Spin Network States in Gauge Theory

TL;DR

This paper constructs a basis of spin network states in the Hilbert space of gauge theory on a manifold, linking graph-based representations with quantum gravity and category theory.

Contribution

It introduces a set of spin network vectors spanning the space of gauge-invariant connections, providing an orthonormal basis and a category-theoretic interpretation.

Findings

Constructed a spanning set of spin network states for L^2(A/G)

Provided an orthonormal basis associated with any fixed graph

Connected spin networks to quantum gravity and category theory

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.