Black Hole's Quantum N-Portrait

TL;DR

The paper introduces the occupation number $N$ of gravitons as a universal measure of classicality in gravity and recasts classical backgrounds as Bose-condensates of soft gravitons. It argues that black holes maximize $N$ for a given size and can be described quantum-mechanically as leaky, self-sustained condensates with $\lambda=\sqrt{N}\,L_P$ and $\alpha_{gr}=1/N$, thereby deriving Hawking thermality and Bekenstein entropy from quantum depletion and micro-state degeneracy. The framework unifies black holes with solitons and baryons under a single large-$N$ paradigm and ties entropy to UV completion via classicalization, suggesting gravity is self-complete non-Wilsonianly. It further generalizes to AdS/dS spaces and highlights connections to holography, while emphasizing the foundational role of energy self-sourcing in driving the UV behavior of gravity. The work provides a compact, operator-based quantum portrait in which semi-classical black hole physics emerges in the $N\to\infty$ limit.

Abstract

We establish a quantum measure of classicality in the form of the occupation number, $N$, of gravitons in a gravitational field. This allows us to view classical background geometries as quantum Bose-condensates with large occupation numbers of soft gravitons. We show that among all possible sources of a given physical length, $N$ is maximized by the black hole and coincides with its entropy. The emerging quantum mechanical picture of a black hole is surprisingly simple and fully parameterized by $N$. The black hole is a leaky bound-state in form of a cold Bose-condensate of $N$ weakly-interacting soft gravitons of wave-length $ \sqrt{N}$ times the Planck length and of quantum interaction strength 1/N. Such a bound-state exists for an arbitrary $N$. This picture provides a simple quantum description of the phenomena of Hawking radiation, Bekenstein entropy as well as of non-Wilsonian UV-self-completion of Einstein gravity. We show that Hawking radiation is nothing but a quantum depletion of the graviton Bose-condensate, which despite the zero temperature of the condensate produces a thermal spectrum of temperature $T \, = \, 1/\sqrt{N}$. The Bekenstein entropy originates from the exponentially growing with $N$ number of quantum states. Finally, our quantum picture allows to understand classicalization of deep-UV gravitational scattering as $2 \rightarrow N$ transition. We point out some fundamental similarities between the black holes and solitons, such as a t'Hooft-Polyakov monopole. Both objects represent Bose-condensates of $N$ soft bosons of wavelength $\sqrt{N}$ and interaction strength 1/N. In short, the semi-classical black hole physics is 1/N-coupled large-$N$ quantum physics.