2013/09/04 by Hawkins, Eli
#22A22 #46L65 #53D17 #53D50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1309.1068
This paper is about the role of Planck's constant, ℏ, 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 ℏ. 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 ℏ 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.