Spin Network States in Gauge Theory
John C. Baez
TL;DR
This work constructs a rigorous Hilbert-space framework for gauge theory on a manifold by exploiting generalized measures on the space of connections modulo gauge transformations, $L^2({\cal A}/{\cal G})$. It introduces spin-network states, labelled by embedded graphs with edges carrying irreducible $G$-representations and vertices by intertwiners, and proves these states span $L^2({\cal A}/{\cal G})$, with an orthonormal basis for each fixed graph. The approach leverages the Ashtekar-Lewandowski uniform measure and the loop representation, relating gauge theory on graphs to gauge theory on manifolds and revealing a natural category-theoretic interpretation via holonomy groupoids and functors to $G$. The results provide a rigorous foundation for kinematical states in loop quantum gravity, illuminate diffeomorphism-invariant constructions, and connect to TQFTs and BF theory through a unifying spin-network/categorical perspective.
Abstract
Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P -> M, there is a canonical `generalized measure' on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L^2(A/G). Here we construct a set of vectors spanning L^2(A/G). These vectors are described in terms of `spin networks': graphs phi embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin networks associated to any fixed graph phi. We conclude with a discussion of spin networks in the loop representation of quantum gravity, and give a category-theoretic interpretation of the spin network states.