The black hole S-Matrix from quantum mechanics

Abstract

We revisit the old black hole S-Matrix construction and its new partial wave expansion of 't Hooft. Inspired by old ideas from non-critical string theory & c = 1 Matrix Quantum Mechanics, we reformulate the scattering in terms of a quantum mechanical model -of waves scattering off inverted harmonic oscillator potentials -that exactly reproduces the unitary black hole S-Matrix for all spherical harmonics; each partial wave corresponds to an inverted harmonic oscillator with ground state energy that is shifted relative to the s-wave oscillator. Identifying a connection to 2d string theory allows us to show that there is an exponential degeneracy in how a given total initial energy may be distributed among many partial waves of the 4d black hole.

Antipodal correlation on the meron wormhole and a bang-crunch universe

“…As we move away from the center of the throat into the single exterior, the correlation on antipodal points of the three sphere diminishes, rendering the maximal entanglement a local effect near the center of the throat. This is a concrete realisation of the antipodal identification suggested in [22,23]. As in that case, the spectrum is naturally halved owing to the following relation between three-sphere harmonics on antipodal points: Y j 1lm ðΩÞ ¼ ð−1Þ j 1 Y j 1lm ðΩÞ.…”

“…In the r coordinates, it reads r → −r. This is an antipodal mapping when appended with appropriate maps on the three-sphere, similar to the one considered in [22,23] in the context of 't Hooft's black hole S-Matrix.…”

Virtual Black Holes and Space–Time Structure

“…(7.11) acts as a boundary condition, bouncing the in-going wave back as an out-going wave. During the entire evolution, the Hamiltonian is just the dilation operator [23]:…”

The Firewall Transformation for Black Holes and Some of Its Implications

“…It was observed by Betzios et al [34], that this dilaton Hamiltonian in the variables u ± and p ± can be transformed into an apparently more conventional form, being the inverted harmonic oscillator. Rather than ellipses, the classical orbits in this potential are hyperbolas in phase space.…”

“…The dominating parts of tunnelling amplitudes can be derived from classical equations in Euclidean space, which is why classical solutions in Euclidean space, in particular those that obey topologically non-trivial boundary conditions, are often considered with interest in particle physics. The instanton that would be of relevance for the black hole would be one where a region is excavated from a topologically trivial domain of Euclidean space, after which antipodal points on its boundary are identified [34].…”