Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus

M. H. Al-Hashimi; U. -J. Wiese

doi:10.1016/j.aop.2008.07.006

paper

Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus

arXiv:0807.0630 · doi:10.1016/j.aop.2008.07.006

Abstract

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the $1/ r$ and $r^{2}$ potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the $θ$ -vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters $θ_{x}$ and $θ_{y}$ explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.

25 pages, 2 figures

Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus

Abstract

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