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An Effective Field Theory of Quantum Mechanical Black Hole Horizons

Walter D. Goldberger, Ira Z. Rothstein

TL;DR

This work develops a worldline effective field theory for black holes with quantum mechanical horizons, valid at distances much larger than the Schwarzschild radius but shorter than the black hole lifetime. By matching bulk propagators and transition amplitudes to the full Schwarzschild background theory, the authors extract a universal set of horizon correlators $A_+(\omega)$ that encode Hawking radiation in the Unruh state and classical absorption in the Boulware state. They show that Hawking radiation is not Planck-suppressed in the Wightman function, but its effects on classical observables arise only at higher orders and through the retarded Green's function, which remains state-independent to leading order. The EFT correctly reproduces the Beckenstein–Wald transition probabilities, exposes IR divergences in forward processes that map to the black hole lifetime, and predicts a consistent next-to-leading-order structure in Unruh state correlators, laying groundwork for extending to gauge, fermion, and graviton fields and to rotating/charged black holes.

Abstract

We develop an effective theory which describes black holes with quantum mechanical horizons that is valid at scales long compared to the Schwarzschild radius but short compared to the lifetime of the black hole. Our formalism allows one to calculate the quantum mechanical effects in scattering processes involving black hole asymptotic states. We point out that the EFT Wightman functions which describe Hawking radiation in the Unruh vacuum are not Planck suppressed and are actually {\it enhanced} relative to those in the Boulware vacuum, for which such radiation is absent. We elaborate on this point showing how the non-Planck suppressed effects of Hawking radiation cancel in classical observables.

An Effective Field Theory of Quantum Mechanical Black Hole Horizons

TL;DR

This work develops a worldline effective field theory for black holes with quantum mechanical horizons, valid at distances much larger than the Schwarzschild radius but shorter than the black hole lifetime. By matching bulk propagators and transition amplitudes to the full Schwarzschild background theory, the authors extract a universal set of horizon correlators that encode Hawking radiation in the Unruh state and classical absorption in the Boulware state. They show that Hawking radiation is not Planck-suppressed in the Wightman function, but its effects on classical observables arise only at higher orders and through the retarded Green's function, which remains state-independent to leading order. The EFT correctly reproduces the Beckenstein–Wald transition probabilities, exposes IR divergences in forward processes that map to the black hole lifetime, and predicts a consistent next-to-leading-order structure in Unruh state correlators, laying groundwork for extending to gauge, fermion, and graviton fields and to rotating/charged black holes.

Abstract

We develop an effective theory which describes black holes with quantum mechanical horizons that is valid at scales long compared to the Schwarzschild radius but short compared to the lifetime of the black hole. Our formalism allows one to calculate the quantum mechanical effects in scattering processes involving black hole asymptotic states. We point out that the EFT Wightman functions which describe Hawking radiation in the Unruh vacuum are not Planck suppressed and are actually {\it enhanced} relative to those in the Boulware vacuum, for which such radiation is absent. We elaborate on this point showing how the non-Planck suppressed effects of Hawking radiation cancel in classical observables.

Paper Structure

This paper contains 13 sections, 79 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Matching the bulk propagators
  3. The full theory
  4. Leading order EFT calculation
  5. Matching Transition Probabilities
  6. The full theory
  7. The EFT Calculation
  8. Forward transition probabilities and IR divergences
  9. Multi-particle probabilities and Gaussianity of EFT correlators
  10. The classical response function
  11. NLO matching in the Unruh state
  12. Conclusions
  13. Acknowledgments

Figures (5)

  • Figure 1: Corrections to the Wightman function $\langle \phi(x) \phi(x')\rangle$ in the EFT. Diagram (a) is leading order. Diagrams (b), (c) are next-to-leading order in the power counting.
  • Figure 2: Vacuum bubble contribution to the persistence amplitude
  • Figure 3: Leading order potential graviton correction to bulk scalar Feynman propagator.
  • Figure 4: Leading order correction to the in-in two-point functions $\langle O_a(\tau) O_b(\tau')\rangle$.
  • Figure 5: The diagram responsible for inclusive photon production induced by the deflection of a black hole due to the interactions with an external current. The soft emission factorizes from the hard scattering off of the source.