The Segal–Bargmann Transform for Path-Groups

LetKbe a connected Lie group of compact type and letW(K) denote the set of continuous paths inK, starting at the identity and with time-interval [0, 1]. ThenW(K) forms an infinite-dimensional group under the operation of pointwise multiplication. Letρdenote the Wiener measure onW(K). We construct an analog of the Segal–Bargmann transform forW(K). LetKCbe the complexification ofK,W(KC) the set of continuous paths inKCstarting at the identity, andμthe Wiener measure onW(KC). Our transform is a unitary map ofL2(W(K), ρ) onto the “holomorphic” subspace ofL2(W(KC),μ). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform forW(K) is nicely related by means of the Itô map to the classical Segal–Bargmann transform for the path-space in the Lie algebra ofK.