Functional Integration on Spaces of Connections

Abstract

Let $G$ be a compact connected Lie group and $P\to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity on the base space $M$. Using `spin networks' we construct explicit functions spanning the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge transformations.

24 pages, LaTeX with epsfig, 6 ps figures included via uufiles. PS file available at http://web.mit.edu/org/m/mathdept/www/

References in corpus (3)

• Projective Techniques and Functional Integration
• Generalized Measures in Gauge Theory
• Topological Measure and Graph-Differential Geometry on the Quotient Space of Connections

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• On the relation between the connection and the loop representation of quantum gravity
• Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables
• A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications