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

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 $J_0\mathrm{}\left(0\right)\mathrm{}$ is taken to be unity, in which case the content of the current commutators is essentially 1. $J_{\mu }\mathrm{}\left(k\right)=J_{\mu }\mathrm{}\left(k\right)\mathrm{}$, i.e., empty. Furthermore, it is shown explicitly in an example that, starting from the covariant wave equation, this condition $J_0\mathrm{}\left(0\right)=1\mathrm{}$ 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.