Quantum Mechanics on the Torus, Klein Bottle and Projective Sphere

Abstract

The Borel quantization shows that there is a topological dependence of the “free” dynamics on the configuration space M,on which a quantum mechanical system is localized. Unitarily inequivalent quantization mappings are classified by elements (α, D) in π _l* (M) × R. In the frame work of Borel quantization the quantization parameter D gives rise to a non-linear Schrödinger equation [1, 2], which reduces to a linear one for D = 0. Our procedure is motivated by the isomorphism of elements in π ₁*(M) in the set of equivalence classes of complex line bundles with flat connection [3]. Using these flat connections we will construct a Laplacian on the complex line bundle.