The Hilbert space of gauge theories: group averaging and the quantization of Jackiw-Teitelboim gravity

TL;DR

This paper addresses the challenge of defining a proper Hilbert space for gauge theories with infinite volume gauge groups, proposing a modified group averaging method to quantize Jackiw-Teitelboim gravity with positive cosmological constant.

Contribution

It introduces a renormalized group averaging procedure to handle infinite gauge volumes and provides the first complete Dirac quantization of Jackiw-Teitelboim gravity with a positive cosmological constant.

Findings

Successfully quantized JT gravity with positive cosmological constant.

Hilbert space splits into infinite superselection sectors.

Inner product is positive-definite and well-defined.

Abstract

When the gauge group of a theory has infinite volume, defining the inner product on physical states becomes subtle. This is the case for gravity, even in exactly solvable models such as minisuperspace or low-dimensional theories: the physical states do not inherit an inner product in a straightforward manner, and different quantization procedures yield a priori inequivalent prescriptions. This is one of the main challenges when constructing gravitational Hilbert spaces. In this paper we study a quantization procedure known as group averaging, which is a special case of the BRST/BV formalism and has gained popularity as a promising connection between Dirac quantization and gravitational path integrals. We identify a large class of theories for which group averaging is ill-defined due to isometry groups with infinite volume, which includes Jackiw-Teitelboim gravity. We propose a modification of group averaging to renormalize these infinite volumes and use it to quantize Jackiw-Teitelboim gravity with a positive cosmological constant in closed universes. The resulting Hilbert space naturally splits into infinite-dimensional superselection sectors and has a positive-definite inner product. This is the first complete Dirac quantization of this theory, as we are able to capture all the physical states for the first time.