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Quantum mechanics in magnetic backgrounds with manifest symmetry and locality.

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Davighi, Joe 
Gripaios, Ben 
Tooby-Smith, Joseph 


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).



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Journal of Physics A: Mathematical and Theoretical

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Science and Technology Facilities Council (ST/S505316/1)
Science and Technology Facilities Council (ST/P000681/1)
JD is supported by The Cambridge Trust and STFC consolidated grant ST/P000681/1. BG is partially supported by STFC consolidated grant ST/P000681/1 and King’s College, Cambridge. JTS is supported by STFC consolidated