Discrete accidental symmetry for a particle in a constant magnetic field on a torus

摘要

A classical particle in a constant magnetic field undergoes cyclotror motion on a circular orbit. At the quantum level, the fact that al classical orbits are closed gives rise to degeneracies in the spectrum It is well-known that the spectrum of a charged particle in a con. stant magnetic field consists of infinitely degenerate Landau levels Just as for the 1 /r and r2 potentials, one thus expects some hidder accidental symmetry, in this case with infinite-dimensional repre. sentations. Indeed, the position of the center of the cyclotron circic plays the role of a Runge-Lenz vector. After identifying the corre. sponding 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 0-vacuum angle in non-Abelian gauge theories, quantum effect due to two self-adjoint extension parameters Ox and Oy explicitly break the continuous translation invariance of the classical theory This reduces the symmetry to a discrete magnetic translation grout and leads to finite degeneracy. Similar to a particle moving on cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.