The Complex-Time Segal-Bargmann Transform

Abstract

We introduce a new form of the Segal--Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $({\rho }_t(x){)}_{t>0,x\in K}$ has a space-time analytic continuation to a holomorphic function \[ (ρ_{\mathbb{C}}(τ,z))_{\mathrm{Re}\,τ>0,z\in K_{\mathbb{C}}} \] where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral \[ (B_τf)(z)=\int_{K}ρ_{\mathbb{C}}(τ,zk^{-1})f(k)\,dk,\quad z\in K_{\mathbb{C}}. \] If $s>0$ and $\tau \in \mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\mathbb{C}}$ for each $f\in L^2(K,{\rho }_s)$. We construct a heat kernel density ${\mu }_{s,\tau }$ on $K_{\mathbb{C}}$ such that, for all $s,\tau$ as above, $B_{s,\tau }:=B_{\tau }{|}_{L^2(K,{\rho }_s)}$ is an isometric isomorphism from $L^2(K,{\rho }_s)$ onto the space of holomorphic functions in $L^2(K_{\mathbb{C}},{\mu }_{s,\tau })$. When $\tau =t=s$, the transform $B_{t,t}$ coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When $\tau =t\in (0,2s)$, the transform $B_{s,t}$ coincides with the one introduced by the first two authors.

32 pages