Spin Networks in Nonperturbative Quantum Gravity

TL;DR

Baez develops a rigorous nonperturbative framework for quantum gravity by exploiting spin networks as a basis of gauge-invariant functions on the space of connections. He shows that spin-network states span the gauge-invariant Hilbert space $L^2(\mathcal{A}/\mathcal{G})$ and connects this kinematical construction to the loop representation. The paper then presents the Ashtekar new variables, recasting general relativity in a gauge-theoretic form with a complex connection $A_+$ and a densitized triad $\tilde{E}_+$, where the constraints become polynomial and quantization can proceed via $L^2(\mathcal{A}_+)$ and coherent-state techniques. Finally, it discusses the canonical quantization program, the geometric interpretation of Gauss and diffeomorphism constraints, and the emergence of discrete spectra for area and volume operators, highlighting both the promise and the outstanding challenge of the Hamiltonian constraint in producing a full dynamics of 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.