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.
Highlights
MonizQuantization is the process of constructing a description that incorporates the principles of Quantum Mechanics starting with a given classical system
Since the loop quantum gravity (LQG) holonomy-flux algebra is again a ?-algebra with identity, it follows from previous comments that, in order to achieve the required unitary implementation of the group of diffeomorphisms, it is sufficient that the quantization be defined by a diffeomorphism invariant state
We reviewed and discussed several results concerning the quantization of systems with relevant applications in cosmology
Summary
Quantization is the process of constructing a description that incorporates the principles of Quantum Mechanics starting with a given classical system. Certain canonical transformations typically stand out in a given classical system, e.g., dynamics or symmetries, and these require, as well, a proper quantum treatment (these two aspects are, related since observables of physical interest often emerge as generators of some special groups of canonical transformations) This poses no problem as far as one considers the standard quantization of linear canonical transformations and the corresponding generators in R2n , precisely because of the above uniqueness theorem. Such criteria are based on remaining symmetries, present in the cosmological system, and crucially on the unitary implementation of the dynamics, which can be seen as a weaker version of the requirement of invariance under time-translations, which can be applied only in stationary settings. We include an Appendix A sketching the proof of the uniqueness of the representation results mentioned in Section 4, in the simplest case of a scalar field in S1 with time-dependent mass, with the purpose of providing the main steps and typical arguments of the proofs to the interested reader, without overloading the main text
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