Quantization of Planck's Constant

This paper is about the role of Planck's constant, $\hbar$, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single ``Heisenberg-Poisson'' manifold. The new coordinate on this manifold is identified with $\hbar$. I present an explicit construction for a symplectic groupoid integrating a Heisenberg-Poisson manifold and discuss its geometric quantization. I show that in cases where $\hbar$ cannot take arbitrary values, this is enforced by Bohr-Sommerfeld conditions in geometric quantization. A Heisenberg-Poisson manifold is defined by linearly rescaling the Poisson structure, so I also discuss nonlinear variations and give an example of quantization of a nonintegrable Poisson manifold using a presymplectic groupoid. In appendices, I construct symplectic groupoids integrating a more general class of Heisenberg-Poisson manifolds constructed from Jacobi manifolds and discuss the parabolic tangent groupoid.