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The Fock quantization of free fields propagating in cosmological backgrounds is in general not unambiguously defined due to the nonstationarity of the space-time. For the case of a scalar field in cosmological scenarios, it is known that the criterion of unitary implementation of the dynamics serves to remove the ambiguity in the choice of Fock representation (up to unitary equivalence). Here, applying the same type of arguments and methods previously used for the scalar field case, we discuss the issue of the uniqueness of the Fock quantization of the Dirac field in the closed FRW space-time proposed by D’Eath and Halliwell.

The physics of the very early universe, and in particular relevant quantum phenomena, can nowadays be tested, comparing the predictions of theoretical models against quite accurate observational data. Besides scalar fields, it is therefore important to explore the impact of different matter sources as well (in this respect, see [

As it is typically the case regarding the quantization of systems with an infinite number of degrees of freedom, also in the quantization of the Dirac field in curved space-time, one must face the issue of ambiguity, or lack of uniqueness in the quantization procedure. Just like in the scalar field case, the ambiguity in the quantization can be seen to lie in the choice of a so-called

In the previous work [

Let us mention that there are similarities between the unitary evolution requirement and the so-called Hadamard condition, in the sense that some type of ultraviolet regularity is imposed in both cases. Moreover, for space-time with compact spatial sections, all (pure Fock) Hadamard states lead to unitarily equivalent quantizations [

In this section, we present a very brief review of the quantum treatment of the Dirac field in the unit three-sphere

Let us consider the closed FRW cosmological model, with metric

Explicitly, the two-component spinors (and their Hermitian conjugates) can be expanded in modes as follows:

The mode components of the Dirac equation, deduced from the Einstein-Dirac action, can be summarized in the following set of differential equations (and corresponding complex conjugates) [

As mentioned in the Introduction, the ambiguity in the Fock quantization of the Dirac field resides in the choice of a complex structure in the space of solutions of the Dirac equation. This in turn amounts to a choice of a set of classical annihilation- and creation-like variables, to be quantized as annihilation and creation operators (see [

The preferred choice of annihilation- and creation-like variables considered in [

The creation-like variables

Let us call

The main feature of this quantization is that it allows a unitary implementation of the classical field dynamics in the quantum theory. Let us see exactly what this means. Since both the equations of motion and the relations (

It turns out that a classical transformation of type (

The question that we now want to answer is the following: is the quantization described in the previous section unique? In what sense? We have learned from the analysis of similar cases involving scalar fields that the requirement of unitary implementation of the dynamics is a successful criterion in the selection of quantum representations. In fact, together with the requirement of invariance of the complex structure under remaining spatial symmetries, the unitary dynamics condition selects a unique quantization (modulo unitary equivalence) for the scalar field propagating in a nonstationary homogeneous and isotropic space-time background [

As shown in detail in [

Among the large class of quantum representations of the Dirac field determined by the invariant complex structures as above, there are certainly an infinite number of representations that are not unitarily equivalent to our reference quantization. In fact, the condition for unitary equivalence reads [

At this point, it should be mentioned that there is an important difference—first noted in [

We are going to show, precisely, that once unitary implementation of the dynamics is imposed as a requirement, only those complex structures that satisfy (

One can easily show (see [

A detailed asymptotic analysis of the evolution matrices

Taking into account the asymptotic limits of the coefficients

However, since we are assuming that the sequence

These two conditions are therefore consequences of (

Luzin’s theorem [

Let us introduce the above result in the last inequality of (

At this point, let us recall from the discussion in the previous section that, without loss of physical generality, the sequence

We have analyzed the issue of the uniqueness of the Fock quantization of the Dirac field in the closed FRW space-time proposed by D’Eath and Halliwell [

A question that deserves discussion is the following. It has been argued by several authors that the so-called Hadamard states and corresponding quantum representations are physically privileged for quantum field theory in curved space-time since, in particular, they allow a regularization of the stress-energy tensor and a well-defined (perturbative) construction of interacting theories. Moreover, it is known—as a consequence of more general results in [

A similar question emerged in the previous studies of analogous (e.g., using the criteria of unitary implementation of the dynamics) uniqueness of quantization results concerning the scalar field [

If one believes, as we do, that preserving unitarity of the dynamics as much as possible is a desirable aspect in quantum physics, the fact that this perspective actually leads to a quantum theory that is equivalent to the one associated with the celebrated Hadamard states appears by itself as an interesting and reassuring result.

Based on the previous experience with the scalar field, we likewise expect that, in the current context of the Dirac field in the closed FRW space-time, both Hadamard states and the requirement of unitary implementation of the dynamics would lead to essentially equivalent quantizations. However, the detailed analysis of the relation between the two different approaches in the Dirac field case seems rather involved, in comparison with the previous study concerning the scalar field [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are immensely grateful to B. Elizaga Navascués and G. A. Mena Marugán for their help, insight, and constant support. This work was partially supported by the Research Grants MICINN/MINECO Project no. FIS2014-54800-C2-2-P from Spain and DGAPA-UNAM IN113115 and CONACyT 237351 from Mexico. José Velhinho would like to acknowledge the COST Action CA16104 GWverse, supported by COST (European Cooperation in Science and Technology). In addition, Mercedes Martín-Benito would like to acknowledge the financial support from the Portuguese Foundation for Science and Technology (FCT, Grant no. IF/00431/2015).

^{3}model: a uniqueness result

^{1}× S

^{2}and S

^{3}Gowdy models coupled to massless scalar fields