Quantum mechanics in magnetic backgrounds with manifest symmetry and locality. (18th March 2020)

• Record Type: Journal Article
• Authors: Davighi, Joe; Gripaios, Ben; Tooby-Smith, Joseph
• Is Part Of: Journal of physics. Volume 53:Number 14(2020)
• Publication Date: 2020-03-18
• Subjects: quantum mechanics -- symmetry -- magnetic fields -- differential geometry -- harmonic analysis -- topological terms; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab78ce
• Languages: English
• ISSNs: 1751-8113

Abstract

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).