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

The Segal–Bargmann transform on a sphere with a positive time parameter is a unitary map from the Hilbert space of square-integrable functions on the sphere with respect to its volume measure onto the Hilbert space of holomorphic square-integrable functions with respect to a certain heat-kernel measure on the quadric corresponding to the sphere. In this thesis, I study the limiting behavior of the Segal–Bargmann transforms on high dimensional spheres. As the dimension of a sphere tends to infinity, with a proper scaling of its radius, the normalized spherical volume measure converges to the infinite-dimensional unit-time Gaussian measure; the heat-kernel measure on the quadric likewise converges to a certain Gaussian measure determined by the time parameter; and, the Segal--Bargmann transform tends to an exponential operator defined via the Hermite differential operator.

This thesis aims to provide an explicit formulation and describe the geometric models for these convergence phenomena. It turns out that the limiting transform is still a unitary map from the limiting domain Hilbert space onto the limiting range Hilbert space.