UC San Diego

Discovered by Segal and Bargmann in 1960s, the Segal--Bargmann trasnform is an important tool in mathematical physics, intertwining the Heisenberg and Fock pictures in quantum mechanics. We will discuss representations of the two-parameter {\em free} unitary Segal--Bargmann transform, which is the large-$N$ limit of Segal--Bargmann--Hall transform on unitary group. Motivated by a conditional expectation interpretation of the Segal-Bargmann transform, we derive the integral kernel for the large-$N$ limit of the two-parameter Segal-Bargmann-Hall transform over the unitary group, and explore its limiting behavior. We also extend the notion of circular systems to more general elliptic systems, giving an alternate construction of our new two-parameter {\em free} unitary Segal-Bargmann-Hall transform via a Biane-Gross-Malliavin type theorem.