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A classical particle in a constant magnetic field undergoes cyclotron motionon a circular orbit. At the quantum level, the fact that all classical orbitsare closed gives rise to degeneracies in the spectrum. It is well-known thatthe spectrum of a charged particle in a constant magnetic field consists ofinfinitely degenerate Landau levels. Just as for the $1/r$ and $r^2$potentials, one thus expects some hidden accidental symmetry, in this case withinfinite-dimensional representations. Indeed, the position of the center of thecyclotron circle plays the role of a Runge-Lenz vector. After identifying thecorresponding accidental symmetry algebra, we re-analyze the system in a finiteperiodic volume. Interestingly, similar to the quantum mechanical breaking ofCP invariance due to the $\theta$-vacuum angle in non-Abelian gauge theories,quantum effects due to two self-adjoint extension parameters $\theta_x$ and$\theta_y$ explicitly break the continuous translation invariance of theclassical theory. This reduces the symmetry to a discrete magnetic translationgroup and leads to finite degeneracy. Similar to a particle moving on a cone, aparticle in a constant magnetic field shows a very peculiar realization ofaccidental symmetry in quantum mechanics.

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