A Brief Overview of Results about Uniqueness of the Quantization in Cosmology

Abstract

The purpose of this review is to provide a brief overview of recent conceptual developments regarding possible criteria to guarantee the uniqueness of the quantization in a variety of situations that are found in cosmological systems. These criteria impose certain conditions on the representation of a group of physically relevant linear transformations. Generally, this group contains any existing symmetry of the spatial sections. These symmetries may or may not be sufficient for the purpose of uniqueness and may have to be complemented with other remaining symmetries that affect the time direction or with dynamical transformations that are, in fact, not symmetries. We discuss the extent to which a unitary implementation of the resulting group suffices to fix the quantization—a demand that can be seen as a weaker version of the requirement of invariance. In particular, a strict invariance under certain transformations may eliminate some physically interesting possibilities in the passage to the quantum theory. This is the first review in which this unified perspective is adopted to discuss otherwise different uniqueness criteria proposed either in homogeneous loop quantum cosmology or in the Fock quantization of inhomogeneous cosmologies.

Generalized quantum Vlasov equation for particle creation and unitary dynamics

“…When it no longer possesses Poincaré symmetry, as it is the case, this requirement is not restrictive enough to select particular functions (ζ k (t), ρ k (t)) in general and ambiguities emerge in the canonical quantization (see section 3). In order to reduce these ambiguities, previous studies for both scalar and fermionic fields in homogeneous cosmological settings [13,[24][25][26] as well as in the context of the Schwinger effect [27,28] impose that the canonical time evolution of the fields be unitarily implemented in the quantum theory. Physically, this translates into a well-defined total number of created particles throughout the evolution of fields at all finite times.…”

“…This physical condition imposes a restriction on the large wave vector k-functions ζ k (t) and ρ k (t). The main consequence of demanding a unitary implementation of the quantum field dynamics, as proven in references [13,[24][25][26][27][28], is its uniqueness: quantizations compatible with this requirement form a unique unitarily equivalent family. One of the primary objectives in this section is to emphasize and generalize the procedures from references [13,[24][25][26][27][28] in order to extract relevant physical properties of the generalized QVE (4.9) when particularized to this unique family of quantizations.…”

“…The main consequence of demanding a unitary implementation of the quantum field dynamics, as proven in references [13,[24][25][26][27][28], is its uniqueness: quantizations compatible with this requirement form a unique unitarily equivalent family. One of the primary objectives in this section is to emphasize and generalize the procedures from references [13,[24][25][26][27][28] in order to extract relevant physical properties of the generalized QVE (4.9) when particularized to this unique family of quantizations. More precisely, once ζ k (t) and ρ k (t) allowing for a unitary implementation of the dynamics are characterized in terms of their asymptotic ultraviolet behavior, we will see that the usual QVE (4.11) is in most cases (but not all) the leading order of the generalized QVE (4.9).…”

“…In order to reduce the ambiguities in the quantization of classical theories with no time-translational invariance, a criterion has been proposed in various settings, from homogeneous cosmologies [13,[24][25][26] to the Schwinger effect [27,28]: the unitary implementation in the quantum theory of matter fields dynamics. Weaker conditions are also found in the literature, imposing that only the in and out states (at asymptotic past and future times) are related by a unitary S-matrix [29,30].…”

Quantum unitary dynamics of a charged fermionic field and Schwinger effect

“…This, in particular, applies to time translations. For a deeper discussion on this point, we refer the reader to [9]. If the unitarity of the dynamics is imposed, the quantizations at each time, which might not be a priori equivalent, are assured to provide the same physics.…”