Uniqueness Criteria for the Fock Quantization of Dirac Fields and Applications in Hybrid Loop Quantum Cosmology

Abstract

In generic curved spacetimes, the unavailability of a natural choice of vacuum state introduces a serious ambiguity in the Fock quantization of fields. In this review, we study the case of fermions described by a Dirac field in non-stationary spacetimes, and present recent results obtained by us and our collaborators about well-motivated criteria capable to ensure the uniqueness in the selection of a vacuum up to unitary transformations, at least in certain situations of interest in cosmology. These criteria are based on two reasonable requirements. First, the invariance of the vacuum under the symmetries of the Dirac equations in the considered spacetime. These symmetries include the spatial isometries. Second, the unitary implementability of the Heisenberg dynamics of the annihilation and creation operators when the curved spacetime is treated as a fixed background. This last requirement not only permits the uniqueness of the Fock quantization but, remarkably, it also allows us to determine an essentially unique splitting between the phase space variables assigned to the background and the fermionic annihilation and creation variables. We first consider Dirac fields in 2 + 1 dimensions and then discuss the more relevant case of 3 + 1 dimensions, particularizing the analysis to cosmological spacetimes with spatial sections of spherical or toroidal topology. We use this analysis to investigate the combined, hybrid quantization of the Dirac field and a flat homogeneous and isotropic background cosmology when the latter is treated as a quantum entity, and the former as a perturbation. Specifically, we focus our study on a background quantization along the lines of loop quantum cosmology. Among the Fock quantizations for the fermionic perturbations admissible according to our criteria, we discuss the possibility of further restricting the choice of a vacuum by the requisite of a finite fermionic backreaction and, moreover, by the diagonalization of the fermionic contribution to the total Hamiltonian in the asymptotic limit of large wave numbers of the Dirac modes. Finally, we argue in support of the uniqueness of the vacuum state selected by the extension of this diagonalization condition beyond the commented asymptotic region, in particular proving that it picks out the standard Poincaré and Bunch–Davies vacua for fixed flat and de Sitter background spacetimes, respectively.

Quantum unitary dynamics of a charged fermionic field and Schwinger effect

“…Therefore, we can interpret that the inequivalence between these two representations is due exclusively to artificially choosing two different conventions (s = s) for the concepts of particle and antiparticle. Analogous results also appear in the study of the uniqueness of the Fock quantization of free Dirac fields in non-stationary curved spacetimes [8].…”

“…In addition, we also impose that they should allow the unitary implementation of the dynamics with two main objectives: assuring throughout the evolution of the vacuum both the physical equivalence of the quantizations and the finiteness of the number of created particles at finite times. This work aims to translate the analysis carried out in non-stationary curved spacetimes for free fields [7,8] to the Schwinger effect. This was first done for a charged scalar field on [13].…”

“…Thus, we need to impose physical criteria on the complex structure in order to reduce the ambiguity in the quantization. Motivated by previous studies in cosmology [7,8] and in the Schwinger effect for a charged scalar field [13], our central work in section 5 will be to characterize the complex structures which preserve the symmetries of the system and unitarily implement the dynamical evolution of a charged fermionic field in presence of a homogeneous time-dependent electromagnetic background. Therefore, let us describe how can time evolution be treated as a Bogoliubov transformation.…”

“…Let us review here some results which will be useful in our later discussion. For more details, see [7,8,23].…”

“…With the aim of reducing the ambiguity in the choice of vacuum, it is desirable to find other physically reasonable requirements to impose on the quantizations. In the context of fields propagating through homogeneous cosmologies a proposal has been put forward which allows to select a unique family of unitarily equivalent Fock representations and, therefore, physically equivalent quantizations [7,8]: the unitary implementation of both the classical symmetries of the equations of motion and the quantum field dynamics at all times.…”