Statistical mechanics of a two-dimensional black hole
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Springer
Received: March 5, 2019 Accepted: May 20, 2019 Published: May 29, 2019
Alexei Kitaev and S. Josephine Suh California Institute of Technology, Pasadena, CA 91125, U.S.A.
E-mail: [email protected], [email protected] 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. Keywords: 2D Gravity, Black Holes, Models of Quantum Gravity, AdS-CFT Correspondence ArXiv ePrint: 1808.07032
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)198
JHEP05(2019)198
Statistical mechanics of a two-dimensional black hole
Contents 1 Introduction
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2 Geometry and classical trajectories
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3 Euclidean path integral
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5 Correlation functions of external operators 5.1 Statement of the problem and some results 5.2 Evaluation of Lorentzian correlators
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6 Summary and discussion
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g 2 spinors f A Representation of SL(2, R) by AdS A.1 Continuous series components A.1.1 Basis functions and asymptotic coefficients f R)-invariant two-point functions A.1.2 SL(2, A.1.3 Some special cases A.2 Discrete series components f R)-invariant two-point functions A.3 The algebra of SL(2,
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1
Introduction
Dilaton gravity in 1 + 1 dimensions is free of UV divergences and therefore should allow a fully quantum treatment. A particularly simple model, due to Jackiw [1], Teitelboim [2], and Almheiri and Polchinski [3], is well-studied semiclassically and represents a whole universality class. Its vacuum solution describes an eternal black hole. The spacetime is rigid g 2 . The entire dynamics with constant negative curvature, and thus can be embedded in AdS is associated with two time-like boundaries that are close to the spatial infinities. They may be regarded as particles moving in the anti-de Sitter space, see figure 1. However, the quantization of this system and the construction of a canonical ensemble pose a challenge because the phase volume is infinite. This issue is also pertinent to higher-dimensional black holes and to the early Universe [4]. Completely resolving it in the simplest case might help to make progress in the more realistic settings.
–1–
JHEP05(2019)198
4 Hilbert space and statistical mechanics 4.1 Single-particle wavefunctions 4.2 Two-sided wavefunctions and density matrices f R)-invariant operators 4.2.1 Single-particle Hilbert space and SL(2, 4.2.2 Main results
b)
Figure 1. The Euclidean (a) and Lorentzian (b) geometries in the Jackiw-Teitelboim theory. The physical spacetime (shaded) is embedded in the Poincare disk or the global anti-de Sitter space.
There are in fact several rel
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