Relativistic Wave Equations Admitting Only Positive Energy

Abstract

The representation of the Poincaré group can be given a great many forms because the principal vectors of the Hilbert space on which the representation’s operators act can be chosen quite arbitrarily. The variables specifying these vectors in the wave equation form of the representation are supposed to have some relation to the position of the particle the motion of which the wave equation describes. Dirac and, somewhat later, Biedenharn, have given very appealing wave equation forms to the simplest representations. The condition that the wave equation suggest the probability ${\left| {\psi (\overrightarrow x )} \right|^2}$for the position x of the particle can be satisfied most simply without the introduction of extraneous variables. The equation for a zero mass, zero spin particle, for instance, can be given in the form

$$\partial \psi (\overrightarrow x )/\partial t = (c/{\pi ^2})\int {[\psi (\overrightarrow x ) - \psi (\overrightarrow x + \overrightarrow \rho )]} d\overrightarrow \rho /{\rho ^4}$$

and similar, but more complicated equations apply in the more general cases.