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The Segal-Bargmann transform for path-groups. (English) Zbl 0894.22005

The purpose of this work is to construct a version of the Segal-Bargmann transform which applies to the group \(W(K)\) of continuous path starting at the identity in a connected Lie group \(K\) of compact type. If \(\rho\) is the Wiener measure on \(W(K)\), this transform is a unitary map of \(L^2(W(K), \rho)\) onto some holomorphic subspace of \(L^2 (W(K_\mathbb{C}), \mu)\) where \(K_\mathbb{C}\) is the complexification of \(K\) and \(\mu\) the Wiener measure on \(W(K_\mathbb{C})\), given by convolution with \(\rho\) followed by analytic continuation. Another way to obtain this transform is to transport the classical Segal-Bargmann transform on \(W(L)\) (where \(L\) is the Lie algebra of \(K)\) onto \(W(K)\) by using the Itô map.

MSC:

22E30 Analysis on real and complex Lie groups
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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