Quaternion symmetry of the Dirac equation

Abstract

Following van der Waerden, the Dirac equation is derived from linearization of the Klein-Gordon equation using the algebraic properties of the Pauli spin matrices. As the algebra of these matrices is identical to that of quaternions, the Dirac equation can be reformulated in terms of quaternion algebra and therefore without reference to a specific spin quantization axis. In this paper we consider the symmetry content of the Dirac equation. It is found that the basic binary symmetry operations in spin space map onto the unit vectors of complex quaternions. We argue that a consistent choice of the inversion operator in spin space is of order four. We furthermore show that quaternion algebra is the natural language for time reversal symmetry. These considerations lead to the formulation of a symmetry scheme that automatically provides maximum point group and time reversal symmetry reduction in the solution of the Dirac equation in the finite basis approximation.