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The usual methods for formulating and solving the quantum mechanics of aparticle moving in a magnetic field respect neither locality nor any globalsymmetries which happen to be present. For example, Landau's solution for aparticle moving in a uniform magnetic field in the plane involves choosing agauge in which neither translation nor rotation invariance are manifest. Weshow that locality can be made manifest by passing to a redundant descriptionin which the particle moves on a $U(1)$-principal bundle over the originalconfiguration space and that symmetry can be made manifest by passing to acorresponding central extension of the original symmetry group by $U(1)$. Withthe symmetry manifest, one can attempt to solve the problem by using harmonicanalysis and we provide a number of examples where this succeeds. One is asolution of the Landau problem in an arbitrary gauge (with either translationinvariance or the full Euclidean group manifest). Another example is the motionof a fermionic rigid body, which can be formulated and solved in a manifestlylocal and symmetric way via a flat connection on the non-trivial $U(1)$-centralextension of the configuration space $SO(3)$ given by $U(2)$.

Quantum mechanics in magnetic backgrounds with manifest symmetry and locality