Midi-superspace Models of Canonical Quantum Gravity

TL;DR

This paper reviews recent advances in the canonical quantization of midi-superspace models of vacuum general relativity, focusing on symmetry reductions, Hamiltonian formulations, and quantum field theory constructions for these models.

Contribution

It provides necessary and sufficient conditions for symmetry reductions at the Lagrangian and Hamiltonian levels, and explores quantization strategies for models with cylindrical and toroidal symmetry.

Findings

Conditions for symmetry reduction at Lagrangian and Hamiltonian levels

Hamiltonian formulation of cylindrically and toroidally symmetric models

Discussion on quantum field theory construction for midi-superspace models

Abstract

A midi-superspace model is a field theory obtained by symmetry reduction of a parent gravitational theory. Such models have proven useful for exploring the classical and quantum dynamics of the gravitational field. I present 3 recent classes of results pertinent to canonical quantization of vacuum general relativity in the context of midi-superspace models. (1) I give necessary and sufficient conditions such that a given symmetry reduction can be performed at the level of the Lagrangian or Hamiltonian. (2) I discuss the Hamiltonian formulation of models based upon cylindrical and toroidal symmetry. In particular, I explain how these models can be identified with parametrized field theories of wave maps, thus a natural strategy for canonical quantization is available. (3) The quantization of a parametrized field theory, such as the midi-superspace models considered in (2), requires construction of a quantum field theory on a fixed (flat) spacetime that allows for time evolution along arbitrary foliations of spacetime. I discuss some recent results on the possibility of finding such a quantum field theory.