Can theSmatrix be defined in relativistic quantum field theories with nonlocal interaction?

Relativistic field theories with nonlocal interaction are investigated. This is essentially done in terms of the action principle with a generalized variation method previously developed by the author. That a classical theory of the above type has a canonical structure, at least for the perturbative solutions in terms of the asymptotic free fields, was once shown by Pauli. The in and out fields, for example, are equally good choices of canonical variables. A fundamental quantization in the Heisenberg picture is, therefore, Yang-Feldman-Källén quantization method. However, since the interacting fields never can be canonical variables, owing to the nonlocality in time, the classical expressions of the Poincaré generators (properly symmetrized) are no longer conserved and have no longer the right generator properties in the quantum case for any nonlinear model. There exist, however, new generators which are conserved and which have the right generator properties. These generators may be derived from the action principle by means of $q$-number variations instead of $c$-number variations. However, these generators are not uniquely determined. In particular it is shown that any nonlinear model has an in and an out representation of the generators which are different in terms of the interacting fields. They can, therefore, only be equal because of some particular property of the field equations. Explicit calculations are performed for a particular model due to Kristensen and Møller (a nonlocalized Yukawa coupling). For this model it is shown that the in and out representations indeed are different (in the fourth order in the coupling constant) even for the charge-symmetric version of the model. (The properties under the discrete transformations $C$, $P$, and $T$ are also considered.) This implies that the field equations do not yield unique quantum solutions and in particular that the solutions with canonical incoming free fields are different from the solutions with canonical outgoing free fields, and none of these solutions render the total action stationary. No meaningful $S$ matrix can therefore be defined. It is also shown that this deficiency cannot be corrected either by restricting the form function or by adding correction terms to the perturbation expansions.