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: ${\psi }^{(n)}(p)$ at an interior point $p$ and ${\psi }^{(n+m)}(q)$ 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 $\psi (x_1,...,x_N)$ 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.

arXiv comment39 pages, 2 figures

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Cited by in corpus (11)

• Another Proof of Born's Rule on Arbitrary Cauchy Surfaces
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• A Lorentz-Covariant Interacting Electron-Photon System in One Space Dimension
• Hamiltonians for polaron models with subcritical ultraviolet singularities
• Interior-Boundary Conditions for Schrodinger Operators on Codimension-1 Boundaries
• Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions
• Extended State Space for describing renormalized Fock spaces in QFT
• Boundary Conditions that Remove Certain Ultraviolet Divergences
• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition
• Creation Rate of Dirac Particles at a Point Source