The spectrum of states of Ba\~nados--Teitelboim--Zanelli black hole formed by a collapsing dust shell

A. A. Andrianov ; D. A. Lyozin ; A. N. Starodubtsev

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 29, Tome 520 (2023), pp. 5-16

Résumé

We perform the canonical analysis of an action $l = 0,\ldots, N$, $ n = -N,\ldots, - l, l,\ldots, N$, in which $2+1$-dimensional gravity with a negative cosmological constant is coupled to a cylindrically symmetric dust shell. The resulting phase space is finite dimensional having geometry of $SO(2,2)$ group manifold. Representing the Poisson brackets by commutators results in the algebra of observables which is a quantum double $D(\mathrm{SL}(2)_q)$. Deformation parameter $q$ is real when the total energy of the system is below the threshold of a black hole formation, and a root of unity when it is above. Inside the black hole the spectra of the shell radius and time operator are discrete and take on a finite set of values. The Hilbert space of the black hole is thus finite-dimensional.