Spin networks in Gauge theory

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 module 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 network states associated to any fixed graph phi. We conclude with a discussion of spill networks in the loop representation of quantum gravity and give a category-theoretic interpretation of the spin network slates. (C) 1996 Academic Press, Inc.