ar X iv : h ep - t h / 93 10 20 1 v 1 3 1 O ct 1 99 3 Generalized Measures in Gauge Theory
نویسنده
چکیده
Let P → M be a principal G-bundle. We construct well-defined substitutes for “Lebesgue measure” on the space A of connections on P and for “Haar measure” on the group G of gauge transformations. More precisely, we define algebras of “cylinder functions” on the spaces A, G, and A/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 on A, G, and A/G in terms of graphs embedded in M . We use this characterization to construct generalized measures on A and G, respectively. The “uniform” generalized measure on A is invariant under the group of automorphisms of P . It projects down to the generalized measure on A/G considered by Ashtekar and Lewandowski in the case G = SU(n). The “generalized Haar measure” on G is rightand left-invariant as well as Aut(P )-invariant. We show that averaging any generalized measure on A against generalized Haar measure gives a G-invariant generalized measure on A.
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