Statistical mechanics of a two-dimensional black hole

Alexei Kitaev,S Josephine Suh

doi:10.1007/jhep05(2019)198

Abstract

The dynamics of a nearly-AdS2 spacetime with boundaries is reduced to that of two particles in the anti-de Sitter space. We determine the class of physically meaningful wavefunctions, and prescribe the statistical mechanics of a black hole. We demonstrate how wavefunctions for a two-sided black hole and a regularized notion of trace can be used to construct thermal partition functions, and more generally, arbitrary density matrices. We also obtain correlation functions of external operators.

Highlights

The Sachdev-Ye-Kitaev (SYK) model [5,6,7] is a well-defined quantum system with a finite-dimensional Hilbert space
(The unspecified coefficients of proportionality depend on the normalization of the integration measure.) This result was derived in several ways, in particular, by solving the SYK model in the double-scaling limit [9], by evaluating the Schwarzian path integral exactly [10], and by reducing the problem to Liouville quantum mechanics [11, 12]
In the first line of (5.22), we replace X (x)Y(x ) → Y(x )X (x) using that Φ has space-like support, note the rest of the integrand Φ†E(x ; x)ΦE (x; x ) = −ΦE(x ; x)ΦE (x; x ) is invariant under x ↔ x. These symmetries imply an emergent time-reversal symmetry in our correlators FXν,−Yν(T, 0) and FXν,Yν (T, 0) (the function WX,Y determines the latter via (5.10) which we prove below), in the sense that in a generic quantum mechanical system, analogous correlators (5.13), (5.14) will be invariant under X ↔ Y only if there the Hamiltonian H and operators X, Y are invariant under time-reversal

Summary

Introduction

The Sachdev-Ye-Kitaev (SYK) model [5,6,7] is a well-defined quantum system with a finite-dimensional Hilbert space At low temperatures, it exhibits a collective soft mode with gravity-like behavior, whose effective action involves the Schwarzian derivative, Sch f (x), x. The Schwarzian partition function and density of states are as follows: ˆ∞ ZSch(β) = e−βESch ρSch(ESch) dESch ∝ β−3/2e2π2/β , ρSch(ESch) ∝ sinh 2π (The unspecified coefficients of proportionality depend on the normalization of the integration measure.) This result was derived in several ways, in particular, by solving the SYK model in the double-scaling limit [9], by evaluating the Schwarzian path integral exactly [10], and by reducing the problem to Liouville quantum mechanics [11, 12]. The last method is the most powerful one as it can be used for calculation of matrix elements

Objectives

Results

Conclusion