On the Geometric Quantization of Canonical Gravity

TL;DR

This paper explores the challenges of geometric quantization in canonical quantum gravity, focusing on the ambiguity caused by refoliation invariance and examining Shape Dynamics as a case study for deriving a quantum evolution equation.

Contribution

It analyzes the impact of non-uniqueness in the vector field choice on quantum gravity quantization and investigates Shape Dynamics to derive a potential time-dependent Schrödinger equation.

Findings

Non-uniqueness in vector field choice leads to ambiguous quantum evolution.

Shape Dynamics allows for a derivation of a time-dependent Schrödinger equation.

Refoliation invariance causes fundamental ambiguities in quantum gravity quantization.

Abstract

One of the hardest problems to tackle in the dynamics of canonical approaches to quantum gravity is that of the Hamiltonian constraint. We investigate said problem in the context of formal geometric quantization. We study the implications of the non uniqueness in the choice of the vector field which satisfies the presymplectic equation for the Hamiltonian constraint, and study the implication of the same in the quantization of the theory. Our aim is to show that this non uniqueness in the choice of said vector field, which really stems from refoliation invariance leads to a very ambiguous notion of quantum evolution. We then investigate the case of a theory where the problem of the Hamiltonian constraint has been dealt with at the classical level, namely Shape Dynamics, and attempt to derive a time dependent Schrodinger equation for the quantum dynamics of this theory.