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Discrete accidental symmetry for a particle in a constant magnetic field on a torus. (English) Zbl 1159.81044
Summary: 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 $$\theta$$-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 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.

##### MSC:
 81V70 Many-body theory; quantum Hall effect 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 22E70 Applications of Lie groups to the sciences; explicit representations
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##### References:
 [1] Fock, V., Z. physik, 98, 145, (1935) [2] Bargmann, V., Z. physik, 99, 576, (1936) [3] Lenz, W., Z. physik, 24, 197, (1924) [4] H.V. McIntosh, in: Group Theory and its Applications, vol. 2, Academic Press, Inc., New York and London, 1971, p. 75. [5] Al-Hashimi, M.H.; Wiese, U.-J., Ann. phys., 323, 82, (2008) [6] Landau, L.D., Z. physik, 64, 629, (1930) [7] Johnson, M.H.; Lippmann, B.A., Phys. rev., 76, 828, (1949) [8] Dulock, V.A.; McIntosh, H.V., J. math. phys., 7, 1401, (1966) [9] Zak, J., Phys. rev., 134, A1602, (1964) [10] Stern, A., Ann. phys., 323, 204, (2008) [11] Chen, G.-H.; Kuang, L.-M.; Ge, M.-L., Phys. rev., B53, 9540, (1996) [12] Zainuddin, H., Phys. rev., D40, 636, (1989) [13] Feldman, A.; Kahn, A.H., Phys. rev., B12, 4584, (1970) [14] Kowalski, K.; Rembielinski, J., J. phys. A: math. gen., 38, 8247, (2005) · Zbl 1081.81058 [15] ’t Hooft, G., Nucl. phys., B153, 141, (1979) [16] ’t Hooft, G., Commun. math. phys., 81, 267, (1981) [17] Kowalski, K.; Rembielinski, J., Phys. rev. A, 75, (2007), 052102-1
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