# Consistency of multi-time Dirac equations with general interaction potentials

## 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 suitable assumptions on the differentiability 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 Poincaré 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.

- Authors:

- Mathematisches Institut, Ludwig-Maximilians-Universität Theresienstr. 39, 80333 München (Germany)

- Publication Date:

- OSTI Identifier:
- 22596651

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIRAC EQUATION; MATHEMATICAL SOLUTIONS; QUANTUM MECHANICS; RELATIVISTIC RANGE; WAVE FUNCTIONS

### Citation Formats

```
Deckert, Dirk-André, E-mail: deckert@math.lmu.de, and Nickel, Lukas, E-mail: nickel@math.lmu.de.
```*Consistency of multi-time Dirac equations with general interaction potentials*. United States: N. p., 2016.
Web. doi:10.1063/1.4954947.

```
Deckert, Dirk-André, E-mail: deckert@math.lmu.de, & Nickel, Lukas, E-mail: nickel@math.lmu.de.
```*Consistency of multi-time Dirac equations with general interaction potentials*. United States. doi:10.1063/1.4954947.

```
Deckert, Dirk-André, E-mail: deckert@math.lmu.de, and Nickel, Lukas, E-mail: nickel@math.lmu.de. Fri .
"Consistency of multi-time Dirac equations with general interaction potentials". United States.
doi:10.1063/1.4954947.
```

```
@article{osti_22596651,
```

title = {Consistency of multi-time Dirac equations with general interaction potentials},

author = {Deckert, Dirk-André, E-mail: deckert@math.lmu.de and Nickel, Lukas, E-mail: nickel@math.lmu.de},

abstractNote = {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 suitable assumptions on the differentiability 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 Poincaré 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.},

doi = {10.1063/1.4954947},

journal = {Journal of Mathematical Physics},

number = 7,

volume = 57,

place = {United States},

year = {Fri Jul 15 00:00:00 EDT 2016},

month = {Fri Jul 15 00:00:00 EDT 2016}

}