Spin Networks in Gauge Theory

✍ Scribed by John C. Baez

• Book ID: 102965624
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 735 KB
• Volume: 117
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1996.0012

✦ Synopsis

Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P Ä M, there is a, canonical generalized measure'' on the space AÂG of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L 2 (AÂG). Here we construct a set of vectors spanning L 2 (AÂG). These vectors are described in terms of spin networks'': graphs , 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 ,. We conclude with a discussion of spin networks in the loop representation of quantum gravity and give a categorytheoretic interpretation of the spin network states.