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 $T_M$, which is often the same thing as holomorphic $L^2$ functions on $T_M$. For this to be meaningful, one needs to choose a complex structure on $T_M$ and a weight function (because $L^2$ refers to a weighted $L^2$ 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.