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Spin networks in gauge theory. (English) Zbl 0843.58012
Summary: Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle PM, 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 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.

MSC:
58D27Moduli problems for differential geometric structures on spaces of mappings
81T13Yang-Mills and other gauge theories
83D05Relativistic gravitational theories other than Einstein’s