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

• Published in: Journal of geometry and physics 1999-08, Vol.31 (1), p.35-50
• Main Authors: Doebner, H.-D., Groth, W., Hennig, J.D.
• Format: Article
• Language: English
• Subjects: Braids; Quantization; Quantum mechanics; Two-dimensional manifolds
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/S0393-0440(98)00070-9

A system of n particles localized on a smooth manifold P has a topologically nontrivial configuration space M if one assumes that M is built from P via an n-fold product, and that the particles cannot be located at the same point 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 unique: there are unitary inequivalent descriptions of its kinematics and dynamics. If the particles are assumed to be identical, further topological effects appear. We study these situations in a unified and strictly geometrical approach and use as an adequate quantization on manifolds M the Borel quantization which is based on Hilbert spaces of square integrable sections of Hermitian line bundles with flat connections. The manifolds M built from P = R 2 or compact-2-manifolds P are discussed in detail for distinguishable and identical particles; the unitarily inequivalent quantizations are classified; for P = R 2 we calculate the flat connections, the kinematics and the Schrödinger equations for the different quantizations. In Appendix A the situation for P = R m, m ≥ 3 is given.