Unitary evolution and uniqueness of the Fock representation of Dirac fields in cosmological spacetimes

Abstract

We present a privileged Fock quantization of a massive Dirac field in a closed Friedmann-Robertson-Walker cosmology, partially selected by the criteria of invariance of the vacuum under the symmetries of the field equations, and unitary implementation of the dynamics. When quantizing free scalar fields in homogeneous and isotropic spacetimes with compact spatial sections, these criteria have been shown to pick out a unique Fock representation (up to unitary equivalence). Here, we employ the same criteria for fermion fields and explore whether that uniqueness result can be extended to the case of the Fock quantization of fermions. For the massive Dirac field, we start by introducing a specific choice of the complex structure that determines the Fock representation. Such structure is invariant under the symmetries of the equations of motion. We then prove that the corresponding representation of the canonical anticommutation relations admits a unitary implementation of the dynamics. Moreover, we construct a rather general class of representations that satisfy the above criteria, and we demonstrate that they are all unitarily equivalent to our previous choice. The complex structures in this class are restricted only by certain conditions on their asymptotic behavior for modes in the ultraviolet sector of the Dirac operator. We finally show that, if one assumes that these asymptotic conditions are in fact trivial once our criteria are fulfilled, then the time-dependent scaling in the definition of the fermionic annihilation and creation-like variables is essentially unique.

Fermions in hybrid loop quantum cosmology

“…Based on recent results about criteria to remove the ambiguity in the choice of a Fock quantization on cosmological backgrounds, first proposed for scalar fields [68][69][70][71] and then extended to fermions [20,21,72], and in particular to Dirac fields in flat FLRW spacetimes [22], we can minimize the physical consequences of our freedom of choice of creation and annihilation variables. Under the requirements of: i) unitary implementability of the dynamics of the chosen variables in the quantum theory, ii) invariance of this theory under the Killing isometries of the toroidal sections and the spin rotations generated by the helicity, and iii) a convention for the concepts of particles and antiparticles that connects smoothly in the massless limit with the standard one, the analysis of Ref.…”

“…[20], we search for two independent solutions of the corresponding classical equation with the formz…”

“…Besides, we will also use recent results on the Fock representation of fermions [20][21][22] with the aim at improving some properties of the quantization of the Dirac field related with the unitarity of the evolution and the backreaction effects. Owing to its physical relevance, we will adopt a flat topology for the spatial sections of our model.…”

“…A detailed calculation of the coefficients of this Bogoliubov transformation [20] leads finally to the following formula (where we obviate the time dependence and the limits of integration over (η, η 0 ) to simplify the notation):…”

Quantum Linear Scalar Fields with Time Dependent Potentials: Overview and Applications to Cosmology

“…Apart from addressing the uniqueness of the quantization of free real scalar fields in FLRW spacetimes [13][14][15][41][42][43], the criteria were successfully employed to remove the ambiguities in the quantization of free (test) KG fields minimally coupled to a de Sitter background [44], as well as in anisotropic Bianchi I spacetimes [45]. It is worth remarking that the criteria of invariance and of unitarity have also been fruitfully exploited to single out a unique preferred quantum description for fermion fields in cosmological scenarios [46][47][48][49][50][51][52][53].…”