Inconsistency in an External Field of Dirac's Positive-Energy Wave Equation with Generalized Internal Variables

Sudarshan, Mukunda & Chiang (1982) (see also Mukunda, Chiang & Sudarshan 1982) have generalized Dirac's (1971) positive-energy relativistic wave equation by the adoption of internal variables satisfying trilinear relations rather than bilinear commutation relations. They have obtained an equation that also describes a spinless particle and admits only positive-energy solutions, and have argued that, unlike the equation proposed by Dirac, their equation remains consistent with minimal coupling to an external electromagnetic field. However, a relation between this equation and the equation for an electron is established here, and exploited to provide a proof of inconsistency for a constant and uniform external magnetic field. Analogous results are obtained in 2 + 1-dimensional and 1 + 1-dimensional space-time, again contrary to the conclusions of Sudarshan et al.