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

Abstract

Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: ψ (n) (p) at an interior point p and ψ (n+m) (q) at a boundary point q, typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac's concept of multi-time wave functions, i.e., wave functions ψ(x 1 , ..., x N ) depending on N space-time coordinates x i for N particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that apart from a constant matrix (which is required to transform in a certain way) the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.

Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off

“…The commutator condition ( 9) is closely related to the integrability condition of the Tomonaga-Schwinger equation, which in turn is closely related to the causality axiom of the Wightman axioms [44, p. 65]. Since interaction potentials violate (9) [35,38], interaction needs to be implemented in a different way, such as through particle creation [25,39,41], zero-range interaction [17,20,22,24], interaction along light cones [23,26,31,32], or other ways [9][10][11]36]. In this paper, we focus on particle creation.…”

Hamiltonians without ultraviolet divergence for quantum field theories

“…analogous to (20) for Model 1. Here, the Dirac delta function δ 3 has been replaced by ϕ : R 3 → C, a square-integrable function describing the charge density of the x-particle, and the limit ϕ → δ 3 would correspond to removing the UV cut-off.…”

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

“…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.…”