Spinor Equations for the Meson and Their Solution When No Field Is Present

Abstract

The spinor equations equivalent to the Proca equations for the meson are found. They differ from those proposed by Dirac in three ways: (1) They contain two symmetric spinors ${A}_{1AB}$, ${A}_{2AB}$ and a spinor ${{B}^{{A}^{\ensuremath{'}}}}_{B}$, whereas the Dirac ones contain only one symmetric spinor and the spinor ${{B}^{{A}^{\ensuremath{'}}}}_{B}$. (2) They are invariant under spin transformations corresponding to both proper and improper Lorentz transformations whereas the Dirac ones are invariant under the former alone. (3) When a field is present they contain terms which cannot be obtained from the equations for free particles by replacing the operator ${p}_{\ensuremath{\nu}}$ by ${p}_{\ensuremath{\nu}}\ensuremath{-}\frac{e{\ensuremath{\phi}}_{\ensuremath{\nu}}}{c}$ as was proposed by Dirac. In case there is no field present the equations can be solved by introducing two pairs of simple spinors (${\ensuremath{\psi}}^{A}$, ${\ensuremath{\phi}}^{A}$) and (${\ensuremath{\chi}}^{A}$, ${\ensuremath{\eta}}^{A}$) each of which satisfies a Dirac equation for a free particle of spin one-half. It is proposed to interpet each pair of spinors as representing a free particle of mass ${m}_{1}$ and ${m}_{2}$, respectively. With this interpretation it follows that a free particle of mass $m$ and spin one satisfying the Proca equations is equivalent to a pair of particles of masses ${m}_{1}$ and ${m}_{2}$ satisfying the Dirac equations in the following sense: If the three particles of masses $m,{m}_{1}$ and ${m}_{2}$ are in states determined by the energy momentum vectors ${p}_{\ensuremath{\nu}}$, ${p}_{1\ensuremath{\nu}}$, and ${p}_{2\ensuremath{\nu}}$, respectively and if $m\ensuremath{\ne}0$, then ${p}_{1\ensuremath{\nu}}=\frac{{m}_{1}}{m{p}_{\ensuremath{\nu}}}$ and ${p}_{2\ensuremath{\nu}}=\frac{{m}_{2}}{m{p}_{\ensuremath{\nu}}}$, $m={m}_{1}+{m}_{2}$; and the spin of the Proca particle is the vector sum of the spins of the two Dirac particles.