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

Jerónimo Cortez, Guillermo A. Mena Marugán, José M. Velhinho

The purpose of this review is to provide a brief overview of some recent
conceptual developments about possible criteria to guarantee the uniqueness of
the quantization in a variety of situations that are found in cosmological
systems. These criteria impose some 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 in fact are 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 rather different uniqueness
criteria proposed either in homogeneous loop quantum cosmology or in the Fock
quantization of inhomogeneous cosmologies.