Abstract
We show that the Ashtekar-Isham extension\(\overline {A/G}\) of the configuration space of Yang-Mills theories\(A/G\) is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.
These results are then used to prove that\(A/G\) is contained in a zero measure subset of\(\overline {A/G}\) with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on\(\overline {A/G}\). Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the “extended” configuration space\(\overline {A/G}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ashtekar, A.: Non-perturbative canonical quantum gravity. (Notes prepared in collaboration with R.S. Tate), Singapore: World Scientific, 1991
Ashtekar, A., Isham, C.: Class. Quant. Grav.9, 1433–85 (1992)
Glimm, J., and Jaffe, A.: Quantum physics. New York: Springer Berlin, Heidelberg, 1987
Baez, J., Segal, I., Zhou Z.: Introduction to algebraic and constructive quantum field theory Princeton, NJ; Princeton University Press, 1992
Gel'fand, I.M., Vilenkin, N.: Generalized functions. Vol.IV, New York: Academic Press, 1964
Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomyC *-algebras. Preprint CGPG-93/8-1. To appear in Proceedings of the Conference “Knots and Quantum Gravity” Baez, J. Oxford U.P. (ed.)
Baez, J.: Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations. Preprint hep-th/9305045, To appear in Proceedings of the Conference “Quantum Topology” Crane, L., Yetter, D. (eds.)
Yamasaki, Y.: Measures on infinite dimensional spaces, Singapore: World Scientific, 1985
Rudin, W.: Functional analysis. New York: McGraw-Hill, 1973
Rendall, A.: Class. Quant. Grav.10, 605–608 (1993)
Rudin, W.: Real and complex analysis. New York: McGraw-Hill, 1987
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Englewood Cliff NJ: Prentice-Hall Inc., 1970
Dalecky, Yu.L., Fomin, S.V.: Measures and differential equations in infinite-dimensional space. Dordrecht: Kluwer Academic Pub., 1991
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Rights and permissions
About this article
Cite this article
Marolf, D., Mourão, J.M. On the support of the Ashtekar-Lewandowski measure. Commun.Math. Phys. 170, 583–605 (1995). https://doi.org/10.1007/BF02099150
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02099150