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Generalized Measures in Gauge Theory | | John C. Baez
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31 Oct 1993 | | Journal: | Lett.Math.Phys. 31 (1994) 213-224 | | Subject: | High Energy Physics - Theory; Operator Algebras | hep-th math.OA | | Abstract: | Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure’’ on the space A of connections on P and for ``Haar measure’’ on the group Ga of gauge transformations. More precisely, we define algebras of ``cylinder functions’’ on the spaces A, Ga, and A/Ga, 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, Ga, and A/Ga in terms of graphs embedded in M. We use this characterization to construct generalized measures on A and Ga, 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/Ga considered by Ashtekar and Lewandowski in the case G = SU(n). The ``generalized Haar measure’’ on Ga is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure on A against generalized Haar measure gives a gauge-invariant generalized measure on A. | | Source: | arXiv, hep-th/9310201 | | Services: | Forum | Review | PDF | Favorites | |
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