New Loop Representations for 2+1 Gravity

TL;DR

This paper demonstrates that loop representations for 2+1 gravity with a non-compact gauge group can be constructed explicitly by choosing an appropriate measure, allowing all quantum states to be represented as functions of loops.

Contribution

It shows how to handle measure-related difficulties in loop representations for 2+1 gravity with non-compact gauge groups, enabling explicit construction of all quantum states.

Findings

Loop representations exist for 2+1 gravity with non-compact gauge groups.

Quantum states can be represented as functions of loops on a two-torus.

Scalar product and observables are explicitly defined in loop variables.

Abstract

Since the gauge group underlying 2+1-dimensional general relativity is non-compact, certain difficulties arise in the passage from the connection to the loop representations. It is shown that these problems can be handled by appropriately choosing the measure that features in the definition of the loop transform. Thus, ``old-fashioned'' loop representations - based on ordinary loops - do exist. In the case when the spatial topology is that of a two-torus, these can be constructed explicitly; {\it all} quantum states can be represented as functions of (homotopy classes of) loops and the scalar product and the action of the basic observables can be given directly in terms of loops.