2+1 quantum gravity as a toy model for the 3+1 theory

Abstract

2+1 Einstein gravity is used as a toy model for testing a program for nonperturbative canonical quantisation of the 3+1 theory. The program can be successfully implemented in the model and leads to a surprisingly rich quantum theory.

Representations of the holonomy algebras of gravity and nonAbelian gauge theories

“…The configuration variable in this model is a real, SU(2)-valued connection. In its connection-dynamics version, 2 + 1-dimensional gravity is structurally very similar to the 3 + 1-dimensional theory in the new canonical variables [12,13]. However, in the 2 + 1-dimensional case, the connection is real and flat and takes its values in the Lie algebra of SO (2,1).…”

The canonical structure of maximally extended supergravity in three dimensions

“…In contrast to the pure gravity and supergravity theories in three dimensions, which have received much attention in the past few 1-Pars [6][7][8][9], extended supergravity theories possess not only topological but also propagating degrees of freedom. It is therefore more difficult to solve the constraint equations and to construct observables (in the sense of Dirac) for both the classical and the quantum theories .…”

“…(This problem is relevant to the question of whether the representations of the rigid symmetry to which the observables belong are unitary and hence infinite-dimensional .) On the other hand, a suitable scalar product can be constructed for three-dimensional pure gravity [8]. This is one of the reasons why the study of three-dimensional gravity in this formalism has proved to be a useful exercise .…”

“…[1], I will here follow refs. [7][8][9] and use the dual (and first-order) spin connection 2. N = 16 supergravity and its symmetries Observe that Aa and its canonical conjugate constitute the analogue of Ashtekar's variables in three dimensions .…”

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

“…[45][46][47] and [48] for a more recent analysis of the model within the spin foam picture). The most general expression for a state sum in (2+1) dimensions takes, for a given spin foam φ, the form (27) where f, e, v denote the faces, edges and vertices respectively.…”