Spin Networks in Nonperturbative Quantum Gravity

TL;DR

This paper reviews the role of spin networks in nonperturbative quantum gravity, highlighting their mathematical structure and applications in canonical quantization of 4D general relativity.

Contribution

It provides a rigorous approach to functional integration in quantum gravity and demonstrates how spin networks serve as fundamental states in this formalism.

Findings

Spin networks form a basis for states in quantum gravity.

Canonical quantization using new variables leads to applications of spin networks.

The approach connects topological quantum field theory with quantum gravity.

Abstract

A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3-dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L^2(A/G) is spanned by states labelled by spin networks. Then we explain the `new variables' for general relativity in 4-dimensional spacetime and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states.