Quantum Mechanics and Nilpotent Groups I. The Curved Magnetic Field

Abstract

The quantum mechanics of massive spinless particles in an external magnetic field polynomial in position variables is shown to be related to nilpotent Lie groups. By using the known representation structure of such groups, the Hilbert spaces of the quantum mechanical systems can be decomposed into irreducible representations of the nilpotent groups. Such a decomposition is given explicitly for constant and curved magnetic fields.
Since the Hamiltonian for polynomial magnetic fields is quadratic in the Lie algebra elements, its spectrum can be found using the representation structure of nilpotent groups. The explicit time dependence of the system can also be found by solving the heat equation on nilpotent groups. These ideas are worked out for the constant magnetic field, where the solution is well known, and the curved magnetic field, where it is not. Generalizations to other systems whose interaction terms are polynomial are also given.