Canonical quantization approach to 2d gravity coupled to c<1 matter

TL;DR

This paper demonstrates that 2D gravity coupled to matter with central charge less than one can be understood through canonical quantization, revealing a free-field structure and addressing gauge independence and unitarity issues.

Contribution

It constructs a canonical transformation to a free-field form, confirming the free-field assumption and gauge independence in 2D gravity with c<1 matter.

Findings

Canonical transformation maps the theory to free-field Virasoro generators.

Confirms gauge independence of DDK results.

Highlights issues with scalar products of discrete states in Fock space.

Abstract

We show that all important features of 2d gravity coupled to $c<1$ matter can be easily understood from the canonical quantization approach a la Dirac. Furthermore, we construct a canonical transformation which maps the theory into a free-field form, i.e. the constraints become free-field Virasoro generators with background charges. This implies the gauge independence of the David-Distler-Kawai results, 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 unitarity of the physical spectrum is presented and we point out that the scalar products of the discrete states are not well defined in the standard Fock space framework.