On quantum mechanics of n-particle systems on 2-manifolds - a case study in topology

A system of n particles localized on a smooth manifold P has a topologicall y nontrivial configuration space M if one assumes that M is built from P vi a an n-fold product, and that the particles cannot be located at the same p oint in P at the same time. Because of this property of M, which holds even if P is topologically trivial, the quantization of the system is not uniqu e: there are unitary inequivalent descriptions of its kinematics and dynami cs. If the particles are assumed to be identical, further topological effec ts appear. We study these situations in a unified and strictly geometrical approach and use as an adequate quantization on manifolds M the Borel quant ization which is based on Hilbert spaces of square integrable sections of H ermitian line bundles with flat connections. The manifolds M built from P = R-2 or compact 2-manifolds P are discussed in detail for distinguishable a nd identical particles; the unitarily inequivalent quantizations are classi fied; for P = R-2 we calculate the flat connections, the kinematics and the Schrodinger equations for the different quantizations. In Appendix A the s ituation for P = R-m, m greater than or equal to 3, is given. (C) 1999 Else vier Science B.V. All rights reserved.