On the support of the Ashtekar-Lewandowski measure

Abstract

We show that the Ashtekar-Isham extension 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 A/G with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on 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 A/G.

A groupoid approach to spaces of generalized connections

“…Using this formalism, we show in section 3 that the quotient ofĀ by the action of the gauge group is homeomorphic to A/G. This new proof, establishing directly the equivalence at the projective limit level, seems to us more transparent than the proof one can obtain by combining results from [AL1,MM,AL2,B1,AL3].…”

“…It turns out that Hom [EG, G], when equipped with an appropriate topology, is homeomorphic to the spaceĀ of generalized connections [MM,AL2,B3]. This identification can be proved using the fact that Hom [EG, G] is the projective limit of a projective family labeled by graphs in the manifold Σ [ALMMT,AL3].…”

A manifestly gauge-invariant approach to quantum theories of gauge fields

“…Finally, every measure dμ on A/G p defined as in (3.10) can be extended to a σ-additive measure on A/G p and is thus a genuine, infinite dimensional measure [9,12,14]. From a physical point of view, however, we should still factor out by the gauge freedom at the base point p. A generic gauge transformation g(.)…”

“…Note that (3.13) is the natural non-linear analog of the Fourier transform (2.13). The generating function χ is again a function on the space of probes which now happens to be the hoop group HG rather than the Schwarz space S. From the normalization and the positivity of dμ it is easy to see that χ(α) satisfies 14) where p denotes the trivial (i.e., identity) hoop, c i ∈ C, are arbitrary complex numbers and N ∈ IN is an arbitrary integer. Finally, the Riesz-Markov theorem [17] implies that every generating functional χ satisfying i c i χ(β i ) = 0 whenever i c i T β i = 0 is the loop transform of a measure dμ so that there is a one to one correspondence between positive, normalized (regular, Borel) measures on A/G and generating functionals on HG p [11].…”

Overview and outlook

“…Thus, what we have is a set of "floating lattices," each associated with a finitely generated subgroup of the hoop group. The space A/G can be rigorously recovered as a projective limit of the configuration spaces of lattice theories (Marolf & Mourão 1993). This construction is potentially quite powerful; it may enable one to take continuum limits of operators of lattice theories in a completely new fashion.…”