Generalized measures in gauge theory (original) (raw)

Abstract

Let_P_ →M be a principal_G_-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections on_P_ and Haar measure on the groupG of gauge transformations. More precisely, we define algebras of ‘cylinder functions’ on the spacesA,G, andA/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures onA,G, andA/G in terms of graphs embedded in_M._ We use this characterization to construct generalized measures onA andG when_G_ is compact. The ‘uniform’ generalized measure onA is invariant under the group of automorphisms of_P._ It projects down to the generalized measure onA/G considered by Ashtekar and Lewandowski in the case_G_ = SU(n). The ‘generalized Haar measure’ onG is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure onA against generalized Haar measure gives aG-invariant generalized measure onA.

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Authors and Affiliations

1. Department of Mathematics, University of California, 92521, Riverside, CA, USA
   John C. Baez

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Baez, J.C. Generalized measures in gauge theory.Lett Math Phys 31, 213–223 (1994). https://doi.org/10.1007/BF00761713

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• Received: 22 December 1993
• Issue date: July 1994
• DOI: https://doi.org/10.1007/BF00761713

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