Abstract
The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).
Highlights
Consider a particle moving on a smooth, connected, manifold M in the presence of some background magnetic eld
We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original con guration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1)
We have formulated the quantum mechanics of a particle moving on a manifold M, with dynamics invariant under the action of a Lie group G, in the presence of a background magnetic eld
Summary
Consider a particle moving on a smooth, connected, manifold M in the presence of some background magnetic eld. We complete the discussion of rigid body rotation (section 4.1) and give a series of other examples which illustrate the method: the Dirac monopole (section 4.2), a charged particle in the electromagnetic eld of a dyon (section 4.3), a repeat of Landau levels on a plane, but using the full Euclidean group (section 4.4), motion on the Heisenberg group manifold (section 4.5), and motion in a uniform magnetic eld with a mass that varies with position (section 4.6), the last of which gives a completely solvable example in the case where the action of G on M is not transitive.
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