New Loop Representations for 2+1 Gravity

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.

28pp, 1 figure (postscript, compressed and uuencoded), TeX, Pennsylvania State University, CGPG-94/5-1

References in corpus (1)

• Solution space of 2+1 gravity on $\mathbf{R}\times T^2$ in Witten's connection formulation

Cited by in corpus (18)

• Loop Quantum Gravity
• Quantum Gravity and Regge Calculus
• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• Spin Foams and Canonical Quantization
• Spin Networks for Non-Compact Groups
• Three Dimensional Canonical Quantum Gravity
• The Extended Loop Representation of Quantum Gravity
• Large Diffeomorphisms in (2+1)-Quantum Gravity on the Torus
• Combinatorial quantisation of the Euclidean torus universe
• QUANTUM ASPECTS OF 2+1 GRAVITY
• Polygon model from first order gravity
• (2+1) gravity for higher genus in the polygon model
• Independent Loop Invariants for 2+1 Gravity
• Geometric flux formula for the gravitational Wilson loop
• A Modular Invariant Quantum Theory From the Connection Formulation of (2+1)-Gravity on the Torus
• Perturbative Analysis of the Two-body Problem in (2+1)-AdS gravity
• (2+1)-AdS Gravity on Riemann Surfaces
• $\La \to 0$ limit of 2+1 Quantum Gravity for arbitrary genus