Hawking Radiation from Tunneling in Black Hole Quantum Mechanics

TL;DR

This paper models black hole decay via Hawking radiation as a tunneling process in a quantum fuzzy sphere framework, reproducing known semi-classical results while maintaining unitarity.

Contribution

It introduces a quantum tunneling model using fuzzy spheres and monopoles to describe black hole decay and Hawking radiation, ensuring unitarity.

Findings

Tunneling rate matches Page's semi-classical decay rate.

Hawking radiation follows a Boltzmann distribution at Hawking temperature.

Full wave function of Hawking quanta can be determined with real-time formulation.

Abstract

It was proposed in \cite{Chu:2024qil} that a quantum black hole can be described by a quantum space configuration of a fuzzy sphere together with a half-filled Fermi sea. In this paper we propose that the tunneling of the fuzzy sphere system to a small one describes the quantum decay of black hole by Hawking radiation. Since the Fermi sea shrinks and the quantum mechanical Hamiltonian conserves fermion number, the amplitude of transition naively vanishes unless the tunneling path provides exact number of zero modes to soak up the excess fermi states. We show that a monopole on fuzzy sphere does exactly that. This fixes the tunneling path. The resulting tunneling rate reproduces Page's result for the semi-classical decay rate of black hole. The quantum states released by the monopole corresponds to gravitational Hawking radiation. At the level of probability, the Hawking radiation is found to be given by a Boltzmann distribution at the Hawking temperature. One can go beyond the probabilistic description by determining the full wave function of the multi-partite Hawking quanta. This is possible with a real time formulation of the tunneling process. Unitarity is manifest in our quantum mechanics.