Two-Component Alternative to Dirac's Equation

Abstract

An alternative to Dirac's factorization of the Klein-Gordon equation is demonstrated which yields two-component, $m\ne 0$, equations. Explicit two-component solutions for the Coulomb field, $\frac{\alpha Z}{r}$, are given in detail; the bound energy levels are found to be precisely the same as for the Dirac equation, differences in the spectrum are discussed. The case for general electromagnetic fields is also discussed. The Poincaré invariance of this alternative equation is proved in two distinct ways: (1) by the standard method using Poincaré generators, and (2) by a group-theoretic analysis based upon Wigner's classic work. The existence of an invertible 1-1 mapping of Dirac's equation for general electromagnetic fields into and onto the alternative equation is demonstrated. That the two alternatives are not necessarily identical (for example, if chirality has a fixed significance) is discussed.