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

Joscha Henheik, Bipul Poudyal, Roderich Tumulka

TL;DR

This work addresses ultraviolet divergences in quantum particle creation at point sources by introducing interior-boundary conditions (IBCs) and extends the approach to curved space-time with a naked singularity, specifically the super-critical Reissner-Nordström geometry. It proves the existence of a self-adjoint Hamiltonian on a mini-Fock space (0- and 1-particle sectors) that couples creation/annihilation at the singularity via IBCs, thereby providing a rigorous relativistic framework for boundary-driven particle dynamics. The authors also outline and partially construct a Bohm-Bell Markov process describing particle creation/annihilation and analyze the asymptotic behavior of Bohmian trajectories near the singularity, showing that a naked singularity can regulate UV behavior through boundary laws. These results suggest gravity can enable well-defined quantum dynamics with particle exchange at singularities, offering a pathway toward realistic, UV-safe models of quantum fields in curved space-times.

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-Nordström 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.

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

TL;DR

This work addresses ultraviolet divergences in quantum particle creation at point sources by introducing interior-boundary conditions (IBCs) and extends the approach to curved space-time with a naked singularity, specifically the super-critical Reissner-Nordström geometry. It proves the existence of a self-adjoint Hamiltonian on a mini-Fock space (0- and 1-particle sectors) that couples creation/annihilation at the singularity via IBCs, thereby providing a rigorous relativistic framework for boundary-driven particle dynamics. The authors also outline and partially construct a Bohm-Bell Markov process describing particle creation/annihilation and analyze the asymptotic behavior of Bohmian trajectories near the singularity, showing that a naked singularity can regulate UV behavior through boundary laws. These results suggest gravity can enable well-defined quantum dynamics with particle exchange at singularities, offering a pathway toward realistic, UV-safe models of quantum fields in curved space-times.

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-Nordström 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.
Paper Structure (36 sections, 7 theorems, 99 equations, 4 figures, 1 table)

This paper contains 36 sections, 7 theorems, 99 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of Our Main Results
  3. IBC Hamiltonian with Particle Creation
  4. Bohmian Trajectories and Bohm-Bell Jump Process
  5. Structure of the Paper
  6. Motivation, Significance, and Background
  7. Space-Time Singularities
  8. Ultraviolet Divergence
  9. Self-adjoint Extensions on Fock Space
  10. Trajectories
  11. On the Structure of the Model
  12. Dirac Equation in Curved Space-Time
  13. Coordinate-free Form
  14. Expression in Spherical Coordinates
  15. Main Results
  16. ...and 21 more sections

Key Result

Theorem 1

For every choice of the parameters $(\widetilde{m}_j,\widetilde{\kappa}_j)\in\mathscr{A}$ and $g\in\mathbb{C}\setminus\{0\}$, there is a self-adjoint operator $H$ with domain $D\subset\mathscr{H}$ such that

Figures (4)

  • Figure 1: Qualitative depiction of the setup in this paper: A relativistic quantum mechanical spin-1/2 particle of charge $q$ and mass $m$ moves in a curved space representing the gravitational field of a "source particle" with charge $Q$ and mass $M$ (and fixed location, which then is a curvature singularity). The quantum particle can be absorbed and emitted by the source particle. The trajectory shown is a Bohmian trajectory of the quantum particle shortly before absorption or after emission by the source particle.
  • Figure 2: Penrose conformal diagram of sRN space-time $\mathscr{M}$, shown with the spacelike coordinate surface $\Sigma_{t_x}=\{t=t_x\}$ bordering on the point $(t_x,r=0)$ on the singularity $\partial \mathscr{M}=\{r=0\}$ (shown as the vertical double line); the value of $t_x$ was chosen arbitrarily; $\mathscr{I}^{\pm}$ is the future (past) null infinity, $i^0$ is the spacelike infinity, and the shaded region comprises the points spacelike separated from $(t_x,r=0)$.
  • Figure 3: Illustrated is a Bohmian trajectory shortly before/after absorption/emission, asymptotically obeying \ref{['eq:asymp']}; the figure is analogous to HT22 but shows quite different behavior. LEFT: Drawn in spherical coordinates, with only the azimuthal angle $\varphi$ shown; to leading order near $r=0$, $\varphi(r)=\varphi_0 + Cr$ as in \ref{['phi(r)']}; the dot marks $(r=0,\varphi_0)$. MIDDLE: The curve $\varphi(r)=\varphi_0 + Cr$ drawn in 2d cartesian coordinates. RIGHT: The curve $\varphi(r)=\varphi_0+Cr$, $\vartheta=\vartheta_0$ drawn in 3d cartesian coordinates, seen along the $y$-axis. Dashed is the cone $\{\vartheta = \vartheta_0\}$.
  • Figure 4: Graph of the function $R(r)$ defined in \ref{['Rdef']} and given explicitly in \ref{['Rsolution']} for $M=1$ and $Q=2$; in this case, $C \approx -0.601$.

Theorems & Definitions (21)

  • Theorem 1: IBC Hamiltonian with particle creation
  • Remark 1: Boundary conditions for the Dirac equation
  • Remark 2: Comparison to timelike
  • Remark 3: Comparison to HT20
  • Remark 4: Full Fock space
  • Remark 5: Multi-time wave functions
  • Proposition 1: Asymptotics of Bohmian trajectories
  • Remark 6: Negative times
  • Remark 7: Foliation
  • Lemma 1: The $R$-coordinate transformation
  • ...and 11 more