On the question of current conservation for the Two-Body Dirac equations of constraint theory

TL;DR

This paper investigates the conservation of current in Two-Body Dirac equations within the multi-time formalism, establishing conditions for relativistic interaction and probability current positivity, with implications for both applications and foundational physics.

Contribution

It provides a comprehensive analysis of current conservation in Two-Body Dirac equations, identifying conditions for positive probability currents and integrating these equations into the multi-time formalism for the first time.

Findings

Conserved tensor currents with positive components can be constructed under certain conditions.

Restrictions on function space or interaction terms can ensure current positivity.

The results have implications for relativistic quantum mechanics and phenomenological modeling.

Abstract

The Two-Body Dirac equations of constraint theory are of special interest not only in view of applications for phenomenological calculations of mesonic spectra but also because they avoid no-go theorems about relativistic interactions. Furthermore, they provide a quantum mechanical description in a manifestly Lorentz invariant way using the concept of a multi-time wave function. In this paper, we place them into the context of the multi-time formalism of Dirac, Tomonaga and Schwinger for the first time. A general physical and mathematical framework is outlined and the mechanism which permits relativistic interaction is identified. The main requirement derived from the general framework is the existence of conserved tensor currents with a positive component which can play the role of a probability density. We analyze this question for a general class of Two-Body Dirac equations thoroughly and comprehensively. While the free Dirac current is not conserved, it is possible to find replacements. Improving on previous research, we achieve definite conclusions whether restrictions of the function space or of the interaction terms can guarantee the positive definiteness of the currents -- and whether such restrictions are physically adequate. The consequences of the results are drawn, with respect to both applied and foundational perspectives.