Quantization of Field Theories Generalizing Gravity-Yang-Mills Systems on the Cylinder

TL;DR

This paper develops a general quantization scheme for a broad class of 2D field theories, including gravity and gauge theories, highlighting differences in phase space and spectra due to gauge transformations and metric conditions.

Contribution

It introduces a unified framework for quantizing 2D gravity and gauge theories via Poisson structures, with detailed examples and analysis of quantum spectra and dynamics.

Findings

Differences in reduced phase spaces due to gauge transformations and metric conditions.

Explicit quantization examples for gravity and gauge theories.

Discussion of the quantum 'problem of time' in $R^2$ gravity coupled to SU(2) Yang-Mills.

Abstract

Pure gravity and gauge theories in two dimensions are shown to be special cases of a much more general class of field theories each of which is characterized by a Poisson structure on a finite dimensional target space. A general scheme for the quantization of these theories is formulated. Explicit examples are studied in some detail. In particular gravity and gauge theories with equivalent actions are compared. Big gauge transformations as well as the condition of metric nondegeneracy in gravity turn out to cause significant differences in the structure of the corresponding reduced phase spaces and the quantum spectra of Dirac observables. For $R^2$ gravity coupled to SU(2) Yang Mills the question of quantum dynamics (`problem of time') is addressed. [This article is a contribution to the proceedings (to appear in LNP) of the 3rd Baltic RIM Student Seminar (1993). Importance is attached to concrete examples. A more abstract presentation of the ideas underlying this article (including new developments) is found in hep-th/9405110.]