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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 \(P\to M\), there is a canonical “generalized measure” on the space \({\mathcal A}/{\mathcal G}\) of smooth connections on \(P\) modulo gauge transformations. This allows one to define a Hilbert space \(L^2 ({\mathcal A}/ {\mathcal G})\). Here we construct a set of vectors spanning \(L^2 ({\mathcal A}/ {\mathcal G})\). These vectors are described in terms of “spin networks”: graphs \(\varphi\) 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 \(\varphi\). 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:

58D27 Moduli problems for differential geometric structures
81T13 Yang-Mills and other gauge theories in quantum field theory
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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