The relationship between causality, in a classical relativistic field theory, and Lorentz invariance in the corresponding quantum theory, for a massive spin-one field in external potentials

Abstract

A massive spin-one field, interacting with external potentials, is described in terms of the Stuckelberg formalism (1938). Some points, concerning necessary and sufficient conditions for the classical theory to be causal, in the sense of Velo and Zwanziger (1969 and 1971) are clarified. It is then shown that with a simple restriction on the type of interaction considered, the classical theory is causal, in the above sense, if and only if the corresponding quantum theory is Lorentz invariant, in that the S operator in the interaction picture is normal independent. Incidental in this demonstration is the result that the characteristic and Lee-Yang determinants share the same form. It is indicated how this result generalized to interactions between Stuckelberg, Klein-Gordon and electromagnetic fields and the validity of the result, in the more usual vector formalism is established.