Holographic excited states in AdS black holes
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Springer
Received: January 17, 2019 Accepted: March 29, 2019 Published: April 3, 2019
Marcelo Botta-Cantcheff, Pedro J. Mart´ınez and Guillermo A. Silva Instituto de F´ısica de La Plata, CCT La Plata — CONICET, and Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
E-mail: [email protected], [email protected], [email protected] Abstract: We have recently presented a geometry dual to a Schwinger-Keldysh closed time contour, with two equal β/2 length Euclidean sections, which can be thought of as dual to the Thermo Field Dynamics formulation of the boundary CFT. In this work we study non-perturbative holographic excitations of the thermal vacuum by turning on asymptotic Euclidean sources. In the large-N approximation the states are found to be thermal coherent states and we manage to compute its eigenvalues. We pay special attention to the high temperature regime where the manifold is built from pieces of Euclidean and Lorentzian black hole geometries. In this case, the real time segments of the Schwinger-Keldysh contour get connected by an Einstein-Rosen wormhole through the bulk, which we identify as the exterior of a single maximally extended black hole. The Thermal-AdS case is also considered but, the Lorentzian regions become disconnected, its results mostly follows from the zero temperature case. Keywords: AdS-CFT Correspondence, Black Holes, Thermal Field Theory ArXiv ePrint: 1901.00505
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)028
JHEP04(2019)028
Holographic excited states in AdS black holes
Contents 1
2 The SvR approach, excited states, and In-Out formalism 2.1 Brief review of the In-Out formalism: open paths 2.2 Closed paths: the Schwinger-Keldysh contour and TFD 2.2.1 TFD evolution and transition amplitudes 2.2.2 Piecewise holographic map
3 3 6 8 9
3 Excited states from the bulk perspective 3.1 Bulk geometry and gluing conditions 3.2 Scalar field solution 3.2.1 Lorentzian sources 3.2.2 Euclidean sources 3.2.3 Analyticity through the wormhole 3.3 Results from bulk analysis
11 11 12 13 15 16 17
4 Canonical quantization of the bulk fields and BDHM dictionary 4.1 Canonical quantization of scalar fields in a BH geometry 4.2 BDHM at finite temperature, TFD doubling and coherence 4.3 On the Unruh’s trick in the TFD formulation
19 19 20 22
5 Discussion and conclusions
23
A Low temperature excited states: thermal AdS
25
1
Introduction
AdS/CFT [1] is mostly developed in Euclidean time [2, 3]. Conceptually, there is no fundamental principle forcing an Euclidean formulation of the duality. However, a direct approach to real time holography give raise to subtleties [4]. In particular, real time evolution demands initial and final conditions which are not immediate to characterize from both sides of the duality, in conflict with a strict holographic viewpoint. The Skenderis and van Rees (SvR) prescription [5, 6] provides a completely holographic real time extension of the GKPW standar
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