The first and second order equations of the quantum theory

Source abstract

Abstract The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator (h/2πi ∂/∂xα - eϕα) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc. Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.