The Segal–Bargmann Transform for Path-Groups

✍ Scribed by Brian C Hall; Ambar N Sengupta

• Book ID: 102972364
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 504 KB
• Volume: 152
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3159

✦ Synopsis

Let K be a connected Lie group of compact type and let W(K ) denote the set of continuous paths in K, starting at the identity and with time-interval [0, 1]. Then W(K ) forms an infinite-dimensional group under the operation of pointwise multiplication. Let \ denote the Wiener measure on W(K ). We construct an analog of the Segal Bargmann transform for W(K ). Let K C be the complexification of K, W(K C ) the set of continuous paths in K C starting at the identity, and + the Wiener measure on W(K C ). Our transform is a unitary map of L 2 (W(K ), ) onto the ``holomorphic'' subspace of L 2 (W(K C ), +). 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 for W(K ) is nicely related by means of the Ito^map to the classical Segal Bargmann transform for the path-space in the Lie algebra of K.

1998 Academic Press

Here x=(x 1 , x 2 , ..., x n ) and x 2 =x 2 1 +x 2 2 + } } } +x 2 n . Clearly, \ t has a (unique) analytic continuation to C n . The finite-dimensional Segal Bargmann transform is a map B t from L 2 (R n , \ t (x) dx) into H(C n ), where article no.