A simple explicitly solvable interacting relativistic N-particle model

TL;DR

This paper extends a relativistic two-particle model to N particles using multi-time wave functions, ensuring Lorentz invariance, antisymmetry, and probability conservation, and explores interactions via delta potentials.

Contribution

It introduces a generalized N-particle relativistic model with explicit solutions, based on multi-time wave functions, and analyzes its properties and potential extensions.

Findings

Model is explicitly solvable and Lorentz invariant.

Interaction is effectively described by delta potentials.

Solution uniqueness follows from probability conservation.

Abstract

In this paper, we generalize a previous relativistic $1+1$-dimensional model for two mass-less Dirac particles with relativistic contact interactions to the $N$-particle case. Our model is based on the notion of a multi-time wave function which, according to Dirac, is the central object in a relativistic multi-particle quantum theory in the Schr\"odinger picture. Consequently, we achieve a manifestly Lorentz invariant formulation on configuration space-time. Our model is constructed to be compatible with antisymmetry and probability conservation in a relativistic sense. On the mathematical side, we further develop the method of multi-time characteristics and show that uniqueness of solutions follows from probability conservation. We prove that the model is interacting and outline how one can understand the interaction as effectively given by a $\delta$-potential at equal times. Finally, we answer the question whether Lorentz invariant and probability-conserving dynamics can also be obtained when the particles are confined in a region with a non-zero minimal space-like distance, a question relevant for an extension to higher dimensions.