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

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.

Interacting relativistic quantum dynamics for multi-time wave functions

“…defined by z 1 − t 1 = c 1 , z 2 − t 2 = c 2 for ψ 1 and constants c 1 , c 2 ∈ R. In analogy to the method of characteristics in PDE theory, we call these surfaces the multi-time characteristics of ψ. They allows for a powerful method to study the existence and uniqueness theory of our model [9]. To see whether ψ i is uniquely determined at a point p ∈ Ω 1 , one considers the multi-time characteristic which includes p. Then one calculates whether it intersects either the boundary set C or the initial data surface t 1 = t 2 = 0 in a single point q.…”

“…By treating dσ 1,μ dσ 2,ν j μν (x 1 , x 2 ) as an 2d-form (d = 1 here), constructing a closed surface including (Σ i × Σ i ) ∩ Ω 1 for two different space-like hypersurfaces Σ i , i = 1, 2 and using Stokes' theorem in its differential geometric form, one can show [9]:…”

“…Therefore, we can express probability conservation on a general space-like hypersurface Σ (and in the domain Ω 1 ) as [9]:…”

“…In [6,9] it was shown that a both physically preferred and mathematically admissible class of boundary conditions is given by (using the basis in spin space corresponding to (11), (12)):…”

“…In this paper, we report on recent progress about a rigorous and manifestly covariant interacting model for two Dirac particles in 1+1 dimensions [9,10]. It is formulated using the multi-time formalism of Dirac, Tomonaga and Schwinger.…”