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.Comment: 19 pages, submitted for publication to Annals of Physic

Fermions in hybrid loop quantum cosmology

“…The effect on the spin structure is a restriction that can be reinterpreted as a spin structure on each of the toroidal spatial sections. The two component spinors of the Dirac field can be seen as families of cross sections of the resulting spinor bundle, parametrized by the time coordinate [22]. On the other hand, the anticommutation canonical relations of the Dirac field are provided by the symmetric Dirac brackets at coincident time 10) where I is the identity matrix in four dimensions.…”

“…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.…”

“…Then, the Dirac equation (2.8) leads to the following set of dynamical equations for all k = τ [22]: 16) where the prime stands for the derivative with respect to a conformal time η defined via dη = e −α dt, andM = M σ. Introducing the representation of the Dirac field in terms of two-component spinors and the mode expansion of the latter, one then arrives to a expression of the action in terms of mode coefficients [17]:…”

“…All these perturbations are included up to quadratic contributions in the action, which is the order of truncation in our perturbative approximation [17,18,22,39]. Since the fermionic part of the Einstein-Dirac action is already quadratic in the fermionic contributions, and they are treated as perturbations, including a possible homogeneous component of the Dirac field, the considered truncation is tantamount to couple the Dirac field directly to the tetrad of the homogeneous unperturbed spacetime.…”

“…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.…”