Holography and unitarity
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Springer
Received: July 11, 2020 Accepted: September 28, 2020 Published: November 11, 2020
Steven B. Giddings Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A.
E-mail: [email protected] Abstract: If holography is an equivalence between quantum theories, one might expect it to be described by a map that is a bijective isometry between bulk and boundary Hilbert spaces, preserving the hamiltonian and symmetries. Holography has been believed to be a property of gravitational (or string) theories, but not of non-gravitational theories; specifically Marolf has argued that it originates from the gauge symmetries and constraints of gravity. These observations suggest study of the assumed holographic map as a function of the gravitational coupling G. The zero coupling limit gives ordinary quantum field theory, and is therefore not necessarily expected to be holographic. This, and the structure of gravity at non-zero G, raises important questions about the full map. In particular, construction of a holographic map appears to require as input a solution of the nonperturbative analog of the bulk gravitational constraints, that is, the unitary bulk evolution. Moreover, examination of the candidate boundary algebra, including the boundary hamiltonian, reveals commutators that don’t close in the usual fashion expected for a boundary theory. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Models of Quantum Gravity ArXiv ePrint: 2004.07843
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)056
JHEP11(2020)056
Holography and unitarity
Contents 1
2 What is holography? (Formalizing beliefs, and some questions)
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3 Expectations for bulk quantum gravity
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4 What holography isn’t
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5 Holography at nonzero GD ? 5.1 Bulk and boundary hamiltonians 5.2 Dressed operators 5.3 Exploring gravitational holography 5.4 Holography and unitary evolution 5.5 A boundary algebra? 5.6 Entanglement wedge reconstruction
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8 8 9 10 11 12 14
Introduction
Holography, and in particular the AdS/CFT correspondence [1], has become a dominant theme in quantum gravity, but nonetheless its precise formulation and explanation remains controversial. The purpose of this paper is to more carefully examine possible properties of a holographic correspondence, and in particular to investigate the role of gravitational effects. Gravity has been argued to play an essential role in holography [2–6], via gauge invariance and the gravitational constraints. This question can be examined systematically, beginning with scattering of states with weak gravitational fields, in large radius AdS. Specifically, for such states, we expect to be able to study properties of the correspondence in an expansion in small Newton’s constant G. One interesting point of comparison, to infer the possible structure of holography, is the G = 0 limit, which we expect to correspond to ordinary local quantum field theory (QFT) and not be holographic. Then, the structure of gravitational effects can
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