Relativistic Wave Equations for Particles with Arbitrary Spin

Abstract

Relativistic wave equations are derived which generalize the recently obtained Galilei-covariant wave equations for massive particles with any integer or half-integer spin. Imposing a minimality condition on the number of components possessed by the relativistic wave function, it is shown that the index transformation properties of the wave function may be either those of the $(s,0)\ensuremath{\bigoplus}(s\ensuremath{-}\frac{1}{2},\frac{1}{2})$ representation of $\mathrm{SL}(2,C)$ or of the representation $(0,s)\ensuremath{\bigoplus}(\frac{1}{2},s\ensuremath{-}\frac{1}{2})$. The minimal extension of these representations which accommodates reflection symmetry yields the Dirac equation for $s=\frac{1}{2}$, the Duffin-Kemmer equation for $s=1$, and an equation for particles with $s>1$ whose wave-function indices transform according to the $(s,0)\ensuremath{\bigoplus}(s\ensuremath{-}\frac{1}{2},\frac{1}{2})\ensuremath{\bigoplus}(\frac{1}{2},s\ensuremath{-}\frac{1}{2})\ensuremath{\bigoplus}(0,s)$ representation of $\mathrm{SL}(2,C)$. The latter theory possesses $4(2s+1)$ independent components, has no subsidiary conditions, and describes a unique mass, $m\ensuremath{\ne}0$, and a unique spin. The theory admits a simple Lagrangian and Hamiltonian formulation and yields a conserved current. Finally, it is shown that for any spin the equation remains consistent and causal in the presence of a minimally coupled external-electromagnetic-field interaction.