# Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes

## Abstract

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 inequivalent 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 Fockmore »

- Authors:

- Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid (Spain)
- Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico D.F. 04510 (Mexico)
- Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid (Spain)
- Departamento de Física, Faculdade de Ciências, Universidade da Beira Interior, R. Marquês D'Ávila e Bolama, 6201-001 Covilhã (Portugal)

- Publication Date:

- OSTI Identifier:
- 22279671

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Cosmology and Astroparticle Physics

- Additional Journal Information:
- Journal Volume: 2012; Journal Issue: 11; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1475-7516

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ASTROPHYSICS; COMMUTATION RELATIONS; COSMOLOGY; FOCK REPRESENTATION; KLEIN-GORDON EQUATION; PERTURBATION THEORY; QUANTIZATION; SCALAR FIELDS; SPACE-TIME; SYMMETRY; TIME DEPENDENCE; TOPOLOGY

### Citation Formats

```
Gomar, Laura Castelló, Cortez, Jerónimo, Blas, Daniel Martín-de, Marugán, Guillermo A. Mena, and Velhinho, José M., E-mail: laucaste@estumail.ucm.es, E-mail: jacq@ciencias.unam.mx, E-mail: daniel.martin@iem.cfmac.csic.es, E-mail: jvelhi@ubi.pt.
```*Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes*. United States: N. p., 2012.
Web. doi:10.1088/1475-7516/2012/11/001.

```
Gomar, Laura Castelló, Cortez, Jerónimo, Blas, Daniel Martín-de, Marugán, Guillermo A. Mena, & Velhinho, José M., E-mail: laucaste@estumail.ucm.es, E-mail: jacq@ciencias.unam.mx, E-mail: daniel.martin@iem.cfmac.csic.es, E-mail: jvelhi@ubi.pt.
```*Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes*. United States. doi:10.1088/1475-7516/2012/11/001.

```
Gomar, Laura Castelló, Cortez, Jerónimo, Blas, Daniel Martín-de, Marugán, Guillermo A. Mena, and Velhinho, José M., E-mail: laucaste@estumail.ucm.es, E-mail: jacq@ciencias.unam.mx, E-mail: daniel.martin@iem.cfmac.csic.es, E-mail: jvelhi@ubi.pt. Thu .
"Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes". United States. doi:10.1088/1475-7516/2012/11/001.
```

```
@article{osti_22279671,
```

title = {Uniqueness of the Fock quantization of scalar fields in spatially flat cosmological spacetimes},

author = {Gomar, Laura Castelló and Cortez, Jerónimo and Blas, Daniel Martín-de and Marugán, Guillermo A. Mena and Velhinho, José M., E-mail: laucaste@estumail.ucm.es, E-mail: jacq@ciencias.unam.mx, E-mail: daniel.martin@iem.cfmac.csic.es, E-mail: jvelhi@ubi.pt},

abstractNote = {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 inequivalent 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.},

doi = {10.1088/1475-7516/2012/11/001},

journal = {Journal of Cosmology and Astroparticle Physics},

issn = {1475-7516},

number = 11,

volume = 2012,

place = {United States},

year = {2012},

month = {11}

}