A simple explicitly solvable interacting relativistic N-particle model

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 Schrödinger 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 δ-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.

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

“…It consists of the N electrons as sources whose point-like nature is smeared out by the form factors ρ comprising the ultraviolet cut-off. The two occurrences of ρ in the double convolution ρ * * δ arise like this: In the computation, the source term is introduced by means of the commutation relation (8). The latter features two occurrences of ϕ whereas each ϕ bares one ρ in its definition in (27).…”

“…In recent years, there has been a renewed interest in constructing mathematically rigoros multi-time models, see [5] for an overview. Some of the current efforts to understand Dirac's multi-time models focus on the well-posedness of the corresponding initial value problems [6,7,8,9,10], other works also ask the question how the multi-time formalism could be exploited to avoid the infamous ultraviolet divergence of relativistic QFT and how a varying number of particles by means of creation and annihilation processes can be addressed [11,12,13]. Beside being candidate models for fundamental formulations of relativistic wave mechanics, a better mathematical understanding of such multi-time evolutions may also be beneficial regarding more technical discussions, such as the control of scattering estimates on vacuum expectation values of products of interacting field operators; see e.g.…”

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

“…We shall now work out a suitable local condition which guarantees exactly that. Before coming to the main result of the section, we introduce for every n ∈ N a certain nd-form, the current form ω (n) which is constructed from the tensor currents (see [13,15] and [14, chap. 1.3]).…”

Interacting relativistic quantum dynamics for multi-time wave functions

“…As shown in [10], the extension to N ≥ 3 particles is straightforward. The formulation of a similar model for higher dimensions, however, does not seem feasible, as the set of coincidence points then is too low-dimensional to have impact on the time evolution [9].…”

“…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.…”