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Zbl 0843.58012
Baez, John C.
Spin networks in gauge theory. (English)
[J] Adv. Math. 117, No.2, 253-272 (1996). ISSN 0001-8708

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 ${\cal A}/{\cal G}$ of smooth connections on $P$ modulo gauge transformations. This allows one to define a Hilbert space $L^2 ({\cal A}/ {\cal G})$. Here we construct a set of vectors spanning $L^2 ({\cal A}/ {\cal G})$. 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 2000:: *58D27 Moduli problems for diff.geometric structures on spaces of mappings
81T13 Gauge theories
83D05 Relativistic gravitational theories other than Einstein's

Keywords: gauge theory; spin networks

Cited in: Zbl 1119.53058

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Zentralblatt MATH Berlin [Germany]