Teaching the hidden symmetry of the Kepler problem in relativistic quantum mechanics - from Pauli to Dirac electron

Abstract

Hidden symmetry in Coulomb interaction is one of the mysterious problems of modern physics. Additional conserved quantities associated with extra symmetry govern wide variety of physics problems, from planetary motion till fine and hyperfine structures of atomic spectra. In this paper we present a simple derivation of hidden symmetry operator in relativistic quantum mechanics for the Dirac equation in the Coulomb field. We established that this operator may be reduced to the one introduced by Johnson and Lippmann. It is worthwhile to notice that this operator was discussed in literature very rarely and so is not known well among physicists and was omitted even in the recent textbooks on relativistic quantum mechanics and/or quantum electrodynamics.