Realization of Current-Algebra Commutation Relations on the Space of Solutions of Finite- or Infinite-Component Wave Equations

Abstract

The relation between the algebra of current densities and the finite- or infinite-component wave equations is critically investigated. It is found that, at any arbitrary momentum, the charge-current density commutation relations can be satisfied by the solutions of the wave equation, but only in a trivial sense, viz., if ${J}_{0}(0)$ is taken to be unity, in which case the content of the current commutators is essentially 1. ${J}_{\ensuremath{\mu}}(k)={J}_{\ensuremath{\mu}}(k)$, i.e., empty. Furthermore, it is shown explicitly in an example that, starting from the covariant wave equation, this condition ${J}_{0}(0)=1$ can be satisfied only if it is made true by definition. The precise connection between the current algebra and infinite-component wave equations is discussed by the introduction of translation operators in momentum space.