TY - JOUR

T1 - Differential geometry on the space of connections via graphs and projective limits

AU - Ashtekar, Abhay

AU - Lewandowski, Jerzy

N1 - Funding Information:
We would like to thank John Baez, David Groisser, Piotr Hajac, Witold Kondracki, Donald Marolf, Jose Mourgo, Tomasz Mrowka, Alan Rendall, Carlo Rovelli, Lee Smolin, Clifford Taubes, and Thomas Thiemann for discussions. Jerzy Lewandowski is grateful to Center for Gravitational Physics at Penn State and Erwin S&r&linger Institute for Mathematical Physics in Vienna, where most of this work was completed, for warm hospitality. This work was supported in part by the NSF grants 93-96246 and PHY91-07007, the Eberly Research Fund of Penn State University, the Isaac Newton Institute, the Erwin Shrijdinger Institute and by the KBN grant 2-P302 11207.

PY - 1995/11

Y1 - 1995/11

N2 - In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a "universal home" for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as "floating lattices" in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G: differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.

AB - In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion A G of the space A G of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory A G is a very large is a very large space and serves as a "universal home" for measures in theories in which the Wilson loop observables are well defined. In this paper, A G is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as "floating lattices" in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A G: differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A G is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.

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U2 - 10.1016/0393-0440(95)00028-G

DO - 10.1016/0393-0440(95)00028-G

M3 - Article

AN - SCOPUS:1542552153

VL - 17

SP - 191

EP - 230

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 3

ER -