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.