In this work we construct a superalgebra for potential problems in the Dirac equation. We show that this superalgebra is closed for the large class of interactions that leave the Dirac sea stable. In this case, we find that it is possible to perform an exact SU(2) transformation for the Dirac Hamiltonian of the problem. This procedure is formally equivalent to obtaining an exact form for the related Foldy-Wouthuysen Hamiltonian. The potential problems that can be exactly diagonalized include the Dirac oscillator, an arbitrary, time-independent magnetic field, and all the odd potentials, as well as its non-Abelian generalizations. We find that the internal group generators do not destroy the structure of the superalgebra. We work out explicitly the example of the Dirac oscillator.

• Received 19 March 1990

DOI:https://doi.org/10.1103/PhysRevD.43.2036