Canonical Quantization of 2d Gravity Coupled to c<1 Matter

TL;DR

This paper employs canonical quantization to analyze 2d gravity coupled to c<1 matter, revealing a free-field structure and providing gauge-independent insights into the theory's spectrum and unitarity.

Contribution

It demonstrates that the theory can be transformed into a free-field form, confirming the free-field assumption and deriving key results in a gauge-independent manner.

Findings

The theory can be expressed as a free-field Virasoro system.

Gauge independence of the path-integral results is established.

The physical spectrum and unitarity of states are analyzed.

Abstract

We study 2d gravity coupled to $c,1$ matter through canonical quantization of a free scalar field, with background charge, coupled to gravity. Various features of the theory can be more easily understood in the canonical approach, like gauge indipendence of the path-integral results and the absence of the local physical degrees of freedom. By performing a non-canonical transformation of the phase space variables, we show that the theory takes a free-field form, i.e. the constraints become the free-field Virasoro constraints. This implies that the David-Distler-Kawai results can be derived in the gauge indipendent way, and also proves the free-field assumption which was used for obtaining the spectrum of the theory in the conformal gauge. A discussion of the physical spectrum of the theory is presented, with an analysis of the unitarity of the discrete momentum states.