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

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.

27 pages, no figures; v2: comparison with norm of Crater and Van Alstine and discussion of results on decay rates added

References in corpus (7)

• Can Bohmian mechanics be made relativistic?
• Multi-Time Wave Functions for Quantum Field Theory
• Multi-Time Schrödinger Equations Cannot Contain Interaction Potentials
• Applications of Two-Body Dirac Equations to the Meson Spectrum with Three versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach
• A relativistically interacting exactly solvable multi-time model for two mass-less Dirac particles in 1+1 dimensions
• Multi-Time Equations, Classical and Quantum
• On the description of subsystems in relativistic hypersurface Bohmian mechanics

Cited by in corpus (6)

• Consistency of multi-time Dirac equations with general interaction potentials
• Born's Rule for Arbitrary Cauchy Surfaces
• Multi-Time Wave Functions versus Multiple Timelike Dimensions
• Consistency Proof for Multi-Time Schrodinger Equations with Particle Creation and Ultraviolet Cut-Off
• Fundamental constraints on two-time physics
• Spin Angular Momentum of Proton Spin Puzzle in Complex Octonion Spaces