1994/11/02 by John C. Baez · 263 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Black Holes and Theoretical Physics #Combinatorics #Discrete mathematics #Hilbert space #Homotopy and Cohomology in Algebraic Topology #Lie group #Mathematics #Noncommutative and Quantum Gravity Theories #Orthonormal basis #Physics #Pure mathematics #Quantum mechanics #Tensor product #Unitary representation #gr-qc #hep-th
paper · pdf · doi:10.1006/aima.1996.0012
published in Advances in Mathematics 117(2), 253-272 (Elsevier BV) · 19 pages, LaTeX
arxiv created 1994/11/02 · openalex publication_date 1996/02/01 · arxiv updated 2010/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 L2(A/G). Here we construct a set of vectors spanning L2(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.