Spin Network States in Gauge Theory

Details Spin Network States in Gauge Theory Spin Network States in Gauge Theory

1994-11-02

arxiv.org/abs/gr-qc/9411007v1

Adv. Math.117:253-272,1996

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

19 pages, LaTeX

Scientific paper

10.1006/aima.1996.0012

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.

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