A symmetrical treatment of the wave equation

Source abstract

Abstract 1. A certain type of matrix has been brought into prominence by P. A. M. Dirac in his theory of the wave mechanics of an electron. C. G. Darwin has commented on the very unsymmetrical form of Dirac’s equations, and on the fact that although they are invariant for Lorentz transformations they are not constructed in tensor form. The invariance seems to be of a kind undiscoverable by the usual methods of the tensor calculus and “it is rather dis­concerting that apparently something has slipped through the net.” In order to throw light on this the following theory of the matrices has been developed; it has several points of interest independently of the application to wave-mechanics. The matrix theory leads to a very simple derivation of the first order wave equation, equivalent to Dirac’s but expressed in symmetrical form. It leads also to a wave equation which we can identify as relating to a system containing electrons with opposite spin. This symmetrical method appears to be advantageous in deducing general properties such as the angular momen­tum of an electron and the conservation of the supposed charge-current vector. 2. Four-Point Matrices.—We consider matrices and tensors in four dimen­sions. The axes are grouped in pairs in three ways denoted by α = 12, 34 β = 13, 42 γ = 14, 23.