Problems of Stability for Quantum Fields in External Time-Dependent Potentials

We discuss the algebraic aspect of the time evolution for various types of fields interacting with external potentials. This is a pure $c$-number problem for the classical wave equations. In the case of interaction, the equation for $s>~\frac{3}{2}$ exhibits the well-known consistency troubles and the breakdown of causality studied by Velo and Zwanziger. Other equations, for example, a modified form of the Joos-Weinberg equations, are shown to lead to different troubles. As for the question of a unitary operator implementing the time-evolution automorphism in the Fock space of the in-fields, we encounter a peculiar property: The scalar coupling of $x=0\mathrm{}$ fields (superrenormalizable) leads to the existence of a time-evolution operator, whereas electromagnetic-type couplings (renormalizable) possess at most an $S$ matrix and no evolution operator in the interaction region. This Haag's phenomenon holds for arbitrary smooth and short-range external fields.