Evaluating the validity of parametrized relativistic wave equations

Abstract

We wish to determine the correct partial differential equation(s) for describing a relativistic particle. A physical foundation is presented for using a parametrized wave equation with the general form

$$i\frac{{\partial \psi }}{{\partial s}} = K\psi$$

where s is the invariant evolution parameter. Several forms have been proposed for the generator K of evolution parameter translations. Of the proposed generators, only the generator K ₂ which is proportional to the inner product P _μ P ^μ of fourmomentum operators can be derived from first principles, notably a probabilistic basis. Although experimental tests must be made to establish the validity of K ₂,we conclude that K ₂ is the leading theoretical candidate for the form of a generator of evolution parameter translations.