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 $(ρ_{t}(x))_{t>0,x\in K}$ 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 $τ\in\mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\mathbb{C}}$ for each $f\in L^{2}(K,ρ_{s})$. We construct a heat kernel density $μ_{s,τ}$ on $K_{\mathbb{C}}$ such that, for all $s,τ$ as above, $B_{s,τ}:=B_τ|_{L^{2}(K,ρ_{s})}$ is an isometric isomorphism from $L^{2}(K,ρ_{s})$ onto the space of holomorphic functions in $L^{2}(K_{\mathbb{C}},μ_{s,τ})$. When $τ=t=s$, 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 $τ=t\in (0,2s)$, the transform $B_{s,t}$ coincides with the one introduced by the first two authors.