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In 1932, Dirac proposed a formulation in terms of multi-time wave functionsas candidate for relativistic many-particle quantum mechanics. A well-knownconsistency condition that is necessary for existence of solutions stronglyrestricts the possible interaction types between the particles. It wasconjectured by Petrat and Tumulka that interactions described by multiplicationoperators are generally excluded by this condition, and they gave a proof ofthis claim for potentials without spin-coupling. Under smoothness assumptionsof possible solutions we show that there are potentials which are admissible,give an explicit example, however, show that none of them fulfills thephysically desirable Poincar\'e invariance. We conclude that in this senseDirac's multi-time formalism does not allow to model interaction bymultiplication operators, and briefly point out several promising approaches tointeracting models one can instead pursue.

Consistency of multi-time Dirac equations with general interaction potentials