# 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:

- Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, México D.F. 04510 (Mexico)
- Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid (Spain)
- Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, NL-6525 AJ Nijmegen (Netherlands)
- 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}

}