Abstract
We consider four-point correlators in an excited quantum state of a field theory. We show that, when the theory and state are holographic, a judiciously applied Fourier transform produces high-quality images of point-like bulk particles, revealing the geometry in which they move. For translation-invariant states, the bulk Einstein’s equations amount to local differential equations on correlator data. In theories or states that are not holographic, images are too blurry to extract a bulk geometry. We verify this for gauge theories at various couplings and the 3D Ising model by adapting formulas from conformal Regge theory.
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References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
P. Arnold and D. Vaman, Jet quenching in hot strongly coupled gauge theories simplified, JHEP 04 (2011) 027 [arXiv:1101.2689] [INSPIRE].
K. Hashimoto, S. Kinoshita and K. Murata, Imaging black holes through the AdS/CFT correspondence, Phys. Rev. D 101 (2020) 066018 [arXiv:1811.12617] [INSPIRE].
K. Hashimoto, S. Kinoshita and K. Murata, Einstein Rings in Holography, Phys. Rev. Lett. 123 (2019) 031602 [arXiv:1906.09113] [INSPIRE].
Y. Kaku, K. Murata and J. Tsujimura, Observing black holes through superconductors, JHEP 09 (2021) 138 [arXiv:2106.00304] [INSPIRE].
J. Polchinski, L. Susskind and N. Toumbas, Negative energy, superluminosity and holography, Phys. Rev. D 60 (1999) 084006 [hep-th/9903228] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
N. Kajuri, Lectures on Bulk Reconstruction, SciPost Phys. Lect. Notes 22 (2021) 1 [arXiv:2003.00587] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
N. Engelhardt and G.T. Horowitz, Towards a Reconstruction of General Bulk Metrics, Class. Quant. Grav. 34 (2017) 015004 [arXiv:1605.01070] [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal Methods in AdS/CFT: BFKL Pomeron at Weak Coupling, JHEP 06 (2008) 048 [arXiv:0801.3002] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (2007) 019 [hep-th/0611122] [INSPIRE].
I. Balitsky, High-energy QCD and Wilson lines, JLAB-THY-00-44 (2001) [DOI] [hep-ph/0101042] [INSPIRE].
Y.V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy, Oxford University Press (2013) [DOI] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Shockwaves from the Operator Product Expansion, JHEP 03 (2019) 201 [arXiv:1709.03597] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, DIAGRAMMAR, NATO Sci. Ser. B 4 (1974) 177 [INSPIRE].
F.M. Haehl, R. Loganayagam, P. Narayan and M. Rangamani, Classification of out-of-time-order correlators, SciPost Phys. 6 (2019) 001 [arXiv:1701.02820] [INSPIRE].
L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].
G. Mack, Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory, Commun. Math. Phys. 53 (1977) 155 [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
S. Caron-Huot, When does the gluon reggeize?, JHEP 05 (2015) 093 [arXiv:1309.6521] [INSPIRE].
S. Caron-Huot, M. Kologlu, P. Kravchuk, D. Meltzer and D. Simmons-Duffin, Detectors in weakly-coupled field theories, CALT-TH 2022-31 (2022) [arXiv:2209.00008] [INSPIRE].
A.H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B 425 (1994) 471 [hep-ph/9403256] [INSPIRE].
I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality: Prescription, Renormalization and Examples, JHEP 05 (2009) 085 [arXiv:0812.2909] [INSPIRE].
P. Glorioso, M. Crossley and H. Liu, A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems, MIT-CTP/5095, EFI-18-21 (2018) [arXiv:1812.08785] [INSPIRE].
B. Chakrabarty, J. Chakravarty, S. Chaudhuri, C. Jana, R. Loganayagam and A. Sivakumar, Nonlinear Langevin dynamics via holography, JHEP 01 (2020) 165 [arXiv:1906.07762] [INSPIRE].
E. Witten, Light Rays, Singularities, and All That, Rev. Mod. Phys. 92 (2020) 045004 [arXiv:1901.03928] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP 07 (2018) 157 [arXiv:1712.04861] [INSPIRE].
M. Kulaxizi, A. Parnachev and A. Zhiboedov, Bulk Phase Shift, CFT Regge Limit and Einstein Gravity, JHEP 06 (2018) 121 [arXiv:1705.02934] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
Y. Hatta, Relating e+ e- annihilation to high energy scattering at weak and strong coupling, JHEP 11 (2008) 057 [arXiv:0810.0889] [INSPIRE].
S. Caron-Huot, Resummation of non-global logarithms and the BFKL equation, JHEP 03 (2018) 036 [arXiv:1501.03754] [INSPIRE].
A.A. Vladimirov, Correspondence between Soft and Rapidity Anomalous Dimensions, Phys. Rev. Lett. 118 (2017) 062001 [arXiv:1610.05791] [INSPIRE].
A.H. Mueller, Conformal spacelike-timelike correspondence in QCD, JHEP 08 (2018) 139 [arXiv:1804.07249] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
S. Caron-Huot and J. Sandor, Conformal Regge Theory at Finite Boost, JHEP 05 (2021) 059 [arXiv:2008.11759] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, AdS bulk locality from sharp CFT bounds, JHEP 11 (2021) 164 [arXiv:2106.10274] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
I. Balitsky and G.A. Chirilli, High-energy amplitudes in N = 4 SYM in the next-to-leading order, Phys. Lett. B 687 (2010) 204 [arXiv:0911.5192] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The Pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].
L.F. Alday and A. Zhiboedov, Conformal Bootstrap With Slightly Broken Higher Spin Symmetry, JHEP 06 (2016) 091 [arXiv:1506.04659] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
N. Su, The Hybrid Bootstrap, arXiv:2202.07607 [INSPIRE].
S. Caron-Huot, Y. Gobeil and Z. Zahraee, The leading trajectory in the 2 + 1D Ising CFT, arXiv:2007.11647 [INSPIRE].
J. Liu, D. Meltzer, D. Poland and D. Simmons-Duffin, The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT, JHEP 09 (2020) 115 [arXiv:2007.07914] [INSPIRE].
J.L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17 (1984) L385 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].
Y. Kusuki, Y. Suzuki, T. Takayanagi and K. Umemoto, Looking at Shadows of Entanglement Wedges, PTEP 2020 (2020) 11B105 [arXiv:1912.08423] [INSPIRE].
N. Jokela and A. Pönni, Towards precision holography, Phys. Rev. D 103 (2021) 026010 [arXiv:2007.00010] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, MIT-CTP/5382 (2021) [arXiv:2112.12156] [INSPIRE].
V. Chandrasekaran, G. Penington and E. Witten, Large N algebras and generalized entropy, arXiv:2209.10454 [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Causality constraints on corrections to Einstein gravity, CALT-TH 2021-003 (2022) [arXiv:2201.06602] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Graviton partial waves and causality in higher dimensions, CALT-TH 2022-17 (2022) [arXiv:2205.01495] [INSPIRE].
S. Weinberg, Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].
X. Liu and H.X. Zhu, The Nucleon Energy Correlators, arXiv:2209.02080 [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic Analysis and Mean Field Theory, JHEP 10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 superYang-Mills, D = 5 gauged supergravity, and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
J. Henriksson and M. Van Loon, Critical O(N) model to order ϵ4 from analytic bootstrap, J. Phys. A 52 (2019) 025401 [arXiv:1801.03512] [INSPIRE].
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Caron-Huot, S. Holographic cameras: an eye for the bulk. J. High Energ. Phys. 2023, 47 (2023). https://doi.org/10.1007/JHEP03(2023)047
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DOI: https://doi.org/10.1007/JHEP03(2023)047