Quantum mechanics in magnetic backgrounds with manifest symmetry and locality

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). Contents1 For the sake of simplicity, we will consider here only examples where G and M are connected.in the lagrangian representing the magnetic field that is valid globally on M . Instead, the best that one can do is to cover M by overlapping patches and to use multiple lagrangians, each of which is valid only locally on some patch. The most famous example, due to Dirac [1], 2 is given by the motion of an electrically-charged particle in the presence of a magnetic monopole, but we will see that there exists an example that is arguably even simpler (and certainly more prevalent in everyday life!), given by the motion of a rigid body which happens to be a fermion. 3The second complication is that the corresponding lagrangian (or lagrangians) will not be invariant under the action of G, but rather will shift by a total derivative. Perhaps the simplest example, made famous by Landau [5], is given by the motion of a particle in a plane in the presence of a uniform magnetic field, where there is no choice of gauge such that the lagrangian is invariant under translations in more than one direction. 4At the classical level, neither of these complications causes any problems, since they disappear once we pass from the lagrangian to the classical equations of motion. Indeed, the equations of motion are both globally valid and invariant (or rather covariant) under G. Thus, we can attempt to solve for the classical dynamics using our usual arsenal of techniques. But this is not the case at the quantum level. There, our usual technique is to convert the hamiltonian into an operator on L 2 (M ) and to exploit the conserved charges corresponding to G to solve, at least partially, the resulting Schrödinger equation. Here though, we do not have a unique hamiltonian, but rather several; even if we did have a unique hamiltonian, we would, in g...

Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

“…Magnetic translations on T d have been studied in e.g. [Fio13,DGTS20] (for constant magnetic fields B), but our treatment is more general and also fits in with expectations from string theory.…”