Coherent states, quantum gravity and the Born-Oppenheimer approximation, II: Compact Lie Groups

TL;DR

This paper develops a Weyl quantisation framework for compact Lie groups to advance loop quantum gravity models, introduces a conjecture on a new coherent state transform, and explores its connection to Berezin quantisation.

Contribution

It proposes a Weyl quantisation for compact Lie groups tailored for loop quantum gravity and conjectures a new coherent state transform with supporting evidence.

Findings

Conjecture of a new Segal-Bargmann-Hall coherent state transform for compact Lie groups.

Support for the conjecture provided by numerical evidence for SU(2).

Analysis of the connection between coherent state and Berezin quantisation for compact Lie groups.

Abstract

In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall "coherent state" transform for compact Lie groups $G$, which we prove for $G=U(1)^{n}$ and support by numerical evidence for $G=SU(2)$. The reason for conjoining this conjecture with the main topic of this article originates in the observation, that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups $G$. But, as Weyl and Berezin quantisation for $\mathbb{R}^{2d}$ are intimately related by heat kernel evolution, it is natural to ask, whether a similar connection exists for compact Lie groups, as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former.