Spacelike Solutions of Infinite-Component Wave Functions

Abstract

A necessary and sufficient condition for the existence of spacelike solutions is given for infinite-component wave equations which are linear in the momentum and for which the charge operator is positive definite. The condition is used to establish the existence of spacelike solutions for wave equations of the Abers-Grodsky-Norton type, i.e., wave equations of the form $({\ensuremath{\gamma}}_{\ensuremath{\mu}}{p}_{\ensuremath{\mu}}\ensuremath{-}M)\ensuremath{\psi}=0$, where ${\ensuremath{\gamma}}_{\ensuremath{\mu}}$ are the Dirac $\ensuremath{\gamma}$'s, and $M$ is an $\mathrm{SL}(2,C)$ invariant with nontrivial dependence on ${\ensuremath{\gamma}}_{\ensuremath{\mu}}$. An interpretation of the spacelike solutions is given for one simple model and suggests that the occurrence of the spacelike solutions in general is simply a reflection of the composite nature of the particles being described. It is also shown that if the current operator has no explicit momentum dependence and the particles are assigned to a representation of $\mathrm{SL}(2,C)$, then current conservation is equivalent to a wave equation.