Spin networks in gauge 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 L2(A/G). Here we construct a set of vectors spanning L2(A/G). These vectors are described in terms of "spin networks": graphs Φ 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 network states associated to any fixed graph Φ. 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. © 1996 Academic Press, Inc.