# Quantum mechanics in magnetic backgrounds with manifest symmetry and locality

## TL;DR

This paper develops a framework for quantum mechanics in magnetic backgrounds that makes locality and symmetry manifest by using principal bundles and central extensions, enabling more straightforward solutions.

## Contribution

It introduces a method to reformulate quantum mechanics in magnetic fields with explicit locality and symmetry using principal bundles and group extensions.

## Key findings

- Successfully solves the Landau problem in arbitrary gauges.
- Provides a local and symmetric formulation of fermionic rigid body motion.
- Demonstrates the effectiveness of harmonic analysis in these reformulations.

## 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 $U(2)$.