On the support of the Ashtekar-Lewandowski measure

TL;DR

This paper demonstrates that the classical configuration space of Yang-Mills theories is a measure-zero subset within the Ashtekar-Lewandowski extended space, enabling quantum states to be represented as functions on this larger space.

Contribution

It establishes the topological and measure-theoretic structure of the Ashtekar-Lewandowski extension as a projective limit of finite lattice spaces, and shows the classical space's measure-zero embedding.

Findings

Classical configuration space is measure-zero in the extended space.

The extended space is a projective limit of finite lattice spaces.

Quantum states can be realized as functions on the extended space.

Abstract

We show that the Ashtekar-Isham extension of the classical configuration space of Yang-Mills theories (i.e. the moduli space of connections) 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 the classical configuration space is contained in a zero measure subset of this extension with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure. 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.