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Title: Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization

Abstract

We address the issue of the infinite ambiguity that affects the construction of a Fock quantization of a Dirac field propagating in a cosmological spacetime with flat compact sections. In particular, we discuss a physical criterion that restricts to a unique possibility (up to unitary equivalence) the infinite set of available vacua. We prove that this desired uniqueness is guaranteed, for any possible choice of spin structure on the spatial sections, if we impose two conditions. The first one is that the symmetries of the classical system must be implemented quantum mechanically, so that the vacuum is invariant under the symmetry transformations. The second and more important condition is that the constructed theory must have a quantum dynamics that is implementable as a (non-trivial) unitary operator in Fock space. Actually, this unitarity of the quantum dynamics leads us to identify as explicitly time dependent some very specific contributions of the Dirac field. In doing that, we essentially characterize the part of the dynamics governed by the Dirac equation that is unitarily implementable. The uniqueness of the Fock vacuum is attained then once a physically motivated convention for the concepts of particles and antiparticles is fixed.

Authors:
 [1];  [2];  [3];  [2];  [4]
  1. Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, México D.F. 04510 (Mexico)
  2. Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid (Spain)
  3. Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, NL-6525 AJ Nijmegen (Netherlands)
  4. Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001, Covilhã (Portugal)
Publication Date:
OSTI Identifier:
22617452
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 376; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COSMOLOGY; DIRAC EQUATION; QUANTIZATION; QUANTUM FIELD THEORY; QUANTUM MECHANICS; SPACE-TIME; SYMMETRY

Citation Formats

Cortez, Jerónimo, E-mail: jacq@ciencias.unam.mx, Elizaga Navascués, Beatriz, E-mail: beatriz.elizaga@iem.cfmac.csic.es, Martín-Benito, Mercedes, E-mail: m.martin@hef.ru.nl, Mena Marugán, Guillermo A., E-mail: mena@iem.cfmac.csic.es, and Velhinho, José M., E-mail: jvelhi@ubi.pt. Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization. United States: N. p., 2017. Web. doi:10.1016/J.AOP.2016.11.005.
Cortez, Jerónimo, E-mail: jacq@ciencias.unam.mx, Elizaga Navascués, Beatriz, E-mail: beatriz.elizaga@iem.cfmac.csic.es, Martín-Benito, Mercedes, E-mail: m.martin@hef.ru.nl, Mena Marugán, Guillermo A., E-mail: mena@iem.cfmac.csic.es, & Velhinho, José M., E-mail: jvelhi@ubi.pt. Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization. United States. doi:10.1016/J.AOP.2016.11.005.
Cortez, Jerónimo, E-mail: jacq@ciencias.unam.mx, Elizaga Navascués, Beatriz, E-mail: beatriz.elizaga@iem.cfmac.csic.es, Martín-Benito, Mercedes, E-mail: m.martin@hef.ru.nl, Mena Marugán, Guillermo A., E-mail: mena@iem.cfmac.csic.es, and Velhinho, José M., E-mail: jvelhi@ubi.pt. Sun . "Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization". United States. doi:10.1016/J.AOP.2016.11.005.
@article{osti_22617452,
title = {Dirac fields in flat FLRW cosmology: Uniqueness of the Fock quantization},
author = {Cortez, Jerónimo, E-mail: jacq@ciencias.unam.mx and Elizaga Navascués, Beatriz, E-mail: beatriz.elizaga@iem.cfmac.csic.es and Martín-Benito, Mercedes, E-mail: m.martin@hef.ru.nl and Mena Marugán, Guillermo A., E-mail: mena@iem.cfmac.csic.es and Velhinho, José M., E-mail: jvelhi@ubi.pt},
abstractNote = {We address the issue of the infinite ambiguity that affects the construction of a Fock quantization of a Dirac field propagating in a cosmological spacetime with flat compact sections. In particular, we discuss a physical criterion that restricts to a unique possibility (up to unitary equivalence) the infinite set of available vacua. We prove that this desired uniqueness is guaranteed, for any possible choice of spin structure on the spatial sections, if we impose two conditions. The first one is that the symmetries of the classical system must be implemented quantum mechanically, so that the vacuum is invariant under the symmetry transformations. The second and more important condition is that the constructed theory must have a quantum dynamics that is implementable as a (non-trivial) unitary operator in Fock space. Actually, this unitarity of the quantum dynamics leads us to identify as explicitly time dependent some very specific contributions of the Dirac field. In doing that, we essentially characterize the part of the dynamics governed by the Dirac equation that is unitarily implementable. The uniqueness of the Fock vacuum is attained then once a physically motivated convention for the concepts of particles and antiparticles is fixed.},
doi = {10.1016/J.AOP.2016.11.005},
journal = {Annals of Physics},
number = ,
volume = 376,
place = {United States},
year = {Sun Jan 15 00:00:00 EST 2017},
month = {Sun Jan 15 00:00:00 EST 2017}
}
  • We study the Fock quantization of scalar fields in (generically) time dependent scenarios, focusing on the case in which the field propagation occurs in –either a background or effective– spacetime with spatial sections of flat compact topology. The discussion finds important applications in cosmology, like e.g. in the description of test Klein-Gordon fields and scalar perturbations in Friedmann-Robertson-Walker spacetime in the observationally favored flat case. Two types of ambiguities in the quantization are analyzed. First, the infinite ambiguity existing in the choice of a Fock representation for the canonical commutation relations, understandable as the freedom in the choice of inequivalentmore » vacua for a given field. Besides, in cosmological situations, it is customary to scale the fields by time dependent functions, which absorb part of the evolution arising from the spacetime, which is treated classically. This leads to an additional ambiguity, this time in the choice of a canonical pair of field variables. We show that both types of ambiguities are removed by the requirements of (a) invariance of the vacuum under the symmetries of the three-torus, and (b) unitary implementation of the dynamics in the quantum theory. In this way, one arrives at a unique class of unitarily equivalent Fock quantizations for the system. This result provides considerable robustness to the quantum predictions and renders meaningful the confrontation with observation.« less
  • The Fock quantization of fields propagating in cosmological spacetimes is not uniquely determined because of several reasons. Apart from the ambiguity in the choice of the quantum representation of the canonical commutation relations, there also exists a certain freedom in the choice of field: one can scale it arbitrarily absorbing background functions, which are spatially homogeneous but depend on time. Each nontrivial scaling turns out into a different dynamics and, in general, into an inequivalent quantum field theory. In this work we analyze this freedom at the quantum level for a scalar field in a nonstationary, homogeneous spacetime whose spatialmore » sections have S{sup 3} topology. A scaling of the configuration variable is introduced as part of a linear, time dependent canonical transformation in phase space. In this context, we prove in full detail a uniqueness result about the Fock quantization requiring that the dynamics be unitary and the spatial symmetries of the field equations have a natural unitary implementation. The main conclusion is that, with those requirements, only one particular canonical transformation is allowed, and thus only one choice of the field-momentum pair (up to irrelevant constant scalings). This complements another previous uniqueness result for scalar fields with a time varying mass on S{sup 3}, which selects a specific equivalence class of Fock representations of the canonical commutation relations under the conditions of a unitary evolution and the invariance of the vacuum under the background symmetries. In total, the combination of these two different statements of uniqueness picks up a unique Fock quantization for the system. We also extend our proof of uniqueness to other compact topologies and spacetime dimensions.« less
  • After its reduction by a gauge-fixing procedure, the family of linearly polarized Gowdy T{sup 3} cosmologies admits a scalar field description whose evolution is governed by a Klein-Gordon type equation in a flat background in 1+1 dimensions with the spatial topology of S{sup 1}, though in the presence of a time-dependent potential. The model is still subject to a homogeneous constraint, which generates S{sup 1}-translations. Recently, a Fock quantization of this scalar field was introduced and shown to be unique under the requirements of unitarity of the dynamics and invariance under the gauge group of S{sup 1}-translations. In this work,more » we extend and complete this uniqueness result by considering other possible scalar field descriptions, resulting from reasonable field reparametrizations of the induced metric of the reduced model. In the reduced phase space, these alternate descriptions can be obtained by means of a time-dependent scaling of the field, the inverse scaling of its canonical momentum, and the possible addition of a time-dependent, linear contribution of the field to this momentum. Demanding again unitarity of the field dynamics and invariance under the gauge group, we prove that the alternate canonical pairs of fieldlike variables admit a Fock representation if and only if the scaling of the field is constant in time. In this case, there exists essentially a unique Fock representation, provided by the quantization constructed by Corichi, Cortez, and Mena Marugan. In particular, our analysis shows that the scalar field description proposed by Pierri does not admit a Fock quantization with the above unitarity and invariance properties.« less
  • We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time-dependent potential, also interpretable as a time-dependent mass. Classically, the field satisfies a linear wave equation of the form {xi}-dot-dot-{xi}{sup ''}+f(t){xi}=0. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field (f=0) provides a unitary implementation of the dynamics for sufficiently general mass terms, f(t). Furthermore, this representation is uniquely specified, among the class of representations determined by S{sup 1}-invariant complex structures, as the only one allowing a unitary dynamics. Thesemore » conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.« less
  • We study the Fock description of a quantum free field on the three-sphere with a mass that depends explicitly on time, also interpretable as an explicitly time dependent quadratic potential. We show that, under quite mild restrictions on the time dependence of the mass, the specific Fock representation of the canonical commutation relations which is naturally associated with a massless free field provides a unitary dynamics even when the time varying mass is present. Moreover, we demonstrate that this Fock representation is the only acceptable one, up to unitary equivalence, if the vacuum has to be SO(4)-invariant (i.e., invariant undermore » the symmetries of the field equation) and the dynamics is required to be unitary. In particular, the analysis and uniqueness of the quantization can be applied to the treatment of cosmological perturbations around Friedmann-Robertson-Walker spacetimes with the spatial topology of the three-sphere, like e.g. for gravitational waves (tensor perturbations). In addition, we analyze the extension of our results to free fields with a time dependent mass defined on other compact spatial manifolds. We prove the uniqueness of the Fock representation in the case of a two-sphere as well, and discuss the case of a three-torus.« less