brackets instead of Poisson brackets, the Darboux transformations (5.23) preserve the functional form of their algebra, so they can still be made anti-homomorphic to an algebra of deformed space-time diffeomorphisms, by a pro- cedure analogous to the one described here. Also note, in particular, that the original fields zA = (Xα, Aa, ψ;Pα, π a, π) will now transform according to the twisted diffeomorphisms of the theory. Thus, while the electric field πa will no longer be gauge invariant, the new field π̃a will be, under the gauge transformation associated with the modified Gauss constraint. Note also that the last equation in (5.22)implies that the Drinfeld deformation of the algebra of functions of the fields involves a ⋆-product which is a composition of (5.4) with ⋆β := exp [ i 2 βab ∫ dσ′′ ←− δ δπa(τ,σ′′) −→ δ δπb(τ,σ′′) ] . Acknowledgements The authors are grateful to Prof. Karel Kuchař for fruitful discussions and clarifications concern- ing his work on parametrized canonical quantization. They are also grateful to the referees for some very pertinent comments and suggestions which helped to clarify considerably some points in the manuscript. The authors also acknowledge partial support from CONACyT projects UA7899-F (M.R.) and 47211-F (J.D.V.) and DGAPA-UNAM grant IN109107 (J.D.V.). References [1] Rosenbaum M., Vergara J.D., Juarez L.R., Canonical quantization, space-time noncommutativity and de- formed symmetries in field theory, J. Phys. A: Math. Theor. 40 (2007), 10367–10382, hep-th/0611160. 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[31] Giller S., Gonera C., Kosinski P., Maslanka P., On the consistency of twisted gauge theory, Phys. Lett. B 655 (2007), 80–83, hep-th/0701014. [32] Kassel Ch., Quantum groups, Springer-Verlag, New York – Berlin – Heidelberg, 1995. [33] Vassilevich D.V., Symmetries in noncommutative field theories: Hopf versus Lie, arXiv:0711.4091. [34] Duenas-Vidal A., Vazquez-Mozo M.A., Twisted invariances of noncommutative gauge theories, arXiv:0802.4201. ar X iv :0 9 0 2 .1 2 0 1 v 1 [m at h -p h ] 6 F eb 2 0 0 9 Lattice vortices induced by noncommutativity A.A. Minzonia, L.R. Juárezb, M. Rosenbaum∗,b aFENOMEC-IIMAS, Universidad Nacional Autónoma de México bInstituto de Ciencias Nucleares-FENOMEC, Universidad Nacional Autónoma de México Abstract We show that the Moyal ⋆-product on the algebra of fields induces an effective lattice structure on vortex dynamics which can be explicitly constructed using recent asymptotic results. Key words: Lattice-Vortices, Dynamics, Noncommutativity PACS: 02.40.Gh, 11.10.Nx, 05.45.Yv, 04.20.Ha, 03.65.Ta 1. Introduction The study of the behavior of classical fields defined as functions of noncom- mutative spatial variables has received a great deal of attention in the last few years (see e.g. [1],[2],[3],[4],[5] and [6] for a review). In particular the study of coherent structures in the form of noncommutative solitons or noncommu- tative vortices has shown that the noncommutative version of ϕ4-type models in two spatial dimensions with polynomial nonlinearities sustain non-collapsing soliton plateau and uncharged static vortex solutions, unlike what occurs in the commutative case. It has also been shown that the confining mechanism is provided by the ⋆-product which acts like a projection. Using this fact it has been further shown that large vortices can be produced by taking static suitable combinations of projections. In this process vortex-like structures with no angular dependence appear, and which have quantized radii R ∼ √n for n ∗Corresponding author Email addresses: tim@mym.iimas.unam.mx (A.A. Minzoni), roman.juarez@nucleares.unam.mx (L.R. Juárez), mrosen@nucleares.unam.mx (M. Rosenbaum) Preprint submitted to Elsevier February 7, 2009 integer. It is also known [7],[8] that spatial noncommutativity in 2-spheres in R3 induces a lattice structure in the radial variable, where the quantized radii scale as R ∼ n. Moreover this lattice can confine radial solutions. On the other hand, recent work in non-linear optics (see e.g. [9], [10], [11], [12] and [13] for a review and background) has shown that optical lattices can trap coherent structures in the form of plateaus and vortices, with and without charge. The dynamics of these structures has been described successfully using modern asymptotics in terms of the Peierls-Nabarro (P-N) potential [12],[14] produced by the interaction between the coherent structure and the physical lattice. The purpose of this letter is to show by means of an asymptotic analysis how the ⋆-product combined with the vortex structure induces a lattice in the spatial dimensions via a P-N like potential in the radial variable which, in turn, confines the vortex itself avoiding the collapse. We show, making use of a coherent state vortex-type solution with varying parameters, how the average Lagrangian [15] for the noncommutative Nonlinear Schrödinger equation is equivalent to the known average Lagrangian for a vortex on a discrete lattice. The difference between the two Lagrangians being the actual form of the P-N potential. For the lattice the P-N potential is periodic with the lattice period, while on the noncommutative case considered here the P-N potential depends on R2, which is the square of the vortex radius. This dependence gives a scalingR ∼ √n which is expected from the exact result described for the ϕ4-type models. We also exhibit how for vortices in two and three spatial dimensions, whose charge is of the same order of their large radius, the ⋆-product induces a P-N potential which results in an equispaced radial lattice. This shows how the splitting of R3 in terms of fuzzy spheres placed on an equispaced radial lattice, as proposed in [7],[8], arises naturally from ⋆-product confining a vortex with large charge, large radius and small width. This results show how the dynamics of the coherent structures sustained by the noncommutativity can be described asymptotically in terms of the classical dynamics of coherent structures in lattices. Furthermore, the present analysis shows that the qualitative behavior for large vortices does not 2 depend on the details of the noncommutative model chosen. 2. Formulation We consider the Nonlinear Schrödinger equation where the nonlinearity is given by the Moyal ⋆-product between the complex functions u(x, t), x is the 2-dimensional spatial position and t is the time. The equation is given by the usual local Lagrangian [15], suitably modified by the nonlocal product in the form: L = ∫ ∫ d xdt[i(utū− ūtu) + |∇u|2 − ū ⋆ u ⋆ ū ⋆ u], (1) where the Fourier transform of the ⋆-product of two complex fields u and v in the Rieffel formula for the Moyal ⋆-product [16] is given by: û ⋆ v(k) = 1 (2π)2 ∫ û(k− p)v̂(p)ei(k−p)∧pdp. (2) Here û(k, t) = ∫ e−ik·xu(x, t)dx and a ∧ b = Θ 2 (a1, a2)   0 1 −1 0     b1 b2  , with Θ fixing the square of the length given by the noncommutativity of the spatial variables. The equation associated with the Lagrangian (1) is the nonlocal Nonlinear Schrödinger equation: iut = ∆u+ 2u ⋆ ū ⋆ u. (3) This is the usual Schrödinger equation where the real potential U(x) is now dependent on the solution ū ⋆ u. In previous studies [1] of the Sine-Gordon equation, static (with no angular momentum) exact vortex-type solutions were found in the limit of Θ→∞. Analogous solutions of (3) can be obtained in the form u(x, t) = eiσtζ(x), (4) where v is a real valued function of the position. Substitution into (3) readily gives − σζ = ∆ζ + ζ ⋆ ζ ⋆ ζ. (5) 3 In the large noncommutativity limit (5) becomes the same as the equation con- sidered in [1]. Specially interesting solutions to (5) in this range of values of the noncommutativity parameter discussed in the above cited work are given (using scaled variables) by: ϕn(r) = 2(−1)ne−r2 Ln(2r2), (6) with r2 = x2 + y2 and σ = 4Θ and where Ln are the Laguerre polynomials. It is known that sums of ϕn(r) are also solutions. For example, taking Pn(r) = n ∑ j=0 ϕj(r) and Wn = ϕn(r) + ϕn−1(r). Thus, using the well established fact that the Laguerre polynomials have caustics when r2 ∼ n [17], we obtain that the plateau Pn(r) is practically flat up to r ∼ √n and confined to that region. On the other hand, Wn has a peak at r ∼ √n. Such a behavior suggests trapping by an annular lattice with radii Rn ∼ √ n, for large n. Moreover, it is known [9],[13] both from analytical as well as numerical cal- culations that the usual Nonlinear Schrödinger equation on the usual square lattice supports stable vortices due to the trapping of the vortex by the P-N po- tential, which prevents their collapse as it occurs in the limit of the continuum. We will show below, using the asymptotic analysis developed in [14], that the ⋆-product in equation (3) indeed generates, via the field, a P-N like potential responsible for the trapping of the vortex. We will further show how this P-N potential generates lattices with radii growing as √ n or as n, depending on the coherent state used for the averaging of the Lagrangian. 3. Asymptotic solutions On the basis of the above considerations, we shall derive next an asymptotic solution to (3) in the form of a vortex with angular momentum (and charge one) given by the local behavior reiθ at the origin, and an envelope suggested by the 4 exact solution with no angular momentum. Thus, we take as the coherent state trial function to average the Lagrangian (1) the expression u(r, θ, t) = a(t)re−( r−R(t) ω(t) )2ei(θ+σ)ei(r−R(t))V (t), (7) where the amplitude a, the width ω and the phase σ are functions of time. The radius R represents the location of the peak of the vortex and the velocity V is the radial velocity of the maximum of the vortex.We will also consider charged vortices with charge m where the angular dependence is rmeimθ close to r = 0. Let us begin by studying the charge one vortex by substituting the trial function (7) into the Lagrangian (1) and performing the spatial integration we obtain an averaged Lagrangian for the parameters a, ω,R, V and σ. This Lagrangian is then varied and the modulation equations obtained for the parameters which give the approximate evolution of the vortex. The aver- aged Lagrangian has the form L̄ = L0 + L∗, (8) where L0 arises from the local terms in (1) and L∗ is the contribution of the ⋆-product. We have, using the results in [14], that L0 2π = −2a2ωR3σ̇ − 2a2R3ω(V Ṙ− V 2 2 )− 2a2R3 3ω . (9) To calculate L∗ we begin by transforming the potential term with the ⋆- product in (1) into the Fourier space. We have, using the Parseval relation, that ∫ ∫ ū ⋆ u ⋆ ū ⋆ u dx = ∫ ∫ |ū ⋆ u|2 dx = ∫ ∫ |̂̄u ⋆ u|2(k) dk. (10) The calculation of the last integrand is performed by making use of (2) and the Fourier transform of coherent state (7) which is given by: û(p) = a(t)eiσ(t) ∫ ∞ 0 ∫ 2π 0 eipr cos(θ−ϕ)eiϕre−( r−R ω )2ei(r−R)V rdϕdr, (11) where p = p(cos(θ), sin(θ)) and ϕ is the polar angle of x. 5 In the highly noncommutative limit the width ω is small. Hence the inte- grand in r is peaked at r = R. Making the change of variables r = R + ωξ we obtain, to leading order in ω, û(p) = ωa(t)eiσ(t)eiθR ∫ ∞ −R ∫ 2π 0 eipR cos ϕeiϕeikωξ cos ϕe−ξ2 eiωξV rdϕdr. (12) Since we are interested in large vortices, we use the stationary phase approx- imation [18] on the angular integral. There are two stationary phase points at ϕ = 0, ϕ = π which give an oscillatory contribution. The radial integral is then calculated extending the lower limit to minus infinity to obtain: û(p) = aωR3/2 (2π)3/2p1/2 sin ( pR+ π 4 ) e−p2ω2(1+V )2ei(θ+σ). (13) In the same way we obtain ˆ̄u(p) = aωR3/2 (2π)3/2 p1/2 sin ( pR+ π 4 ) e−p2ω2(1−V )2e−i(θ+σ). (14) Consequently, using (13) and (14) together with (2) allows us to arrive at an approximation for the integrand in (10) in the form (̂̄u ⋆ u)(k) = a2ω2R3 ∫ ∞ 0 ∫ 2π 0 eik∧p e −p2ω2(1+V )2 p1/2 e−|k−p|2ω2(1−V )2 |k− p|1/2 × sin ( pR+ π 4 ) sin ( |k− p|R+ π 4 ) eiθ1eiθ2pdpdθ1dθ2, (15) where θ1 and θ2 correspond to the angular coordinates of p and k− p, respec- tively. Again (15) will be evaluated approximately in the strongly noncommu- tative limit using the stationary phase method in the angular integral. To this end recall that both k and p are large since the vortex is narrow because ω is small. Using p = q ω we obtain k ∧ p = Θ ω kq sin θ, |p− k| = √ q2 w2 − 2 q ω k cos θ + k2 (16) Since in the strongly noncommutative limit k is also large we know that the points of the stationary phase are for θ = π/2 and θ = 3π/2. Using again the same type of calculation as in the derivation of equation (12) we obtain: 6 (̂̄u ⋆ u)(k) = ω1/2e−ω2k2 a2ω2 (2π)3 (ω Θ )1/2 R ∫ ∞ 0 sin ( kq Θ ω ) e−q2(1+V )2 × sin ( R ω q ) sin ( R √ q2 ω2 + k2 ) e−(q2+k2ω2)(1−V )2 √ q2 ω2 + k2 q1/2dq. (17) This integral is again evaluated asymptotically for larger R using the method of stationary phase. We need to observe that for small ω and to leading order of this width the stationary phase point is q = kω 2 . We then have (̂̄u ⋆ u)(k) = R5/2 (2π)3ω1/2 e−ω2k2 a2ω2 (ω Θ )1/2 × sin ( k2 2 Θ ) sin2(kR)e− ω 2 4 k2((1+V )2+ 5 4 (1−V )2). (18) Finally, integrating this last expression over k we obtain that the Lagrangian term L∗ is given by: L∗ = a4R5/2ω2 (2π)3Θ ∫ ∞ 0 e−ω2k2[2+ 1 2 (1+V )2+ 5 8 (1−V )2] sin2 ( k2 2 Θ ) sin4(kR)kdk, (19) and, evaluating once more with the method of stationary phase for large R we obtain: L∗ = − a4 (8π)3Θ ω2R5/2 ( 1 2ω2 + F (R,ω, V ) ) , (20) where the function F is the analogue of the Peierls-Nabarro potential for the lattice generated by the noncommutative self interaction of the field, and takes the form: F (R,ω, V ) = 1 4R1/2 cos ( R2 2Θ + π 4 ) e−ω2R2[2+ (1+V )2 2 + 5(1−V )2 8 ]. (21) Hence the final average Lagrangian is: L =2a2ωR3σ̇ + 2a2ωR+ 2a2ωR3(V Ṙ− V 2 2 ) + 2 a2R3 3ω − a4R (8π)3Θ − F (R,ω, V ). (22) 7 This Lagrangian is, except for the R2 dependance in the potential, the same average Lagrangian obtained in [cite] for large Nonlinear Schrödinger commu- tative vortices on a discrete lattice. However in the present case the lattice is generated by the ⋆-product and it manifests itself in the R2 dependance of the Peierls-Nabarro potential. 4. Vortex solutions and radial stability The approximate dynamics of the vortex is obtained from the variational equations of (22). These are δσ : d dt (2a2ωR) = 0, δa : 4aωR3dσ dt + 4aR3 3ω − Ra3 2Θ + ∂aF = 0, δω : 2a2R3 dσ dt − 2a2R3 3ω2 + ∂ωF = 0, (23) δV : Ṙ− V − ∂V F = 0, δR : d dt 2a2ωR3V + ∂RL = 0. The last expression above is the equation of motion for the peak of the vortex, analogous to a particle in the Peierels-Nabarro potential. The dynamics of the solutions to (23) simplifies for large vortices in lattices, with low kinetic energy, i.e. for V ≪ 1. In this case, since ω is small, the dominating terms for the δa and δω equations can be readily solved to give a steady vortex amplitude width relation in the form a = 32 √ Θ 3ω . (24) For large vortices with Rω still small and with small kinetic energy V ≪ 1 the dynamics of the peak is described by the simple equation 8 dR dt = V, a2ω d(RV ) dt = a4ω4 2Θ2 R3 sin ( R2 2Θ + π 4 ) . (25) Equation (25) shows that the vortex moves in the lattice which was generated by the ⋆-product. The fixed points are the possible equilibrium positions and are given by Rn = √ 2nπΘ. (26) The odd values of n give stable vortices, while the even values of n give insta- bility. If a vortex starts at an unstable value it will shrink radiating until it is trapped at the lower minimum of the potential. It is to be noted that the scaling of the radius is the same as the one obtained by the exact trapped solutions of ϕ4-type noncommutative models. This result has a simple interpretation in terms of the Landau cells used in [7]. In fact the P-N potential generated by the ⋆-product induces an annular lattice, where the n-th annulus has an area An = 2πRn(Rn+1 − Rn) and where Rn is given by (25). As n → ∞ so that An ∼ π2Θ, which gives the constraint area of the Landau cell. We thus can say that the lattice induced by the ⋆-product is a lattice of Landau cells. The same calculation when performed for a charge m vortex, with R of the same order as m, gives an equispaced lattice. In fact the asymptotic evolution of the integrals in Eq.(12) replaces the term sin2(kR) of (18) by sin2(k + 1 k )R. This results in a P-N potential of the form (21) where the term (cos R2 2θ + π 4 ) is replaced by cos(R+ π 4 ). this induces an equispaced lattice. The area of the Landau cells is again constant, as can be readily verified. Finally, when we go to R3 and use in the averaging of the Lagrangian coherent structures with no angular depen- dence, we obtain shell lattices with Rn ∼ √ n; while when including an angular dependence given by the spherical function Ylm(θ, ϕ), induces - for large values of l and as a consequence of the form of the coherent state - equispaced lattices, thus recovering in a natural way the noncommutative structure proposed in [7]. 9 5. Discussion and Conclusions We have shown that the effect of noncommutativity of the spatial variables, when averaged on the appropriate coherent vortex or plateau-like states, induces an effective spatial lattice of Landau cells whose distribution and sizes depend on the coherent states in question. This shows that the effect of noncommutativity on coherent structures whose local width is comparable to the spatial scale Θ of the ⋆-product behave as classical structures on a physical lattice and allows us to calculate the lattice and the corresponding dynamics of the noncommutative coherent structure. It is to be remarked that the lattice structures in three space dimensions have been constructed using a group on the sphere and taking the Casimir values as the possible values of the radii inducing a lattice. We have shown that in the appropriate coherent states the ⋆-product induces the same lattice in the radial variable. We also observe that unlike physical lattices which are not translation in- variant, the lattices induced by the ⋆-product are translation invariant. Because of this reason coherent states can move with uniform velocity. We thus see that the effect of the ⋆-product has no classical analogy since it is capable of forming a lattice to support the coherent structure without loosing the uniform transla- tional motion. We end by remarking that in the weak noncommutative limit, which is the opposite limit to the one considered here, equation (1) resembles a high order non-linear system of equations which incorporates Raman scattering effects [13]. However at the present time the existence of an optical analogue of equation (1) in the strongly noncommutative limit is not known. References [1] R. Gopakumar, S. Minwalla, and A. Stromiger, JHEP05:020 (2000). [2] M. G. Jackson, JHEP0109:004 (2001). [3] D. Bak, K. Lee, and J. Park, Phys. Rev. D 63, 125010 (2001). 10 [4] R. Gopakumar, M. Headrick, and M. Spradlin, Commun. Math. Phys. 233, 355 (2003). [5] D. Tong, J. Math. Phys. 44, 3509 (2002). [6] O. Lechtenfeld (2006), lectures given at International Workshop on Non- commutative Geometry and Physics 2005, Tohoku, Japan, 1-4 Nov 2005., hep-th/0605034. [7] J. Madore, Annals Phys. 219, 187 (1992). [8] H. Grosse, M. Maceda, J. Madore, and H. Steinacker, Int. J. Mod. Phys. A 17, 2095 (2002). [9] D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzekakis, Physica D 212, 20 (2005). [10] J. Wang and J. Yang, Phys. Rev. A 77, 033834 (2008). [11] B. M. C. Chong, R. Carreter-Gonzalez and P. G. Kevrekidis, to appear in Physica D (2008). [12] L. Cisneros, A. A. Minzoni, and J. Ize, in print in Physica D. [13] Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003). [14] L. A. Cisneros, A. A. Minzoni, P. Panayotaros, and N. F. Smyth, Phys. Rev E 78, 036604 (2008). [15] G. B. Whitham, Linear and nonlinear waves (Wiley-Interscience, 1974). [16] M. A. Rieffel, Bull. London Math. Soc. 506, 305 (1993). [17] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964). [18] N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals (Dover, 1986). 11 Adv. appl. Clifford alg. Online First c© 2010 Birkhäuser / Springer Basel AG, Switzerland DOI 10.1007/s00006-010-0217-9 Advances in Applied Clifford Algebras Noncommutativity and Parametrization of Fields: The Scalar Electrodynamics Case L. Roman Juarez, Marcos Rosenbaum and J. David Vergara Abstract. The aim of this paper is to review the formalism of noncommuta- tivity using canonical parametrization theory. In the first part we present the formalism for the case of Quantum Mechanics, and we show that using this ap- proach and an appropriate basis we can get the noncommutativity expressed in terms of the Moyal product from the Dirac brackets of an extended phase space. We generalize our formalism to the context of Quantum Field Theory where we discuss the case of scalar electrodynamics. The interesting result is that our approach works correctly when we consider an interaction term be- tween the gauge field and the scalar field. Finally, we present an argument that shows that gauge theories are not deformed if we use only noncommutativity of the coordinates. Mathematics Subject Classification (2000). Primary 70S10, 70S05; Secondary 81T75, 20C20. Keywords. Noncommutativity, star products. 1. Introduction The present paper deals with the noncommutativity of particles and fields in the context of canonical parametrization theory. There are several reasons for writing a review article in this topic. Firstly, in recent years, space-time noncommutativity has become the subject of increasing interest. Furthermore, it has been considered as common wisdom among practitioners of noncommutative field theory that at the first quantization level, fields are elements of an algebra where multiplication is deformed by means of the Moyal ⋆-product [1]. This anzatz, which originated in a basically heuristic fashion from some results in string theory [2], is based on an analogy with the Weyl-Wigner-Groenewold-Moyal (WWGM) formalism of Quantum Mechanics. But, in Quantum Mechanics time is a parameter of the theory and, in order for space-time to have a truly noncommutativity physical meaning we need to consider both space and time as observables represented by 2 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA noncommutative operators and include them as dynamical variables in an extended Heisenberg algebra. Moreover, since particle quantum mechanics can be viewed, in the free field or weak coupling limit, as a minisuperspace sector of quantum field theory where most of the degrees of freedom have been frozen, it is a very convenient arena for further investigating the implications of the quantum mechanical space-time noncommutativity in the formulation of field theories, as well as for evaluating the justification of some statements that are generally accepted in noncommutative field theories. As our starting point we shall consider a parametrization invariant system. This means that if the system is not naturally invariant under parametrizations we promote the original parameters of the theory, for example the time in the case of particle dynamics, to the level of canonical variables. The second step is to perform the canonical analysis of this theory. One point that we must be care- ful with is that, since we add new variables to the system, we have to introduce constraints associated to the parametrization invariance symmetry of the theory in order that the number of degrees of freedom are preserved. The third step is to introduce an arbitrary canonical potential that allows us to realize the required noncommutativity, and to show that under the Dirac brackets the first class con- straint generate the symmetry. This means that the constraints will probably need to be modified. At this point, if we have several constraints, we need to check that the algebra of these first class constraints closes. Once we finish this procedure we obtain the quantum evolution equations for our system. Alternatively, we can introduce the canonical potential in the action and select an appropriate basis in order to quantize the system using the path integral formalism. For certain choices of the potentials that generate a given symplectic structure, the phase of the quantum transition function between the admissible bases corresponds to a linear canonical transformation, by means of which the actions associated to each of these bases may be related and hence lead to equivalent quantizations. We must stress however that in contradistinction to the case when time plays the role of a parameter, the canonical transformation here is implemented in an extended phase space, where the time and its conjugate momentum are included. With the purpose of examining all the above mentioned facets of the space- time noncommutativity, our presentation has been structured as follows: In Section 2 we consider the canonical formalism of parametrization invariant systems. In subsection 2.1 we introduce an arbitrary symplectic structure in the action, and after the canonical analysis we construct the Dirac brackets associated to the theory and also obtain the action for the reduced system. In Section 3, we quantize the noncommutative theory using the Dirac’s method. In Sec. 4 we review the essential aspects of the construction of canonical parametrized field theories and representations of space-time diffeomorphisms, following [3, 4, 5, 6, 7]. In Sec. 5 we show how the formalism can be extended to the case of parametrized gauge field theories including scalar matter. In subsection 5.1 we combine the results of the previous sections in order to extend the formalism to the noncommutative Noncommutativity and Parametrization of Fields 3 space-time case, by deforming the symplectic structure of the theory to account for the noncommutativity of the space-time embedding coordinates. We thus derive a deformed algebra of constraints in terms of Dirac-brackets which functionally satisfy the same Dirac relations as those for the commutative case and can therefore be related anti-homomorphically to a Lie algebra of generators of twisted space- time diffeomorphisms. In Sec. 6 we study the gauge symmetries and we conclude in Secs. 7 and 8 with an analysis of noncommutativity in a case that illustrates such theories. 2. Parametrization Invariant Systems We review here the essentials of the canonical analysis of parametrized systems following the approach in [8, 9, 10]. To this end, consider the action for a particle in a N -dimensional configuration space, with an arbitrary potential: S = t2 ∫ t1 dt ( 1 2 m ( dqi dt )2 − V ( qi, t ) ) , (2.1) where i = 1, . . . , N . In this action the time t plays the role of a parameter in the theory. This means that it makes little sense to consider the noncommutativity of the coordinates qi and the time t. It is therefore necessary to promote the time to the level of another coordinate of our theory, i.e. we extend our configuration space with one extra dimension t = q0. To do this, we parametrize the action by introducing a new parameter τ and assume that the coordinates qi(t) are scalars under this parametrization, i.e., t → τ, qi (t) → qi (τ) . (2.2) The action (2.1) then takes the form S = τ2 ∫ τ1 dτ ( 1 2 m ( dqi dτ )2 ( dτ dq0 ) − V ( qi, q0 ) ( dq0 dτ ) ) , (2.3) where t = q0 now plays the role of a new coordinate in the theory. Making the identifications q̇i ≡ ( dqi dτ ) and q̇0 ≡ dt dτ , we can rewrite (2.3) in the form S = τ2 ∫ τ1 dτ ( 1 2 m (q̇i)2 q̇0 − V (qi, q0)q̇0 ) . (2.4) Now, because of the fact that by introducing a new variable in the theory we need in the Hamiltonian formalism to add that restriction to the physical evolution of the system that account for the fact that the new N + 1 coordinates are not all 4 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA independent. This action (2.4) reads S = τ2 ∫ τ1 dτ ( p0q̇ 0 + piq̇ i − λϕ ) , (2.5) where ϕ = p0 + H ≈ 0 is the first class primary constraint associated to the symmetry under parametrizations, and H is the canonical Hamiltonian of the action (2.1), and λ(τ) is a Lagrange multiplier. The action (2.5) is invariant up to a total derivative under the transformations δq0 = {q0, εϕ} , δp0 = {p0, εϕ} , δpi = {pi, εϕ} , δqi = { qi, εϕ } δλ = ε̇, (2.6) generated by the constraint ϕ, where the variation of the Lagrange multiplier is imposed in such way that when varying the action it should vanish up to a boundary term. Following Dirac [8], we propose that at the quantum level the physical states of the theory are invariant under the above transformations, i.e., eiεϕ̂ |ψ〉P = |ψ〉P . (2.7) So in infinitesimal form we get ϕ̂ |ψ〉P = 0. (2.8) We thus see that the constraint leads to a supplementary condition on the physical states, and is another way to reduce the quantum theory to its physical sector without imposing a gauge condition. Now if we consider the configuration representation with basis |q0, qi〉, equa- tion (2.8) yields, ϕ̂ |ψ〉P = 0 ⇒ ( −i ∂ ∂t − 2 2m ∇2 + V (qi, t) ) ψ(qi, t) = 0, (2.9) where we have identified t = q0. We therefore obtain the Schrödinger equation as a result of imposing at the quantum level the classical invariance under parametriza- tions of the theory. In the following subsection we shall apply the same procedure to the case of arbitrary symplectic structures. 2.1. Non-commutativity and Dirac Brackets Let za = ( q0, qi, p0, pi ) , with a = 1, ..., 2N + 2, denote the 2N + 2 phase-space variables of a parametrized system in the Hamiltonian formulation. In this case we don’t have a second order action to begin with as in (2.1). We can however consider a general first order action, equivalent to (2.5), given by S = τ2 ∫ τ1 dτ (Aa(z)ża − λϕ(z)) , (2.10) where Aa(z) is a vector potential which we shall use to generate an arbitrary symplectic structure associated to the Poisson brackets in the Hamiltonian formu- lation. Noncommutativity and Parametrization of Fields 5 Applying the Dirac’s method for constrained systems, we have from (2.10) that the corresponding canonical Hamiltonian is given by Hc = λϕ(z), (2.11) and the canonical momenta lead to the set of primary constraints, χa = pza − Aa (z) . (2.12) Consequently, the total Hamiltonian for this theory is HT = λϕ + µaχa. (2.13) Moreover, from the evolution of the constraints we obtain the following consistency conditions χ̇a = {pza − Aa (z) , HT } = −λ ∂φ ∂za + µbωab ≈ 0, (2.14) where ωab := ∂aAb − ∂bAa = {χa, χb}. (2.15) This antisymmetric matrix will play the role of the symplectic structure of the theory. Assuming further that ωab is invertible so all the Lagrange’s multipliers µa in (2.14) can be determined, it then follows from (2.15) that the constraints χa are second class. Note that in the case where the symplectic structure is degenerate, at least one of the χa’s will be first class, but in this case the number of degrees of freedom of the generalized theory will not correspond to the degrees of freedom of the original theory. Hence in what follows we will assume that all the constraints χa are second class. Now, in order to impose these constraints as strong conditions when quantizing, we construct the associated Dirac brackets which are given by {A, B}∗ = {A, B} − {A, χa}ωab {χb, B} , (2.16) where ωab, is the inverse matrix of ωab. Computing the Dirac’s brackets of the coordinates with the above expression we obtain { za, zb }∗ = ωab. (2.17) Thus, quantizing a theory constrained by symmetries under parametrization re- sults in the noncommutativity of the quantum operators corresponding to the phase space coordinates: [ ẑa, ẑb ] = iωab. (2.18) The simplest case corresponds to the usual Heisenberg algebra of ordinary Quan- tum Mechanics, for which the inverse matrix of the canonical symplectic structure takes the form Jab := ωab|θ=0 = ( 0 I −I 0 ) . (2.19) 6 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA 3. Non-Commutative Quantum Mechanics In the previous section we have considered a general procedure for quantizing a theory with an arbitrary symplectic structure. One interesting feature of this formalism is that by including time as a canonical variable allows us to consider also noncommutativity between the time and the spatial coordinates. In the case of field theory a similar extension will allow us to consider noncommutativity between the coordinates. Now, given a symplectic structure we can quantize the system either by using the Dirac’s procedure where the first class constraints act as operators on the physical states, or by imposing supplementary conditions on them, and replacing the Dirac brackets of the second class constraints by commutators. Alternatively, we can also quantize by first evaluating the generating potentials of the symplectic structure and then applying path integral methods in order to derive the Feynman propagators. Her we discuss the first approach. It should be noted, however, that for a given symplectic structure the solution for the potentials Aa is not unique, although all the possible resulting actions and resulting classical theories are related by canonical transformations. Furthermore, in the Dirac quantization the commutators (2.18) of the gen- erators of the extended Heisenberg algebra define the possible complete sets of commuting observables of the theory and the correlative admissible bases (labeled by the eigenvalues of these sets). For each of these admissible bases, we obtain a realization of the Heisenberg algebra and of the subsidiary condition (2.8). Note finally that there are also actions originating from solutions of (2.15) for which no fixed end-points, corresponding to one of the admissible bases in the Dirac quantization exits. A quantization can be defined however by using a Dar- boux map, by means of which new dynamical variables, given in terms of linear combinations of the original ones are introduced and that, consequently implies a change in the initial symplectic structure to a canonical one. The Dirac quanti- zation results then from promoting to the rank of operators these new variables, which will satisfy the Heisenberg algebra of ordinary quantum mechanics. So in this case the deformation of the symplectic structure at the classical level is reflected at the quantum level in a deformed Hamiltonian while the standard Heisenberg algebra of the usual quantum mechanics is preserved. To further illustrate the above observations, following [10], we next consider an example of quantum noncommutativity in the context of the Dirac formalism. For analytical simplicity we assume a 1+1 space-time, generalization to higher order dimensions is fairly straightforward. Noncommutativity and Parametrization of Fields 7 3.1. Space-time noncommutativity Let us consider first the case where the Dirac brackets (2.17) determine a sym- plectic structure of the form ωab =     0 θ 1 0 −θ 0 0 1 −1 0 0 0 0 −1 0 0     , ωab =     0 0 −1 0 0 0 0 −1 1 0 0 θ 0 1 −θ 0     . (3.1) Quantizing according to Dirac’s prescription by using (2.18) leads to the commu- tators [t̂, x̂] = iθ, [x̂, p̂x] = i, [t̂, p̂t] = i, [p̂t, p̂x] = 0, (3.2) and, using (2.8), to the supplementary condition ϕ̂|ψ〉 = 0, (3.3) where ϕ̂ is given by ϕ̂ = p̂t + H(t̂, x̂, p̂x). (3.4) It is obvious from (3.2) that for a mechanical Hamiltonian the sets of complete commuting observables in this case are {x̂, p̂t}, {t̂, p̂x} and {p̂t, p̂x}. The admissible bases in Hilbert space are then {|x, pt〉}, {|t, px〉} and {|pt, px〉}, respectively. Here, we only consider the example of the basis |x, pt〉, see [10] for full treat- ment. For this basis the algebra (3.2) is realized by t̂ψ(x, pt) = i (∂pt + θ∂x)ψ(x, pt), p̂xψ(x, pt) = −i∂xψ(x, pt), (3.5) while the remaining generators of the extended Heisenberg algebra are just mul- tiplicative quantities. Also projecting on (3.3) with 〈x, pt| and substituting (3.5) into (3.4), with a Hamiltonian of the form H = p2 x 2m +V (x, t), yields the subsidiary condition ( pt − 2 2m ∂2 x + V (x, i(∂pt + θ∂x)) ) ψ(x, pt) = 0 (3.6) on the wave function ψ(x, pt). One interesting feature of the Dirac quantization resulting from the use of this basis is that for a t independent potential, equation (3.6) becomes ( pt − 2 2m ∂2 x + V (x) ) ψ(x, pt) = 0. (3.7) For such a time independent Hamiltonian, (3.7) may be interpreted as an eigen- value equation, with −pt the energy eigenvalues of the system and ψ(x, pt) the corresponding eigenvectors. Note that the energy spectrum of the resulting theory does not have any corrections from the noncommutativity of the space-time. A similar result was obtained by Balachandran, et al [11] by means of a very differ- ent approach. Also, we must notice that, using the basis |x− θ 2pt, t + θ 2px〉 we will recover directly the results of the usual Moyal noncommutativity [10]. In the following section we consider how to generalize the above concepts to the case of parametrized field theory. 8 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA 4. Spacetime Diffeomorphisms in Parametrized Gauge Theories As it is well known, see e.g. [3, 4], for Poincaré invariant field theory on a flat Minkowskian background, each generator of the Poincaré Lie algebra, represented by a dynamical variable on the phase-space of the field, is mapped homomorphi- cally into the Poisson bracket algebra of these dynamical variables. On a curved spacetime background field theories are not Poincaré invariant but, by a parametri- zation consisting of extending the phase-space by adjoining to it the embedding variables, they can be made invariant under arbitrary spacetime diffeomorphisms [8, 12]. Hence spacetime parameters are raised to the level of fields on the same footing as the original fields in the theory. Moreover, in this case it can also be shown [3] that: a) An anti-homomorphic mapping can be established from the Poisson al- gebra of dynamical variables on the extended phase-space and the Lie algebra £diff M of arbitrary spacetime diffeomorphisms. Thus, {Hτ [ξ], Hτ [η]} = −Hτ [£ξη], (4.1) where ξ, η ∈ £diff M are two complete spacetime Hamiltonian vector fields on M, Hτ [ξ] := ∫ Σ dσ ξαHα, and Hα are the constraints (supermomenta and super- Hamiltonian) of the theory, satisfying the Dirac vanishing Poisson bracket algebra {Hα(σ),Hβ(σ′)} ≃ 0. (4.2) b) The Poisson brackets of the canonical variables representing the £diff M correctly induce the displacements of embeddings accompanied by the evolution of the field variables, predicted by the field equations. For the prescribed pseudo-Riemannian background M, equipped with coor- dinates Xα, reparametrization involves a foliation Σ×R of this spacetime, where R is a temporal direction labeled by a parameter τ and Σ is a space-like hyper- surface of constant τ , equipped with coordinates σa (a = 1, 2, 3), and embedded in the spacetime 4-manifold by means of the mapping Xα = Xα(σa). (4.3) This hypersurface is assumed to be spacelike with respect to the metric gαβ on M, with signature (-,+,+,+). Let now the embedding functionals Xα a(σ, X) := ∂Xα(σ) ∂σa and nα(σ, X), defined by gαβXα anβ = 0, and gαβnαnβ = −1, (4.4) be an anholonomic basis consisting of tangent vectors to the hypersurface and unit normal, respectively. We can therefore write the constraints Hα as Hα = −H⊥nα + HaXα a, (4.5) where H⊥ and Ha are the super-Hamiltonian and super-momenta constraints, respectively. Using this decomposition the Dirac relations (4.2) can be written Noncommutativity and Parametrization of Fields 9 equivalently as {H⊥(σ),H⊥(σ′)} = 3 ∑ a=1 γabHb(σ)∂σaδ(σ − σ ′) − (σ ↔ σ ′), {Ha(σ),Hb(σ ′)} = Hb(σ)∂σaδ(σ − σ ′) + Ha(σ′)∂σbδ(σ − σ ′), (4.6) {Ha(σ),H⊥(σ′)} = H⊥(σ)∂σaδ(σ − σ ′), where γab is the inverse of the spatial metric γab(σ, X) := gαβ(X(σ))Xα aXβ b. (4.7) Also, as a consequence of the antihomomorphism between the Poisson algebra of the constraints and £diff M we can write Hτ [ξ]  Ĥτ [ξ] ≡ δξ = ξα(X(τ, σ)) ∂ ∂Xα ∣ ∣ ∣ ∣ X(τ,σ) . (4.8) Indeed, since [η, ρ] = £ηρ we have [δη, δρ]φ = δ£ηρφ = Ĥτ [£ηρ] ⊲ φ ∼= {φ, Hτ [£ηρ]} = Ĥτ [η] ⊲ [Ĥτ [ρ] ⊲ φ] − Ĥτ [ρ] ⊲ [Ĥτ [η] ⊲ φ] (4.9) ∼= {{φ, Hτ [ρ]}, Hτ [η]} − {{φ, Hτ [η]}, Hτ [ρ]} = −{φ, {Hτ [η], Hτ [ρ]}} after resorting to the Jacobi identity and where φ is some field function in the theory. Making use of this antihomomorphism as well as of the dynamical origin of ⋆-noncommutativity in field theory from quantum mechanics exhibited in [13], the extension of the reparametrization formalism and the canonical representation of spacetime diffeomorphisms to the study of field theories on noncommutative space- times has been presented in [7]. More specifically, in that paper we discussed the particular case of a Poincaré invariant scalar field immersed on a flat Minkowskian background, and showed that the deformation of the algebra of constraints due to the incorporation of a symplectic structure in the theory originated the Drinfeld twisting of that isometry. However, although the formalism developed there can be extended straightforwardly to any field theory with no internal symmetries, for the case of parametrized gauge theories some additional complications arise, as pointed out in [4] and [6], due to the fact that the components of the gauge field perpendicular to the embedding are not dynamical but play instead the role of Lagrange multipliers which are not elements of the extended phase space and therefore can not be turned into dynamical variables by canonical transformations. To do so, and recover the anti-homomorphism between the algebra of spacetime diffeomorphisms and the Poisson algebra of constraints it is necessary to impose additional Gaussian conditions. The simplest case where such a procedure can be exhibited is the parametrized electromagnetic field. This has been very clearly elaborated in [6], and applied to the formulation of a noncommutative theory for the free electromagnetic gauge field in [14]. Here, we will extend the analysis [6], by include in the formalism scalar matter and the interaction between this field 10 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA and the electromagnetic field. So, we will study the parametrized scalar electro- dynamics. 5. Parametrized Scalar Electrodynamics and Space-Time Diffeomorphisms Consider a massive charged scalar field in interaction with an electromagnetic field in a prescribed pseudo-Riemannian spacetime, the action for this system is given by S = ∫ d4X √−g ( −gµνgαβFµαFνβ 4 + gµνφ∗ ,µφ,ν − m2φ∗φ + LI ) , (5.1) where Fµα = A[α,µ] := Aα,µ − Aµ,α and the interaction Lagrangian is LI = iegµν ( φ∗ ,µφAν − φ∗φ,νAµ ) + e2gµνφ∗φAµAν (5.2) As pointed out in the previous section, in the canonical treatment of the evolution of a field one assumes it to be defined on a space-like 3-hypersurface Σ, equipped with coordinates σ, which is embedded in the spacetime manifold M by the mapping given by (4.3). Here, by adjoining the embedding variables to the phase space of the field results in a parametrized field theory where the spacetime coordinates have been promoted to the rank of fields. In terms of the spacetime coordinates σα = (τ, σ) determined by the foliation M = R × Σ, the action (5.1) becomes S = ∫ dτd3σ √−ḡ ( − ḡµν ḡαβFµαFνβ 4 + ḡµνφ∗ ,µφ,ν − m2φ∗φ + L̄I ) , τ ∈ R, (5.3) with the inverse metric ḡαβ given by ḡαβ = ∂σα ∂Xµ ∂σβ ∂Xµ , (5.4) which can be therefore seen as a function of the coordinate fields. In (5.3) ḡ := det(ḡµν) and √−ḡ = J is the Jacobian of the transformation. Also, L̄I is written in terms of the embedding metric (5.4). In order to carry out the Hamiltonian analysis of the action (5.3), we define in similar way to (4.4) the tangent vectors to Σ, Xα a and the unit normal nα = −(−ḡ00)− 1 2 ḡ0ρ ∂Xα ∂σρ . We thus arrive at S[Xµ, Pµ, Aa, πa, A⊥, φ, π, φ∗, π∗] = ∫ dτd3σ ( PαẊα + πaȦa + πφ̇ + π∗φ̇∗ − NΦ0 − NaΦa − MG ) , (5.5) where N and Na are the lapse and shift components of the deformation vector Nα := ∂Xα/∂τ , M = NA⊥ − NaAa, and Aa := Xα aAα, A⊥ := −nβAβ are the Noncommutativity and Parametrization of Fields 11 tangent and normal projections of the gauge potential. The constraints Φ0, Φa and G in (5.5) are defined by: Φ0 = Pαnα + 1 2 γ−1/2γabπ aπb + 1 4 γ1/2γacγbdFabFcd + γ−1/2ππ∗ + γ1/2γabφ∗ ,aφ,b + γ1/2m2φφ∗ − ieγ1/2γab ( φφ∗ ,aAb − φ∗φ,aAb ) + e2γ1/2γabAaAb, Φa = PαXα ,a + Fabπ b + πφ,a + π∗φ∗ ,a, G = πa ,a + ie (φ∗π∗ − φπ) , (5.6) where γab, γ are the metric components on Σ and their determinant, respectively. By a fairly lengthly calculation which makes repeated use of the various Poisson brackets derived in [3, 4], it can be shown that the above constraints satisfy the relations: {Φ0(σ) + A⊥(σ)G(σ), Φ0(σ ′) + A⊥(σ′)G(σ′)} = [ γab(σ)Φb(σ) +γab(σ′)Φb(σ ′) ] δ,a(σ, σ′), {Φa(σ) − Aa(σ)G(σ), Φb(σ ′) − Ab(σ ′)G(σ′)} = (Φb(σ) − Ab(σ)G(σ))δ,a(σ, σ′) + (Φa(σ′) − Aa(σ′)G(σ′)) δ,b(σ, σ′), {Φa(σ) − Aa(σ)G(σ), Φ0(σ ′) + A⊥(σ′)G(σ′)} = (Φ0(σ) + A⊥(σ)G(σ)) δ,a(σ, σ′), {Φ0(σ), G(σ′)} = 0, {Φa(σ), G(σ′)} = 0. (5.7) We have to stress here that the Gauss constraint G, needed to achieve the closure of the algebra of the super-Hamiltonian and super- momenta constraints, Φ0, Φa, under the Poisson-brackets, has to be modified according to the last equation in (5.6) to account for the interaction of the fields. Note however, that because of the gauge invariance implied by the Gauss constraint G ≈ 0, the scalar potential A⊥ occurs in (5.7) not as a dynamical variable but as a Lagrange multiplier. The end result of this mixing of constraints and consequent foliation dependence of the spacetime action in gauge theories, is that the super-Hamiltonian and the supermomenta constraints nαHα =H⊥ := Φ0(σ) + A⊥(σ)G(σ), Xα a Hα =Ha := Φa(σ) − Aa(σ)G(σ), (5.8) do not satisfy the Dirac closure relations (4.2) of section 4 ({Hα(σ),Hβ(σ′)} ≃ 0), so we do not have a direct homomorphic map from the Poisson brackets algebra of constraints into the Lie algebra of spacetime diffeomorphisms for such theories. Nonetheless, this difficulty can be circumvented by turning the scalar potential 12 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA into a canonical momentum Π, as shown in detail in [6] for the free Maxwell field Π := √ γA⊥, (5.9) conjugate to a supplementary scalar field ψ and prescribing their dynamics by imposing the Lorentz gauge condition. The new super-Hamiltonian and super- momenta ∗H⊥ : = H⊥ −√ γγabψ,aAb, ∗Ha : = Ha + Πψ,a, (5.10) of the modified theory satisfy the Dirac closure relations, and the mapping ξ → ∗Hτ [ξ] = ∫ Σ dσ ′ ξα(X(σ′)) ∗Hα (5.11) results in the desired anti-homomorphism : {∗Hτ [ξ], ∗Hτ [ρ]} = −∗Hτ [£ξρ], (5.12) from the Lie algebra £diffM ∋ ξ, ρ into the Poisson algebra of the constraints on the extended phase space Aa, πa, φ, φ∗, π, π∗, ψ, Π, Xα, Pα of the modified electrodynamics with the spacetime action: S(φ, ψ) = ∫ M dτd3σ √−ḡ ( − ḡµν ḡαβFµαFνβ 4 + ḡµνφ∗ ,µφ,ν − m2φ∗φ + L̄I + ψ,αḡαβAβ ) . (5.13) Note however that in order to recover Maxwell’s electrodynamics from the dynam- ically minimal modified action (5.13), one needs to impose the additional primary and secondary constraints C(σ) := ψ(σ) ≈ 0; G(σ) ≈ 0 (5.14) on the phase space data. In this way, the new algebra of constraints leading to scalar electrodynamics from (5.13) is: {∗H⊥(σ),∗ H⊥(σ′)} = γab(σ) ∗Hb(σ)δ,a(σ, σ′) − (σ ↔ σ ′), {∗Ha(σ),∗ H⊥(σ′)} = ∗H⊥(σ)δ,a(σ, σ′), {∗Ha(σ),∗ Hb(σ ′)} = ∗Hb(σ)δ,a(σ, σ′) − (aσ ↔ bσ′) {C(σ),∗ H⊥(σ′)} = (γ)− 1 2 (σ)G(σ)δ(σ, σ′), (5.15) {C(σ),∗ Ha(σ′)} = C,a(σ)δ(σ, σ′), {G(σ),∗ H⊥(σ′)} = ( (γ) 1 2 (σ)γab(σ)C,b(σ)δ(σ, σ′) ) ,a , {G(σ),∗ Ha(σ′)} = (G(σ)δ(σ, σ′)),a . Similarly to what it was noted in [6], this Poisson algebra implies that once the constraints (5.14) are imposed on the initial data they are preserved in the dynamical evolution generated by the total Hamiltonian associated with (5.13), so that if the derivations ∗Ĥτ [ξ] := δξ representing spacetime diffeomorphisms Noncommutativity and Parametrization of Fields 13 start evolving a point of the extended phase space lying on the intersection of the constraint surfaces ∗H⊥(σ) ≈ 0 ≈∗ Ha(σ) and C(σ) := ψ(σ) ≈ 0 ≈ G(σ), (5.16) the point will keep moving along this intersection. In summary, we have seen that for canonically parametrized field theories with gauge symmetries in addition to spacetime symmetries the Poisson algebra of the constraints does not agree with the Dirac relations and, therefore, cannot be directly interpreted as representing the Lie algebra of the generators of spacetime diffeomorphisms. The reason being that because of the gauge invariance there are additional constraints in the theory which cause that not all the relevant vari- ables are canonical variables. Following the arguments in [6] for the case of the electromagnetic field, we have seen that these difficulties can be circumvented by complementing the original action (5.1) with the addition of a term, containing the scalar field ψ, that enforces the Lorentz condition, so the modified action is given by (5.13). Varying this action with respect to the gauge potential Aα gives 1 2 ( √ −g 1 2 Fαβ),β = √ −g 1 2 gαβ ( ψ,β + ie(φ∗ ,βφ − φ∗φ,β) + 2e2φφAβ ) , (5.17) which therefore implies that the modified action introduces an additional source term into the Maxwell equations, so the dynamical theory resulting from (5.13) is not the same as scalar electrodynamics. Parenthetically, it is interesting to observe, that the additional charge source ψ on the right of (5.17) is a real field and not a complex one as one would have expected. The dynamical character of ψ, however, is made evident by differentiating this last equation with respect to Xα whereby, due to the vanishing of the left side, this field must satisfy the wave equation ψ,β ,β = − ( ie(φ∗ ,βφ − φ∗φ,β) + 2e2φφAβ ),β . (5.18) Consequently, in order to recover the scalar electrodynamics it is required that ψ vanish or at least that it is a spacetime constant, since (5.18) is equivalent to the current conservation. This was achieved by simply imposing additional constraints on the phase space data, given by (5.14), which implies loosing the generator of gauge transformations. This procedure, and its generalization to the case of non- Abelian Yang-Mills fields then allows (still within the canonical group theoretical framework) to undo the projection and replace the Poisson bracket relations (5.7) by the genuine Lie algebra £ diff M of spacetime diffeomorphisms. Note that even though the algebra in (5.15) involves derivatives of the con- straints G(σ) and C(σ), these derivatives can be removed by simply using the identity J(σ′)δ,a(σ, σ′) = J(σ)δ,a(σ, σ′) + J,a(σ)δ(σ, σ′), so the algebra does close, as it is to be expected from counting degrees of freedom. As a consequence the elements ∗Hτ [ξ], together with Gτ [ᾱ] := ∫ dσ ᾱ(X(σ))G(σ) and Cτ [β̄] := ∫ dσβ̄(X(σ))C(σ), form a closed algebra under the Poisson brackets. 14 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA On the basis of the above discussion and by argument which parallel those used in [14], we can derive explicit expressions for the generators of the Lie algebra of spacetime diffeomorphisms associated with the anti-homomorphism (5.12) and investigate whether these Lie algebra can be extended with the smeared elements Gτ [ᾱ] and Cτ [β̄] and, if so what would be the interpretation of such an extension. The end result of such a calculation is that, the largest Lie algebra acting on the space-time manifold that we can associate with the Poisson algebra (5.15) is the one of spacetime diffeomorphisms, given by the homomorphism implied by (5.12) and originating from the sub-algebra of the super-Hamiltonian and super- momenta described by the first 3 equations in (5.15). In order to elaborate further on the implications of these results on the possible structures of noncommutative gauge theories we need first to relate them to some basic aspects of gauge theory as formulated from the point of view of principal fiber bundles. 6. Gauge Transformations Recall (cf e.g.[15]) that a gauge transformation of a principal fiber bundle (PFB) π : P → M, with structure Lie group G, is an automorphism f : P → P such that f(pg) = f(p)g and the induced diffeomorphism f̄ : M → M, defined by f̄(π(p)) = π(f(p)), is the identity map f̄ = 1M (i .e.π(p) = π(f(p))). Moreover, if we define f : P → P by f(p) = pζ(p), where ζ is an element of the space C(P,G) of all maps such that ζ(pg) = g−1 · ζ(p) = Adg−1ζ(p) (so G acts on itself by an adjoint action), then C(P,G) is naturally anti-isomorphic to the group of gauge transformations GA(P ). That is, for f, f ′ ∈ GA(P ) and ζ, ζ′ ∈ C(P,G) we have that (f ◦ f ′)(p) = p(ζ′(p)ζ(p)). From the above, it can be readily shown that f∗(σu∗X) = d dt ( Rζ(p)−1◦ζ(σu(γ(t)))f(p) ) |t=0 + Rζ(p)∗(σu∗X), (6.1) where TM ∋ X = d dt (γ(t))|t=0, with γ(t) : R → U ∈ M and σu : U → P is a local section. Writing ζ(p)−1 ◦ ζ(σu(γ(t))) := etb as an element of a one-parameter subgroup of G, we have f∗(σu∗X) = b∗f(p) + Rζ(p)∗(σu∗X), (6.2) where b∗f(p) is the fundamental vector field on f(p) corresponding to b = L−1 ζ(p)∗ζ∗(σu ∗ X). (6.3) Consequently, (σ∗ uf∗ω)(X) = b + Ad(σ∗ uζ)(X)−1(σ∗ uω)(X). (6.4) In the above expressions, ωf(p) is a connection 1-form at f(p) ∈ P , (f∗ω)p is its pull-back to p with the gauge map f and (σ∗ uf∗ω)π(p) is in turn its pull-back with σu, and (σ∗ uζ)(Xµ) is a spacetime-valued element of G. Noncommutativity and Parametrization of Fields 15 Write now ζ as an element of a one-parameter subgroup of C(P,G) by means of the exponential map ζ = exp(−tαBTB), (6.5) where αBTB := α is an element of the gauge algebra space C(P, g), and the TB denote the basis matrices of the Lie algebra g associated with G, and replacing (6.5) into (6.3) and (6.4) we get (σ∗ u(Rexp(−tαB(p)TB)) ∗ω)(X) = d ds [exp(tᾱB(X)TB) exp(−sᾱB(γ(s))TB)]|s=0 + Adexp(tᾱB(X)TB)(σ ∗ uω)(X), (6.6) where ᾱB := (σ∗ uαB). The infinitesimal version of (6.6) follows directly by differ- entiating both sides of the above equation with respect to the parameter t and evaluating at zero. We therefore arrive at δᾱA := d dt (σ∗ u(Rexp(−tαBTB)) ∗ω)|t=0 = −dᾱ − [A, ᾱ] = −Dᾱ ∈ Λ̄1(M, g), (6.7) where Λ1(M, g) denotes the space of 1-forms on M valued in the Lie algebra g. In the particular case where the one-parameter group is Abelian, it immedi- ately follows that (6.7) simplifies to δᾱA = −i dᾱ. (6.8) Moreover, since (6.8) implies that δᾱAµ = −i ∂µᾱ, we obtain, by projecting on the sheet Σ with Xµ a, δᾱAa = −i ∂aᾱ(X(σ)). (6.9) Let us now turn to the Gauss constraint G(σ), introduced in (5.6), and to the smearing map ᾱ → Gτ [ᾱ] = ∫ Σ dσ ′ ᾱ(X(σ′)) G(σ′). (6.10) Clearly, i{Aa, Gτ [ᾱ]} ∼= δᾱAa, (6.11) so the Poisson bracket of the projection Aa of the gauge 4-vector onto the space- like hypersurface Σ with the Gauss constraint smeared with the scalar function ᾱ(X(σ′)) is the same as the pullback to M of the infinitesimal action of the gauge algebra of the PFB with group U(1) on the connection one-form ω evaluated on a tangent vector to Σ . In addition, for f ∈ GA(P ), it is a simple matter to show that if ω is a connection 1-form then the pullback f∗ω is also a connection 1-form. This theorem follows immediately by noting first that the action of f∗ω on a fundamental vector yields its corresponding Lie algebra generator, and second that the requirement ωpg(Rg∗X) = Adg−1ωp(X) in the definition of a connection 1-form is directly satisfied when acting on ω with the pullback of f ◦ Rg = Rg ◦ f , which in turn is equivalent the automorphism condition f(pg) = f(p)g. Let now V be a vector space on which G acts from the left. If Lg : V → V is linear, then the homomorphism G → GL(V ) by g → Lg is a representation of 16 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA G. In this case C(P, V ) will denote the space of all maps ζ : P → V such that ζ(pg) = g−1 · ζ(p) and the elements of C(P, V ) correspond to particle fields. In particular, C(P, V ) = Λ̄0(P, V ), where, in general, Λ̄k(P, V ) is the space of V -valued differential k-forms ϕ on P such that R∗ gϕ = g−1 · ϕ, ϕ(Y1, . . . ,Yk) = 0, if any one of the Y1, . . .Yk ∈ TpP is vertical. (6.12) Making now use of the exponential map (6.5) it readily follows that f∗ϕ = ζ−1 · ϕ. (6.13) Or, differentiating with respect to t and evaluating at t = 0, we arrive at the following infinitesimal version of (6.13): δᾱϕ̄ = ᾱBTB · ϕ̄. (6.14) Furthermore, related to our discussion in the following sections, note that from the definition of diffeomorphisms we have that Rg ◦ f = f ◦ Rg, thus acting with the pull-back of this equality on any element κ ∈ Λ̄k(P, V ), and recalling that the action of the differential f∗ on a fundamental field b∗ is again a fundamental field, it then immediately follows that (f∗κ)(b∗) = κ(b∗) = 0. Hence f∗κ ∈ Λk(P, V ), k = 0, 1, 2 . . . , and since C(P, V ) = Λ̄0(P, V ) it also follows that the gauge group GA(P ) acts on particle fields via pull-back, so that f∗ϕ(p) = ϕ(f(p)), (6.15) i.e. if ϕ is a particle field, so is also f∗ϕ. Using the above results we can now formulate the multiplication rules for gauge and particle fields under gauge transformations, when pulled-back to the base space M. Thus, given two g-valued potential 1-forms A, A′ ∈ Λ1(M, g), their product is defined by [A, A′] := ( Aa ∧ A′b ) ⊗ [Ta, Tb], (6.16) while the product of two particle fields ϕ1, ϕ2 ∈ C(P, V ) is by simple point mul- tiplication. Now, as shown previously, the action of an element f ∈ GA(P ) on a connection 1-form and on a particle field is via pull-back (c.f. Eqs.(6.4) and (6.15)) and since the pull-back of a connection is a connection and the pull-back of a particle field is a particle field, it therefore follows that f : [A, A′]  [(σ∗ uf∗ω1), (σ ∗ uf∗ω2)], (6.17) f : (σ∗ uϕ1)(π(p)) · (σ∗ uϕ2)(π(p))  (σ∗ uf∗ϕ1)(π(p)) · (σ∗ uf∗ϕ2)(π(p)). (6.18) By (6.7) and (6.14), the infinitesimal expression for the above is: δᾱ ([A, A′](X1,X2)) := µ[(δᾱ ⊗ 1 + 1 ⊗ δᾱ) ( Aa(X1) ⊗ A′b(X2) − Aa(X2) ⊗ A′b(X1) ) ] ⊗ [Ta, Tb] = (δᾱAa ∧ A′b − Aa ∧ δᾱA′b)(X1,X2) ⊗ [Ta, Tb], (6.19) Noncommutativity and Parametrization of Fields 17 and δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p))) = δᾱ(ϕ̄1(π(p)))·ϕ̄2(π(p))+ϕ̄1(π(p))·δᾱ(ϕ̄2(π(p))), (6.20) respectively. This last result implies that under an infinitesimal gauge transforma- tion the product of two particle fields transforms according to the Leibniz rule. We can therefore give this infinitesimal transformations the structure of a Hopf algebra with coproduct ∆δᾱ = δᾱ ⊗ 1 + 1 ⊗ δᾱ, so that δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p))) = µ[∆δᾱ (ϕ̄1(π(p)) · ϕ̄2(π(p)))]. (6.21) From the above discussion we can derive some additional insight into the implications of the PFB point of view of gauge transformations on our previous results. We thus see that since gauge transformations are automorphisms on the fibers that project to the identity on the base space, the Gauss constraint - which we have seen here to be related to the pull-back of the infinitesimal gauge trans- formations, and which was shown in Sec. 5 to be needed in order to close the algebra in (5.7) - occurs in the extended algebra (5.15) primarily as part of the super-Hamiltonian and super-momenta associated with the Lie algebra of space- time diffeomorphisms. Its independent appearance is then only as a constraint which, together with C(σ) ≃ 0, have to be implemented at the end as strong conditions in order to recover the Maxwell theory. This provides an additional natural explanation for why these two constraints can not be mapped into deriva- tions that could lead to an enlarged Lie algebra beyond the one of the spacetime diffeomorphisms. 7. Noncommutative Gauge Theories With these results in hand, let us now consider an approach for extending the theory of gauge fields to the noncommutative space-time case, following the for- malism established in Sec. 3. We will consider the case of the scalar electrody- namics discussed in the last two sections. Recall, in particular, that - because of the anti-homomorphism that can be established between the Poisson sub-algebra of the constraints occurring in the first 3 lines of (5.15), for the modified the- ory in extended phase space, and the Lie algebra £diff M -we can use the latter to investigate the deformed spacetime isometries of the system by requiring that this sub-algebra of constraints, modified by the noncommutativity of space-time, should continue obeying the Dirac relations, relative to the Dirac brackets result- ing from admitting an arbitrary symplectic structure in the action (5.5). This, as shown in [7], was needed in turn in order to incorporate into the parametrized canonical formalism the dynamical origin of star-noncommutativity from quan- tum mechanics [13]. Moreover, since the constraints depend on the metric of the embedding space-time, this last step would require in general a well developed the- ory of quantum mechanics in curved spaces and knowledge of the commutators of the operators representing the phase space coordinates. We shall defer such more general considerations for some future presentation, and concentrate here only on 18 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA the case of fields on flat Minkowski space-time and the corresponding quantum mechanics for the extended Weyl-Heisenberg group. Consequently, admitting a symplectic structure in the action (5.13) we have S[z] = ∫ d4σ ( B(z)AżA − Nα(∗H̃α) − MG̃(σ) − T C̃(σ) ) , (7.1) with the symplectic variables zA = (Xα, Aa, φ, φ∗, ψ; Pα, πa, π, π∗, Π) and sym- plectic potentials B(z)A to be determined by a prescribed symplectic structure. Here M, T are the additional Lagrange multipliers needed to recover scalar elec- trodynamics and the tildes on the formerly introduced constraints are needed, in order that their Dirac-bracket algebra, originated by the new symplectic structure, be identical to their sub-algebra in (5.15). That is, we want to maintain the al- gebra of these constraints invariant by utilizing new twisted generators. Observe however, that since the G̃(σ) and C̃(σ) can not form part of our Lie algebra of space-time isometries, but are strictly constraints to be implemented in order to obtain the scalar electrodynamics. As noted in [7], the symplectic structure is defined by, ωAB := ∂BB ∂zA − ∂BA ∂zB , (7.2) from where we can solve for the symplectic potentials, which are defined up to a canonical transformation. The resulting second-class constraints can then be elim- inated by introducing Dirac brackets, according to a scheme analogous to the one described in Sec. 3, from where the inverse of the symplectic structure is addi- tionally defined through the Dirac-brackets for the symplectic variables zA. More specifically, based on the premise that quantum mechanics is a minisuperspace of field theory and for a quantum mechanics on flat Minkowski space-time based on the extended Weyl-Heisenberg group, we have shown in [13] that the WWGM for- malism implies that, for the phase space variables to have a dynamical character, we need to modify their algebra by twisting their product according to µ(Xα ⊗ Xβ)  µθ(X α ⊗ Xβ) := Xα(τ, σ) ⋆θ Xβ(τ, σ′), (7.3) where ⋆θ := exp [ i 2 θµν ∫ dσ′′ ←− δ δXµ(τ, σ′′) −→ δ δXν(τ, σ′′) ] , (7.4) and where, since the embedding space-time variables are functionals of the fo- liation, we use functional derivatives. Also, since fields are in turn functions of the embedding space-time variables their multiplication in the noncommutative case is inherited from (7.3). Moreover, using this ⋆-product we can now define the commutator [Xα(τ, σ), Xβ(τ, σ′)]θ := Xα(τ, σ) ⋆θ Xβ(τ, σ′) − Xβ(τ, σ′) ⋆θ Xα(τ, σ) = iθαβδ(σ, σ′), (7.5) Noncommutativity and Parametrization of Fields 19 and let {Xα, Xβ}∗ = [Xα(τ, σ), Xβ(τ, σ′)]⋆θ = iθαβδ(σ, σ′). (7.6) On the other hand, defining the map X̃α = Xα + θαβ 2 Pβ , (7.7) it follows from (7.6) that {X̃α, X̃β}∗ = 0, (7.8) and {∗H̃α(σ),∗ H̃β(σ′)}∗ = 0. (7.9) Thus, in parallel to (5.12), we here have {∗H̃τ [ξ], ∗H̃τ [ρ]}∗ = −∗H̃τ [£ξρ]. (7.10) Furthermore, making the identification Pβ = −i δ δXβ in the Darboux map (7.7) we can write X̃α  ˆ̃Xα = (Xα) ⋆−1 θ := (Xα) exp [ − i 2 θµν ∫ dσ′′ ←− δ δXµ(τ, σ′′) −→ δ δXν(τ, σ′′) ] , (7.11) where the bi-differential acting from the right on the embedding coordinates Xα is the inverse of (7.4). Hence {X̃α, X̃β}∗ ∼= [ ˆ̃Xα, ˆ̃Xβ]⋆θ = [Xα, Xβ] ⋆−1 θ = 0, (7.12) since under point multiplication the embedding coordinates commute. So the map (7.11) retrieves (7.8). In addition, since multiplication in the algebra of the operators ˆ̃Xα is by the ⋆θ-product we can generalize the last result to {(X̃α)m, (X̃β)n}∗ ∼= [( ˆ̃Xα)m ⋆ , ( ˆ̃Xβ)n ⋆ ]⋆θ = [(Xα)m, (Xβ)n] ⋆−1 θ = 0. (7.13) We can therefore conclude from the above that, when replacing the func- tional dependence on the embedding variables in the constraints in (5.15) by the “tilde” variables (7.7) and the point multiplication of fields by their ⋆-product, the functional form of their algebra is evidently preserved for the noncommutative case. That is, {∗H̃τ [ξ], ∗H̃τ [η]}∗ ∼= [∗Ĥτ [ξ], ∗Ĥτ [η]]⋆ ⋆−1 θ , (7.14) and ∗Ĥτ [ξ] = δξ  ∗Ĥτ [ξ] ⋆−1 θ = δ⋆ ξ , (7.15) where the multiplication µθ of the algebra of generators of diffeomorphisms δ⋆ ξ ∈ £diff M is via the ⋆θ-product. Consequently, by using the example of the modified scalar electrodynamics within the context of canonical parametrized field theory, it was shown that, by includ- ing additional constraints, Maxwell’s equations could be recovered as well as the 20 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA possibility of also establishing for gauge field theories the anti-homomorphism be- tween Dirac-brackets of the modified constraints and space-time diffeomorphisms. Furthermore using our previous results in [7] where it was shown that noncom- mutativity in field theory - manifested as the twisting of the algebra of fields - has a dynamical origin in the quantum mechanical mini-superspace which, for flat Minkowski space-time, is related to an extended Weyl-Heisenberg group, and including these results into the symplectic structure of the parametrized field the- ory then allowed us to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms, as shown by (7.14) and (7.15) above. We turn now to the derivation of the explicit form for the modified first-class constraints ∗H̃τ [ξ], by observing that the formalism requires that their algebra should now close relative to the Dirac-brackets. Now, taking into account that the Dirac-bracket algebra of the variables zA = (X̃α, Aa, φ, φ∗ψ; Pα, πa, π, π∗, Π), is the same as the Poisson algebra of zA = (Xα, Aa, φ, φ∗ψ; Pα, πa, π, π∗, Π), it therefore follows that ∗H̃⊥ = Pαñα + 1 2 γ̃−1/2γ̃abπ aπb + 1 4 γ̃1/2γ̃acγ̃bdFabFcd + γ̃−1/2ππ∗ + γ̃1/2γ̃abφ∗ ,aφ,b + γ̃1/2m2φφ∗ − √ γ̃γ̃abψ,aAb − ieγ̃1/2γ̃ab ( φφ∗ ,aAb − φ∗φ,aAb ) + e2γ̃1/2γ̃abAaAb + A⊥G, ∗H̃a = PαX̃α ,a + Fabπ b + πφ,a + π∗φ∗ ,a − AaG + Πψ,a, G = πa ,a + ie (φ∗π∗ − φπ) , C = ψ (7.16) We notice, that the constraints G and C are not changed and in consequence the gauge transformations are not deformed. We will rename the full set of deformed constraints as CA = (∗H̃⊥,∗ H̃a, G, C). (7.17) Then, when quantizing, the constraints CA are promoted to the rank of operators satisfying, in the same way that in Quantum Mechanics, the subsidiary conditions ĈA|Ψ〉 = 0, (7.18) but now this equations are functional equations instead of differential equations like in Quantum Mechanics. To make quantum mechanical sense of Eqs.(7.18) we need to select a basis. The most useful basis will be | X̃µ, φ, φ∗, Aa, Π〉. (7.19) For this basis the action of the constraint C = ψ is trivial, and we get that the states are independent of Π. On the other hand the action of the Gauss law constraint G, implies that the states only depend on two polarization states of the electromagnetic field. The interesting point of the basis (7.19) is the fact that the deformed coordinate fields X̃α(σ), have a trivial action, then the metric γ̃ab have exactly the same action over the states that in the commutative case. So for Noncommutativity and Parametrization of Fields 21 this basis the noncommutative quantum theory is completely equivalent to the commutative one. Now, if we select a different kind of basis, of course with respect to the noncommutative coordinates, we will get a completely different quantum field theory. So, for noncommutative theories, different basis are not physically equivalents. An additional point that we must check for consistency at the quantum level are that the conditions [ĈA, ĈB]|Ψ〉 = 0, (7.20) be satisfied. This implies that the commutator of the first class constraint operators has to be of the form [ĈA(τ, σ), ĈB(τ, σ′)] = ĉAB C(σ, σ′)ĈC , (7.21) where, in general, the ĉAB C are functions of the field operators that need to appear to the left of the ĈC . This, in turn, involves finding the operator ordering needed to achieve this requirement in order to have an appropriate quantum theory. In our four dimensional case this problem has no obvious solution. In the two dimensional case we were able to manage this issue in [7]. See also [16] where this problem has been treated using the so called Polymer parametrized field theory. It would be interesting to compare the relation of both approaches. Now, as observed in [14], by making use of (7.15) we can summarize the action of space-time diffeomorphisms on particle fields associated with gauge theories, and the transition of the theory to the noncommutative space-time case by means of the following functorial diagrams: ∗Hτ [ξ] ∈ V θ−−−−→ V⋆ ∋ ∗H̃τ [ξ] = ∫ dσ(ξ̃⊥ ∗H̃⊥ + ξ̃a ∗H̃a) C    C    ∗Ĥτ [ξ] ∈ V̂ C(θ)−−−−→ V̂⋆ ∋ ∗Ĥτ [ξ] ⋆−1 θ ≡ δ⋆ ξ ; (7.22) (where V denotes the space of constraints satisfying the algebra (5.15), V⋆ is the corresponding space of constraints for the space-time noncommutative case with the embedding coordinates mapped according to (7.7) and V̂ , V̂⋆ denote the spaces of the Lie algebra of diffeomorphisms and their corresponding twisted form, respectively) and ϕ̄ ∈ A δξ−−−−→ A ∋ δξ ⊲ ϕ̄ D    D    ϕ̄ ∈ Aθ D(δ⋆ ξ ) −−−−→ Aθ ∋ δ⋆ ξ ⊲ ϕ̄ = δ⋆ ξ ⋆θ ϕ̄(X(τ, σ)); (7.23) (here A denotes the module algebra of particle fields ϕ̄ ∈ C(M, V ) with point multiplication µ and Aθ is its noncommutative twisting with ⋆-multiplication µθ := µ ◦ e i 2 θµν∂µ⊗∂ν ). 22 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA It then follows from these two diagrams that {ϕ̄, ∗Ĥτ [ξ]} ∼= δξ ⊲ ϕ̄ → δ⋆ ξ ⋆θ ϕ̄(X(τ, σ)) = ∗Ĥτ [ξ] ⊲ ϕ̄. (7.24) Note that the diagrams (7.22), (7.23) and Eq.(7.24) provide an explicit ex- pression for the mappings δρ → δ⋆ ρ, which in turn imply [ δ⋆ ρ, δ⋆ η ] ⋆θ = δ⋆ £ρη, (7.25) and δ⋆ ρ ⋆θ (ϕ̄1 ⋆θ ϕ̄2) = δρ(ϕ̄1 ⋆θ ϕ̄2), (7.26) where ϕ̄1, ϕ̄2 ∈ Aθ. Note also that the universal envelopes U(V̂) and U(V̂⋆) of the derivations δξ and twisted derivations δ⋆ ξ can be given the structure of Hopf algebras. Thus, in particular, we can obtain an explicit expression for the coproduct in U(V̂⋆) by making use of the duality between product and coproduct, followed by the application of equation (7.26). We get µθ ◦ ∆(δ⋆ ρ)(ϕ̄1 ⊗ ϕ̄2) = δ⋆ ρ ⋆θ (ϕ̄1 ⋆θ ϕ̄2) = δρ(ϕ̄1 ⋆θ ϕ̄2) = µ(δρ ⊗ 1 + 1 ⊗ δρ)(e i 2 θµν∂µ⊗∂ν ϕ̄1 ⊗ ϕ̄2) = ∑ n 1 n! ( i 2 )nθµ1ν1 . . . θµnνn [ (δ⋆ ρ ⋆θ ∂µ1...µn ϕ̄1)e − i 2 θµν←−∂ µ −→ ∂ ν ⋆θ ∂ν1...νn ϕ̄2 + (∂µ1...µn ϕ̄1)e − i 2 θµν←−∂ µ −→ ∂ ν ⋆θ (δ⋆ ρ ⋆θ ∂ν1...νn ϕ̄2) ] = µθ ◦ [ e− i 2 θµν∂µ⊗∂ν (δ⋆ ρ ⊗ 1 + 1 ⊗ δ⋆ ρ)e i 2 θµν∂µ⊗∂ν ] (ϕ̄1 ⊗ ϕ̄2). (7.27) This result compares with the Leibniz rule given in [17]. Furthermore, if we let F = e− i 2 θµν∂µ⊗∂ν ∈ U(V̂) ⊗ U(V̂), and define ϕ̄1 ⋆θ ϕ̄2 = µθ(ϕ̄1 ⊗ ϕ̄2) := µ(F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2)), we then have [18]: δρ(ϕ̄1 ⋆θ ϕ̄2) = δρ ⊲ µ(F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2)) = µ[(∆δρ)F−1 ⊲ (ϕ̄1 ⊗ ϕ̄2))] = µF−1[(F(∆δρ)F−1)((ϕ̄1 ⊗ ϕ̄2)))] (7.28) = µθ[(F(∆δρ)F−1)((ϕ̄1 ⊗ ϕ̄2)))]. Thus, the undeformed coproduct of the symmetry Hopf algebra U(V̂) is related to the Drinfeld twist ∆F by the inner endomorphism ∆Fδρ := (F(∆δρ)F−1) and, by virtue of (7.28), it preserves the covariance: δρ ⊲ ((ϕ̄1 · ϕ̄2))) = µ ◦ [∆(δρ)(ϕ̄1 ⊗ ϕ̄2))] = (δρ(1) ⊲ ϕ̄1) · (δρ(2) ⊲ ϕ̄2) θ→ δ⋆ ρ ⊲ (ϕ̄1 ⋆θ ϕ̄2) = (δ⋆ ρ(1) ⊲ ϕ̄1) ⋆θ (δ⋆ ρ(2) ⊲ ϕ̄2), (7.29) where we have used the Sweedler notation for the coproduct. Consequently, the twisting of the coproduct is tied to the deformation µ → µθ of the product when the last one is defined by ϕ̄1 ⋆θ ϕ̄2 := (F−1 (1) ⊲ ϕ̄1)(F−1 (2) ⊲ ϕ̄2). (7.30) Noncommutativity and Parametrization of Fields 23 We want to reiterate at this point that the ⋆-product, associated with the algebra Aθ, that we have been considering here is the one originated when con- sidering in turn the flat-Minkowski space-time quantum mechanics generated by the extended Weyl-Heisenberg group H5, for the even more particular case of an extension of the Lie algebra of H5 by the commutator [Xµ, Xν] = iθµν , in the simplest case when θµν = constant. In this case the generators δρ of isometries become the infinitesimal generators of the Poincaré group of transformations, and the coproduct defined in this equation reduces to the twisted coproduct consid- ered by e.g. [19]. Since the embedding coordinates in the canonical parametrized theory can in general be associated to a curved space-time manifold and, since the constraints and related diffeomorphisms are constructed for such spaces, it seems possible in principle that our formalism could be extended to curved space-time backgrounds with a ⋆-product determined by the Lie algebra associated with, for instance, a given homogeneous space. This would imply finding first the equiva- lent of the mapping (7.7) and also, of course, the realization of this map in terms of the ⋆-product, perhaps by a procedure based on the deformation quantization formalism developed by Stratonovich [20]. A fairly simple example of the above is the Darboux map given in [21], for the case of the Snyder algebra [22]. However, finding a full realization of the ⋆-product is a more difficult job. In Eq. (6.20) of the previous section we derived the expression for the in- finitesimal gauge transformation on a product of particle fields in A. Let us now consider the effect of such a gauge transformation on the product of two particle fields in Aθ when we have space-time noncommutativity. For this purpose we first recall Eq.(6.15) which shows that if ϕ is a particle field, so is its gauge transforma- tion by pull-back, i.e. ϕ ∈ C(P, V ) ⇒ ϕ′ := f∗ϕ ∈ C(P, V ). From this it follows that to a given element of C(P, V ) we can always associate another one which is the pull-back of the former, thus the twisted product of the pull-back with the section σu of any pair of particle fields can be written as ϕ̄′ 1 ⋆θ ϕ̄′ 2 = (σ∗ u(f∗ϕ1)) ⋆θ (σ∗ u(f∗ϕ2)). (7.31) Observe however that, because of the noncommutativity that the algebra (7.5) of the embedding coordinates is required to satisfy, the pull-back to M of the gauge transformation (6.13) now should be understood as σ∗ uf∗ϕ = ζ̄−1 ⋆ (X) ⋆θ ϕ̄(X); so that ϕ̄′ 1 ⋆θ ϕ̄′ 2 = (ζ̄−1 ⋆ ⋆θ ϕ̄1) ⋆θ (ζ̄−1 ⋆ ⋆θ ϕ̄2), (7.32) where, due to the noncommutativity, Eq.(6.5) is replaced by ζ̄−1  ζ̄−1 ⋆ = exp⋆(tᾱ(X)) := 1 + tᾱ + t2 2 ᾱ ⋆θ ᾱ + . . . (7.33) Using the infinitesimal version of this map we have that ϕ̄′ 1 = ϕ̄ + ᾱ ⋆θ ϕ̄, so that (7.32) becomes δᾱ : (ϕ̄1 ⋆θ ϕ̄2) = (ᾱ(X) ⋆θ ϕ̄1(X)) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ (ᾱ(X) ⋆θ ϕ̄2(X)). (7.34) 24 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA By a similar argument, since f ∈ GA(P ) also maps connections into connections, its infinitesimal action on the ⋆-product of two gauge fields (cf. (6.19)) goes into δᾱ : ([A, A′]⋆θ (X1,X2)) := − [( dᾱA(X1) + 1 2 cA CD[AC(X1), ᾱ D(X1)]⋆θ ) ⋆θ A′B(X2)− − ( dᾱA(X2) + 1 2 cA CD[AC(X2), ᾱ D(X2)]⋆θ ) ⋆θ A′B(X1) + AA(X1) ⋆θ ( dᾱB(X2) + 1 2 cB CD[A′C(X2), ᾱ D(X2)]⋆θ ) − AA(X2) ⋆θ ( dᾱB(X1) + 1 2 cB CD[A′C(X1), ᾱ D(X1)]⋆θ )] ⊗ [TA, TB]. (7.35) Note that we have written the last two equations for the general case of any group of gauge transformations, where ᾱ(X) = ᾱBTB, in order to underline the fact that, because of the ⋆-product in the multiplication of the fields one needs to apply the constraint that these NC gauge groups have to be in the fundamental or adjoint unitary representation (i.e. TA ∈ U(n)), since only in this representation the gauge group closes (cf. e.g. [23, 24] ). See however also [25] for arguments tending to circumvent this constraint. Hence, in the NC case the generators of gauge symmetry act on particle fields with the fundamental representation ϕ̄  ϕ̄′ = ζ−1 ⋆ ⋆θ ϕ̄ = exp⋆(tᾱ(X)) ⋆θ ϕ̄, (7.36) while on gauge fields the action is via the adjoint representation A(X)  A′(X) = ζ−1 ⋆ ⋆θ A(X) ⋆θ ζ⋆ + ζ−1 ⋆ ⋆θ (dζ⋆)(X). (7.37) Equations (7.36) and (7.37) agree with those on which [26] is based when remarking on some of the conclusions on deformed gauge theories arrived at in [27, 28, 29, 30]. Indeed, one basic idea in this other approach of gauge twisted theories is the assumption that the gauge generators δᾱ := ᾱ(X) = ᾱB(X)TB act on particle and gauge fields with the usual point product, so instead of (7.34) they define δᾱ(ϕ̄1 ⋆θ ϕ̄2) := (δᾱϕ̄1) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ (δᾱϕ̄2). (7.38) Moreover, by assuming that the algebra of the gauge generators can be given an additional Hopf bialgebra structure, and that the derivatives of any order of the gauge and particle fields are, as noted in [26], in the same representation of the gauge algebra as the fields themselves, one could further write δᾱ(ϕ̄1 ⋆θ ϕ̄2) = (ᾱ(X)ϕ̄1) ⋆θ ϕ̄2 + ϕ̄1 ⋆θ ᾱ(X)ϕ̄2. = µ ◦ (δᾱ ⊗ 1 + 1 ⊗ δᾱ) ◦ (e i 2 θµν∂µ⊗∂ν ϕ̄1 ⊗ ϕ̄2) (7.39) = µθ[(∆ Fδᾱ) ◦ (ϕ̄1 ⊗ ϕ̄2)]. Assuming a scalar particle field for simplicity and setting ϕ̄2 = ∂µϕ̄ and ϕ̄1 = ∂µϕ̄†, it can be readily seen that one immediate consequence of the extra assumption Noncommutativity and Parametrization of Fields 25 leading to equating the last two lines in (7.39) with the first one is that the latter then yields: δᾱ(∂µϕ̄† ⋆θ ∂µϕ̄) = 0, (7.40) which implies that the kinetic terms in the Lagrangian of the particle fields are invariant by themselves, so there would be no need to introduce the gauge po- tentials to achieve gauge invariance of the theory. Consequently, since (7.39) only fully agrees with (7.34) when ᾱ is coordinate independent, there appears to be a discrepancy as a consequence of local internal symmetry between assuming the validity of (7.38) and some essential aspects of the theory of gauge invariance. Recall furthermore, that a Drinfeld twist (cf. e.g. [18, 31, 32] ) involves a simultaneous and covariant deformation of the product of an algebra A of func- tions and the coproduct of a bialgebra H . More specifically, the algebra A is a module algebra (H-module algebra) over a Hopf bialgebra whose elements are in the universal enveloping algebra U(L) of a Lie algebra L, such that if x ∈ L then ∆(x) = x⊗1+1⊗x, and x(ab) = x(a)b+ax(b) ∀ a, b ∈ A, so that x acts as a deriva- tion. On the other hand, as shown by equations (6.10) and (6.11), the infinitesimal gauge transformation of the gauge potential is given by the Poisson bracket of the smeared Gauss constraint Gτ [ᾱ] with the gauge potential; but, as it was also shown in Sec.5 of this paper, the δᾱ can not be made isomorphic to a derivation operator acting as such on the gauge potentials or particle fields, contrary to the case of the smeared super-Hamiltonian and super-momenta constraints. Consequently the algebra of the infinitesimal gauge transformations can not be considered as part of the Hopf algebra of the space-time diffeomorfisms δξ, associated with Lie algebra L and its universal envelope, from which a Drinfeld twist could be properly con- structed. Note also that in the context of the canonical parametrized formalism, the Gauss constraint is defined on the spacelike hypersurface Σ and, again con- trary to the super-Hamiltonian and super-momenta constraints, does not depend on the embedding variables. This translates in the fact that for the NC case the space-time diffeomorphisms δξ, on the one hand, and the infinitesimal gauge trans- formations δᾱ, on the other, act quite differently on the gauge and particle fields. This is clearly seen when comparing the actions (7.15) and (7.36) on the gauge and particle fields, as well as their actions (7.28) and (7.34) on their respective products. 8. Gauge Invariance, Space-time Diffeomorphisms and Twist Symmetries on PFB. As we noted in Sec. 6 space-time diffeomorphisms act on the base space of a PFB while gauge transformations act on the fibers. As a consequence of this the infinitesimal generator of gauge transformations δα defined by the right side of the Poisson bracket in (6.11) can not viewed as an element of a Lie algebra that would enlarge the Lie algebra of space-time diffeomorphisms £diff M. Space-time and gauge diffeomorphisms act on different spaces. However instead of pulling back 26 L. R. Juarez, M. Rosenbaum and J. D. Vergara AACA gauge transformations to the base space of the bundle as done in Se. 6, we could ask if a more general Lie algebra including the gauge fields could be derived by horizontally lifting the space-time diffeomorphisms to the bundle space P . The answer to this is also negative as may be seen from the following argument: Consider the Lie commutator [ δα, δξ̃ ] = £αξ̃, (8.1) where α(p) = αB(p)TB, and ξ̃ is the horizontal lift of the generator ξ = ξµ(X) ∂ ∂Xµ of space-time diffeomorphisms. But, as we have shown previously C(P, g) ∋ α(p) = d dt exp(tα)|t=0, so the element ζ(t) = exp(tα) of the one parameter subgroup of C(P,G) describes a curve along the fiber π−1(x). It then follows that [ δα, δξ̃ ] = d dt ( R−1 ζ ∗ ξ̃ ) = 0, (8.2) since ξ̃ is horizontal. Hence the two algebras are independent of one another. It thus appears from our present results as well as from those in [7] (where the noncom- mutative reparametrized scalar field was considered and its respective constraints together with their anti-homomorphic relation to space-time diffeomorphisms was explicitly established), that it might not be possible to extend the concept of a Drinfeld twist symmetry to include gauge symmetries, when considering the min- imal coupling of gauge and particle fields in order to investigate a full model of NC theory in the context of the canonical reparametrized theory [33]. However, if one were to consider relaxing the concept of twisted symmetries and modify the definition of a deformed Leibniz rule (such as the one exhibited in (7.38)), several different twists and gauge invariants may be constructed that would lead to alternate formulations for NC gauge theories. Some new ideas in this context that might help to remove some of the inconsistencies pointed out here as well as elsewhere, are discussed in [34]. This would involve, essentially, assuming different deformations of products of elements in the same algebra of space-time functions A, when considering different transformation groups. Such an assumption how- ever, would be hard to reconcile with the point of view that the product in this algebra of functions is inherited from the deformation of the algebra of space-time coordinates and its dynamical origin in the quantum mechanical mini-superspace. As it was remarked previously the ⋆-product considered so far applies to an underlying flat Minkowski space-time, and the corresponding twisted isometries refer then to the Poincaré group. It is interesting to observe, however, that our formalism admits a natural extension of (7.4) which allows us to consider much more general symplectic structures than (7.2) that would imply noncommutativ- ity among all the symplectic variables zA = (Xα, Aa, φ, φ∗, ψ; Pα, πa, π, π∗, Π). Moreover, because of the appearance of the embedding metric in the canonical parametrized formalism, this could lead in turn to the possibility of extending our analysis to the case of twisted isometries on curved space backgrounds. As an ad- ditional remark related to future work, one can consider an approach similar to the one discussed in [35], where the quantum mechanical Groenewold-Moyal ⋆-product Noncommutativity and Parametrization of Fields 27 is formulated in terms of Clifford algebras, in order to extend the Grassmann alge- bra of spinor fields to Clifford algebras. In this way, it could be possible to extend our formalism to the case of non-commutative spinors. References [1] R. J. 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Acknowledgment The authors are grateful to Prof. Karel Kuchař for fruitful discussions and clarifi- cations concerning his work on parametrized canonical quantization. The authors also acknowledge partial support from CONACyT projects UA7899-F (M.R.) and 47211-F(J.D.V.) and DGAPA-UNAM grant IN109107 (J.D.V.). L. Roman Juarez, Marcos Rosenbaum and J. David Vergara Instituto de Ciencias Nucleares Universidad Nacional Autónoma de México A. Postal 70-543 , México D.F. México e-mail: roman.juarez@nucleares.unam.mx mrosen@nucleares.unam.mx vergara@nucleares.unam.mx Received: October 15, 2008. Accepted: December 17, 2008. Ashdin Publishing Journal of Physical Mathematics Vol. 2 (2010), Article ID P100803, 22 pages doi:10.4303/jpm/P100803 On deformed quantum mechanical schemes and ⋆-value equations based on the space-space noncommutative Heisenberg-Weyl group L. ROMÁN JUÁREZ and Marcos ROSENBAUM Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D.F., Mexico E-mails: roman.juarez@nucleares.unam.mx, mrosen@nucleares.unam.mx Abstract We investigate the Weyl-Wigner-Gröenewold-Moyal, the Stratonovich, and the Berezin group quantization schemes for the space-space noncommutative Heisenberg- Weyl group. We show that the ⋆-product for the deformed algebra of Weyl functions for the first scheme is different than that for the other two, even though their respective quantum mechanics’ are equivalent as far as expectation values are concerned, provided that some additional criteria are imposed on the implementation of this process. We also show that it is the ⋆-product associated with the Stratonovich and the Berezin formalisms that correctly gives the Weyl symbol of a product of operators in terms of the deformed product of their corresponding Weyl symbols. To conclude, we derive the stronger ⋆-valued equations for the 3 quantization schemes considered and discuss the criteria that are also needed for them to exist. 2000 MSC: 81Q99, 81R60, 81S30 1 Introduction It is well known [15, 16, 27, 34, 35, 36] that for nonrelativistic standard quantum mechan- ics, the expectation value of an operator on Hilbert space can be formally represented as a statistical-like average of the corresponding Weyl phase-space function with the statistical density given by the Wigner function associated with the density matrix of the quantum state. Moreover, when applying this scheme to a product of two arbitrary operator functions of the quantum position and momentum operators, their corresponding Weyl phase-space function was given by the exponential of the Poisson bidifferential acting on the Weyl equiv- alent of each of the two operators. This correspondence between the product of quantum operators and the twisted product of their classical phase-space equivalents can be viewed as a deformation of the point product in the algebra A of C∞ phase-space functions with the Gröenewold-Moyal multidifferential operator: ⋆~ := exp [ i~ 2 Λ ] := exp [ i~ 2 (←−∇q · −→∇p − ←−∇p · −→∇q ) ] , inducing this deformation. This concept of a twisted product was given a more general mathe- matical framework by Bayen et al. in [5], whose proposed deformation quantization paradigm and noncommutative symbol calculus led to an autonomous reformulation of quantum theory 2 L. Román Juárez and M. Rosenbaum directly in terms of phase-space functions, composed via the twisted or ⋆-product, instead of operators and Hilbert space states. While applications of the original Weyl-Wigner-Gröenewold-Moyal (WWGM) formalism were restricted to the description of systems in flat phase space, the systems under con- sideration in the more general deformation quantization scheme possess an intrinsic group of symmetries with the phase-space being an homogenous manifold on which the group of transformations acts transitively [2, 4, 12, 17, 18, 22, 23, 24, 25, 26]. This implies then the possibility of extending the phase space approach to the “quantization” of curved spaces. However, for the various known versions of deformation theory, there are a large variety of ⋆-products which in turn imply, in general, different quantum mechanical theories for the same problem. In order to deal with such nonuniqueness and arrive at a ⋆-product that would ensure the physical equivalence of deformation quantization with the ordinary quantum mechan- ics, the need for supplementary conditions has been suggested, so that the linear bijective mapping between operators on Hilbert space and classical functions on phase space can be implemented by a kernel operator which satisfies a number of physically sensible postulates thus hopefully providing a scheme to single out the most adequate symbol calculus from the many that have and could be proposed. Moreover, such nonuniqueness becomes manifest even for quantum deformation schemes with known equivalent ⋆-products in flat space-time standard quantum mechanics, when space-space and/or space-time noncommutativity is incorporated into the formalism. This noncommutative quantum mechanics and the behavior of classical fields, defined as functions of noncommutative spatial variables, have been the object of a great deal of attention in the last years. Physicists became attracted to the more mathematical aspects of deformation quantization with the hope that such theories would provide the tools needed to remove the singularities in physical field theories without the need of renormalization. Although these expectations have not materialized up to now, noncommutative field theory and its quantum mechanical minisuperspace have led to many new and interesting results. In particular, in the context of string theory, there has been a lot of interest in studying solitonic solutions of noncommutative field theory [3, 13, 14, 19, 30]. Also motivated by that work, but in a somewhat different direction, coherent structures in the form of noncommutative solitons and vortices were studied by the authors in a recent collaboration [21]. It was shown there that the noncommutativity of the spatial variables, when averaged with vortex or plateau-type coherent states, induced an effective lattice structure of Landau cells whose distribution and size depended on the coherent states considered. This shows that the effect of the noncom- mutativity on coherent structures, with an amplitude comparable to the scale parameter θ of noncommutativity of the ⋆-product, is to induce a behavior of classical structures in a physical lattice whose dynamics can be described in terms of a Peierls-Nabarro potential. It would not be unreasonable to expect that such dynamical creation of lattice structures as an effect of the noncommutativity on coherent states, which mathematically would be reflected in the replacement of differential field equations by equations of differences, could be related to another important quantization scheme known as loop quantum gravity. This final objective forms part of an ongoing program initiated in [21], and it is within that much wider context that the present work is intended. Thus, in order to arrive at an identification of the ⋆-product appropriate for the above mentioned program, we will here specifically start by extending the WWGM procedure in order to analyze a space-space noncommutative Heisenberg-Weyl algebra (again, noncom- mutativity being understood here as a nonvanishing commutator between the operators of On deformed quantum mechanical schemes and ⋆-value equations 3 spatial coordinates or momenta) in order to obtain the generalization of the well-known ex- pressions of the Heisenberg-Weyl algebra of usual quantum mechanics. Afterwards, we will apply to this same Lie algebra two quantization formalisms which are purportedly more general and that were developed to provide a quantization scheme even for curved spaces. The first one started with the work of Stratonovich [31] and was further developed elsewhere [8, 11, 33]. The second corresponds to the Berezin geometric quantization program of co- variant and contravariant symbols for Kähler manifolds [6]. Finally, we derive the additional specific requirements that need to be imposed on these different schemes, in order to ob- tain ⋆-valued equations which constitute a stronger quantization requirement, as they relate eigenvalues of the physical states appearing in the density matrix to the Weyl equivalents of the operator observables. 2 The WWGM phase-space quantum mechanics based on the space-space noncommutative Heisenberg-Weyl Lie algebra By a space-space (and/or momentum-momentum) noncommutative Heisenberg-Weyl alge- bra, we understand [29] the algebra of position and momentum operators satisfying the commutation relations: [ R̂i, R̂j ] = iθij Î , [ P̂i, P̂j ] = i~θ̄ij Î , [ R̂i, P̂j ] = i~δij Î , (2.1) where R̂i, P̂i, i = 1, . . . , d are the components of the position and momentum quantum operators, respectively, with component eigenvalues on R d, the identity Î is the central element of the algebra, and θij and θ̄ij are evidently antisymmetric matrices, which in the most general case can be functions of the generators of the above algebra. For our present purposes and algebraic simplicity, in what follows, we will set θ̄ij = 0 and d = 2 and consider only the zeroth order constant term of the Taylor expansion of θ12 ≡ θ. From an intrinsically noncommutative operator point of view, the development of a formu- lation for the quantum mechanics based on the above Heisenberg-Weyl algebra of operators requires first a specification of a representation for the generators of the algebra, second a specification of the Hamiltonian which governs the time evolution of the system, and last a specification of the Hilbert space on which these operators and the other observables of the theory act. As for the choice of the Hilbert space, a reasonable assumption is that it can be taken to be the same as that for the corresponding system in the usual quantum mechanics, but for a realization of the space-space noncommutative Heisenberg-Weyl algebra, because of the noncommutativity (2.1), we cannot use configuration space as a basis. We can use, however, for a basis either of the eigenkets |p1, p2〉, |q1, p2〉, |q2, p1〉, of the commuting pairs of observables (P̂1, P̂2), (R̂1, P̂2), or (R̂2, P̂1), respectively, or any combination of the (R,P ) such that they form a complete set of commuting observables. Specifically, we choose as the realization of our Heisenberg-Weyl algebra the one based on |q1, p2〉. The construction follows standard procedures (cf., e.g., [20]) and it is detailed in [29]. We then have that R̂2 in this basis is realized by R̂2 = −iθ∂q1 + i~∂p2 , (2.2) P̂1 = −i~∂q1 . (2.3) The representations for the remainder of the generators R̂1 and P̂2 of the algebra are obviously just multiplicative. Note that the change of basis |q1, p2〉 → |q2, p1〉 follows directly from the 4 L. Román Juárez and M. Rosenbaum transition function 〈q1, p2|q2, p1〉, which is derived [1] by noting that 〈 q1, p2|R̂2|q2, p1 〉 = q2 〈 q1, p2 ∣ ∣q2, p1 〉 = i ( ~∂p2 − θ∂q1 )〈 q1, p2|q2, p1 〉 , 〈 q1, p2|P̂1|q2, p1 〉 = p1 〈 q1, p2|q2, p1 〉 = −i~∂q1 〈 q1, p2|q2, p1 〉 . Combining these two expressions yields ( ~q2 − θp1 )〈 q1, p2|q2, p1 〉 = i~∂p2 〈 q1, p2|q2, p1 〉 , which can be readily solved to give, after normalization, the following: 〈 q1, p2|q2, p1 〉 = 1 2π~ exp [ − i ~ ( q2p2 − θ ~ p1p2 − q1p1 )] . (2.4) Since the displacement operators {(2π~)−1 exp[ i ~ (y · R̂ + x · P̂)]}, where x = (x1, x2), y = (y1, y2), form a complete orthonormal set in the space-space noncommutative Heisenberg algebra any Schrödinger operator (which may depend explicitly on time), A(P̂, R̂, t) can be written as follows: A ( P̂, R̂, t ) = ∫∫ dx dyα(x,y, t) exp [ i ~ ( x · P̂ + y · R̂ )] , (2.5) where the c-function α(x,y, t) is determined by α(x,y, t) = (2π~)−2 Tr { A ( P̂, R̂, t ) exp [ − i ~ ( x · P̂ + y · R̂ ) ]} . (2.6) The Weyl function corresponding to the quantum operator A(P̂, R̂, t) is then given by WA ( p,q, t ) = ∫∫ dx dyα(x,y, t) exp [ i ~ (x · p + y · q) ] = ∫∫ dx1dy2e i ~ (x1p1+y2q2) × 〈 q1 − x1 2 − θy2 2~ , p2 + y2 2 |Â|q1 + x1 2 + θy2 2~ , p2 − y2 2 〉 . (2.7) To derive the expectation value of a product of two Schrödinger operators, one writes the expectation value of the product in terms of the von Neumann density matrix ρ as follows: 〈 Â1Â2 〉 = Tr [ ρÂ1Â2 ] , (2.8) and evaluates the trace in the above chosen basis. Thus by using completeness of the basis |q1, p2〉 and substituting (2.5) for the operators Â1 and Â2, equation (2.8) then becomes 〈 Â1Â2 〉 = ∫ dx dy du dv dq1 dp2 dq ′ 1 dp ′ 2 dq ′′ 1 dp ′′ 2 〈 q1, p2|ρ|q′1, p′2 〉 α1(x,y, t)α2(u,v, t) × 〈 q′1, p ′ 2|e i ~ (x·P̂+y·R̂)|q′′1 , p′′2 〉〈 q′′1 , p ′′ 2|e i ~ (u·P̂+v·R̂)|q1, p2 〉 . On deformed quantum mechanical schemes and ⋆-value equations 5 Moreover, resorting to the Baker-Campbell-Hausdorff theorem, making use of (2.4), and performing the integrals over q′1, p ′ 2, q ′′ 1 and p′′2, we obtain 〈 Â1Â2 〉 = ∫ dx dy du dv dq1 dp2 〈 q1, p2|ρ|q1 − x1 − u1 − v2θ ~ − y2θ ~ , p2+y2+v2 〉 α1(x,y, t)α2(u,v, t) × exp [ i ~ ( y1q1 − y1u1 + v1q1 + x2p2 + x2v2 + u2p2 − y1x1 2 + y2x2 2 − v1u1 2 + u2v2 2 )] × exp [ i ~ ( − θ ~ y1v2 − θ 2~ y1y2 − θ 2~ v1v2 )] . (2.9) Making now the change of variables q1 = ξ, p2 = η and substituting α1(x,y, t) and α2(u,v, t) in terms of their corresponding Weyl functions, equation (2.9) becomes 〈 Â1Â2 〉 = ( 1 2π~ )8 ∫ dp dq dp′ dq′ dx dy du dv dξ dη × 〈 ξ, η|ρ|ξ − x1 − u1 − v2θ ~ − y2θ ~ , η + y2 + v2 〉 ×WA1 (p,q, t)WA2 (p′,q′, t) exp [ i ~ y1 ( ξ − u1 − θ ~ v2 − x1 2 − θ 2~ y2 − q1 )] × exp [ i ~ v1 ( ξ − u1 2 − θ 2~ v2 − q′1 )] e i ~ v2(x2+ u2 2 −q′ 2 )e i ~ y2( x2 2 −q2) × e− i ~ x1p1e− i ~ u1p′ 1e− i ~ x2(p2−η)e− i ~ u2(p′ 2 −η). Next, we integrate over y1, x2, v1, u2, u1, v2, ξ, and η to get 〈Â1Â2〉 = 4 (2π~)4 ∫ dp dq dp′ dq′ dx1 dy2 × 〈 2q′1 − q1 − x1 2 − θy2 2~ , 2p′2 − p2 + y2 2 |ρ|q1 − x1 2 − θy2 2~ , p2 + y2 2 〉 ×WA1 (p,q, t)WA2 (p′,q′, t)e− i ~ y2q2e− i ~ x1p1 × e− i ~ q′ 2 (2p2−2p′ 2 −y2)e− i ~ p′ 1 (2q′ 1 −2q1− 2θ ~ p2+ 2θ ~ p′ 2 −x1). Observe now that this expression can also be written as follows: 〈 Â1Â2 〉 = 4 (2π~)4 ∫ dp dq dp′ dq′ dx1 dy2 × [ e θy2 ~ ∂x1 〈 2q′1 − q1 − x1 2 , 2p′2 − p2 + y2 2 |ρ|q1 − x1 2 , p2 + y2 2 〉] ×WA1 (p,q, t)WA2 (p′,q′, t)e− i ~ y2q2e− i ~ x1p1 × e− i ~ q′ 2 (2p2−2p′ 2 −y2)e− i ~ p′ 1 (2q′ 1 −2q1− 2θ ~ p2+ 2θ ~ p′ 2 −x1), 6 L. Román Juárez and M. Rosenbaum and after integrating by parts, we obtain 〈 Â1Â2 〉 = 4 (2π~)4 ∫ dp dq dp′ dq′ dx1 dy2 × 〈 2q′1 − q1 − x1 2 , 2p′2 − p2 + y2 2 |ρ|q1 − x1 2 , p2 + y2 2 〉 ×WA1 (p,q, t)WA2 (p′,q′, t)e− i ~ y2q2e− i ~ q′ 2 (2p2−2p′ 2 −y2)e− i ~ p′ 1 (2q′ 1 −2q1) × e i ~ x1(p′ 1 −p1)e− i ~2 θy2(p′ 1 −p1)e 2i ~2 θp′ 1 (p2−p′ 2 ). (2.10) To reconstruct the star product that should arise from this formulation, we use the following identities: e − θ ~ p′ 1 ∂q′ 2e i ~ q′ 2 y2 = e i ~ q′ 2 y2e− iθ ~2 y2p′ 1 , e− θ ~ p1∂q2e− i ~ q2y2 = e− i ~ q2y2e iθ ~2 y2p1 , e − θ ~ p′ 1 ∂q′ 2e− 2i ~ q′ 2 (p2−p′ 2 ) = e− 2i ~ (p2−p′ 2 )(q′ 2 − θ ~ p′ 1 ), so that (2.10) becomes 〈Â1Â2〉 = 4 (2π~)4 ∫ dp dq dp′ dq′ dx1 dy2 × 〈 2q′1 − q1 − x1 2 , 2p′2 − p2 + y2 2 |ρ|q1 − x1 2 , p2 + y2 2 〉 ×WA1 (p,q, t)WA2 (p′,q′, t)e− i ~ p′ 1 (2q′ 1 −2q1)e i ~ x1(p′ 1 −p1) × e− θ ~ p′ 1 ∂q′ 2 ( e i ~ q′ 2 y2e− 2i ~ q′ 2 (p2−p′ 2 ) )( e− θ ~ p1∂q2e− i ~ q2y2 ) . After integrating by parts, the above equation reads 〈 Â1Â2 〉 = 4 (2π~)4 ∫ dpdqdp′dq′dx1dy2 × 〈 2q′1 − q1 − x1 2 , 2p′2 − p2 + y2 2 |ρ|q1 − x1 2 , p2 + y2 2 〉 ×WA1 ( p, q1, q2 + θ ~ p1, T ) WA2 ( p′, q′1, q ′ 2 + θ ~ p′1, T ) e− i ~ p′ 1 (2q′ 1 −2q1) × e i ~ x1(p′ 1 −p1)e i ~ y2(q′ 2 −q2)e− 2i ~ q′ 2 (p2−p′ 2 ). Now make the following change of variables: x1 = 2q1 − 2z1, y2 = 2z2 − 2p2, q′1 = q1 + µ1, q′2 = q2 + µ2, p′1 = p1 + ν1, p′2 = p2 + ν2 to obtain 〈 Â1Â2 〉 = 16 (2π~)4 ∫ dp dq dµ1 dµ2 dν1 dν2 dz1 dz2 × 〈 z1 + 2µ1, z2 + 2ν2|ρ|z1, z2 〉 e− 2i ~ µ1p1e 2i ~ ν2q2 × e− 2i ~ ν1(µ1−q1+z1)e− 2i ~ µ2(p2−ν2−z2)WA1 ( p, q1, q2 + θ ~ p1, t ) × eν1∂p1eν2∂p2eµ1∂q1eµ2∂q2WA2 ( p, q1, q2 + θ ~ p1, t ) . (2.11) On deformed quantum mechanical schemes and ⋆-value equations 7 But e 2i ~ q1ν1eν1 −→ ∂ p1WA2 = e 2i ~ q1ν1e− i~ 2 ←− ∂ q1 −→ ∂ p1WA2 , e 2i ~ q2ν2eν2 −→ ∂ p2WA2 = e 2i ~ q2ν2e− i~ 2 ←− ∂ q2 −→ ∂ p2WA2 , e− 2i ~ p1µ1eµ1 −→ ∂ q1WA2 = e− 2i ~ p1µ1e i~ 2 ←− ∂ p1 −→ ∂ q1WA2 , e− 2i ~ p2µ2eµ2 −→ ∂ q2WA2 = e− 2i ~ p2µ2e i~ 2 ←− ∂ p2 −→ ∂ q2WA2 , which, when substituted into (2.11) and integrated by parts, results in 〈 Â1Â2 〉 = 16 (2π~)4 ∫ dp dq dµ1 dµ2 dν1 dν2 dz1 dz2 × 〈 z1 + 2µ1, z2 + 2ν2|ρ|z1, z2 〉 e− 2i ~ µ1p1e 2i ~ ν2q2 × e− 2i ~ ν1(µ1−q1+z1)e− 2i ~ µ2(p2−ν2−z2) × [ WA1 ( p, q1, q2 + θ ~ p1, t ) ⋆~ WA2 ( p, q1, q2 + θ ~ p1, t )] . (2.12) Last, integrating over ν1, µ2, µ1, and ν2 and performing the final change of variables z1 = q1 + s1 2 , z2 = p2 + s2 2 , equation (2.12) takes the following form: 〈 Â1Â2 〉 = 1 (2π~)2 ∫ dp dq ds1 ds2 〈 q1 − s1 2 , p2 − s2 2 |ρ|q1 + s1 2 , p2 + s2 2 〉 e i ~ s1p1 × e− i ~ s2q2 [ WA1 ( p, q1, q2 + θ ~ p1, t ) ⋆~ WA2 ( p, q1, q2 + θ ~ p1, t )] . (2.13) Recalling the definition of the Wigner function: ρw(p,q) := 1 (2π~)2 ∫ ds1 ds2 〈 q1− s1 2 , p2− s2 2 |ρ|q1+ s1 2 , p2+ s2 2 〉 e i ~ s1p1e− i ~ s2q2 , (2.14) equation (2.13) may be expressed in the following compact form: 〈 Â1Â2 〉 = ∫ dp dq ρw(p,q) [ WA1 ( p, q1, q2+ θ ~ p1, t ) ⋆~ WA2 ( p, q1, q2+ θ ~ p1, t )] , (2.15) where ⋆~ := exp [ ∑ i=1,2 i~ 2 (←− ∂ qi −→ ∂ pi −←−∂ pi −→ ∂ qi ) ] . (2.16) Consequently, in the phase-space formulation of quantum mechanics based on the algebra (2.1), the algebra of Weyl functions is deformed by a ⋆-product defined by WA1 ⋆ WA2 := m ◦ [ e ∑ i=1,2 i~ 2 (∂qi ⊗ ∂p′ i −∂q′ i ⊗ ∂pi ) ◦ e θ ~ p1∂q2 ⊗ e θ ~ p′ 1 ∂q′ 2WA1 (p,q)⊗WA2 (p′,q′) ] q,p=q′,p′ . (2.17) In addition, by a similar calculation to the one above, we can show that the Weyl symbol: Wρ(p,q) = (2π~)−2 ∫ dx dy Tr [ ρe− i ~ (x·P+y·R) ] e i ~ (x·p+y·q) (2.18) 8 L. Román Juárez and M. Rosenbaum associated with the density matrix ρ is related to the Wigner function by Wρ(p,q) = e− θ ~ p1∂q2ρw(p,q). (2.19) Hence for the space-space noncommutative Heisenberg-Weyl algebra, the Weyl symbol of the density matrix and the Wigner function as defined in (2.14) are not the same, contrary from what is the case for the usual quantum mechanics Heisenberg algebra: Wρ(p,q) θ→0−−−−→ ρw(p,q). Note now that if we substitute (2.19) into (2.15) and integrate by parts, we get 〈 Â1Â2 〉 = ∫ dp dqWρ(p,q)e− θ ~ p1 −→ ∂ q2 [ WA1 ( p, q1, q2+ θ ~ p1, t ) ⋆~ WA2 ( p, q1, q2+ θ ~ p1, t )] = ∫ dp dqWρ(p,q)e− θ ~ p1 −→ ∂ q2 × [ WA1 ( p1 − i~ 2 −→ ∂ q1 , p2 − i~ 2 −→ ∂ q2 , q1, q2 + i~ 2 −→ ∂ p2 + θ ~ ( p1 − i~ 2 −→ ∂ q1 ) , t ) ×WA2 ( p, q1 − i~ 2 ←− ∂ p1 , q2 + θ ~ p1, t )] = ∫ dp dqWρ(p,q) [ WA1 (p,q, t) ⋆θ ◦ ⋆~ WA2 (p,q, t) ] , (2.20) where ⋆θ ◦ ⋆~ := e iθ 2 ( ←− ∂ q1 −→ ∂ q2 − ←− ∂ q2 −→ ∂ q1 ) ◦ exp [ ∑ i=1,2 i~ 2 (←− ∂ qi −→ ∂ pi −←−∂ pi −→ ∂ qi ) ] . (2.21) Clearly, the expectation values obtained from (2.13) and (2.20) are the same. However, since for the space-space noncommutative Heisenberg-Weyl algebra the Wigner function associated with the density matrix ρ̂ and its corresponding Weyl symbol are not the same, the twistings in (2.18) and (2.20) of the product of Weyl symbols of two arbitrary operators do not agree in general. Their explicit forms are obviously basis dependent as well as dependent on whether averaging is done relative to the Wigner function or the Weyl symbol of the density matrix. Furthermore, given the two different ⋆-products (2.17) and (2.21) of a pair of Weyl- symbols, it is pertinent to inquire which of them corresponds to the Weyl-symbol of a product of two operators. To answer this question univocally, we need to make use of (2.4), (2.6), and (2.7). After a rather lengthy but fairly direct calculation, one can show that WA1A2 = WA1 ⋆θ ◦ ⋆~ WA2 . (2.22) So, for the quantum mechanics based on the space-space noncommutative Heisenberg-Weyl Lie group, we need to make iterative use of (2.22) for the calculation of Weyl-symbols corre- sponding to quantum operators. In particular, note that the Weyl-symbol corresponding to an operator Â1 = Â1(P̂) which is a function only of the momenta operators is given by the c-function WA1 (p) having the same functional form as the quantum operator, as it is the case in the usual WWGM quantum mechanics. However, for q-functions of the position operators, this is not always true for the space-space noncommutative Heisenberg-Weyl group, as can On deformed quantum mechanical schemes and ⋆-value equations 9 be easily seen, when consider, for example, the Weyl-symbol associated with the operator R̂1R̂2, for which (2.22) yields WR1R2 = (q1 + i θ2∂q2 )q2 = q1q2 + i θ2 . From a statistical point of view, both the Wigner function (2.14) and the Weyl symbol (2.18) for the density matrix admit a quasiprobabilistic interpretation, although the projected density probabilities are not all the same. Indeed, projecting (2.14) onto the plane q1 − p2 (i.e., integrating over q2, p1) immediately yields ∫ dp1 dq2ρw(p,q) = 〈 q1, p2|ρ̂|q1, p2 〉 , while projecting onto the q2 − p1 plane by making use of (2.4) results in ∫ dp2 dq1ρw(p,q) = 〈 q2 + (θ/~)p1, p1|ρ̂|q2 + (θ/~)p1, p1 〉 . However, if we perform the same calculations for the corresponding Weyl symbol, we find ∫ dp1 dq2Wρ(p,q) = 〈 q1, p2|ρ̂|q1, p2 〉 , ∫ dp2 dq1Wρ(p,q) = 〈 q2, p1|ρ̂|q2, p1 〉 . Let us now see how the above results compare with the ones resulting from applying the Stratonovich-Weyl correspondence and the Berezin geometric quantization to the space-space noncommutative Heisenberg-Weyl Lie group. 3 The Stratonovich-Weyl correspondence for the space-space noncommutative Heisenberg-Weyl Lie group In order to make our discussion self-contained and fix notation, we begin by summarizing the essential elements of the Stratonovich-Weyl correspondence. For a considerably more ample presentation of this formalism, we refer the reader to the work in [8, 11, 31, 33]. Let X be an even dimensional homogenous space given by the quotient G/H, where G is a simply connected Lie group (of finite dimension n) describing the dynamical symmetry of a given quantum system, and H ⊂ G its isotropy subgroup. If X is given a Kählerian structure, then it can be interpreted as the phase space of a classical dynamical system. The mapping Ω → |Ω〉〈Ω|, where Ω = Ω(g) is a point in X and g ∈ G, is the geometric quantization for this system [6]. The Stratonovich generalization of the standard Gröenewold-Moyal quantization to quan- tum systems possessing and intrinsic group G of symmetries is based on the following pos- tulates: (i) linearity: there is a one-to-one map Â→WA(Ω); (ii) reality: WA†(Ω) = [WA(Ω)]∗; (iii) standardization: ∫ X dµ(Ω)WA(Ω) = Tr Â, where dµ(Ω) is the invariant space measure; (iv) traciality: ∫ X dµ(Ω)WA1 (Ω)WA2 (Ω) = Tr(Â1Â2). (v) covariance: Wg·A(Ω) = WA(g−1 ·Ω), where g ·A denotes the adjoint action of a unitary irreducible representation π of G on Â. A functionWA(Ω) satisfying these five properties is known as the Stratonovich-Weyl (SW) symbol associated with a quantum operator  acting on Hilbert space. The linearity map is implemented by means of the generalized Weyl rule: WA(Ω) = Tr [ Â∆(Ω) ] , (3.1) 10 L. Román Juárez and M. Rosenbaum where ∆(Ω) is the Stratonovich-Weyl Kernel which is an operator-valued function on X. By virtue of the tracial property, we have that Tr [ Â∆(Ω) ] = ∫ x dµ ( Ω′ ) WA(Ω′)W∆(Ω)(Ω ′) = ∫ X dµ ( Ω′ ) Tr [ Â∆(Ω′) ] W∆(Ω)(Ω ′), (3.2) where W∆(Ω)(Ω ′) is the Weyl-equivalent of the Stratonovich Kernel. From (3.2), we infer that ∆(Ω) = ∫ X dµ ( Ω′ ) ∆(Ω′)W∆(Ω)(Ω ′), (3.3) so that the function K(Ω,Ω′) := W∆(Ω)(Ω ′) = Tr [ ∆(Ω)∆(Ω′) ] (3.4) behaves as a Dirac delta function on the manifold X. Consequently, making use of this property, the Weyl rule (3.1) may be inverted to give the following:  = ∫ X dµ(Ω)WA(Ω)∆(Ω). (3.5) Furthermore, from (3.1), (3.3), and (3.4), the SW-postulates (ii)–(v) translate to the following conditions on the SW-kernel operator: (iib) ∆(Ω) = [∆(Ω)]†, ∀Ω ∈ X; (iiib) ∫ X dµ(Ω)∆(Ω) = I; (ivb) ∫ X dµ(Ω′) Tr[∆(Ω)∆(Ω′)]∆(Ω′) = ∆(Ω); (vb) ∆(g · Ω) = π(g)∆(Ω)π(g)−1. In terms of the formalism of coherent states [9, 10, 28], we have that, whenever the Peter- Weyl theorem applies [11, 33], the SW kernel ∆(Ω), satisfying the above conditions, can be given explicitly as [8] ∆(Ω) = ∑ ν Y ∗ν (Ω)Dν = ∑ ν Yν(Ω)D†ν . (3.6) Here, Dν := ∫ X dµ(Ω)Yν(Ω)|Ω〉〈Ω| (3.7) denotes a set of operators acting on the Hilbert space H. The harmonic functions Yν(Ω), which form a complete orthonormal basis in L2(X,µ), are eigenfunctions of the Laplace- Beltrami operator (δd+ dδ) associated with the space X, while the index ν is, in general, a composite label. We would like to stress here, as it should have already become evident from our previous considerations, that since we are always going from the quantum mechanics of operators and Hilbert space to classical phase space averages, our Weyl correspondences are surjective and therefore unique maps (to a given quantum operator there corresponds a unique Weyl function, which corresponds to the case s = 0 for the families of operators and functions considered in [8]). Note now that when substituting (3.6) and (3.7) in (3.1), we get WA(Ω) = ∑ ν Y ∗ν (Ω)Aν = ∑ ν Yν(Ω)Ãν , On deformed quantum mechanical schemes and ⋆-value equations 11 where Aν = Tr(ÂDν), Ãν = Tr ( ÂD†ν ) . The generalized twisted product of two SW-symbols follows directly from (3.5) and the above and is given by WA(Ω) ⋆S WB(Ω) := WAB(Ω) := Tr [ ÂB̂ ∆(Ω) ] = ∫ X dµ(Ω′) ∫ X dµ(Ω′′)WA(Ω′)WB(Ω′′)L(Ω,Ω′,Ω′′), (3.8) where the tri-kernel L(Ω,Ω′,Ω′′) is defined by L(Ω,Ω′,Ω′′) := Tr [ ∆(Ω)∆(Ω′)∆(Ω′′) ] . (3.9) We are now ready to apply these results of the general formalism to the space-space noncommutative Heisenberg-Weyl alggroup H5, defined by the nilpotent Lie algebra (2.1), for the particular case (d = 2, θ̄ij = 0) considered in the previous section. In terms of bosonic creation and destruction operators and holomorphic coordinates, appropriate for calculating the SW kernel, and symbols in terms of coherent states, the Lie algebra of the generators of H5 is given by [ âi, â † j ] = δij , i = 1, 2, [ âi, âj ] = [ â†i , â † j ] = 0, i = 1, 2, (3.10) where â1 = ( √ 2~ )−1 ( R̂1 + θ 2~ P̂2 + iP̂1 ) , â†1 = ( √ 2~ )−1 ( R̂1 + θ 2~ P̂2 − iP̂1 ) , â2 = ( √ 2~ )−1 ( R̂2 − θ 2~ P̂1 + iP̂2 ) , â†2 = ( √ 2~ )−1 ( R̂2 − θ 2~ P̂1 − iP̂2 ) . (3.11) The group elements are therefore of the following form: g(s, α, β) = e(isI+αâ † 1 −ᾱâ1+βâ † 2 −β̄â2), where α, β ∈ C, and ᾱ, β̄ denotes complex conjugation. Clearly, here X = H5/U(1) = C 2, and the invariant measure is dµ(Ω) = π−2d2αd2β. The Glauber coherent states are |Ω〉 := |α, β〉 = D(α, β)|0〉 with D(α, β) denoting the displacement operator: D(α, β) := e(αâ † 1 −ᾱâ1+βâ † 2 −β̄â2). (3.12) Since the harmonic functions in this case are the exponentials: Yν(Ω) := Y(ξ,η)(α, β) = exp ( ξᾱ− ξ̄α+ ηβ̄ − η̄β ) , (3.13) 12 L. Román Juárez and M. Rosenbaum so that ∆(α, β) = 1 π2 ∫ C d2ξ ∫ C d2ηD(ξ, η) exp ( ξ̄α− ξᾱ+ η̄β − ηβ̄ ) ; (3.14) the expectation value of a quantum operator  is given by 〈Â〉 = Tr [ ρ̂ ] = 1 π2 ∫ C d2α ∫ C d2βWρ(α, β)WA(α, β), (3.15) where Wρ(α, β) = Tr [ ∆(α, β)ρ̂ ] (3.16) is the SW-symbol corresponding to the density matrix operator ρ̂. We can now make use of (3.8) and (3.9) together with (3.12) and (3.14) to get an explicit expression for the twisted product of two SW-symbols based on the quotient space C 2 = H5/U(1). Thus, noting that since the â1, â † 1 commute with the â2, â † 2, we can write the displacement operator as D(α, β) = D(α)D(β), and the tri-kernel as L(α, α′, α′′;β, β′, β′′) = L(α, α′, α′′)L(β, β′, β′′). Moreover, using also repeatedly the coherent states properties: D(ξ)|β〉 = ei Im(ξβ̄)|ξ + β〉, (3.17) 〈α|α′〉 = e− 1 2 (|α|2+|α′|2−2ᾱα′), (3.18) we find L(α, α′, α′′) = 4 exp [ 4i ( α′2α1 − α′1α2 + α′1α ′′ 2 − α′2α′′1 + α′′1α2 − α′′2α1 )] , and an analogous expression for L(β, β′, β′′). Consequently, WA(α, β) ⋆S WB(α, β) = 16 π4 ∫ C d2α′′ ∫ C d2α′e4iα′ 1 (α′′ 2 −α2)e4iα′ 2 (α1−α′′ 1 )e4i(α′′ 1 α2−α′′ 2 α1) × ∫ C d2β′′ ∫ C d2β′e4iβ′ 1 (β′′ 2 −β2)e4iβ′ 2 (β1−β′′ 1 )e4i(β′′ 1 β2−β′′ 2 β1)WA(α′, β′)WB(α′′, β′′). (3.19) Making next the change of variables α′′1 = α1 + η1, α ′′ 2 = α2 + η2, β ′′ 1 = β1 + ξ1, β ′′ 2 = β2 + ξ2, we can write WA(α, β) ⋆S WB(α, β) = 16 π4 ∫ C . . . ∫ C dη1dη2dξ1dξ2dα ′ 1dα ′ 2dβ ′ 1dβ ′ 2e 4i(α′ 1 −α1)η2 × e−4i(α′ 2 −α2)η1e4i(β′ 1 −β1)ξ2e−4i(β′ 2 −β2)ξ1WA(α1, α2, β1, β2) × e(η1 ~∂α1 +η2 ~∂α2 +ξ1~∂β1 +ξ2~∂β2 )WB(α1, α2, β1, β2). (3.20) We can change the last exponential in the above equation into a bidifferential by noting that e4i(α′ 1 −α1)η2eη2 ~∂α2WB ( α1, α2, β1, β2 ) = e4i(α′ 1 −α1)η2e − i 4 ←− ∂ α′ 1 −→ ∂ α2WB ( α1, α2, β1, β2 ) , On deformed quantum mechanical schemes and ⋆-value equations 13 and similarly for the other terms. Hence, substituting the results in (3.19), integrating by parts, and integrating over the remaining variables in the integrand, we finally arrive at WA(α, β) ⋆S WB(α, β) := WA(α, β)e i 4 ( ←− ∂ α1 −→ ∂ α2 − ←− ∂ α2 −→ ∂ α1 + ←− ∂ β1 −→ ∂ β2 − ←− ∂ β2 −→ ∂ β1 )WB(α, β). (3.21) Now, substituting this result into (3.15), we obtain the expectation value of a product of quantum operators derived according to the Stratonovich-Weyl correspondence in the context of the space-space noncommutative Heisenberg-Weyl group. Moreover, since the alternate calculation in the previous section was done based on the Lie algebra of the same group and since the Stratonovich phase-space formulation was purported to be a generalization of the later to physical systems with Lie group symmetries which, evidently include the one common to the two approaches, a coincidence of results would then appear natural. In order to verify this conjecture we first need to convert the holomorphic variables in (3.15), (3.16), and (3.21) into phase-space variables. That is, we need to make the substitutions: α1 −→ 1√ 2~ ( q1 + θ 2~ p2 ) , α2 −→ 1√ 2~ p1, β1 −→ 1√ 2~ ( q2 − θ 2~ p1 ) , β2 −→ 1√ 2~ p2. (3.22) Hence, ∂α1 = √ 2~∂q1 , ∂α2 = √ 2~ ( θ 2~ ∂q2 + ∂p1 ) , ∂β1 = √ 2~∂q2 , ∂β2 = √ 2~ ( − θ 2~ ∂q1 + ∂p2 ) , (3.23) from where the Stratonovich twist bidifferential expressed in terms of phase-space variables takes the following form: ⋆S = ⋆θ ◦ ⋆~. (3.24) Furthermore, making use of (3.12), (3.13), (3.16), and (3.14), we have Wρ(α, β) = Tr [ ∆(α, β)ρ̂ ] = 1 π2 ∫ C d2ξ ∫ C d2ηTr [ e(ξâ † 1 −ξ̄â1+ηâ † 2 −η̄â2) ρ̂ ] exp ( ξ̄α− ξᾱ+ η̄β − ηβ̄ ) . Evaluating now the trace in the above expression relative to the mixed phase-space basis {|q1, p2〉} and after a fairly lengthy but straightforward calculation, we arrive at Wρ(α, β) = 4 ∫∫ dq′1dp ′ 2e 2iα2(2α1− √ 2 ~ q′ 1 − θ ~ √ 2~ p′ 2 ) e −2iβ1(2β2− √ 2 ~ p′ 2 ) × 〈 q′1, p ′ 2|ρ̂|2 √ 2~α1 − q′1 − 2θ√ 2~ β2,−p′2 + 2 √ 2~β2 〉 . Finally, making the change of variables: q′1 = √ 2~α1 − λ1 2 − θ√ 2~ β2, p′2 = β2 − λ2 2 14 L. Román Juárez and M. Rosenbaum yields Wρ(α, β) = ∫∫ dλ1dλ2e 2iα2√ 2~ (λ1+ θ 2~ λ2) e − 2iβ1λ2√ 2~ × 〈√ 2~α1 − λ1 2 − θ√ 2~ β2, β2 − λ2 2 |ρ̂| √ 2~α1 + λ1 2 − θ√ 2~ β2, β2 + λ2 2 〉 . In terms of phase-space variables, this result reads Wρ ( α ( p1, q2 ) , β ( q1, p2 )) = e− θ ~ p1∂q2 ∫∫ dλ1dλ2e i ~ (p1λ1−q2λ2) 〈 q1 − λ1 2 , p2 − λ2 2 |ρ̂|q1 + λ1 2 , p2 + λ2 2 〉 . (3.25) If we now compare (3.21), (3.24), and (3.25) with (2.20), (2.21), (2.14), and (2.19) of the previous section, we see that for the space-space noncommutative Weyl-Heisenberg Lie group the quantum mechanics resulting from both formalisms are equivalent provided that in the calculation of the expectation values, we derive the phase-space averages by combining the appropriate ⋆-product for the evaluation of Weyl-symbols with the appropriate Wigner function or Weyl-symbol associated with the density matrix for the problem, according to the above referred formulas. 4 The Berezin quantization procedure by means of involution operators and its application to the space-space noncommu- tative Heisenberg-Weyl algebra This quantization scheme arises from the basic property that for homogenous symmetric spaces, there is an involutive automorphism of G acting on them. Such is the case for X = H5/U(1), where the involution automorphisms are reflections around each point. Re- calling equations (2.5), (2.6) in Section 2, we see that the Weyl function is the Fourier transform of the α function in (2.5), while the Fourier transform of the unitary displacement operators {(2π~)−1 exp[ i ~ (y · R̂ + x · P̂)]} is indeed reflections. It is thus natural to write [6, 22, 23, 24, 25, 26]  = ∫ X dµ(x)wA(x)Û(x) (4.1) as a generalization of (2.5). Here, Û(x) is the unitary operator corresponding to the group element that performs reflections around the point x ∈ X. As noted by the authors in [22, 23, 24, 25, 26], the use of the reflection operator provides a way to circumvent the situation when a Fourier transform on X cannot be consistently defined. The function wA(x) appearing in (4.1) corresponds to the Weyl contravariant symbol which is, in general, different from the Weyl covariant symbol defined as: w̃A(x) := Tr [ ÂÛ(x) ] . Berezin also showed that there exists a bijective map relating wA, w̃A to the usual contravari- ant and covariant symbols PA, QA, respectively, whose expressions are given by  = ∫ X dµ(x)PA(x)|x〉〈x|, QA(x) = 〈x|Â|x〉, where {|x〉} corresponds to an overcomplete basis of normalized states tagged by points in X. On deformed quantum mechanical schemes and ⋆-value equations 15 Thus in order to implement this quantization formalism, we must first determine what will be in our case the reflection operator Û(x). To this end, we will make use of the Hilbert space spanned by the coherent states of the last section, which in fact constitute an overcomplete basis. Each coherent state |α, β〉 = |α〉 ⊗ |β〉 is tagged by a point (α, β) ∈ C 2 = X. We may now construct the reflection operator Û(α, β) by acting transitively on the re- flection operator around the origin Û(0, 0) with the unitary operator associated to g ∈ G. From the properties of the algebra (3.10), it is clear that Û(α, β) = Û(α)⊗ Û(β), where each Û(α) acts on a copy of C. Then for simplicity, we will reduce the calculation to one copy of C and obtain the final result just by taking the direct product of the two copies. Thus, following Berezin, consider a complex line bundle L over C with fiber metric e−K(v,v̄), where K(v, v̄) = vv̄ is the Kähler potential. The Hilbert space H consists of holomorphic sections of L with inner product: 〈f |g〉 = 1 π ∫ C d2v f̄(v)g(v)e−vv̄, where the holomorphic section f(v) denotes the evaluation: f(v) = 〈v|f〉. The coherent state |α〉, expressed in the Fock-Bargmann representation F, is given by |α〉 = e− 1 2 |α|2 ∞ ∑ n=0 αn √ n! |n〉. Hence, 〈v|α〉 = e− 1 2 |α|2 ∞ ∑ n=0 αnv̄n n! = e− 1 2 |α|2+αv̄. (4.2) Making use of the identity resolution: I = 1 π ∫ C d2v e−|v| 2 |v〉〈v|, (4.3) we can write the left hand of (4.2) as follows: α(v) = 1 π ∫ C d2v′〈v|v′〉e−|v′|2α(v′). It is easy to show that this last expression becomes an identity if we set 〈v|v′〉 := B(v′, v̄) = ev ′v̄ and make use of (4.2) on both sides of the equation. Moreover, it also follows that B(v′, v̄) satisfies the following properties: 1 π ∫ C d2v′ e−|v ′|2B(v′, v̄)f(v′)=f(v), 1 π ∫ C d2v′ e−|v ′|2B(v, v̄′)B(v′, ū)=B(v, ū). (4.4) Thus B(v′, v̄) is the Bergman reproducing kernel [7], and in the F representation space, the quantity πδ(v, v′) := B(v′, v̄)e−|v ′|2 acts as a Dirac delta function under integration. Let us now define the operator Û(0) by Û(0) := 1 π ∫ C d2v e−|v| 2 | − v〉〈v|. 16 L. Román Juárez and M. Rosenbaum To show that this is the reflection operator around the origin, we take the action of Û(0) over any arbitrary state |v′〉 and use the above definition of the delta function action: Û(0)|v′〉 = 1 π ∫ C d2ve−|v| 2 | − v〉〈v|v′〉 = 1 π ∫ C d2v e−|v| 2 B(v′, v̄)| − v〉 = | − v′〉. With the above results, we are now in a position to calculate the more general operator Û(ζ). This is done by noticing that by taking the unitary transformation D̂(ζ)Û(0)D̂†(ζ), where D̂(ζ) is the unitary displacement operator representation of the H3 group acting on coherent states according to (3.17). Since Û(0) is an involution, D̂(ζ) induces displacements and (D̂(ζ)Û(0)D̂†(ζ))2 = I, the operator Û(ζ) must correspond to a reflection around ζ ∈ C. To show this, we first use (3.12) to obtain the explicit form of the operator Û(ζ) := D̂(ζ)Û(0)D̂†(ζ): Û(ζ) = 1 π ∫ C d2ve−|v| 2 D̂(ζ)| − v〉〈v|D̂†(ζ). Making now use of (4.2) in order to express the arbitrary ket |v〉 in terms of the normalized coherent state basis: |v〉 = 1 π ∫ C d2α e(− 1 2 |α|2+ᾱv)|α〉, and applying (3.17) on the coherent state |α〉 yields D̂(ζ)|v〉 = 1 π ∫ C d2α e(− 1 2 |α|2+ᾱv+i Im(ζᾱ))|α+ ζ〉. (4.5) Furthermore, making use of (4.5) and the properties of the Bergman kernel in (4.4), we obtain after some fairly straightforward calculations the expression: Û(ζ) = 1 π ∫ C d2α e(ζᾱ−ζ̄α)|α+ ζ〉〈ζ − α|. Finally, making the change of variables ζ − α = ρ yields Û(ζ) = 1 π ∫ C d2ρ eζ̄ρ−ρ̄ζ |2ζ − ρ〉〈ρ|. (4.6) We next use this expression to repeat a similar calculation to the one we did above in order to obtain Û(0). Thus, taking the action of the operator Û(ζ) on an arbitrary state |v〉 and expanding the coherent state |2ζ − ρ〉 in (4.6) in terms of |v〉, by making use of (4.2) and (4.3), we get Û(ζ)|v〉 = 1 π2 e−2|ζ|2 ∫ C d2v′ e−|v ′|2e2v̄′ζ |v′〉 ∫ C d2ρ e−|ρ| 2 evρ̄e(2ζ̄−v̄′)ρ, which when resorting repeatedly to equation (4.4) gives Û(ζ)|v〉 = e2(ζ̄v−|ζ|2)|2ζ − v〉. (4.7) The function inside the ket in the above equation can be rewritten as 2(ζ − v) + v to make evident the fact that this is the reflection of the point v around ζ. To complete the proof, On deformed quantum mechanical schemes and ⋆-value equations 17 we check that Û(ζ) is indeed an involution. This follows directly by once more acting with Û(ζ) on (4.7). Accordingly, we obtain Û(ζ) 2|v〉 = Û(ζ) [ e2(ζ̄v−|ζ|2)|2ζ − v〉 ] = e2(ζ̄v−|ζ|2)e2ζ̄(2ζ−v)e−2|ζ|2 |2ζ − (2ζ − v)〉 = |v〉. As we mentioned at the beginning of this section, the Weyl contravariant and covariant symbols are not the same in general. We will show, however, that for the symmetric homoge- nous space treated here this is not the case. Indeed, making the change Û(ζ)→ 2Û(ζ) ≡ V̂ (ζ) in (4.1), the latter reduces to (3.5) and consequently wA = WA = w̃A in which case both symbols are equal. This follows from equation (4.6) and observing that by using our previous results, we can write the identity as follows: eζ̄ρ−ρ̄ζ |2ζ − ρ〉 = 1 2π ∫ C d2λ eλ̄ζ−ζ̄λe 1 2 (ρ̄λ−λ̄ρ)|λ+ ρ〉. Moreover, the coherent state e 1 2 (ρ̄λ−λ̄ρ)|λ+ ρ〉 is nothing else but D̂(λ)|ρ〉, so we can replace this into (4.6), and the operator V̂ (ζ) = 2Û(ζ) takes now the following form: V̂ (ζ) = 1 π2 ∫ C ∫ C d2λd2ρ eλ̄ζ−ζ̄λD̂(λ)|ρ〉〈ρ|. (4.8) Finally, observe that in this last expression the quantity 1 π ∫ C d2ρ |ρ〉〈ρ| is just the identity operator in terms of normalized coherent states. It is then obvious that (4.8) reduces simply to V̂ (ζ) = 1 π ∫ C d2λ eλ̄ζ−ζ̄λD̂(λ), (4.9) which allows us to conclude that V̂ (α)⊗ V̂ (β) ≡ ∆(α, β) as seen from (3.14). This argument demonstrates that for the Heisenberg-Weyl algebra (2.1), the SW formalism as well as that of Berezin provide the same quantization scheme. It is interesting to observe that because both the SW and the Berezin formalisms are based on complex valued holomorphic states and non-Hermitian operators, defined in turn by means of creation and destruction operators, the noncommutativity of the observables in the algebra (2.1) is hidden in the definition of those creation and destruction operators. So, as long as we remain in the complex domain, their quantum mechanics for the ordinary and the Heisenberg-Weyl algebras (2.1) appear as indistinguishable (see, e.g., (3.19)). It should also be clear from our presentation so far that there are a variety of Bopp maps that can be chosen to construct creation and destruction operators from phase-space operator observables. In our construction (see (3.11)), we have chosen a map that keeps the algebra of â and ↠unchanged, as this choice allows us to use all the machinery of standard WWGM up to the point where we re-express the final results in terms of real dynamical phase-space variables. Moreover, it is known that for the WWGM quantum mechanics, there is a ⋆-value equation which is a result stronger than the one providing the phase-space expectation values for operators and products of operators on Hilbert space. Indeed, it is fairly straightforward to show that (see, e.g., [32]) the star-value equation: WH(p,q) ⋆~ ρw = Eρw is a necessary and sufficient condition for the weaker expectation value relation: ∫∫ dpdqWH(p,q) ρw = ∫∫ dpdqWH(p,q) ⋆~ ρw 18 L. Román Juárez and M. Rosenbaum to follow. Here, WH(p,q) is the Weyl-symbol associated with the Hamiltonian operator Ĥ satisfying the eigenvalue equation Ĥ|Ψ〉 = E|ψ〉, |Ψ〉 is a pure energy state, and ρw is the Wigner function corresponding to the pure state density matrix ρ̂ = |ψ〉〈ψ|. We will investigate next if similar ⋆-valued equations exist for the quantum mechanical formulations on the Weyl-Heisenberg group consider above, and whether their equivalence stands for such stronger equations. 5 Star-value equations for phase-space quantum mechanics based on the space-space noncommutative Heisenberg-Weyl group Given a Hamiltonian Ĥ(P̂, R̂) for a quantum mechanical system where P̂, R̂ satisfy the algebra (2.1) (with i, j = 1, 2 and θ̄ = 0) and the pure state density matrix ρ̂ = |ψ〉〈ψ|, we can consider star-value equations associated with the ⋆-products (2.17) or (2.21). Let us begin by considering first the ⋆-product in (2.17) between the Weyl-symbol corresponding to Ĥ and the Weyl-symbol corresponding to the density matrix ρ̂. We get (after resorting to (2.19) in order to obtain the last equality): WH ⋆ Wρ = m ◦ [ e ∑ i=1,2 i~ 2 (∂qi ⊗∂p′ i −∂q′ i ⊗∂pi ) ◦ e θ ~ p1∂q2 ⊗ e θ ~ p′ 1 ∂q′ 2WH(p,q)⊗Wρ(p ′,q′) ] q,p=q′,p′ = ( e θ ~ p1∂q2WH ) ⋆~ ( e θ ~ p1∂q2Wρ ) = ( e θ ~ p1∂q2WH ) ⋆~ ρw. Note that in general, e θ ~ p1∂q2WH(p,q) = WH ( p, q1, q2 + θ ~ p1 ) , which says the following: calculate first the Weyl-symbol corresponding to the Hamiltonian operator by applying (2.17) repeatedly, followed by the displacement of the q2 argument by the exponential on the left hand side of the above expression. Hence, WH ⋆ Wρ = WH ( p, q1, q2 + θ ~ p1 ) ⋆~ ρw. Substituting now the expression (2.14) for the Wigner function and (2.16) for the ⋆~-product, we have WH ⋆ Wρ = (2π~)−2 ∫∫ ds1ds2ψ ( q1 − s1 2 , p2 − s2 2 ) ψ∗ ( q1 + s1 2 , p2 + s2 2 ) × [ ŴH ( q1, q2 + i~ 2 −→ ∂ p2 + θ ~ ( p1 − i~ 2 −→ ∂ q1 ) ; p1 − i~ 2 −→ ∂ q1 , p2 ) × e i ~ s1(p1+ i~ 2 ←− ∂ q1 )e− i ~ s2(q2− i~ 2 ←− ∂ p2 ) ] = (2π~)−2 ∫∫ ds1ds2ψ ( q1 − s1 2 , p2 − s2 2 ) ψ∗ ( q1 + s1 2 , p2 + s2 2 ) × [ ŴH ( q1 − s1 2 , q2 + i~ 2 −→ ∂ p2 + θ ~ p1 − iθ 2 −→ ∂ q1 ; p1 − i~ 2 −→ ∂ q1 , p2 − s2 2 ) × e i ~ s1p1e− i ~ s2q2 ] . On deformed quantum mechanical schemes and ⋆-value equations 19 If we now note that we can make the following replacement of the q2 and p1 arguments in WH inside the square brackets: q2 −→ i~∂s2 , p1 −→ −i~∂s1 , and integrate by parts, we arrive at WH ⋆ Wρ = (2π~)−2 ∫∫ ds1ds2e i ~ s1p1e− i ~ s2q2 × [ ŴH ( q1− s1 2 ,−i~∂s2 + i~ 2 −→ ∂ p2 +iθ∂s1 − iθ 2 −→ ∂ q1 ; i~∂s1 − i~ 2 −→ ∂ q1 ; p2− s2 2 ) × ψ ( q1 − s1 2 , p2 − s2 2 ) ψ∗ ( q1 + s1 2 , p2 + s2 2 )] . Observe next that making the identifications: Q̂1 := q1 − s1 2 , Π̂1 := i~∂s1 − i~ 2 ∂q1 , Π̂2 := p2 − s2 2 , Q̂2 := −i~∂s2 + i~ 2 ∂p2 + θ ~ Π̂1, (5.1) we obtain a realization for the Heisenberg-Weyl algebra: [ Q̂1, Q̂2 ] = iθ, [ Q̂i, Π̂j ] = i~δij , [ Π̂1, Π̂2 ] = 0. Observe also that the operator ŴH(Q̂1, Q̂2, Π̂1, Π̂2) annihilates any function of q1 + s1 2 and p2 + s2 2 . Hence, WH ⋆ Wρ = (2π~)−2 ∫∫ ds1ds2e i ~ s1p1e− i ~ s2q2ψ∗ ( q1 + s1 2 , p2 + s2 2 ) × [ ŴH ( Q̂1, Q̂2, Π̂1, Π̂2 ) ψ ( q1 − s1 2 , p2 − s2 2 )] . (5.2) Furthermore, consider the eigenvalue equation: Ĥ ( R̂1, R̂2; P̂1, P̂2 ) |ψ〉 = E|ψ〉. (5.3) Since the operators P̂, R̂ satisfy the algebra (2.1) (with i, j = 1, 2 and θ̄ = 0), the projection of (5.3) with the bra 〈R1, P2| yields (making use of (2.2)) Ĥ ( R1,−iθ∂R1 + i~∂P2 ;−i~∂R1 , P2 ) 〈R1, P2|ψ〉 = E〈R1, P2|ψ〉. (5.4) Setting now R1 ≡ Q̂1 = q1 − s1 2 , P2 ≡ Π̂2 = p2 − s2 2 , and comparing the expression for R̂2 = −iθ∂R1 + i~∂P2 in (5.4) with Q̂2 in (5.1), we get ∂R1 = 1 2 ∂q1 − ∂s1 , ∂P2 = 1 2 ∂p2 − ∂s2 . However, also comparing the R̂2 in (5.4) with (2.2) yields ∂q1 = ∂R1 , ∂p2 = ∂P2 , 20 L. Román Juárez and M. Rosenbaum from where it also clearly follows ∂s1 = −1 2 ∂R1 , ∂s2 = −1 2 ∂P2 . Substituting the above into (5.4) and comparing with (5.1), we arrive at Ĥ ( Q̂1, Q̂2; Π̂1, Π̂2 ) 〈Q1,Π2|ψ〉 = E〈Q1,Π2|ψ〉, so, if we could make the identification Ĥ(Q̂1, Q̂2; Π̂1, Π̂2) = WH(Q̂1, Q̂2, Π̂1, Π̂2), we would then have that (5.2) would immediately imply that WH ⋆ Wρ = (2π~)−2E ∫∫ ds1ds2e i ~ s1p1e− i ~ s2q2ψ∗ ( q1+ s1 2 , p2+ s2 2 ) ψ ( q1− s1 2 , p2− s2 2 ) =Eρw, or WH ( p; q1, q2 + θ ~ p1 ) ⋆~ ρw(p,q) = Eρw. (5.5) Note, however, that the feasibility of this identification requires that Ĥ(Q̂1, Q̂2; Π̂1, Π̂2) and WH(Q̂1, Q̂2, Π̂1, Π̂2) should be of the same functional form for their operator arguments, but, according to our discussion following equation (2.22), this will only be possible for Hamiltonians having the Weyl symmetrized ordering of operators. The corresponding expression of the ⋆-value equation for the product WH ⋆θ◦⋆~Wρ follows immediately by recalling (see the argument given in the paragraph following equation (4.9)) that in holomorphic coordinates, the ⋆-value equation does not see the noncommutativity: WH(α, β) ⋆S Wρ(α, β) = EWρ ≡WH(α, β) ⋆~ Wρ(α, β) = EWρ. Thus, when going back to phase-space variables by making use of (3.22) and (3.25) yields WH ( 1√ 2~ ( q1 + θ√ 2~ p2 ) , 1√ 2~ ( q2 − θ√ 2~ p1 ) , 1√ 2~ p1, 1√ 2~ p2 ) × ⋆θ ◦ ⋆~e − θ ~ p1∂q2ρw ( q1, q2, p1, p2 ) = Ee− θ ~ p1∂q2ρw ( q1, q2, p1, p2 ) . 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Rev., 40 (1932), 749–759. Received July 31, 2010 Revised September 27, 2010 ar X iv :1 4 0 1 .1 7 5 9 v 3 [g r- q c] 3 1 M ar 2 0 1 4 ICN 2014-01-10 A Twisted C⋆ - algebra formulation of Quantum Cosmology with application to the Bianchi I model Marcos Rosenbaum,∗ J. David Vergara,† and Román Juárez‡ Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543 , México D.F., México A.A. Minzoni§ Instituto de Investigación en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, A. Postal 70-543 , México D.F., México (Dated: April 2, 2014) A twisted C ⋆- algebra of the extended (noncommutative) Heisenberg-Weyl group has been con- structed which takes into account the Uncertainty Principle for coordinates in the Planck length regime. This general construction is then used to generate an appropriate Hilbert space and ob- servables for the noncommutative theory which, when applied to the Bianchi I Cosmology, leads to a new set of equations that describe the quantum evolution of the universe. We find that this formulation matches theories based on a reticular Heisenberg-Weyl algebra in the bouncing and ex- panding regions of a collapsing Bianchi universe. There is, however, an additional effect introduced by the dynamics generated by the noncommutativity. This is an oscillation in the spectrum of the volume operator of the universe, within the bouncing region of the commutative theories. We show that this effect is generic and produced by the noncommutative momentum exchange between the degrees of freedom in the cosmology. We give asymptotic and numerical solutions which show the above mentioned effects of the noncommutativity. PACS:03.70.+k, 98.80.Qc, 04.60.Pp: ∗ mrosen@nucleares.unam.mx † vergara@nucleares.unam.mx ‡ roman.juarez@nucleares.unam.mx § tim@mym.iimas.unam.mx 2 I. INTRODUCTION Reductionism is an essential concept in Physics which has been validated by experiments involving energies ranging from orders of eV ’s in molecular and atomic physics to a few TeV in the strong interaction regime. This paradigm has led to such successes of quantum unification as the Standard Model, involving Electromagnetic, Weak and Strong Interactions. However the oldest interaction known to man: Gravity, and its most beautiful geometrical formulation: General Relativity, have to this day avoided quantization and even more so, unification with the other three funda- mental forces of Nature. Thus Quantization of Relativity at distances of the order of the Planck length and energies of the order of 1016TeV , still remains to be one of the most compelling problems in the field, mainly due to the lack of experimental data that could help shed some more light on which path should one pursue. Because Quantum Cosmology can be seen as a minisuperspace of Quantum Gravity where most of the degrees of freedom have been frozen and, although there is no a priori reason to assume that the conclusions derived from the former can be readily translated to the later, it is expected that some approaches to Quantum Cosmology can provide a convenient initial framework to investigate quantum processes involving distances of the order of Planck lengths where manifestations of noncommutativity should occur. The main purpose of this paper is to provide what we consider might be one such self-consistent formulation for Quantum Cosmology that could lead to further insights and directives towards Quantum Gravity at scales where the implications of the Uncertainty Principle of Quantum Mechanics and the Principle of Equivalence of Gravitation become commensurate. Indeed, regardless of which will be eventually the final and complete Theory for Quantum Gravity, it seems that the present attempts for its formulation have as a common denominator some concept of noncommutativity ( see e.g. [1], [2], [3], [4], [5], [6] ). Thus, in addition to the fact that Physics is a discipline based on experiment and that a theory needs to be validated or dismissed only on this basis before its ultimate acceptance, it is sensible to expect that the concept of noncommutativity should be a self-consistent part of it. One formulation that appeals to many physicists in the field is String Theory [7]. Several research groups in Relativity on the other hand believe that a more geometrical approach such as Loop Quantum Gravity (LQG) constitutes an equally viable candidate (see e.g. [8]) and, on the other extreme of the theory spectrum, is the Noncommutative Geometry developed by A. Connes and others (see e.g. [9], [10], [11], [12]). As pointed out in the Review by Douglas and Nekrasov [13], some of the strong arguments in favor of noncommutativity and of further support for Noncommutative Geometry originated from these varied approaches has led to a flurry of activities and trends where mathematical clarity and conceptual self-consistency ”appear less central to physical considerations”. Examples of such a case are the earlier quantum cosmology formulations based on a Bopp map deformation of the Wheeler-De Witt equation, resulting from inserting a Moyal ⋆-product between the classical Hamiltonian and the elements of the Hilbert vector space of wave functions. This, from the viewpoint of Deformation Quantization where the Moyal ⋆-product arises as a deformation of the algebra product of the Weyl symbols of quantum operator observables, has no conceptual support. Moreover, as we have shown in [14] (and references therein) a more logical noncommutative replacement for the Schrödinger equation is the ⋆-value equation involving the deformed Moyal ⋆-product of the Weyl symbol of the quantum Hamiltonian operator and the Wigner function. It may be meaningful to notice here also that in a previous work [15] of the type mentioned above, the region close to the singularity has not been explored and the wave functions have branch points which imply an undetermined behavior near the singularity, which could very well be attributed to the authors use of this unsubstantiated Moyal product in the Wheeler-de Witt equation. Alternatively, the C⋆-algebra A, on which our approach is based, is in particular a good example of the strategy of Noncommutative Geometry, and a motivational argument for basing our approach on this formalism hinges, on a nut shell, on the theoretical observations that since physically meaningful quantities should be independent of the choice of a gauge, the concepts of gauge potentials or connections had to be incorporated into the formulation of Action 3 Densities for describing our perception of Nature. This then has led naturally to the formalism of fiber bundles to describe the basic forces of nature and the mathematical physics for dealing with Gauge Theory and Variational Principles in Field Theory. Now, a bundle P (M,F, τ) consists of a topological space P , a baseM , a typical fiber F and a continuous surjection τ : P → M , where in semi-classical physics M is the space-time continuum with a Hausdorff topology. Moreover, it can be shown that a vector bundle over M can be described purely in terms of concepts pertinent to the commutative C⋆-algebra C(M) (see e.g.[16]). Furthermore, by the Gel’fand-Naimark Theorem [17]: “To every commutative C⋆-algebra with unit there corresponds a Hausdorff space, which implies a complete duality between the category of locally compact Hausdorff spaces and the category of commutative C⋆-algebras C(M) and ⋆-homomorphisms. However, at distances of the order of the Planck length, where the Principle of Uncertainty and the Principle of Equivalence become equally important and noncommutativity dominates the dynamics of the system, one needs to generalize the notion of a Hilbert bundle in such a way that the commutative C⋆-algebra C(M) is replaced by an arbitrary C⋆-algebra A, and the dual notion of a Hausdorff topological space M be replaced by the space of all unitary classes of irreducible representations of A ([18], [19],[20],[21]). On the basis of the previous remarks and in order to implement this ideas so as to provide the possibility of calculation for observable quantities in physical models, the material in this paper has been structured as follows: In Section II we introduce a projective unitary realization of the generators of the twisted discrete translation group C⋆-algebra A of bounded operators with unit, ⋆-homomorphic to the Heisenberg-Weyl group of deformed quantization. Thus the noncommutative lattices, generated from the primitive spectrum of A, are the structure spaces of the T0 Jacobson topology and the noncommutative analogue of the Hausdorff topology of the space M of the Gel’fand - Naimark theorem. In Section III we go on to use the homomorphism obtained in the previous section and the Gel‘fand- Naimark-Segal construction to derive the kinematic Hilbert space on which the bounded operators in A will act. In addition, the functions resulting from the Pontryagin duality on this Hilbert vector space yield a complete set of functions which satisfy the same orthogonality and summation completeness relations as the algebra of almost periodic functions [22]. Section IV begins by considering the ADM reduced classical action of the anisotropic Bianchi I model cosmology coupled to a massless scalar to assume the part of an inner time. We then quantize the system following Dirac’s procedure after expressing the observables of the system in terms of the C⋆-algebra of Hermitized bounded operators previously introduced. Using then the Hamiltonian constraints of the system and applying well documented techniques such as the ones summarized and cited in the text, we derive the physical states of the system from the kinematical states constructed in Sec.III. In Section V the so far inherently discrete system of equations is converted to the continuum by making use of the Feynman Path Integral construction for quantization. It should be noted, however, that the symbol of noncommutativity appears in various terms of the action and acquires different levels of relevance for the different possible stages of evolution of the system, as shown in the later sections. This analysis is in fact carried out extensively in Sections VI and VII, after deriving the equations of motion by applying the method of stationary phase to the action derived in Sec.V. In Section VII, in particular, we consider several scenarios for the system evolution which evidence clearly that noncommutativity, in the form that we have introduced here, not only prevents the singularities that occur in the Classical and Wheeler-DeWitt quantization approach to the Bianchi Cosmology, but it also provides the driving force which, under appropriate boundary conditions, allows the system to leave from a stage of oscillatory evolution within Planck length scales, to stages of regions where noncommutativity becomes negligible and the universe growth is monotonical. In Sec. VIII we summarize what we consider are the main results of this work and possible future lines of research that would extend it. II. TWISTED DISCRETE TRANSLATION GROUP C⋆-ALGEBRA AND DEFORMATION QUANTIZATION Let us now consider [23], [24], [25] the twisted (unital, discrete) C∗-dynamical system Σ = (A, G, α, σ) where the algebra A can be related by means of a *-homomorphism to the C∗-algebra A ⊂ B(H) of bounded operators with unit, acting on a Hilbert space H. For this purpose and as a starting point of our analysis we observe that, since the 4 base topological M space in Classical Bianchi I Cosmology is an R3, for which translations are isometries, whereas physical space at the Noncommutative Geometry level is described as a sort of a subjacent discrete noncommutative cellular structure (posets), we let A be the algebra of the noncommutative extended Heisenberg-Weyl group [14], G be the discrete topological group of translations in R3, (α, σ) the twisted action of G on A, with α denoting the map α : G→ Aut(A) and σ : G×G→ T (A) is a normalized 2-cocycle on G with values in the multiplicative group T of all complex numbers of unit modules, such that σ(x1,x2)σ(x1 + x2,x3) = σ(x2,x3)σ(x1,x2 + x3), x1,x2,x3 ∈ G σ(x,0) = σ(0,x) = 1. (II.1) In the above we have identified the discrete Abelian group of translations G with the vector space T3, associated with R 3 as an affine space with a discrete topology and with coset decomposition T3 = ∞ ∑ j1,j2,j3=−∞ (µiji)êi, ji ∈ Z, (II.2) where the êi are the basic translations in R3, the vectors x(l) = ∑3 i=1(µij(l)i)êi ∈ T3 are elements of R3 as a group and the set Γ : {µij(l)i} form a 3-dimensional cell. We then have Definition II.1. A left σ(x1,x2)-projective unitary representation Û of G on a (non-zero) Hilbert space H is a map from the group G into the group U(H) of unitaries on H such that U(x1)U(x2) = σ(x1,x2)U(x1 + x2). (II.3) Taking in particular U(H) ∋ σθ(x1,x2) := σ(x1,x2) = e−iπxT 1 R x2 = e−iπθ·(x1×x2), (II.4) where R is the anti-symmetric matrix R =    0 θ3 −θ2 −θ3 0 θ1 θ2 −θ1 0    , (II.5) where the θi have been assumed to be Poincaré invariant, as shown in [26], when considering a deformation of the universal enveloping Hopf algebra U(P ) of the Poincaré algebra P by means of a Drinfeld twist [27]. Definition II.2. A left projective regular unitary realization of the algebra (II.3) and (II.4) on l2(G) can be defined as 〈x|Ûi|ξ〉 := e−2πiεixi〈x− 1 2 εiêi × θ|ξ〉 = e−2πiεixiξ(x− 1 2 εiêi × θ); ξ(x) ∈ H. (II.6) Identifying x with the corresponding function on T3 which is one at x and zero otherwise, i.e. if we let this function be δx ∈ l2(T3) (the delta function at x) then it readily follows that Ûiδx := e−2πiεixiδ( 1 2 εi êi×θ+x), (II.7) and Ûi|x〉 = e−2πiεixi |x+ 1 2 εiêi × θ〉. (II.8) Thus the unitary Ûi translates the vector x in a direction perpendicular to êi by the amount 1 2εiθ. It is now fairly straightforward to show, by successive applications of (II.6), that ÛiÛj = e−iπεiεjθ·(êi×êj)Ûi+j , (II.9) 5 and interchanging indices and substituting back the result into (II.9) we arrive at ÛiÛj = e−2iπεiεjθ·(êi×êj)ÛjÛi. (II.10) Since the parameter of noncommutativity actually has units of length square the quantities εi must have units of length−1 and εiêi×θ are thus basic vectors in the directions perpendicular to the êi which determine the fundamental lengths of the lattice. Extending now the above algebra with the generators V̂l := V̂ (µlêl) such that V̂l|x〉 = |x+ µlêl〉, (II.11) so we find that V̂l also acts on the kets |x〉 ∈ H as a translation operator on the vector x in the direction of êl by an amount µl. It also follows from (II.11) that V̂iV̂l = V̂lV̂i, (II.12) and commuting with Ûi as given in (II.8), we arrive at ÛiV̂l = e−2πiεiµl(êi·êl)V̂lÛi = e−2πiεiµlδil V̂lÛi. (II.13) This is indeed a *-homomorphism between the C∗-algebra A ⊂ B(H) of operators generated by the unitaries Ûi’s and V̂l’s and the extended noncommutative Heisenberg-Weyl algebraA of the C⋆-dynamical system discussed before. Note also that the quantities µl and εi introduced in the above relations strictly appear so far as independent parameters of the action of the discrete subgroups of the twisted (extended noncommutative) Heisenberg-Weyl group. This would however imply two different simultaneous noncommutative lattices generated by the unitaries Ûi’s and V̂l’s. Clearly in order to avoid this the µl and êl · (εiêi × θ) must be related. We shall show later on that this relation appears naturally when constructing the Hilbert space on which these operators act. We also find it important to point out here that, although the expressions (II.9) and (II.10) for the subalgebra of the Ûi appear to be the same as that used to describe the quantum torus (cf. e.g. [28]), the realization (II.6) (or (II.8)) introduced here has quite different implications. Indeed, as mentioned in the paper cited above, in the quantum torus formulation the Ûi act as Laplacian operators that translate on momentum space, and thus are appropriate to describe noncommutativity in momentum space [29]. On the other hand the realization of the Ûi and V̂l unitaries in (II.8) and (II.11) is geared to generate a Hilbert space by sequential translations, effected by the noncommutation matrix factor, on a cyclic vector. Thus in this case the noncommutativity is associated with the dynamical configuration variables of our formulation. The strong repercussions for our developments of this choice of realization is evidenced in the analysis presented in the last sections of this work. III. GNS-CONSTRUCTION OF THE KINEMATIC HILBERT SPACE Let us now use this homomorphism to derive explicit forms for the elements of the Hilbert space H on which the operators in A act by applying the Gel‘fand -Naimark-Segal (GNS) construction [30],[11]. To this end first note that for any state functional φ we have that ∀ a ∈ A ∃ φ such that φ(a∗ ⋆ a) = 1. Moreover, since any element a in the subjacent algebra A is unitary, we have that this equality is always true here which, in turn, implies that the left ideal I = {a ∈ A |φ(a∗ ⋆ a) = 0} in A is empty, so that the quotient space Nφ = A/Iφ ≡ A ⇒ φ is faithful. Thus, by the GNS construction, we have a pre-Hilbert space with a non-degenerate product defined by A×A → C, 〈a, b〉 7→ φ(a∗ ⋆ b), (III.14) and where Hφ is the completion of A in this norm. Note that the ⋆-homomorphism πφ : A → B(Hφ), defines a representation (A,Hφ) of the C ⋆-algebra A by associating to an element a ∈ A an operator πφ(a)) ∈ A ⊂ B(H) by πφ(a)b = a ⋆ b, (III.15) 6 which is a well defined bounded linear operator in Hφ. Indeed, from the above definition it follows that πφ(a1)πφ(a2)(b) = a1 ⋆ a2 ⋆ b = πφ(a1 ⋆ a2)b, (III.16) which shows that (III.15) is in fact a representation. Note also that in this construction the C⋆-algebra is itself a Hilbert A-module. Now, in order to generate the elements of the Hilbert space we start with a distinguished vector ξφ which is cyclic for πφ, i.e. such that {π(a)ξφ|a ∈ A} is dense in Hφ. Since A is unital we can chose ξφ := 〈x = 0|ξφ〉 = ξφ(0, 0, 0) = I, which is clearly cyclic provided the parameters εi and µl, generated by the operators πφ(a) = Ûi, V̂l ∈ B(Hφ), according to (II.8) and (II.11) and which translate in directions perpendicular to each other, are appropriately related in order that the set of elements generated by the action of the πφ(a) on ξφ is indeed dense in Hφ. It is not difficult to show that such a consistency can be achieved by setting µ1 = n1 2 ε2θ3 µ2 = n2 2 ε1θ3 (III.17) µ3 = n3 2 ε1θ2, where, as we shall show later on in Section VII, the magnitudes ni ∈ N + and ε̄i are scale factors of the µi’s and εi’s determined by the relative relevance of the noncommutative tensor symbol in the different stages of evolution of the dynamical system that we shall consider later on. In fact, we can consider the µi’s and εi’s as introduced in the formalism to effectively represent a family of continuous projections πm,n acting on a family of topological spaces Y n such that πm,n : Y m → Y n, n ≤ m. (III.18) Hence the manifold M with Hausdorff topology (Y∞) can be recovered as the limiting procedure of the inverse of such a sequence of projectors [31]. Moreover, in the limit εi → 0 it readily follows that (II.8) becomes multiplicative and the µl decouple from (III.17) and (III.19), so our twisted Heisenberg-Weyl algebra reduces to that in [32] and the commutative lattices generated by the primitive spectrum of this algebra are now structure spaces of a T1 topology where, as we shall show later on in Sec.VI, the elementary length of the cell induced by the µl’s is of O(λP ). Taking the further limit µl → 0 will then result in the classical Heisenberg-Weyl algebra and a Hausdorff or T2-space. Note also that in some sense the relations (III.17) are an equivalent of the improved dynamics introduced in [33], which in our case appear directly from the consistency required by the translations generated by the noncommutativity. From (III.17), (II.8), and (II.11) we also get ε2θ3 =ε3θ2 ε1θ3 =ε3θ1 (III.19) ε1θ2 =ε2θ1. Consequently, it follows from the above relations that the subset {π(V̂i)ξφ} will be by itself dense in Hφ and, by virtue of (III.15) and (III.14) (and the GNS Theorem), we have that given a vector-state functional φ on {Vl} ⊂ A there is a ⋆-representation with a distinguished cyclic vector ξφ ∈ Hφ with the property 〈ξφ, πφ(Vl)ξφ〉 = 〈I, Vl〉 = φ(Vl). (III.20) Recall now that (II.11) implies that 〈x1 = 0|V̂l|ξφ〉 = ξφ(0+ µlêl) = ξφ(µlêl), (III.21) 7 so, if via the algebra *-homomorphism we associate to the element Vl ∈ A the operator πφ(Vl) = V̂ (−µlêl)), then combining (III.20) with(III.21) allows us to identify φ(Vl) with the character of the discrete translation group, so that ξkφ(xn) = e2πi ∑3 l=1 µl(klj(n)l), j(n)l ∈ Z (III.22) where k ∈ R 3, and µl are quantities whose magnitudes determine the size of the fundamental noncommutative lattice cell. Observe also that, since I is empty, the representation (Hφ, ξφ) is irreducible. The functions ξkφ(x) in (III.22) are a one-dimensional irreducible regular representation of the operator group D̄k(x) of the discrete Abelian group of translations. That is D̄k(xn) = e2πi ∑ l µl(klj(n)l), (III.23) and satisfies the relations of orthogonality and Poisson summation completeness [34] ∫ 1/2µl −1/2µl µldkl D̄ kl(j(1)l)D kl(j(2)l) = δj(1)lj(2)l , l = 1, 2, 3 ∞ ∑ ji−∞ D̄ki (ji)D ki ′ (ji) = ∞ ∑ mi=−∞ δ(µiki − µik ′ i +mi), (III.24) respectively, after noting that the left hand side of the second equation above is a periodic generalized function with period one [35]. Observing that since the representations (III.23) of the translation group are invariant under the reciprocal group, the range of fundamental domain of the components of the vector parameter k is −1/2µi ≤ ki ≤ 1/2µi. Also, making use of the completeness of the ket space {|k〉} we can write D̄kl(j(n)l) = e2iπj(n)lµlkl := 〈µlj(n)l|kl〉 = 〈x(n)l|kl〉, (III.25) with 3 ∏ l=1 ∫ 1 2µl − 1 2µl µldkl〈x(n)l|kl〉〈kl|x(n′)l〉 =: 〈x(n)|x(n′)〉 = δx(n),x(n′) . (III.26) Furthermore, by the Pontryagin duality theorem, the dual of a discrete Abelian group is a compact Abelian group, so by Fourier analysis we can write (for a fixed index i) f̂(ki) = ∞ ∑ j(l)i=−∞ f(j(l)i) e µij(l)i(2iπki), −1/2µi ≤ ki ≤ 1/2µi, i = 1, 2, 3, (III.27) and f(j(l)i) = ∫ 1/2µi −1/2µi dki f̂(ki) e −ki(2iπµij(l)i). (III.28) Denote by Γ = {eki(2iπµij(l)i)} the compact Abelian group of continuous characters dual to the twisted discrete translation group G, and let Ḡ denote the Abelian compact group of all characters, continuous or not, of G. Then Γ is a continuous isomorphism of G onto a dense subgroup β(G) of Ḡ. Thus, since the generators e(2iπki) of the basis of mono-parametric subgroups in (III.27) are isomorphic to the circle group T we have that the f̂(ki) in (III.27) can be regarded as elements of the dense subgroup of the Bohr compactification of the twisted discrete translation group onto the quantum 3-torus =Ḡ. In particular, setting x(l)i := µij(l)i we see that the function e2iπx(l)iki is continuous and periodic in ki, thus the polynomial function ∑N l=1 f(x(l)i) e −2iπx(l)iki is an almost periodic function in the sense of Bohr (cf. [36] [37]). 8 Furthermore if the latter function converges uniformly to the series ∑∞ l=1 f(x(l)i) e 2iπx(l)iki when N → ∞, then the limit function is also almost periodic. Next note that if we now introduce the reciprocal group of the discrete group of translations on the reciprocal lattice LR := {bR = bi/µi, bi ∈ Z}, (III.29) it follows immediately from (III.27) that f̂(ki) = f̂(ki + bi/µi), (III.30) which confirms the statement below equation (III.24) regarding the fundamental domain of ki. In summary, we have seen that the space-space noncommutativity of the Heisenberg algebra can be expressed by a realization of the associated Heisenberg-Weyl group by a C∗-algebra A ⊂ B(H) of bounded unitary operators with unit, acting on a non-separable Hilbert space where an orthonormal basis is the set of almost periodic functions : {ξkφ(x(l)) = D̄k(x(l)) = e2iπx(l)·k}, (III.31) given by the characters in (III.22). IV. QUANTUM COSMOLOGY FOR THE ANISOTROPIC BIANCHI I MODEL As it is well known the classical action function, after ADM reduction to canonical form, for a Bianchi I cosmology describing a gravitational field, with space-time metric gµν =      −N2(t) 0 0 0 0 a21(t) 0 0 0 0 a22(t) 0 0 0 0 a23(t)      , (IV.32) minimally coupled to a massless scalar field ϕ(t) independent of the spatial coordinates, is given by Sgrav + Sϕ = ( c3 G )∫ ( πij ġij − N(t) √ 3g [ − 1 2 (πk k ) 2 + πijπij ] ) d4x + ~ ∫ d4x ( pϕϕ̇− 1 2 N √ 3g p2ϕ ) , (IV.33) where (cf. Chapter 21 of [38]) the tensor densities πij are the canonical momenta conjugate to the metric components gij = a2i (t) (the square of the Universe radii), N(t) is the lapse function and pϕ is the canonical momentum conjugate to ϕ, with pϕ being in units of length and ϕ in units of inverse of length . Moreover, writing the kinematic term in (IV.33) as πij ġij = 2πiiaiȧi and making the definition 2πiiai := πi we can re-express the gravitational action in (IV.33) in the form Sgrav = 1 2 ( c3 G )∫  πiȧi − N(t) 2 √ 3g  − 1 2 ( 3 ∑ i=1 πiai )2 + 3 ∑ i=1 (πiai 2πi)     d4x, (IV.34) or, observing next from equation (21.91) in [38] that πij is unitless and therefore that πi has units of length, we can define a new quantity pi := c3 G~ πi, which has units of inverse of length, so (IV.34) can be written as Sgrav = 1 2 ~ ∫  piȧi − N(t) 2 √ 3g ( G~ c3 )  − 1 2 ( 3 ∑ i=1 piai )2 + 3 ∑ i=1 (piai 2pi)     d4x. (IV.35) 9 In addition, the scalar field action can be re-expressed as: Sϕ = ~ ∫ d4x ( pϕϕ̇− 1 2 N √ 3g ( G~ c3 )( c3 G~ ) p2ϕ ) , (IV.36) and defining pφ := ( c3 G~ ) 1 2 pϕ, and φ̇ := ( G~ c3 ) 1 2 ϕ̇, (IV.37) where both pφ and φ̇ are unitless, we arrive at Sφ = ~ ∫ d4x ( pφφ̇− 1 2 N √ 3g ( G~ c3 ) p2φ ) . (IV.38) Consequently the total classical Hamiltonian constraint is [39], [40]: Cgrav + Cφ = N(t) 2 √ 3g ( G~ c3 )    − 1 2 ( 3 ∑ i=1 piai )2 + 3 ∑ i=1 (piai 2pi)   + 1 2 p2φ   = 0. (IV.39) If we choose the lapse function to be N(t)(4(3g))− 1 2 = ( c3 G~ ) and assume for simplicity the following ordering for the quantum Hamiltonian constraint operator, we therefore have: Ĉ = Ĉgrav + Ĉφ = 1 2  − ∑ i6=j p̂ip̂j âiâj + ∑ i p̂iâ2i p̂ i  + 1 2 p̂2φ = 0̂. (IV.40) Now, since the action of the p̂i and âi operators on our Hilbert space basis of kets is to be derived from the unitary operator representations discussed in the previous section and whose action on the Hilbert space is displayed in equations (II.8) and (II.11). For this purpose it is important to notice that the Hilbert space is constructed from the noncommutative group of operators A. Moreover, due to the noncommutativity, the elements of this group are not exponentials of self adjoint operators. To construct the observables âi we thus take âi := − Ûi − Û † i 2iεi , (IV.41) so that âi|x(n)〉 = − 1 2iεi ( e−2iπεixi |x(n) + 1 2 εiêi × θ〉 − e2iπεixi |x(n) − 1 2 εiêi × θ〉 ) , (IV.42) and p̂l := ( Vl(µl)− V † l (µl) 2iµl ) . (IV.43) so that p̂l|x〉 = 1 2iµl (|x+ µlêl〉 − |x− µlêl〉) . (IV.44) That (IV.41) reproduces the uncertainty principle for mean-square-deviations of the distributions 〈Ψ|âi|Ψ〉 and the noncommutative algebra of the âi for the discrete case, can be seen by substituting (IV.41) in the commutator [âi, âl] and making use of (II.8) and (II.9). We then find that 〈j′|[âi, âl]|j〉 = ( 2i εiεl ) sin(πεiεlθ · (êi × êl)) 3 ∏ m=1 ∫ 1 2 − 1 2 dk̄m e2πik̄·(j ′−j) cos ( 2πεiµi[ji + ( 1 2µi )k · (êi × θ)] ) × cos ( 2πεlµl[jl + ( 1 2µl )k · (êl × θ)] ) where k̄m := µmkm, (IV.45) 10 from where it can be inferred that the quantity ( 2 εiεl ) sin(πεiεlθ · (êi × êl)) cos ( 2πεiµi[ji + ( 1 2µi )k · (êi × θ)] ) × cos ( 2πεlµl[jl + ( 1 2µl )k · (êl × θ)] ) (IV.46) is the symbol of the action of the operator commutator on the spectral representation of the product 〈j′|j〉. In the limit εiεlθ · (êi × êl) << 1 (since by (III.17) and (III.19) also implies εiµi << 1 ) , the above symbol of [âi, âl] is 2πθ · (êi × êl). The expressions (IV.42), (IV.44), are to be substituted into (IV.40) in order to derive the action of the constraint operator on the Hilbert vectors |x(n)〉. To make a detailed connection with other formulations we use the Feynman phase space path integral procedures considered in [32]. The general idea of the group averaging procedure (see e.g. [41]) is that the physical state |Ψphys〉 ∈ Hphys, which is a solution of the constraint equation, is derived by averaging the action of the unitary monoparametric Abelian group exp(iαĈ), α ∈ R, on a state |Ψkin〉 in an auxiliary kinematic Hilbert space Hkin dense in Hphys. Thus |Ψphys〉 = ∫ ∞ −∞ dα exp(iαĈ)|Ψkin〉. (IV.47) Heuristically (IV.47) can be justified as a refined algebraic quantization by observing that the integrand can be viewed as a Fourier Dirac delta representation: ∫ ∞ −∞ dα exp(iαĈ) ∼ δ(Ĉ), (IV.48) and that by acting on (IV.47) with U(β) = exp(iβĈ) we have U(β)|Ψphys〉 = exp(iβĈ)δ(Ĉ)|Ψkin〉 = δ(Ĉ)|Ψkin〉 = ∫ ∞ −∞ dα exp[i(α+ β)Ĉ]|Ψkin〉 = ∫ ∞ −∞ dα′ exp(iα′Ĉ)|Ψkin〉 = |Ψphys〉, (IV.49) therefore the unitaries U(β) ∀β act trivially on the physical states defined as in (IV.47), consistent with Dirac’s requirement that physical states be annihilated by the constraints. however, the physical state defined by (IV.47) is not normalizable. Hence, in order to eliminate one of the deltas in the inner product, this is defined according to (Φphys|Ψphys) := ∫ ∞ −∞ dα 〈Φkin| exp(iαĈ)|Ψkin〉. (IV.50) Clearly this definition of the inner product has the advantage that it remains the same for any two other physical states of the form |Φ′ phys〉 = exp(iuĈ)|Φphys〉. Now, an orthonormal basis of kinematic quantum states are |x, φ〉 := |x〉|φ〉, where |x〉 := |µ1j1, µ2j2, µ3j3〉 and |φ〉 are the eigenvectors of the scalar field, such that 〈x′, φ′|x, φ〉 = δx′,xδ(φ ′, φ). (IV.51) We can therefore write (IV.47) in this basis as 〈x, φ|Ψphys) = ∑ x′ ∫ dφ′ A(x, φ;x′, φ′)Ψkin(x ′, φ′), (IV.52) where the Kernel A(x, φ;x′, φ′) is given by A(x, φ;x′, φ′) = ∫ dα〈x, φ|eiαĈ |x′, φ′〉. (IV.53) 11 V. THE PATH INTEGRAL APPROACH We shall follow here the path integral approach, based on [42] and developed for a timeless framework in [32], which consists essentially in replacing the transition function in Feynman’s formalism by the Kernel A(xf , φf ;xI , φI), where the subscripts f and I denote the final and initial states of the system, and regarding the constraint operator exp(iαĈ) in (IV.53) in a purely mathematical sense as a Hamiltonian with evolution time equal to one. That is, eiαĈ = eitĤ where Ĥ = αĈ and t = 1. Emulating now the standard Feynman construction, we decompose the fictitious evolution into N infinitesimal evolutions of length λ = 1 N+1 . Thus we get 〈xf , φf |e iαĈ |xI , φI〉 = ∑ xN ,...,x1 ∫ dφN . . . dφ1 × 〈xN+1, φN+1|e iλαĈ |xN , φN 〉 . . . 〈x1, φ1|e iλαĈ |x0, φ0〉, (V.54) where 〈xf , φf | ≡ 〈xN+1, φN+1| and |xI , φI〉 ≡ |x0, φ0〉. If we now consider in detail the particular n-th term in (V.54) we can readily derive expressions for the remaining other terms. Thus, with Ĉ as given by (IV.40) we get 〈xn+1, φn+1|e iλαĈ |xn, φn〉 = 〈φn+1|e −iλαp̂2 φ |φn〉〈xn+1|e iλαĈgrav |xn〉 = ( 1 2π ∫ dpne iλαp2 neipn(φn+1−φn) ) 〈xn+1|e iλαĈgrav |xn〉. (V.55) To evaluate the gravitational constraint factor above note that, to order one in λ = 1 N+1 and for N ≫ 1 we have 〈xn+1|e iλαĈgrav |xn〉 ≈ δxn+1,xn + iλα〈xn+1|Ĉgrav|xn〉+O(λ2). (V.56) Making use of (IV.42), (IV.44), as well as of (II.8) -(II.11) we see that there are 16 terms conforming the transition function 〈xn+1|Ĉgrav|xn〉. These terms involve products of the unitaries and/or their conjugates. Let us consider in detail the term of the form 〈x(n+1)|V̂iV̂j ÛiÛj |x(n)〉 = e−iπεiεj(êi×êj)·θ e−2πi(εix(n)i+εjx(n)j)〈x(n+1)−µiêi−µj êj |x(n)+ 1 2 (εiêi+ εj êj)×θ〉. (V.57) Now, as pointed out in Sec.2 we have associated the action of the translation group on itself as leading to an affine space with a discrete topology and with a coset decomposition T3 = ∑∞ j1,j2,j3=−∞(µiji)êi, where j(l)i ∈ Z and the êi are the basic translations in R3. The vectors x(l) = ∑3 i=1(µij(l)i)êi ∈ T3 are elements of R3 as a group and the set Γ : {µij(l)i} form a 3-dimensional cell. This in turn led us (cf eqn. (III.26)) to introduce a Kronecker inner product for the space of these vectors. Moreover, when using the GNS construction to derive the kinematic Hilbert space we were also led to require that the translations induced by the Unitary operators Ûi and V̂l should be related in order that the “reticulations” induced by any of them should coincide. We suggested there that such a coincidence could be achieved by establishing the relations (III.17) and (III.19). This can now be verified directly by noting first that the arguments in the “bra” vectors in (V.57) are clearly integer multiples of the µi and so are the arguments of the “ket” vectors provided the following relations are satisfied: êl · [(εiêi ± εj êj)× θ] 2µl ∈ Z. (V.58) These requirements are indeed identically satisfied by the relations (III.17) and (III.19) for all the entries in the transition function in (V.56). Consequently 〈x(n+1)|V̂iV̂j ÛiÛj |x(n)〉 = e−iπεiεj(êi×êj)·θ e−2πi(εix(n)i+εjx(n)j) 3 ∏ l=1 ∫ 1 2µl − 1 2µl µldk(n)l × e−2πiµlk(n)l(j(n+1)l−j(n)l)e2πik(n)l[µjδlj+µiδli+ 1 2 êl·(εi êi+εj êj)×θ], (V.59) 12 and making use of (IV.41), (IV.43) and (V.59) we find that ∑ i6=j 〈x(n+1)|p̂ ip̂j âiâj |x(n)〉 = 1 2 ∑ i 0. After substituting this in (VI.129) we get χ̇i(0) = π ∑ j 6=i θij µiµj Rj sin(εiµiζi) sin(εjµjζj) sin(δi) sin(δj). (VI.132) Noting from (V.88) that |χi| ≤ 1 εiµi and consequently |Ri| ≤ 3 εiµi and using | sin(α)| ≤ |α|, we can establish an upper bound for the absolute value of χ̇i(0) and using (VI.127) yields |χ̇i(0)| = ∣ ∣ ∣ ∣ 2π2 ∑ j 6=i Rj sin(εiµiζi) sin(εjµjζj) sin(δi) sin(δj) ∣ ∣ ∣ ∣ ≤ 6π2εiµi ∑ j 6=i |ζi||ζj ||δi||δj |, (VI.133) For an upper bound M ∈ R+ such that 6π2εiµi ∑ j 6=i |ζi||ζj ||δi||δj | ≤M, (VI.134) the inequalities can be solved to obtain |ζi||δi| ≤ √ M 12π2εiµi , (VI.135) which can be further relaxed if all the χi’s are chosen to have the same sign and so |Ri| ≤ 2 εiµi , in which case |ζi||δi| ≤ √ M 8π2εiµi . (VI.136) Finally we need to enforce the cutoff condition in the interval of validity of ζi. This is done directly from demanding 1 εi sin(2πεiµiQ̄i(0)) ≥ Li, (VI.137) or equivalently 1 εi cos(εiµi|ζi|) ≥ Li, (VI.138) which for our case where εiµi|ζi| ≤ π 2 also implies |ζi| ≤ 1 εiµi arccos(εiLi). (VI.139) Together, the inequalities (VI.138) and (VI.139) provide the refinement for the admissible intervals of values for ζi and δi expressed now as 0 < |ζi| ≤ 1 εiµi arccos(εiLi), 0 < |δi| ≤ √ M 8π2εiµi 1 |ζi| . (VI.140) This criteria provides with the full description of the system below and above the cutoff where from expression (VI.121) the matching boundary conditions at the cutoff region must satisfy (ai)symb(0) = 1 εi sin(2πµiεiQ̄i(0)) = 2πµiQ̄i(0), χi(0) = 1 εiµi sin(2πµiεiQ̄i(0)) sin(2πk̄i(0)) = 2πQ̄i(0) sin(2πk̄i(0)), (VI.141) 22 which implements the change of physical variables when going from below the cutoff to the region above. In this sense any trajectory governed by the noncommutative algebra evolution of expressions (VI.93) and (VI.95), with boundary values (VI.131) and (VI.140) at the cutoff region, obeys a compatible commutative evolution (to order M) outside the Planckian region determined by (VI.122-VI.125). The results just obtained can be further explained as follows. The system has a 6-dimensional phase-space, of which a suitable parametrization of a projection is the 2-dimensional plot (Vsymb, V̇symb) shown in Fig.(1) (this phase-space diagram applies to the case discussed in section 8 with reference to Fig.(6) ) . This figure shows a monotone orbit followed by an oscillatory behavior emerging into a new expanding orbit. Even though the quantities εi, µj are linked by the fundamental physics θij , strictly from a differential equations point of view we can consider θij = 0 with εi, µj 6= 0. Then when θij = 0, the Ri are constant and the equations (which follow from multiplying (V.88) by Ri) Riχi = ( Ri εiµi ) sin(2πεiµiQ̄i) sin(2πk̄i) = const. (VI.142) provide a family of invariants of the system. thus in this formulation the universe will oscillate in a quasi-periodic way. Now, when θij 6= 0 the tori are subjected to the corresponding Hamiltonian perturbation. 500 1000 1500 2000 2500 3000 3500 VHΦL -40 000 -20 000 20 000 V   @ΦD FIG. 1. Phase-space plot of the volume with visible transition from an open collapsing orbit (lower branch) to periodic orbits connecting various invariant tori ending with an open expanding orbit (upper branch). Consequently the unperturbed orbits have now periods which depend on the amplitude (this can be seen simply by quadrature using (VI.142) for each degree of freedom. Moreover, as the orbits approach the origin in the Q̄i variables the period becomes longer, since this is a hyperbolic point. Then the classical KAM results ([44]) guarantee the existence of nearby invariant tori for a large (in measure) set of unperturbed tori. In the actual behavior of the solutions we have that, generically, the basic periodic solution of the ith degree of freedom pics up two more periods 23 due to the interaction with the two other phases. When the invariant tori come close to the separatrix the basic orbit has a long period. These corrections will cause the oscillations. Furthermore, since the basic solutions have long periods, the resulting orbits become very sensitive (as the numerics in the following Section shows) to the parameters and initial conditions. When considering the implications of this behavior in the evolution of the volume, we would expect a relatively fast contracting orbit away from the saddle point merging with a long period resulting thus in a periodic oscillation caused by the noncommutativity and merging again (due to the integrability of the commutative problem) with the expanding solution. It is important to recall that this behavior is not special but generic and is expected for any noncommutative model with an integrable structure in the commutative limit. We therefore can conclude from the above that generically the noncommutative scenario and its induced evolution of the the invariants (VI.142), produces multiple solutions and effective noncommutative lattice structures as a consequence of the cosmology dynamics. VII. NUMERICAL SOLUTIONS In order to provide consistent values for the parameters in the equations and for appropriate initial conditions in the interesting parameter regimes described qualitatively in the previous section, let us now recall equations (III.17) and (III.19) which may be written as µi = ni 2 εjθk with the indices i, j, k ordered cyclically. Expressing the above equation in units of Planck lengths we have µ̄iλP = ni 2 ε̄j θ̄k L̄j λP , (VII.143) where, as defined previously, bared symbols denote their magnitude and L̄j is the magnitude of the scale factor of the εj . Let us next consider the behavior of the two terms in the right of equation (VI.95). In the Planck region the scale magnitude of L̄j is of the order of a Planck length so also setting the scale magnitude ni of µi equal to a few Planck lengths we have that µi = εjθk ≈ 1λP = O(λp). Consequently µiεi is of the order of one in this case. Applying a similar reasoning to the expression θij µiµj we get that µ1µ2 ≈ 4λ2P ε̄1ε̄2θ̄2θ̄3 = O(λ2P ), (VII.144) which makes it consistent with (VI.127) and, since for calculation simplicity we are taking the tensor of noncommu- tativity to be of the same magnitude for all three planes, the second term on the right of equation (VI.95) turns out to be commensurate with the first. To illustrate the possible scenarios and how markedly they depart in the noncommutative case from classical (and non-classical) solutions, consider then the strongly noncommutative solutions of (VI.109) which occur when the noncommutative force term described above is commensurate with the first term in (VI.95) at all times. As mentioned, this corresponds to values of εi such that εiµi is of order one. Fig.2 and Fig.3 constitute examples of this regime, with evident similar properties, obtained for numerical values of εi = 0.8(λp) −1 and εi = 0.4(λp) −1 respectively. As neither of the solutions can reach the scales that would make noncommutative effects negligible the solutions are confined to Planckian scale volumes. Although similar, the system in Fig.3 is seen to evolve more diversely than in Fig.2 with global minima and maxima now differing by orders of magnitude. The irregular oscillatory behavior is in both cases the product of the noncommutative force term acting as a drive, modulating the frequencies of the solutions of the independent symbols of the radii of the universe, as can be better observed in Fig.4 where the three independent symbols (ai)symb associated to the volume in Fig.3 have been plotted. This shows explicitly that it is the noncommutativity the agent which eventually drives the universe to scales past the Planckian scale through the smooth cutoff. 24 -3 -2 -1 1 2 3 Φ 0.5 1.0 1.5 VHΦL FIG. 2. For εi = 0.8(λP ) −1, solutions for the Volume (with initial conditions for the radii symbols of order λp) display oscillatory behavior. Maxima and minima are always within the same order of magnitude and the system is confined to Planckian volume scales. -15 -10 -5 5 10 15 Φ 2 4 6 8 10 12 14 VHΦL FIG. 3. Solution for ε = 0.4(λP ) −1 . For smaller ǫi the system has access to bigger volumes and constructive interference among the independent symbols of the radii allows the formation of maxima of orders of magnitude greater than the minima. For values of ǫi < 1/Li these maxima eventually reach the cutoff region where the solutions are governed by the commutative regime and Eqs. (7.140)-(7.143). By analyzing the χi variables, which in the commutative case are constants of motion and therefore can be inter- preted as action variables, it is observed from Fig.5 that their behavior in the Planckian regime is not adiabatic and noncommutativity is not simply a perturbation. In fact, the abrupt changes of these variables are associated to min- ima of the volume where noncommutative effects are stronger, whereas approximately adiabatic regions correspond to maxima of the volume and such regions become more and more dominant at larger scales. It is then that the evolution of the system can continue along commutative states, which is the basis for our selection of boundary values 25 Ha1Lsymb Ha2Lsymb Ha3Lsymb -10 -5 0 5 10 Φ 0.5 1.0 1.5 2.0 2.5 HaiLsymb FIG. 4. The independent symbols (a1)symb, (a2)symb, (a3)symb, associated to the volume in Fig.3, display complex evolutions due to the noncommutative force term that mixes interactions in the three independent directions at the cutoff, as confirmed by the following cases. Χ2 Χ1 Χ3 -10 -5 0 5 10 Φ 0.5 1.0 1.5 2.0 Χi FIG. 5. Plot of χ1, χ2, χ3 associated to the volume in Fig.3 where the approximately adiabatic regions around φ ≈ ±3.3 correspond to the global maxima seen for the volume. Thus, let us now consider the evolution when approaching the cutoff from below, i.e. near L̄i = 30 then, by virtue of (VI.121), the first term on the right of (VI.95) becomes πQ̄i cos(2πk̄i)R1 with Ri given by (VI.94) with α = β = γ = 0 and the χi becoming constants of motion. On the other hand, after observing that (VII.144) is independent of scales, and therefore the coefficients of ˙̄kj are again of order one and the second term becomes negligible relative to the first one so the evolution beyond this stage is given by equations (VI.122)-(VI.125); In this case Q̄i ≈ q̄i. Moreover, observe that ∑3 j 6=i θij µiµj ˙̄kj acts as a force with unitless ”mass” θij µiµj and unitless acceleration ˙̄kj driving the 26 canonical variables Q̄i in a direction perpendicular to their ith-components. This is made even more transparent when noting that by setting the tensor of noncommutativity equal to zero in (VI.95) the Ri become constants of motion and the remaining first term becomes strictly oscillatory. To exemplify this kind of solutions consider first the type of bounce depicted in Fig.7. Here we have a scenario where a collapsing trajectory (dashed) enters the noncommutative regime from the left, leading to a noncommutative evolution (solid) below the cutoff, where a number of noncommutative oscillations can be observed, until the effects of the noncommutative force term bring the system to an expansion phase such that it can reach the cutoff region and finally continue along a continuous expansion. Fig.7 provides more insight on the underlying interactions among the independent symbols (ai)symb that, due to the constructive and destructive interferences, lead to the behavior of the volume shown inside the noncommutative region. -6 -5 -4 -2 -1 0 Φ 10 000 20 000 30 000 40 000 V HΦL FIG. 6. Collapsing and expanding solution for εi = 0.031(λP )−1 . The noncommutative evolution (solid), compatible with the boundary values of a collapsing solution (dashed) that enters from the left of the figure, remains inside the noncommutative region for a finite period of time before constructive interference brings the system back to the commutative region expanding away from the cutoff. To finalize the discussion regarding this case compare the corresponding evolution of all the χi’s in Fig.8 with that of Fig.5 which confirms the fact that at larger scales the adiabatic regions become more dominant and, in particular, it is at both extremes of Fig.8 that the system continues evolving for φ ≷ 0 along those constant values of χi. In terms of the stationary phase approximation the solutions so far obtained are for the center of a (gaussian) quantum state moving along classical paths. Thus, in most cases the complete picture of the collapse followed by an expansion is set to occur given decoherence is absent. Our two final examples deal with this possibility. The first case of Fig.9 shows a collapsing solution obtained for boundary conditions with ζi > 0 near the cutoff. Because in the commutative regime (dashed) nothing prevents the system from collapsing all the way down to Planckian scales the system will eventually enter the noncommutative regime with boundary values at the cutoff (dot) compatible with a noncommutative evolution (solid) that, just as the previous solutions, avoids singularities and also displays the irregular oscillatory behavior which is the strong indicator of noncommutative effects taking place. As the center of the quantum state remains oscillating within Planck length scales it can be said the state has dissipated due to decoherence. Time reversing the previous scenario would lead to a situation where the quantum state evolves from decoherence to an expansion. Fig.10 corresponds to the numerical solution for this case characterized by ζi < 0 near the cutoff. 27 Ha1Lsymb Ha2Lsymb Ha3Lsymb -6 -4 -2 0 2 Φ 5 10 15 20 25 30 HaiLsymb FIG. 7. Independent symbols (a1)symb, (a2)symb, (a3)symb for ε = 0.031(λP )−1. The constructive (resp. destructive) interference inside the noncommutative regime region leading to the evolution of the volume above (resp. below) the cutoff in (Fig.6) is evidenced. Χ1 Χ2 Χ3 -6 -5 -4 -3 -2 -1 0 1 Φ 0.5 1.0 1.5 2.0 2.5 3.0 Χi FIG. 8. Plot of χ1, χ2, χ3 associated to the volume in Fig.6 where simultaneous regions of constant χi at the left and right of the figure lead to the asymptotic evolution of the volume beyond the cutoff. Once again the noncommutativity driven oscillations of irregular amplitudes are noted before the system reaches the commutative regime by means of the noncommutative force term discussed previously. Above the cutoff the volume evolves according to (VI.122-VI.125) with boundary values at the cutoff (dot). 28 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Φ 10 000 20 000 30 000 40 000 VHΦL FIG. 9. Collapsing solution for εi = 0.031(λP )−1 . The commutative regime solution (dashed) enters the noncommutative region through the cutoff (dotted) and continues below it along a noncommutative evolution with compatible boundary values (dot). The quantum state undergoes dissipation and cannot bounce back. -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Φ 10 000 20 000 30 000 40 000 VHΦL FIG. 10. Expanding solution for εi = 0.031(λP )−1 . For a fixed cutoff value Li = 30λP the noncommutative regime solution (solid) expands from decoherence reaching the cutoff region (dotted) following a commutative evolution algebra (dashed) compatible with the boundary values (dot). VIII. CONCLUSIONS In this article we approach Quantum Cosmology from the point of view of a minisuperspace of a theory of Quantum Gravity . We employ in particular the noncommutative C⋆-algebra A outlined in Sections II and III which provides a well founded mathematical structure for introducing the concept of noncommutativity, from the point of view of an 29 operational impossibility of measurement at distances smaller than a few orders of the Planck length. This approach also allowed us to relate the C⋆-algebra formulation to some aspects of the Loop Quantum Cosmology, as mentioned in Section III as well as in the discussion of the asymptotics and numerics in Sections VI and VII. In fact, taking εi → 0 in (II.8) reduces our noncommutative C⋆-subalgebra of A to a subalgebra of commutative Ûi’s which, together with (II.11), would lead to essentially the same results as those contained in Ref.[32]. Moreover, when considering the εi as scale factors and acted by a test function of compact support which regularizes them, we have that the limit εi → 0 decouples εi from µi in (III.17) and (III.19). Hence, as shown in (VI.128), the µi are always of the order of a Planck length. This implies that the granularity attributed to space in LQG is induced in our formalism due to noncommutativity. Also the LQG variables involve the concept of holonomies. But holonomies are naturally understood in the theory of principal fiber bundles as integrals of connections between two fibers. Although the trajectories resulting from these integrals are not necessarily closed in the bundle space, they are when projected to the base space. This would suggest the idea of the loops. However, there is nothing in classical differential geometry that says that the loops cannot have infinitesimal radii when the fibers over base space are infinitesimally close. To have a minimal radius one has to assume a discrete underpinning the continuum of base space, which accounts for the ”granularity” of space in LQG and is reflected in the introduction of non-piecewise parameters of the Heisenberg-Weyl group in order to avoid the implications of the Stone-von Neumann Theorem. Thus ”granularity” in LQG corresponds to noncommutativity in our formulation. Moreover, connections (gauge fields) are, according to Connes’ Noncommuative Geometry, a consequence of noncommutativity [46], so all this therefore suggests its underlying presence in the three main approaches mentioned in the Introduction. In Sections IV-VII the quantum collapse of a Bianchi I Universe was studied in the context of noncommutative geometry. The noncommutativity of the space variables (the axes of the Bianchi Universe) was taken into account in a consistent way by representing them in terms of the twisted discrete translation group algebra of Sec.II. This representation is then used to construct the transition amplitude by using the Feynman integral formalism, which was shown to be dominated by an effective action that provides a new set of equations that resulted to have a new dynamical behavior that took into account the effect of the noncommutativity. It was shown asymptotically and numerically in a generic case that the noncommutativity induces an oscillatory motion of the volume due to the nontrivial evolution of the action variables which are constant for reticular space commutative theories. We thus have that the dynamical effects of noncommutative produce an oscillatory behavior of the volume in the region of the quantum bounce of reticular space commutative theories. It will be interesting to study if these oscillations in a full quantum field theory with spatial degrees of freedom can be indeed interpreted as a topological change. The differences mentioned above between our formalism and LQC lead to some additional physical implications which result from our GNS construction of the kinematic Hilbert space. The basic point being that the reticulation induced on the arguments of the Hilbert space contain at each point a tower of states, generated by the consistency conditions required between the twisted translations produced the unitaries Û ’s and the translations due to the V̂ ’s. This implies that our reticulation induced by noncommutativity is not the same as that in Ref.[33] and allows us to have, within the cosmology, a mechanism which could prevent that all the fluctuations in our Bianchi I universe could grow, thus avoiding to have a bounce at low matter densities. This fundamental characteristic is obtained only in the improved version of the polymeric cosmology of LQC, while in our case it occurs naturally because of the way noncommutativity was implemented. Moreover, in spite of the persistent difficulties inherent to this field of research to obtain experimental information, we could hope that phenomena lying in the interface of general relativity and quantum physics, such as those involving quantum entanglement and quantum coherence and which may be accessible to the experiment in the near-term future, could provide further theoretical insights to a full quantum theory of gravitation. This is suggested by the study of noncommutativity in a simpler problem [45] where it was shown that depending on the width of the wave packet of a coherent state one could go from the commutative regime for wide packets to the noncommutative regime for narrow packets. To perform this evolution one needs to find a consistent analogue of the Schrödinger equation in the noncommutative regime, and solve this equation asymptotically as well as numerically in order to understand this transition. This is currently under study. 30 IX. ACKNOWLEDGMENT The authors are grateful to Prof. Michael Ryan for many stimulating discussions on the subject and to Prof. Noel Smyth, from the Department of Mathematics and Statistics, The Kings Buildings, University of Edinburgh, for valuable help with some preliminary numerical analysis of the system. This work was supported in part by CONACyT Project UA7899-F (M.R., A.A.M.) and DGAPA grant UNAM-IN109013 (J.D.V.). [1] D. Bahns, S. Doplicher, K. Fredenhagen, and G. Piacitelli, Commun. Math. Phys. 308, 567 (2011). [2] S. Doplicher, K. Fredenhagen, and J.F. Roberts, Commmun. Math. Phys. 172, 187 (1995). [3] S. Doplicher, K. Fredenhagen, and J.F. Roberts, Phys. Lett. B 331, 39-44 (1994). [4] G. Amelino-Camelia, G. Gubitosi, and F. Mercat, Phys. Lett. B 676, 180 (2009). [5] L. Tomassini and S. Viaggiu, Class. Quantum Grav. 28, 075001 (2011). [6] P. Martinetti, F. Mercati, Tomassini L., Rev. Math. Phys. 24, 1250010 (2012). [7] A. Connes, M. R. Douglas and A. 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Salisbury, J. Helpert, and A. Schmitz, Gen. Relativ. Gravit. 40, 1475 (2008). [41] D. Marolf, in Proceedings of the 9th Marcel Grossmann Conference, Rome, 2000, edited by B.G. Gurzadyan, R.T. Jantzen and R. Ruffini (World Scientific, Singapore, 2002), p. 1348. [42] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets (World Scientific, Singapore, 2004) [43] L. Hörmander, The Analysis of linear Partial Differential Operators I, Second Edition, (Springer, Berlin, Heidelberg, 1990). [44] J.K. Moser, Lectures in Hamiltonian systems, Mem. AMS 81, 1 (1968). [45] A.A. Minzoni, L.R. Juarez, and M. Rosenbaum, Phys.Lett. A 373, 1510 (2009). [46] A. Connes, J. Math. Phys. 36, 6194 (1995). Noncommutative Coherent States and Quantum Cosmology Román Juáreza and David Mart́ınezb Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543 , México D.F., México a)roman.juarez@nucleares.unam.mx Instituto de Investigación en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, b)dmr@mym.iimas.unam.mx ABSTRACT The set of coherent states for a noncommutative quantum Bianchi I anisotropic cosmology were built to circumvent the absence of a simultaneous set of configuration observables. By extending known methods of path integrals with coherent states to their noncommutative analogues allowed us to obtain the formal expression of the propagator of the theory in terms of the covariant Husimi Q-symbol of the quantum constraint. The analysis of the equations of motion resulting from a steepest descent procedure showed the existence of solutions displaying a minimum value of the volume scaling function which are, in turn, compatible with a physically inspired definition for a bounce. More importantly, a lower bound for the volume at the bounce which incorporates quantum mechanical and noncommutativite contributions was established. The asymptotic analysis of the solutions in the vicinity of the bounce was performed by implementing techniques used in boundary-layer problems. The numerical simulations that confirm our results are also presented. PACS: 03.70.+k, 98.80.Qc, 03.65.Sq 2010 MSC: 81Q99, 81R60, 81S30, 81Q20, 81S40, 76M45 1 2 1. Introduction Since their introduction by Glauber [1], coherent states have become fundamental objects in the study of a plethora of physical phenomena. Their properties have profound conceptual implications, such as the case of their non spreading nature which allows to read classical behavior from quantum systems. As normalized overcomplete sets for Hilbert spaces they correctly encode all the probabilistic interpretation of Quantum Mechanics. In a more mathematical context they constitute the appropriate language to study harmonic analysis in Lie groups and, as it was shown in [2], they bridge different deformation quantization schemes. As for the latter it was also demonstrated in [3] that such bridge remains true even for the case of Noncommutative Quantum Mechanics understood as the quantum theory associated to the Lie algebra hθ2n+1 of commutators: (1.1) [Q̂i, Q̂j ] = iθij Î, [Q̂i, P̂j ] = i~δij Î, [P̂i, P̂j ] = 0, i, j = 1, ..., n, where θij is a real antisymmetric constant matrix. The algebra (1.1) can be justified by simultaneously applying the equivalence principle and the uncertainty principle [4]. According to which the measurement of the position of a particle in every spatial coordinate with ever higher precision leads to a minimum possible value, the Planck length ℓp = √ G~ c3 , beyond which smaller distances lack operational meaning due to the presence of horizons that prevent to extract information from such a region.∗ This implies an uncertainty principle among spatial coordinates ∆qi∆qj ≥ ℓ2p that in the language of Quantum Mechanics translates to the first commutator in (1.1) if |θij | ∼ ℓ2p. Research in Loop Quantum Cosmology viewed as the one parameter minisuperspace sector of Loop Quan- tum Gravity has shed light on the implications of implementing a smallest length scale as inherent property of the structure of space-time [5]. There, the spectra of geometrical observables (volumes and areas) become discretized and the classical singularities are replaced by a bounce behavior which connects collapsing solu- tions to expanding ones. This interesting aspect is actually the consequence of recurring to an appropriate Hilbert space for the theory known as the polymeric Hilbert space Hpoly. Even more interesting is the fact that the usual normalizable basis of Hpoly is closely related with the set of coherent states based on a regular lattice (cf. Chap. 15 of [6]). In this direction it has also been shown that noncommutativity of space can dynamically reproduce features of field theories based on lattices by studying the evolution of gaussian (coherent) states [7]. Therefore, it is tempting to implement the full coherent-state formalism in the case of a Noncommutative Quantum Cosmology, where the configuration observables will be described by the algebra (1.1), in the hope that at an effective level the dynamical properties of noncommutativity suffice to generate solutions which are free of singularities. Overcomplete sets of coherent states have been constructed elsewhere for the case of noncommutative theories [3, 8, 9, 10], where it must be emphasized that the representations used in all of these references are not equivalent and consequently the results obtained from their implementation in a given problem should differ. In the present work, however, we will opt for the use of the coherent system developed in [3]. Thus, to investigate the consequences of introducing a noncommutative structure at the level of a Quan- tum Cosmology we organized this work in the following manner: In §2 and §3 we review the basics of Perelomov’s definition of Generalized Coherent States and reproducing kernels [11, 12], with their applica- tion to Feynman integrals of constraint theories by means of projectors, according to Klauder’s procedure of coherent state quantization [13]. We then devote §4 and §5 to construct the generalizations of these concepts ∗ℓp = 1.6162× 10−35m. 3 for Noncommutative Quantum Mechanics, where the operators representing position observables satisfy non trivial commutators. In §6 we briefly summarize the anisotropic Bianchi I model in the ADM separation by selecting the scale factors from the metric and their conjugate momenta as dynamical variables. Then in §7, by promoting these variables to quantum operators that satisfy a noncommutative algebra, we give a notion of a Noncommutative Quantum Cosmology. Here, we construct the quantum equivalent of the Hamiltonian constraint in terms of ladder operators, associated to the observables of the theory, and we obtain the corresponding coherent state Husimi Q-symbol in the path integral action. In §8 we analyze the equations of motion from a stationary phase evaluation of the action. A bounce of the system is defined in §9 and its implications on the solutions are studied. In §10 we provide various numerical solutions that illustrate the bounce-like behavior at Planckian scales. Finally in §11 we discuss our results and present our conclusions. 2. Generalized Coherent States The existing definitions for a coherent state are just as numerous as its applications (see e.g., [14, 15, 6] and references therein). Perelomov’s definition of generalized coherent state [11, 12] is adequate to study physical systems with an underlying symmetry group, as is the case of elementary systems [2]. Definition 1 (Coherent State). An elementary (classical) system corresponds to a symplectic manifold X which is homogeneous under the action of a Lie group G and, in consequence, isomorphic to the orbits generated by a maximal subgroup (the isotropy group) H ⊂ G, meaning X ≃ G/H. If in addition G is semisimple and acts on a Hilbert space via some UIR Û(g) and H is compact, then, the coherent states of H tagged by points of X are the vectors obtained via the transitive action of G on the H invariant subspace H0 ⊂ H. Thus the first step in building the CS system is picking a fiducial normalized state |φ0⟩ ∈ H0, called the ground state, which due to the H-invariance satisfies (2.2) Û(h)|φ0⟩ = eiρ(h)|φ0⟩, h ∈ H, where ρ(h) is a real valued function of h. Now, because G admits a decomposition in terms of left cosets, i.e. G = {gxh| gx ∈ G/H, h ∈ H}, the action of an arbitrary element g ∈ G over |φ0⟩ is given by (2.3) Û(g)|φ0⟩ = Û(gx)Û(h)|φ0⟩ = eiρ(h)Û(gx)|φ0⟩, ∀g ∈ G, where gx is identified with the geometric point x ∈ X. In this way the vector Û(gx)|φ0⟩ from the previous expression is, according to the definition, the generalized coherent state associated to the point x: (2.4) |x⟩ := Û(gx)|φ0⟩, ∀gx ∈ G/H, which, by construction, will also be a normalized state of H. The first and foremost property of the set {|x⟩} is that, due to Schur’s lemmas, there is a measure dµ(x) for which the coherent states form a complete basis of H, in other words, there’s a resolution of unity: (2.5) ∫ X dµ(x) |x⟩⟨x| = ÎH, where dµ(x) is actually the Riemann invariant measure in X. The second property of {|x⟩} is the modulus of the transition function between any two coherent states: (2.6) 0 < |⟨x|x′⟩| ≤ 1, 4 which, along with the resolution of unity, implies the well known fact that the coherent states constitute an overcomplete basis. The bounded and continuous function K(x′, x) := ⟨x′|x⟩ provides an example of a reproducing kernel [16], meaning (2.7) K(x′, x) = ∫ X dµ(y)K(x′, y)K(y, x), ∀x′, x, an identity that may appear trivial simply by making use of (2.5). However, contrary to what happens in the case of Dirac-δ functions, by setting x′ = x and noticing K(x, y) = K∗(y, x) it is then clear that Ky := K(y, ·) ∈ H. Also by evaluating any state in the coherent basis ⟨x|ψ⟩ = ψ(x) it is seen that (2.8) ψ(x) = ∫ X dµ(y)K(x, y)ψ(y), which expresses the non-local nature of K (the integral has actually to be carried out). In what follows we will use the above definition of coherent state to obtain the noncommutative equivalent of the path integral formalism, but first we will review the necessity of a well defined reproducing kernel for the case of constraint theories. 3. Constrained Reproducing Kernel When working with the canonical quantization scheme for a constraint system [17], once all the second– class constraints have been removed, the supplementary quantization condition that unambiguously ensures the evolution of a physical quantum state demands: (3.9) Φ̂a|ψphys⟩ = 0, ∀Φ̂a, where {Φ̂a} is the set of first–class quantum constraints. Expression (3.9) establishes the appropriate Hilbert space Hphys for the theory. Thus if H constitutes a Hilbert space for a suitable representation of the operators that characterize the system, known as the kinematical Hilbert space, then, only the vector subspace Hphys := {|ψ⟩ ∈ H| Φ̂a|ψ⟩ = 0, ∀Φa} will be of main interest. If we decide to use a basis of coherent states for H then there will be an overcomplete subset of {|x⟩} contained in Hphys. This basis and its associated kernel shall be obtained by means of a projector which can be constructed using the method of group averaging with the constraints [13]: (3.10) Ê := ∫ dµ(σ)e−i ∑ a σaΦ̂a , where dµ(σ) is a (normalized) invariant measure on the group manifold generated by the algebra of con- straints.† Due to the measure invariance and the Lie algebra of the constraints, Ê is a self-adjoint and idempotent operator: (3.11) Ê † = Ê, Ê 2 = Ê, so (3.10) defines a projector indeed. †Observe that the σa must have the appropriate units so the argument in the exponential is dimensionless. 5 The operator Ê produces a surjective endomorphism H ։ Hphys which now may be implemented to obtain the corresponding reproducing kernel in Hphys. By noticing that (3.12) ⟨x′|Ê|x⟩ = ⟨x′|Ê2|x⟩ = ∫ X dµ(y)⟨x′|Ê|y⟩⟨y|Ê|x⟩, and after comparing with (2.7) it shows that the projected transition function KΦ(x′, x) := ⟨x′|Ê|x⟩ is a reproducing kernel too. Therefore, similarly to equation (2.8), when projected on the basis {|x⟩}, any state |ψphys⟩ will have the form (3.13) ψphys(x) = ∫ X dµ(y)KΦ(x, y)ψphys(y) which together with (3.12) are expressions purely in Hphys as intended. 4. Noncommutative Coherent States of Hθ 2n+1 By non-commutative coherent states we will mean the states defined in expression (2.4) where the cor- responding symmetry group G is the extended Heisenberg-Weyl group Hθ 2n+1 whose Lie algebra hθ2n+1 generators satisfy the commutators (1.1). Thus hθ2n+1 corresponds to the algebra of Quantum Mechanics, but where position observables do not commute anymore, implying a new uncertainty principle that sets a minimum value in the precision of any attempt to locate a particle in space: (4.14) ∆Qi∆Qj ≥ |θij | 2 . In this case a simultaneous basis for position operators is no longer possible, and this will represent our main motivation behind using coherent states as it should be evident in what follows. It is worth mentioning that other bases are also admissible (see e.g. [18]), for example of eigenstates of P̂ operators, or one spatial coordinate and n−1 momenta, although contrary to what happens with a coherent state basis none of those belong to H. The typical element of Hθ 2n+1 is the exponentiation g(c, q, p) = eχ(c,q,p) where χ(c, q, p) ∈ hθ2n+1 corre- sponds to (4.15) χ(c, q, p) = icÎ+ i ~ n∑ i=1 (piQ̂i − qiP̂i); c ∈ R, q ∈ R n, p ∈ R n. If we now make use of alternate generators (cf. [3]) Âi := 1√ 2~ (Q̂i + 1 2~ n∑ j=1 θijP̂j + iP̂i), † i := 1√ 2~ (Q̂i + 1 2~ n∑ j=1 θijP̂j − iP̂i), (4.16) the original position and momenta operators can be written as Q̂i = 1√ 2~ [ Âi + † i + i 2~ n∑ j=1 θij(Âj − † j) ] , P̂i = 1 i √ 2~ (Âi − † i ). (4.17) 6 The group element g(c, q, p) in terms of the new generators is (4.18) g(c, q, p) = g(c, α, α∗) = eiĉI exp [ n∑ i=1 (αi † i − α∗ i Âi) ] , with the parameters αi, α ∗ i ∈ C. Operators Âi,  † i satisfy the n-dimensional creation–annihilation commutator algebra (4.19) [Âi,  † j ] = δij , [Âi, Âj ] = [† i ,  † j ] = 0. where each pair Âi,  † i spans a Hilbert space Hi in the Fock basis {|mi⟩}, satisfying Âi|mi⟩ = √ mi|mi − 1⟩, † i |mi⟩ = √ mi + 1|mi + 1⟩, |mi⟩ = (A† i ) mi √ mi! |0⟩, (4.20) so the entire Hilbert space H = ⊗n i=1 Hi is spanned by the basis {|m1, ...,mn⟩} From (4.18) it is clear that Î generates the maximal compact subgroup U(1) which trivially leads to the homogeneous space C n = Hθ 2n+1/U(1). Thus according to expressions (2.3) and (2.4) from the general theory, the noncommutative coherent states of H tagged by points of Cn are those obtained from the action (4.21) |α⟩ = exp [ n∑ i=1 (αi † i − α∗ i Âi) ] |0⟩ = n∏ i=1 e(αi † i −α∗ i Âi)|0⟩, which are just the usual Glauber-Sudarshan coherent states corresponding to the eigenstates Âi|α⟩ = αi|α⟩, with Âi given by (4.16). An important consequence, due to the these considerations, is that the set of phase-space variables {qi, pi, i = 1, ..., n} are precisely the expectations ⟨α|Q̂i|α⟩ = qi = 1√ 2~ [ αi + α∗ i + i 2~ n∑ j=1 θij(αj − α∗ j ) ] , ⟨α|P̂i|α⟩ = pi = 1 i √ 2~ (αi − α∗ i ), (4.22) and so they are quantities that can be specified simultaneously and do not constitute a major conflict, contrasting with the case of eigenvalues of Q̂i. The resolution of unity is given by ÎH = ∫ Cn dµ(α, α∗)|α⟩⟨α|, dµ(α, α∗) = 1 (2πi)n n∏ i=1 dαidα ∗ i , (4.23) and the reproducing kernel corresponds to (4.24) K(α∗, β) = ⟨α|β⟩ = e(− 1 2 |α| 2− 1 2 |β| 2+α∗·β). These results show that through definitions (4.16) one recovers the well known properties of the bosonic quantum mechanical coherent states with no explicit presence of the noncommutativity of the spatial coor- dinates, as long as one remains in the holomorphic variables. This property has been studied in the context of star products in [3]. 7 We are thus in position to derive the corresponding path integral, by taking advantage of the fact that at the level of holomorphic coordinates the construction should not differ from the approach in [13], and only when going back to physical variables the differences will become clear. 5. Noncommutative Coherent State Path Integral Following the construction of the path integral for constraint theories in terms of Coherent States de- veloped in [13], and, for clarity sake, we will proceed with a similar calculation in the case of the coherent state system (4.21). Furthermore we will specialize to the case of a purely constrained system with a single constraint, i.e., when the total Hamiltonian is only the constraint Φ̂. As it was discussed in a previous section, the adequate reproducing kernel for the present case will be given by (5.25) ⟨α′′|Ê|α′⟩ = ∫ dµ(σ)⟨α′′|e−iσΦ̂|α′⟩. Because there’s no actual Hamiltonian, there is no evolution and the reproducing kernel will effectively act as the propagator. This can be shown by trying to evolve an arbitrary physical state |ψphys⟩ = Ê|ψ⟩ with the constraint (5.26) |ψphys, τ⟩ = e−iτΦ̂|ψphys⟩ = e−iτΦ̂ Ê|ψ⟩, and observing that (5.27) e−iτΦ̂ Ê = ∫ dµ(σ)e−i(σ+τ)Φ̂ = Ê, we obtain (5.28) |ψphys, τ⟩ = |ψphys⟩, which implies that the physical states are frozen and the transition amplitude of going from an initial physical state to a final physical state will depend only on the reproducing kernel (5.25).  To evaluate the reproducing kernel by the Feynman procedure of infinitesimal time slices we will need to introduce a fictitious interval Nϵ = 1: ⟨α′′|Ê|α′⟩ = ⟨α′′| ∫ dµ(σ)e−iσΦ̂|α′⟩ = ∫ dµ(σ)⟨α′′| e−iϵσΦ̂ × ...× e−iϵσΦ̂ ︸ ︷︷ ︸ N |α′⟩ = ∫ dµ(σ) ∫ N−1∏ k=0 dµ(αk, α ∗ k)⟨αk+1|e−iϵσΦ̂|αk⟩, (5.29) after making use of (4.23) and where αN = α′′ and α0 = α′. If Φ̂ is already a normal ordered operator, meaning it can be formally written as (5.30) Φ̂ = n∏ i=1 ∑ ri,si ξri,si( † i ) ri(Âi) si , then by focusing on the k-th term in expression (5.29) at leading order ϵ we have ⟨αk+1|e−iϵσΦ̂|αk⟩ ≈ ⟨αk+1|αk⟩ − iϵσ⟨αk+1|Φ̂|αk⟩ = ⟨αk+1|αk⟩[1− iϵσΦ(α∗ k+1, αk)] ≈ ⟨αk+1|αk⟩e−iϵσΦ(α∗ k+1,αk), (5.31) 8 with (5.32) Φ(α∗ k+1, αk) = n∏ i=1 ∑ ri,si ξri,si(α ∗ {k+1}i) ri(α{k}i) si , where notation α{k}i implies the i-th component of vector αk. Thus we have the approximation (5.33) ⟨α′′|Ê|α′⟩ ≈ ∫ dµ(σ) ∫ N−1∏ k=0 dµ(αk, α ∗ k)⟨αk+1|αk⟩e−iϵσΦ(α∗ k+1,αk), and lastly by using (4.24) we have ⟨αk+1|αk⟩ = exp [ − α∗ k+1 · αk+1 2 − α∗ k · αk 2 + α∗ k+1 · αk ] = exp [ ϵ ( αk 2 · (α∗ k+1 − α∗ k) ϵ − α∗ k+1 2 · (αk+1 − αk) ϵ )] , (5.34) which when substituted in (5.33) gives ⟨α′′|Ê|α′⟩ ≈ ∫ dµ(σ) ∫ N−1∏ k=0 dµ(αk, α ∗ k) { e−iϵσΦ(α∗ k+1,αk) × exp [ ϵ ( αk 2 · (α∗ k+1 − α∗ k) ϵ − α∗ k+1 2 · (αk+1 − αk) ϵ )]} , (5.35) that in the limit N → ∞ becomes the exact expression ⟨α′′|Ê|α′⟩ = ∫ α′′ α′ Dµ(α, α∗) ∫ dµ(σ)ei ∫ 1 0 dt[ i2 (α ∗(t)·α̇(t)−α(t)·α̇∗(t))−σΦ(α(t),α∗(t))].(5.36) This form for the path integral confirms the properties discussed for coherent states (4.21), in that there is no explicit appearance of the noncommutativity. To recover the noncommutative version of the path integral we make use of expressions (4.22) to replace the holomorphic coordinates in (5.36) by phase-space variables. It can be easily checked that the result is ⟨q′′, p′′|Ê|q′, p′⟩ = ∫ q′′,p′′ q′,p′ Dµ(q, p) ∫ dµ(σ)ei ∫ 1 0 dt [ 1 ~ ∑ i ( piq̇i+ ∑ j θij 2~ piṗj ) −σΦ(q(t),p(t)) ] ,(5.37) from which the noncommutative action associated to this path integral reads (5.38) S = ∫ 1 0 dt [∑ i ( piq̇i + ∑ j θij 2~ piṗj ) − σ̃Φ(q(t), p(t)) ] , σ̃ = ~σ. Formula (5.37) for the noncommutative path integral coincides (in its functional form, although inequiva- lent) with similar versions obtained via different approaches, see, e.g, [19, 20], however, because this represen- tation was obtained by making use of the coherent state kernel (5.25) it has the advantages detailed in [13]. In particular, it is gauge invariant and because the associated Hamiltonian is the covariant Husimi Q-symbol [21] of the quantum constraint, one can expect a stationary phase evaluation (along classical trajectories) to be a fairly good approximation [22]. 6. Classical Anisotropic Bianchi I cosmology In order to be able to introduce a noncommutative relation between the space-like field observables of the quantum theory, of the type discussed in the Introduction and section 4, it is necessary to work with a system of more than one spatial coordinate. Therefore as a classical model we will recur to the anisotropic Bianchi I cosmology, defined by the line element 9 (6.39) ds2 = −N2(τ)dτ2 + gij(τ)dx idxj , gij(τ) = a2i (τ)δij , where N(τ) is the lapse function and the dimensionless quantities ai(τ) are the scales of the cosmology in three independent space-like directions. Because the 3-curvature tensor vanishes, the corresponding Einstein-Hilbert action written in terms of the ADM canonical separation corresponds to S = κ ∫ dτd3x [ πij ġij − N(τ) √ (3)g ( πijπij − 1 2 (πi i) 2 )] = κ ∫ dτd3x [ πij ġij − N(τ) √ (3)g ( πijπklgikgjl − 1 2 (πijgij) 2 )] , (6.40) with (3)g = Det(gij) and π ij being the momenta conjugate to gij . ‡ Substituting the components gij explicitly in the action above and by using the canonical transformation πi := 2πijaj , we can recast (6.40) in terms of the canonical pair (ai, π i) as (6.41) S = κ ∫ dτd3x [ πiȧi − N(τ) 4 √ (3)g ( (πi)2(ai) 2 − 1 2 (πiai) 2 )] , with the Hamiltonian constraint (6.42) Cgrav = N(τ) 4 √ (3)g ( (πi)2(ai) 2 − 1 2 (πiai) 2 ) . The equations of motion may now be obtained by fixing N(τ) 4 √ (3)g = 1 and taking the variations δai and δπ i: ȧi = 2δji π ja2j − (πjaj)ai = [2δji π jaj − (πjaj)]ai, π̇i = −2δij(π j)2aj + πi(πjaj) = −[2δijπ jaj − (πjaj)]π i, (6.43) and because the quantities π1a1, π 2a2, π 3a3 are clearly constants of motion, the solutions are of the form ai(τ) = ai(τ0)e (τ−τ0)ηi , πi(τ) = πi(τ0)e −(τ−τ0)ηi , (6.44) where ηi = (2πiai − πjaj)|τ0 , with no sum over i. Depending on the sign of ηi, solutions (6.44) imply the well known asymptotic collapse (singular) or infinite expansion behaviors for τ → ±∞ in every phase-space variable. Now we will study the way this behavior gets modified (if at all) at the level of the noncommutative coherent state action (5.38). 7. Quantum Noncommutative Bianchi I model Before promoting the classical pair (ai, π i) directly to quantum operators we first make some pertinent considerations on the significance of noncommutativity in a cosmological context. Because the scale factors are arbitrary and only have meaning relative to their values at some other epoch, then we will be interested in their relation with respect to their values during the Planck epoch. Thus, from our introductory remarks we will assume that the corresponding field observables âi will dynamically inherit, as Heisenberg operators (see [20] for details of such mechanism), the noncommutativity of space (1.1), which ‡κ = c3/G. For later convenience during our quantization prescription we decide to retain all the units. 10 in turn should have been relevant when the size of the Universe was comparable to a Planck length ℓp. This can be achieved through the commutator (7.45) [âi, âj ] := iθ̃ij , θ̃ij = λ2θij where λ is a continuous parameter of units L−1 proportional to the inverse of the (mean) radius of the Universe at some epoch and |θij | ≃ ℓ2p as before. Therefore we see that during the Planck epoch θ̃ij ≈ 1, meanwhile in the present epoch θ̃ij ≈ 10−122, which reproduces the notion that noncommutative effects at cosmological scales are no longer observed. Now to complete our algebra of operators we may implement the ordinary commutator between the âi and their conjugate momenta π̂i as (7.46) [âi, π̂ j ] := i~ρλ2δji , where the parameter ρ has units (ML/T )−1, and so λ and ρ are to be related through (ρλ)−1 ≥ ~ in order to be physically consistent magnitudes. The previous commutator is dimensionally correct recalling that the classical πi’s have units L−1. In particular, for (ρλ)−1 = ~ (7.47) [âi, π̂ j ] = iλδji , which for large scales recovers the commutative classical behavior of âi and π̂ i in a similar fashion to (7.45). Therefore during calculations we may regard the parameters λ and ρ simply as factors meant to fix units, and return to their interpretation during the various epochs later. After quantization, the Hamiltonian constraint (6.42) can be identified with the Hermitian operator: (7.48) Ĉgrav = π̂iâ2i π̂ i − 1 2 (π̂iâi)(âj π̂ j), however, it will not be regarded as the quantum constraint. The latter will be obtained, according to expression (5.30), through normal ordering using appropriate noncommutative ladder operators. Similarly to what was done with noncommutative coherent states in §4, we may now define the dimen- sionless operators Γ̂i := 1√ 2~λρ ( âi + 1 2~λ2ρ θ̃ij π̂ j + i λ π̂i ) , Γ̂† i := 1√ 2~λρ ( âi + 1 2~λ2ρ θ̃ij π̂ j − i λ π̂i ) , (7.49) so that by using (7.45) and (7.46) it can be verified that they satisfy the commutators (7.50) [Γ̂i, Γ̂ † j ] = δij , [Γ̂i, Γ̂j ] = [Γ̂† i , Γ̂ † j ] = 0, with the inverse forms âi = √ ~λρ/2 [ Γ̂i + Γ̂† i + ∑ j iθ̃ij 2~λρ (Γ̂j − Γ̂† j) ] , π̂i = −iλ √ ~λρ/2[Γ̂i − Γ̂† i ]. (7.51) After substituting (7.51) into (7.48) and making repeated use of (7.50), along with the antisymmetry of θ̃ij , we can arrive at the normal ordered form of Ĉgrav which will be promoted to act as the quantum 11 constraint Φ̂grav, i.e. : Ĉgrav :=~ 2λ4ρ2 [ 1 8 ∑ i,j (Γ̂2 i Γ̂ 2 j − 2Γ̂†2 i Γ̂2 j + Γ̂†2 i Γ̂†2 j )(1− 2δij) + 1 2 ∑ i Γ̂† i Γ̂i − i 4~λρ ∑ i,j θ̃ij(Γ̂ 3 i − Γ̂†2 i Γ̂i − Γ̂† i Γ̂ 2 i + Γ̂†3 i − Γ̂i − Γ̂† i )(Γ̂j − Γ̂† j) + ( 1 4~λρ )2 ∑ i,j,k θ̃ij θ̃ik(Γ̂ 2 i − 2Γ̂† i Γ̂i + Γ̂†2 i − 1)(Γ̂jΓ̂k − 2Γ̂† jΓ̂k + Γ̂† jΓ̂ † k − δjk) ] = Φ̂grav. (7.52) Now, comparing (7.50) with (4.19) and according to the construction of the coherent states (4.21), the corresponding coherent states for the Bianchi I anisotropic noncommutative quantum model are obtained from the transitive action (7.53) |γ⟩ = exp [ 3∑ i=1 (γiΓ̂ † i − γ∗i Γ̂i) ] |0⟩ = 3∏ i=1 e(γiΓ̂ † i −γ∗ i Γ̂i)|0⟩, characterized by the eigenvalue equation (7.54) Γ̂i|γ⟩ = γi|γ⟩, γi ∈ C. Making use of these elements and general expression (5.36) we can immediately write the corresponding constrained reproducing kernel that will act as the propagator of the theory ⟨γ′′|Ê|γ′⟩ = ∫ γ′′ γ′ Dµ(γ, γ∗) ∫ dµ(σ)ei ∫ dt[ i2 (γ ∗·γ̇−γ·γ̇∗)−σΦgrav(γ,γ ∗)],(7.55) where Φgrav(γ, γ ∗) = ⟨γ|Φ̂grav|γ⟩ is directly computed from (7.52) Φgrav(γ, γ ∗) =~ 2λ4ρ2 [ 1 8 ∑ i,j (γ2i γ 2 j − 2γ∗ 2 i γ2j + γ∗ 2 i γ∗ 2 j )(1− 2δij) + 1 2 ∑ i γ∗i γi − i 4~λρ ∑ i,j θ̃ij(γ 3 i − γ∗ 2 i γi − γ∗i γ 2 i + γ∗ 3 i − γi − γ∗i )(γj − γ∗j ) + ( 1 4~λρ )2 ∑ i,j,k θ̃ij θ̃ik [ (γ2i − 2γ∗i γi + γ∗ 2 i )(γjγk − 2γ∗j γk + γ∗j γ ∗ k) − (γ2i − 2γ∗i γi + γ∗ 2 i )δjk − (γjγk − 2γ∗j γk + γ∗j γ ∗ k) ] ] . (7.56) To recover physical insight we recall from (4.22) and using (7.51) that a pair of phase-space coordinates (ri, pi) is obtained from the expectations ri := ⟨γ|âi|γ⟩ = √ ~λρ/2 [ γi + γ∗i + ∑ j iθ̃ij 2~λρ (γj − γ∗j ) ] , pi := ⟨γ|π̂i|γ⟩ = −iλ √ ~λρ/2[γi − γ∗i ], (7.57) and by inverting them we can express the holomorphic coordinates in terms of physical variables as γi := 1√ 2~λρ ( ri + 1 2~λ2ρ ∑ j θ̃ijpj + i λ pi ) , γ∗i := 1√ 2~λρ ( ri + 1 2~λ2ρ ∑ j θ̃ijpj − i λ pi ) . (7.58) 12 Then, by substituting (7.58) in (7.56) and after lengthy algebraic simplifications we obtain Φgrav(r, p) = ∑ i p2i r 2 i − 1 2 ( ∑ i piri )2 + ~λρ 4 ∑ i (p2i + λ2r2i ) + λ 4 ∑ i,j θ̃ijpirj − 1 16~λρ ∑ i,j,k θ̃ij θ̃ikpjpk(1− 2δjk), (7.59) where the first two terms are identified as the classical Hamiltonian constraint (6.42), whereas the other terms correspond to quantum and noncommutative corrections. An important element to note here is the presence of a purely noncommutative correction linear in θ̃ij , in the form of an angular momentum. According to the results in [7] we can already expect this term to be relevant in the stability of the solutions and central in preventing the collapse in the deep Planckian sector. 8. Path Integral action, dynamics and µ-parameters Similarly to what was done to arrive at expression (5.38), the action in the path integral (7.55) can be rewritten in terms of physical variables using (7.58) (8.60) S = 1 λ2ρ ∫ dτ [∑ i ( piṙi + ∑ j θ̃ij 2~λ2ρ piṗj ) − λ2ρ~σΦgrav(q, p) ] , and noticing that in order to recover the classical equations of motion (6.43), controlled only by the first two terms in (7.59), i.e. when ~ = θ̃ = 0, we need to identify σ = 1 ~λ2ρ . Observe that this involves integrating over the measure dµ( 1 ~λ2ρ ) in the transition amplitude (7.55) which makes the final result independent of scales. As is well known in the theory of Feynman integrals, the trajectories for which δS = 0 are the ones that most contribute to the propagator, such trajectories are solutions to the Euler-Lagrange equations of (8.60) which, by using (7.59), are immediately seen to be ṙi = ri(2piri − ∑ j pjrj) + ~λρ 2 pi + λ 4 ∑ j θ̃ijrj + 1 8~λρ ∑ j,k θ̃ij θ̃jkpj(1− 2δik)− 1 ~λ2ρ ∑ j θ̃ij ṗj , ṗi = −pi(2piri − ∑ j pjrj)− ~λ3ρ 2 ri + λ 4 ∑ j θ̃ijpj . (8.61) Instead of working with the previous equations it will prove more convenient to work with dimensionless expressions. This can be done first by substituting θ̃ij = λ2θij , then removing the units of any pi and any τ derivative through the change of variables pi = λp̃i and ḟ = λf ′, where f is arbitrary. After carrying this out one arrives at the equivalent dimensionless system r′i = ri(2p̃iri − ∑ j p̃jrj) + ~λρ 2 p̃i + λ2 4 ∑ j θijrj + λ3 8~ρ ∑ j,k θijθjkp̃j(1− 2δik)− λ ~ρ ∑ j θij p̃ ′ j , p̃′i = −p̃i(2p̃iri − ∑ j p̃jrj)− ~λρ 2 ri + λ2 4 ∑ j θij p̃j . (8.62) with this normalization of units it is possible to see that the limit λ → 0 properly reproduces the classical equations of motion, whereas this limit was ill defined in equations (8.61). By defining the dimensionless ”µ-parameters” (8.63) µ~(λ) := ~λρ ≤ 1, µθ ij(λ) := λ2θij ≤ 1, 13 which characterize the quantum and Planckian scales respectively, the differential system (8.62) takes the form r′i = ri(2p̃iri − ∑ j p̃jrj) + µ~ 2 p̃i + 3 4 ∑ j µθ ijrj + ∑ j µθ ij µ~ p̃j ( 2p̃jrj − ∑ k p̃krk ) + 1 8 ∑ j,k µθ ijµ θ jk, µ~ p̃j(1 + 2δik), p̃′i = −p̃i(2p̃iri − ∑ j p̃jrj)− µ~ 2 ri + 1 4 ∑ j µθ ij p̃j . (8.64) and the constraint (7.59) now reads Φ̃grav(r, p̃) = ∑ i p̃2i r 2 i − 1 2 ( ∑ i p̃iri )2 + µ~ 4 ∑ i (p̃2i + λ2r2i ) + 1 4 ∑ i,j µθ ij p̃irj − 1 16 ∑ i,j,k µθ ijµ θ ik, µ~ p̃j p̃k(1− 2δjk), (8.65) Observe that the pure quantum µ~ and noncommutative µθ ij sectors can now be clearly distinguished in (8.64) and (8.65), as well as the interplay between them trough the quotient sectors µθ ij µ~ , µθ ijµ θ jk µ~ , which in turn correspond to newly derived µ-parameters. Because of their λ-scale dependence, a strict hierarchy can be established among the different µ-parameters for all epochs, making them analogous to the running couplings of field theories. Thus the renormal- ization group for this quantum noncommutative Bianchi I cosmology is characterized by the couplings µ~, µ θ ij , µθ ij µ~ , µθ ijµ θ jk µ~ . Within this context the fact that this group has an infrared stable point, i.e. when λ→ 0, confirms that the classical Bianchi I cosmology results as an effective theory of our model. 9. Bounce-like Paths As the classical symmetries are broken in the Planckian regime, and are recovered only in the limit λ→ 0, the quantities p̃iri are no longer constants of motion and analytical solutions for this highly coupled nonlinear differential system are not easy to come by. In this section we will show that a specific type of nonsingular solutions, namely a family of bounce-like trajectories for the volume scaling function v := r1r2r3, are allowed by the equations of motion (8.64). To that purpose we give the following: Definition 2 (Bounce). The differential system (8.64) will be said to have a bounce inside an interval B = (τB − ϵ, τB + ϵ) when all the momenta undergo a sign change for values τ ∈B, where for the simplest case this occurs simultaneously meaning that at τ = τB the momenta satisfy p̃i(τB) = 0. The previous definition is not sufficient, however, to guarantee all the properties we look for in the solutions, where for the simplest case the volume scaling function v(τ) = r1(τ)r2(τ)r3(τ) together with its first and second derivatives should also satisfy v(τB) > 0, v′(τB) = 0, v′′(τB) > 0. (9.66) Therefore by using (8.64) to explicitly calculate the above derivatives, where for simplicity sake we fix µθ 12 = µθ 23 = µθ 31 = µθ, and substituting the bounce definition, it is not very difficult to show that at τ = τB the last two expressions in (9.66) yield (9.67) v′(τB) = 3µθ 4 ∑ (ijk) r2i ( rj − rk ) = 0, 14 where (ijk) are the cyclic permutations of indices (123), and (9.68) v′′(τB) = [ µ~ 2 ∑ i r2i − µ2 θ 8 ∑ (ijk) ri rjrk ( 9ri − 14(rj + rk) ) − 3µ2 ~ 4 − 63µ2 θ 8 ] v(τB) > 0. We then observe from (9.67) that the simplest solution corresponds to a bounce for r1 = r2 = r3 = rB that, when substituted in (9.68), leads to the greatly simplified lower bound (9.69) 3µ~ 2 r2B > 3 4 (µ2 ~ + µ2 θ). To better understand the implications of these results observe first that the condition to extremize the volume is of pure noncommutative nature, in fact, it originates from the noncommutative term of angular momentum in (7.59). This confirms our remark in the last paragraph of §7, thus establishing the relevance of the angular momentum in controlling the bounce and providing more evidence that noncommutative effects are manifest dynamically. On the other hand, condition (9.69) expresses the infimum for rB at all scales above which a bounce occurs. But more interesting is the fact that in the Planckian regime when µ~ = µθ = 1 this condition reduces simply to rB > 1, or equivalently that the bounce has to occur for scales greater than one cubic Planck length. For a deeper analysis of the behavior of the system (8.64) around τB and to determine if bounce-like solutions are admissible, we apply the method of matched asymptotic expansions often used in boundary- layer problems [23, 24]. Contrary to the usual procedure, we are not interested in finding the first terms of an asymptotic analytical solution for (8.64), but to show that the zeroth order of the expansion near the Planckian regime is compatible with a bounce-like behavior. To do this we start in a scale one order of magnitude larger than the Planckian i.e λ ≈ 1032cm−1, or µ~ = 10−1 = ε and µθ = 10−2 = ε2, in a manner such that the quantum and noncommutative corrections can be handled perturbatively around the well known classical solutions, but where noncommutative effects from the Planckian regime remain relevant for the evolution. Thus the system (8.64) becomes: r′i = ri(p̃iri − p̃jrj − p̃krk) + εp̃j(p̃jrj − p̃krk − p̃iri)− εp̃k(p̃krk − p̃iri − p̃jrj) + ε 2 p̃i + 3ε2 4 rj − 3ε2 4 rk +O(ε3), p̃′i = −p̃i(p̃iri − p̃jrj − p̃krk)− ε 2 ri + ε2 4 p̃j − ε2 4 p̃k +O(ε3), (9.70) where the indices (i, j, k) are cyclically ordered. The type of solutions for (9.70) that are our main concern are such that their behavior away from Planck- ian scales corresponds to that of a classical collapse in the past which transits to a classical expansion in the future. From the way in which the linear terms appear in the system, an oscillatory behavior is expected in the region where such terms are dominant, namely for small values of ri and pi. Imposing the bounce criteria that we introduced at the beginning of this section requires the solutions in the transition region, when going from collapse to expansion, to allow for the change of sign in the pi while keeping the ri positive, as pictured in Fig(1). Thus as a first step we divide the dynamical phase space (r, p̃, τ) in three regions: (I) S1 = {(r, p̃, τ)|τ < τ1}, (II) S2 = {(r, p̃, τ)|τ1 < τ < τ2} and (III) S3 = {(r, p̃, τ)|τ > τ2}, where we assume that outer (classical) solutions will be in regions S1 and S3 and an inner solution for region S2. To simplify the calculations we will specialize to the symmetrical case around the origin: τ2 = −τ1 = τ̃ . 15 (S1) ri pi ττ1 τ2 ri(O) ri(O) ri(i) (S2) (S3) (S1) τ1 τ2 pi(O) pi(O) pi(i) (S2) (S3) τ ~ ~ ~ ~ Figure 1. The expected behavior of the bounce-like solutions is shown. The evolution of ri (top) and p̃i (bottom) is separated in the three regions of interest, detailing their general asymptotic properties. For each region we have the asymptotic forms: r (A) j (t) = R (A) j +O(ε), p̃ (A) j (t) = P̃ (A) j +O(ε), (9.71) where the superscript (A) denotes whether the solution is outer (O) or the inner (i). Outer solution. For regions S1 and S3, i.e. outside the Planckian region, the solutions are expected to behave as the classical ones. Then to zeroth order the equations (9.70) read: R′ i = Ri(P̃iRi − P̃jRj − P̃kRk), P̃′ i = −P̃i(P̃iRi − P̃jRj − P̃kRk), (9.72) which have the usual classical solutions: Ri(τ) = Ri(τ0)e (χi−χj−χk)(τ−τ0), P̃i(τ) = P̃i(τ0)e −(χi−χj−χk)(τ−τ0) (9.73) where as before the indices (i, j, k) are cyclic and χi := Ri(τ0)P̃i(τ0) with τ0 corresponding to a time in the remote past for S1 and in the distant future for S3. Where a classical collapse (expansion) for region S1 (S3) occurs for: Ri(τ1) > 0 (Ri(τ2) > 0) and P̃i(τ1) > 0 (P̃i(τ2) < 0). Which implies that a necessary condition for a bounce-like solution of the volume scaling function is for the momenta to change sign inside S2. Inner solution. As the noncommutative terms O(ε2) in (9.70) are expected to become important inside S2, it is reasonable to scale down the variables to that same order to assert the relevant dynamics in this region, i.e. (ri, p̃i) → (ε2ri, ε 2pi). To further simplify the equations resulting from this scaling we also introduce the time variable t = ε(τ − τB), thus the system (9.70) in the inner region reduces to: dri dt = 1 2 pi + 3ε 4 rj − 3ε 4 rk +O(ε3), dpi dt = −1 2 ri + ε 4 pj − ε 4 pk +O(ε3), (9.74) where the indices (i, j, k) are again cyclically ordered. The solutions to this linear system can be obtained in closed forms, however, we only present the first terms of their asymptotic expansion. The six orthogonal 16 eigenvectors at leading order ε are: w1(t) =(cos(ν0t), cos(ν0t), cos(ν0t),− sin(ν0t),− sin(ν0t),− sin(ν0t)), w2(t) =(sin(ν0t), sin(ν0t), sin(ν0t), cos(ν0t), cos(ν0t), cos(ν0t)), w3(t) =(cos(ν+t) + √ 3 sin(ν+t), cos(ν+t)− √ 3 sin(ν+t),−2 cos(ν+t), − √ 3ξ− cos(ν+t)− ξ− sin(ν+t), √ 3ξ− cos(ν+t)− ξ− sin(ν+t), 2ξ− sin(ν+t)), w4(t) =(− sin(ν+t) + √ 3 cos(ν+t),− sin(ν+t)− √ 3 cos(ν+t), 2 sin(ν+t), √ 3ξ− sin(ν+t)− ξ− cos(ν+t),− √ 3ξ− sin(ν+t)− ξ− cos(ν+t), 2ξ− cos(ν+t)), w5(t) =(cos(ν−t)− √ 3 sin(ν−t), cos(ν−t) + √ 3 sin(ν−t),−2 cos(ν−t), √ 3ξ+ cos(ν−t)− ξ+ sin(ν−t),− √ 3ξ+ cos(ν−t)− ξ+ sin(ν−t), 2ξ+ sin(ν−t)), w6(t) =(− sin(ν−t)− √ 3 cos(ν−t),− sin(ν−t)− √ 3 cos(ν−t), 2 sin(ν−t), − √ 3ξ+ sin(ν−t)− ξ+ cos(ν−t), √ 3ξ+ sin(ν−t)− ξ+ cos(ν−t), 2ξ+ cos(ν−t)), (9.75) where ξ± = 4± √ 3ε 4∓ √ 3ε , ν± = 1 2 ± √ 3 2 ε and ν0 = 1 2 . A solution that satisfies the prescription that the pi change sign around the origin with ri positive can be obtained with the linear combination: ( r, p )(i) S2 =Aw1 + B ( ξ+w3 + ξ−w5 ) + C ( ξ+w4 − ξ−w6 ) (9.76) where (r, p)(i) = (r (i) 1 , r (i) 2 , r (i) 3 , p (i) 1 , p (i) 2 , p (i) 3 ) and A = 1 3 ( r (i) 1 (0) + r (i) 2 (0) + r (i) 3 (0) ) , B = 1 6 ( r (i) 1 (0) + r (i) 2 (0)− 2r (i) 3 (0) ) , C = 1 2 √ 3 ( r (i) 1 (0)− r (i) 2 (0) ) . (9.77) It is important to stress that the range of validity of this solution is confined to values −ε < t < ε, but that its qualitative properties persist throughout all the three regions on to the exponential regime. From solution (9.76) we have various possible scenarios depending on the values of ri(0). In the isotropic case it simplifies only to the first expression in (9.75) meaning a unique evolution of the scale factors and also their conjugate momenta. In the more general scenario of different values for the three ri(0) the oscillation will be a linear combination of periodic functions with three different frequencies. Also note that for the limit ε→ 0 in (9.74), which corresponds to neglect all the noncommutative terms, the system decouples into three identical oscillators causing the scale factors to evolve independently. The solution of this system can very well describe the behavior of the isotropic case near the bounce as the frequency is the same, but not in the arbitrary anisotropic case. This method could now be extended to match both inner and outer solutions, through an intermediate solution, to obtain an approximate analytical solution valid for all τ , however this goes beyond the scope of this work. To show the validity of our considerations and solutions (9.73) and (9.76), we recur to numerical solutions. 10. Numerical Results In what follows all our simulations were obtained for values of µ~ = µθ 12 = µθ 23 = µθ 31 = 1 which corresponds to the deep Planckian regime. As a first check for our previous assumptions we provide the plot associated to the most symmetric case, when r1 = r2 = r3 = rB at the bounce point. Fig.(2) corresponds to the 17 numerical solution of the volume scale function and the scale factors governed by the equations of motion (8.64) when rB > 1. In this case the expectation of the three scale factors follows the same trajectory and are undistinguishable, which coincides with our analysis of the inner solution (9.76). All the scale factors evolve equally around the bounce before the exponential regime rapidly takes over with a similar situation occurring for the momenta in Fig.(3). Therefore the numerical solution confirms the simplest type of bounce hypothesized in the previous section. -1.0 -0.5 0.0 0.5 1.0 Τ 5 10 15 20 25 30 vHΤL ‡ r1HΤL r2HΤL r3HΤL Figure 2. Plot of the volume scaling function (solid) and scale factors (dashed) in the symmetric case rB = 2. The collapsing solution coming from the left bounces at the origin to evolve along an expanding trajectory to the right. The evolution of the scale factors is the same for the three cases. -1.0 -0.5 0.5 1.0 Τ -1.5 -1.0 -0.5 0.5 1.0 1.5 piHΤL Figure 3. Plot of the momenta conjugate to the ri from Fig.(2). The three momenta follow the same solution with the bounce occuring when they intersect the origin. A more interesting situation can be obtained by allowing anisotropy among the scale factors as exemplified in Fig.(4). While the bounce of the volume is clearly present and not too different from the previous case, a richer scenario occurs with the independent scale factors. Here, noncommutativity induces a secondary effect in the evolution by producing irregular oscillations of the trajectories around the bounce in the way predicted by the inner solutions obtained in (9.76), where contributions from frequencies ν± are now also present. Note that these noncommutative oscillations are such that the solutions for the three scale factors tend towards a single trajectory in both branches until a point where, if the solution is continued in the 18 vertical axis, no remnant of the original anisotropy is noticeable. This is a very appealing feature in the sense that it provides with a plausible mechanism to produce isotropic cosmologies (at large scales) in the far future out of anisotropic conditions in the remote past. -1.0 -0.5 0.0 0.5 1.0 Τ 5 10 15 20 vHΤL ‡ r1HΤL r2HΤL r3HΤL Figure 4. Plot of the volume scaling function (solid) and scale factors r1 (dashed), r2 (dotted), r3 (dotdashed) in the anisotropic case with values at the bounce r1 = 3, r2 = 2, r3 = 1. The heterogenous oscillatory regime generated by the noncommutativity drives the evolution of the scale factors to coalesce into one asymptotic trajectory. -1.0 -0.5 0.5 1.0 Τ -1.5 -1.0 -0.5 0.5 1.0 1.5 piHΤL Figure 5. Plot of the momenta conjugate to the ri of Fig.(4). The amplitude of the oscillations is suppressed as the system evolves away from the bounce. The third numerical example shows that bounce-like trajectories can display more than one minimum. In Fig.(6) we present the bounce-like solution obtained for anisotropic values of ri at the bounce such that the volume scale function remains above one Planck volume, yet the expectation of one scale factor can take sub-Planckian values. The evolution of the volume scaling function presents various minima, however, only the central minimum fulfills the bounce criteria, i.e. pi = 0. Similar to the previous case, the three scale factors are seen to evolve towards an average trajectory. 19 -2 -1 0 1 2 Τ 5 10 15 20 vHΤL ‡ r1HΤL r2HΤL r3HΤL Figure 6. Plot of the volume scaling function (solid) and scale factors r1 (dashed), r2 (dotted), r3 (dotdashed) in the anisotropic case with values at the bounce r1 = 2, r2 = 2, r3 = 0.3. More than one minima of the volume can be observed with only the central minimum satisfying the condition pi = 0. 11. Discussion and Conclusions A set of coherent states were built for a noncommutative quantum Bianchi I anisotropic cosmology, which in turn were used to circumvent the absence of a simultaneous set of configuration observables. Instead, the chosen dynamical variables were the expectation values of the scale factors and momenta operators in the coherent base. By extending known methods of path integrals for constraint systems in terms of coherent states, and taking advantage of the property that noncommutative expressions have the same functional form as their commutative counterparts when written in holomorphic variables [3], we obtained a form for the propagator of the theory and, with it, the corresponding modified noncommutative action. As the dynamical variables (ri, pi) represent the center of gaussian states, a steepest descent analysis of the action was performed to obtain the approximate description of the full quantum states evolution, enhanced by the fact that the covariant Husimi Q-symbol for the path integral is more fit for our semiclassical scenario, cf. [22]. Thus we obtained the corresponding equations of motion from the effective Hamiltonian in the path integral action where, by keeping all the units of the theory and introducing the dimensionless µ-parameters, we were able to clearly differentiate the macroscopic dominant contributions from the pure quantum and noncommutative sectors as well as their mixed sectors. Here we must emphasize an important feature of our Hamiltonian constraint, that is confirmed simply by power counting, which is that the noncommutative corrections coming from our selection of dynamical variables are not originated in a Bopp map of the classical constraint, as is usually the case in other noncommutativity related literature. From the perspective of an effective theory, our final expressions for the Hamiltonian (8.65) and the equations of motion (8.64) were shown to flow towards a stable infrared point, which indeed corresponds to the classical model. By changing the value of the λ-scale in the couplings of the theory, one can analyze the relevance of the noncommutative and quantum effects during different epochs. It is interesting to note that, although the purely noncommutative effects die off relatively fast for scales larger than a few Planck lengths (equivalently, for λ < 1), the mixed sector µθ ij/µ~ induces a noncommutative-quantum mechanical interaction that persists through such scales. This is a novelty of our model which gives hope to probe noncommutative effects at intermediate scales between the quantum regime and the Planckian one, a gap known in particle physics as the great desert for its attributed absence of new physics. 20 By providing a physically admissible definition for the bounce of the system we showed that it was compatible with the existence of a minimum for the volume scaling function. The particular scenario analyzed in detail represented an evolution where the volume collapses for cosmological times τ < τB , reaches its minimum value at τB and then expands for τ > τB . In order to satisfy the supplementary conditions that ensure this behavior, we encountered that the noncommutative angular momentum term determined the relationship among the values of the scale factors of the cosmology at τB . On the other hand, the criteria for a minimum resulted in a more complicated expression that in the general case is rather difficult to interpret in a geometrical sense, however, in the simple isotropic case, where the three scale factors take the same value rB at τB , it greatly reduces to a more insightful form which imposes a lower bound for the value of rB . It was observed that for the Planckian regime a consequence of this condition is that rB > 1, meaning that a bounce cannot occur for volume scales smaller than one Planck volume. It is also remarkable that this expression incorporates the contributions from quantum mechanics and noncommutativity in order for the minimum to acquire physically operational values. In this semiclassical context, proving the existence of bounce-like solutions implies that the probability of the quantum states to have singular behavior is marginal. After implementing notions from solution methods frequently used in boundary-layer problems of fluid mechanics we inferred the asymptotic behavior of the analytical solution of the equations of motion in the vicinity of the bounce as well as far from it. This lead us to confirm the compatibility of the classical collapse and expansion solutions with the bounce mechanism induced by the new terms in the Hamiltonian. The linear approximation around the bounce indicates an interaction between the values of all the dynamical variables via the noncommutative corrections, which in the case where noncommutativity is not present decouples completely in three independent subspaces. It is worth mentioning that a missing piece of our analysis, due to the inherent technical complexities presented by the differential system, is the availability of an intermediate solution that bridges the outer solution with the inner solution, therefore we had to recur to computational resources to obtain a full picture of the evolution. In the numerical results another aspect of the solutions was observed that is not evident from the as- ymptotic analysis, this corresponded to an apparent averaging of the scale factors around a mean trajectory when departing from anisotropic values at the bounce. 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