Large-$N$ Limit of the Segal--Bargmann Transforms on the Spheres

TL;DR

This paper investigates the behavior of the Segal--Bargmann transform on high-dimensional spheres as the dimension grows, providing explicit formulas and geometric models for the limit, and establishing the unitarity of the limiting transform.

Contribution

It offers a detailed analysis of the large-$N$ limit of the Segal--Bargmann transform on spheres, including explicit formulations and geometric interpretations.

Findings

Explicit formulas for the limit of the transform

Geometric models for the limiting domain and range

The limiting transform remains unitary

Abstract

We study the large-$N$ limit of the Segal--Bargmann transform on $S^{N-1}(\sqrt N)$, the $(N-1)$-dimensional sphere of radius $\sqrt N$, as a unitary map from the space of square-integrable functions with respect to the normalized spherical measure onto the space of holomorphic square-integrable functions with respect to a certain measure on the quadric. In particular, we give an explicit formulation and describe the geometric models for the limit of the domain, the limit of the range, and the limit of the transform when $N$ tends to infinity. We show that the limiting transform is still a unitary map from the limiting domain onto the limiting range.