Klein paradox in the Breit equation

Abstract

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r 0)=E there appears a Klein paradox atr=r 0. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm (1)→∞ andm (2)→∞, where we neglect the terms ∼1/m (1) and 1/m (2). In Appendix I we show that in the Breit equation the oscillations accumulating atr=r 0 in the case ofm (1)≠m (2) are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r 0 in the case ofm (1)=m (2).