Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds H P (M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds H P (M) carry both an H 1 and a L 2 type Riemannian structures, 1999 Academic Press
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 e t∆ A /2 followed by analytic continuation. Here ∆ A 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 e t∆ K /2 followed by analytic continuation, where ∆ K is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that ∆ A reduces to ∆ 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, − 1 2 ∆ 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 holmorphic 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.
We study the (two-parameter) Segal-Bargmann transform B N s,t on the unitary group U N , for large N . Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit G s,t as N → ∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of trace polynomials, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal-Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform B N s,t to its limit G s,t . We characterize the operator G s,t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s = t, the limit transform G t,t is the "free Hall transform" G t introduced by Biane.This highlights the fact that the Segal-Bargmann transform does not preserve the space of polynomial functions of a U N -variable; in general, it maps such functions to trace polynomials. Definition 1.7. Let C[u, u −1 ] denote the algebra of Laurent polynomials in a single variable u:C[u, u −1 ] = k∈Z a k u k : a k ∈ C, a k = 0 for all but finitely-many k ,(1.11)
The free multiplicative Brownian motion bt is the large-N limit of the Brownian motion on GL(N ; C), in the sense of * -distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of bt. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region Σt that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density Wt on Σt, which is strictly positive and real analytic on Σt. This density has a simple form in polar coordinates:, where wt is an analytic function determined by the geometry of the region Σt.We show also that the spectral measure of free unitary Brownian motion ut is a "shadow" of the Brown measure of bt, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results. t 4. Properties of Σ t 5. The PDE for S 6. The Hamilton-Jacobi method 6.1. Setting up the method 6.2. Constants of motion 6.3. Solving the equations 6.4. More about the lifetime of the solution 6.5. Surjectivity
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