Uniqueness of the Fock quantization of scalar fields in a Bianchi I cosmology with unitary dynamics

Abstract

The Fock quantization of free scalar fields is subject to an infinite ambiguity when it comes to choosing a set of annihilation and creation operators, a choice that is equivalent to the determination of a vacuum state. In highly symmetric situations, this ambiguity can be removed by asking vacuum invariance under the symmetries of the system. Similarly, in stationary backgrounds, one can demand time-translation invariance plus positivity of the energy. However, in more general situations, additional criteria are needed. For the case of free (test) fields minimally coupled to a homogeneous and isotropic cosmology, it has been proven that the ambiguity is resolved by introducing the criterion of unitary implementability of the quantum dynamics, as an endomorphism in Fock space. This condition determines a specific separation of the time dependence of the field, so that this splits into a very precise background dependence and a genuine quantum evolution. Furthermore, together with the condition of vacuum invariance under the spatial Killing symmetries, unitarity of the dynamics selects a unique Fock representation for the canonical commutation relations, up to unitary equivalence. In this work, we generalize these results to anisotropic spacetimes with shear, which are therefore not conformally symmetric, by considering the case of a free scalar field in a Bianchi I cosmology.

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

“…The criteria of symmetry invariance and of a unitary dynamics have been successfully imposed not only in homogeneous and isotropic backgrounds, but also in anisotropic spacetimes with shear, which are then not conformally symmetric. Specifically, the criteria have been employed to remove the ambiguities in the quantization of scalar fields propagating in Bianchi I spacetimes [45]. This section gives an overview of the result of uniqueness for these anisotropic scenarios.…”

“…where Q k and P k are complex constants, and Θ ε k (τ ) are complex functions (ε = q, p), it can be shown [45] that the solutions with initial conditions Θ ε k (τ 0 ) = 0 andΘ ε k (τ 0 ) = kbk(τ 0 ) have an asymptotic (ultraviolet) behavior given by k ′ } = −iδ ll ′ δ k k ′ . Let us now consider families of representations that are connected by a unitary implementable dynamics.…”

“…This comprises the relevant cosmological case (inasmuch as spatial flatness is favored by current observations [38][39][40]) of compact sections with threetorus topology [41][42][43]. 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].…”

Unitary quantization of a scalar charged field and Schwinger effect

“…[22,23]) and fermionic fields (see e.g. [24,25]), or anisotropic Bianchi I spacetimes with a scalar field [26]. By imposing this condition, one forces not only the initial and final states of the field (asymptotic states) to be related by a unitary transformation given by the asymptotic S-matrix, (whose existence was proven in [18] for general backgrounds) but also that the intermediate states of the field must be related through a unitary operator, namely, the evolution operator.…”

“…We show that the unitary implementation of the quantum field evolution reduces the ambiguity on the choice of complex structure up to unitary equivalence. This is achieved by adapting the analyses of [24][25][26] to our system. Finally, in section V, we provide an explicit example of the application of the procedure developed in the previous sections to the scalar Schwinger effect in a concrete external potential, the Sauter pulse [1].…”