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

Matthias Lienert

Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany

E-mail Address: matthias.lienert@uni-tuebingen.de

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and

Lukas Nickel

Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstr. 39, 80333 München, Germany

E-mail Address: nickel@math.lmu.de

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https://doi.org/10.1142/S0129055X2050004XCited by:13 (Source: Crossref)

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: $ψ^{(n)} (p)$ $ψ^{(n)} (p)$ 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 $ψ (x_{1}, \dots, x_{N})$ 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.

Keywords:

AMSC: 81Qxx, 81Txx, 35Qxx

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