Topological conditions in geometric and Maslov quantization (2020)
Among the most important theories in physics is quantum mechanics which, in contrast to classical mechanics, uses topology in addition to differential geometry. Specifically, in this work, we will study the well-known geometric quantization procedure due mainly to B. Kostant and J-M. Souriau, and then compare it to an alternative geometric quantization method using the Maslov index due to J. Czyz. Given a symplectic manifold, that generally models a classical physical system, we wish to quantize the Poisson algebra of observables on it. The idea is to construct a Hilbert Space associated to the symplectic manifold and associate to each smooth function a self-adjoint operator acting on H. This construction is done in such a way that the Dirac quantization conditions hold. The full process consists of three steps. The first is called prequantization, in which a topological condition (on the cohomology class of the symplectic form) gives rise to a geometric space...