  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 $U(2)$.
