A Canonical Quantization of Poisson Manifolds: a 2-Groupoid Scheme

Abstract

We canonically quantize a Poisson manifold to a Lie 2-groupoid, complete with a quantization map, and show that it relates geometric and deformation quantization: the perturbative expansion in $\hbar$ of the (formal) convolution of two quantized functions yields Kontsevich's star product. Meanwhile, we can push forward this quantization map (by integrating over homotopies of paths) to obtain a quantization map in traditional geometric quantization. This gives a polarization-free, path integral definition of the quantization map, which does not have a prior prescription for Poisson manifolds and which only has a partial prescription for symplectic manifolds. We show that in conventional quantum mechanics our operators are defined on the entire prequantum Hilbert space and that they all preserve polarized sections, contrary to the Kostant-Souriau approach.