A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions

TL;DR

This paper introduces an exactly solvable multi-time model for two massless Dirac particles in 1+1 dimensions, demonstrating Lorentz invariance, probability conservation, and interaction via boundary conditions, advancing relativistic quantum mechanics foundations.

Contribution

It develops a novel multi-time framework with boundary conditions to model relativistic interactions, providing a mathematically rigorous solution method for the first time.

Findings

Model is Lorentz invariant and probability conserving.

Interaction is implemented through boundary conditions at coincidence points.

The method ensures existence and uniqueness of solutions in a non-trivial domain.

Abstract

The question how to Lorentz transform an N-particle wave function naturally leads to the concept of a so-called multi-time wave function, i.e. a map from (space-time)^N to a spin space. This concept was originally proposed by Dirac as the basis of relativistic quantum mechanics. In such a view, interaction potentials are mathematically inconsistent. This fact motivates the search for new mechanisms for relativistic interactions. In this paper, we explore the idea that relativistic interaction can be described by boundary conditions on the set of coincidence points of two particles in space-time. This extends ideas from zero-range physics to a relativistic setting. We illustrate the idea at the simplest model which still possesses essential physical properties like Lorentz invariance and a positive definite density: two-time equations for mass-less Dirac particles in 1+1 dimensions. In order to deal with a spatio-temporally non-trivial domain, a necessity in the multi-time picture, we develop a new method to prove existence and uniqueness of classical solutions: a generalized version of the method of characteristics. Both mathematical and physical considerations are combined to precisely formulate and answer the questions of probability conservation, Lorentz invariance, interaction and antisymmetry.