Yang–Mills Theory and the Segal–Bargmann Transform

Abstract:

We use a variant of the Segal–Bargmann transform to study canonically quantized Yang–Mills theory on a space-time cylinder with a compact structure group K. The non-existent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The Segal–Bargmann transform is then a unitary map from the L ² space over the space of connections to a holomorphic L ² space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by followed by analytic continuation. Here is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory.

On the gauge-trivial subspace, consisting of functions of the holonomy around the spatial circle, the Segal–Bargmann transform becomes followed by analytic continuation, where Δ _K is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that reduces to Δ _K on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized Yang–Mills theory on a space-time cylinder, namely, −½Δ _K is the Hamiltonian and L ²(K) is the Hilbert space. As a byproduct of these considerations, we find a new one-parameter family of unitary transforms from L ²(K) to certain holomorphic L ²-spaces over the complexification of K. This family of transformations interpolates between the two previously known unitary transformations.

Our work is motivated by results of Landsman and Wren and uses probabilistic techniques similar to those of Gross and Malliavin.