How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem

TL;DR

This paper demonstrates that imposing interior-boundary conditions in curved space-time with a naked singularity can rigorously define particle creation processes, effectively resolving ultraviolet divergence issues in quantum Hamiltonians.

Contribution

It extends the interior-boundary condition approach to curved space-time with a naked singularity, proving the existence of well-defined Hamiltonians in this setting.

Findings

Interior-boundary conditions work in curved space-time with singularities.

Existence of self-adjoint Hamiltonians with particle creation established.

Naked singularities can be incorporated without breaking physical laws.

Abstract

Particle creation terms in quantum Hamiltonians are usually ultraviolet divergent and thus mathematically ill defined. A rather novel way of solving this problem is based on imposing so-called interior-boundary conditions on the wave function. Previous papers showed that this approach works in the non-relativistic regime, but particle creation is mostly relevant in the relativistic case after all. In flat relativistic space-time (that is, neglecting gravity), the approach was previously found to work only for certain somewhat artificial cases. Here, as a way of taking gravity into account, we consider curved space-time, specifically the super-critical Reissner-Nordstr\"om space-time, which features a naked timelike singularity. We find that the interior-boundary approach works fully in this setting; in particular, we prove rigorously the existence of well-defined, self-adjoint Hamiltonians with particle creation at the singularity, based on interior-boundary conditions. We also non-rigorously analyze the asymptotic behavior of the Bohmian trajectories and construct the corresponding Bohm-Bell process of particle creation, motion, and annihilation. The upshot is that in quantum physics, a naked space-time singularity need not lead to a breakdown of physical laws, but on the contrary allows for boundary conditions governing what comes out of the singularity and thereby removing the ultraviolet divergence.