Nonclassical Transformation in Special Relativity

Abstract

In a previous article, we were led to consider a Lorentz transformation in the general case in which the velocity which generates the transformation is no longer parallel to one of the axes of the coordinates. It was noticed that the general form of this transformation, which is no longer linear with respect to the components of the velocity, could not be conveniently applied to certain problems of wave mechanics. We give here a general transformation, linear with respect to the components of the velocity, which is connected with Hamilton's quaternions and Dirac's matrices. Although the transformation is complex, it may, in certain problems, replace with advantage the Lorentz transformation.In the realm of wave mechanics, our formalism enables us, by a simple transformation of the wave functions and of the well-known Einstein relation $W=h\ensuremath{\nu}$, to obtain a very general system of equations which contains Proca's equations, de Broglie's equations of the photon, and the Maxwell-Lorentz equations. Our method leads us to consider the equations of electromagnetism as a natural generalization of the Cauchy-Riemann conditions and electromagnetic fields as generalizations of analytic functions, the variable being a quaternion.