Symmetries induced by conserved vector currents in the theory of one scalar field

Abstract

Let us consider a quantum theory of one scalar, real, local, Poincaré covariant field with massive one-particle states and unique vacuum. The asymptotic fields are assumed to be irreducible. Our conjecture is that under some technical assumptions the “charge” of every real, locally conserved local Poincaré covariant (pseudo) vector current relatively local to the original field appearing in this theory — vanishes. The only symmetry groups one can find are generated by global, self-adjoint, Poincaré invariant operators. Our arguments can be extended to a theory of one complex scalar field. In this case the only admissible symmetry induced by a current can be the gauge transformation. Incidentally we show that a similarity transformation which links two real fields together can be replaced by a unitary transformation. Although the complex field we started with does not necessarily satisfy the superselection rule we may define another irreducible field which has the same asymptotic fields as the former and fulfils the superselection rule requirements.