Large-<i>N </i>Limit of the Segal–Bargmann Transforms on the Spheres

<p>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.</p><p> 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. </p>