Dependence of the affine coherent states quantization on the parametrization of the affine group

TL;DR

This paper investigates how different parametrizations of the affine group in affine coherent states quantization lead to unitarily inequivalent quantum theories, highlighting the importance of parametrization choices in quantum gravity models.

Contribution

It demonstrates that the choice of parametrization in affine coherent states quantization affects the resulting quantum theory's equivalence class, which is a novel insight.

Findings

Different parametrizations yield unitarily inequivalent quantum theories.

The dependence on parametrization can be potentially used constructively.

The study emphasizes the significance of parametrization in quantum gravity models.

Abstract

The affine coherent states quantization is a promising integral quantization of Hamiltonian systems when the phase space includes at least one conjugate pair of variables which takes values from a half-plane. Such a situation is common for gravitational systems which include singularities. The construction of the quantization map includes a one-to-one mapping of the half-plane onto the affine group. Particular cases of this mapping define specific parametrizations of the group. Our aim is showing that different such parametrizations lead to unitarily inequivalent quantum theories. Depending on the Hamiltonian system under consideration, this dependence could potentially be used constructively.