László Lempert, Purdue U., USA

Quantizing a Riemannian manifold

Abstract

Typically, the first step in the quantization of a physical system is finding a Hilbert space whose vectors represent the quantum states of the system. Assuming we understand the classical configuration space, a Riemannian manifold M, geometric quantization provides a way to construct this Hilbert space. The Kähler version of geometric quantization constructs the quantum Hilbert space as the space of square integrable holomorphic sections of a certain line bundle over the tangent bundle TM, which is often the same thing as holomorphic L² functions on TM. For this to be meaningful, one needs to choose a complex structure on TM and a weight function (because L² refers to a weighted L² space).

The talk will discuss my joint results with Szöke on how one can make these choices and whether the quantum Hilbert spaces corresponding to different choices are canonically isomorphic.