Current Algebra and Infinite-Component Fields

A second-order infinite-component wave equation is analyzed in detail. It is shown that a complete set of solutions contains timelike momenta (physical states) as well as spacelike and lightlike momenta (ghosts). The timelike solutions describe particles of spin $\frac{1}{2},\frac{3}{2},\frac{5}{2},\cdots$ with a nondegenerate mass spectrum. The wave equation is used to construct a model of local current algebra at infinite momentum, and it is shown that this model coincides with the so-called hydrogen model which has been obtained as a particular solution of the current-algebra relations. The hydrogen model, too, is known to contain unphysical states; they correspond to the ghost solutions of the wave equation. We prove that for finite mass splitting, the current-algebra relations are not satisfied unless transitions to ghosts are taken into account. On the other hand, it is shown that as long as one restricts oneself to an expansion in powers of the mass splitting to arbitrary finite order, the ghosts do not contribute.