Consistency of multi-time Dirac equations with general interaction potentials

TL;DR

This paper investigates the consistency conditions of multi-time Dirac equations with interaction potentials, showing that physically desirable Poincaré invariant interactions cannot be modeled by multiplication operators, thus challenging certain approaches in relativistic quantum mechanics.

Contribution

It demonstrates the non-existence of physically Poincaré invariant interaction potentials as multiplication operators within Dirac's multi-time formalism under smoothness assumptions.

Findings

Multiplication operator interactions are generally inconsistent with multi-time Dirac equations.

Explicit examples of admissible potentials are constructed, but they lack Poincaré invariance.

Dirac's multi-time formalism cannot model interactions via multiplication operators for physically relevant cases.

Abstract

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