Abstract
We discuss the relationship between the bulk-boundary correspondence in Rehren’s algebraic holography (and in other ‘fixed-background’, QFT-based, approaches to holography) and in mainstream string-theoretic ‘Maldacena AdS/CFT’. Especially, we contrast the understanding of black-hole entropy from the point of view of QFT in curved spacetime—in the framework of ’t Hooft’s ‘brick wall’ model—with the understanding based on Maldacena AdS/CFT. We show that the brick-wall modification of a Klein–Gordon field in the Hartle–Hawking–Israel state on \(1+2\) dimensional Schwarzschild AdS has a well-defined boundary limit with the same temperature and entropy as the brick-wall-modified bulk theory. One of our main purposes is to point out a close connection, for general AdS/CFT situations, between the puzzle raised by Arnsdorf and Smolin regarding the relationship between Rehren’s algebraic holography and mainstream AdS/CFT and the puzzle embodied in the ‘complementarity principle’ proposed by Mukohyama and Israel in their work on the brick-wall approach to black hole entropy. Working on the assumption that similar results will hold for bulk QFT other than the Klein–Gordon field and for Schwarzschild AdS in other dimensions, and recalling the first author’s proposed resolution to the Mukohyama–Israel puzzle based on his ‘matter–gravity entanglement hypothesis’, we argue that, in Maldacena AdS/CFT, the algebra of the boundary CFT is isomorphic only to a proper subalgebra of the bulk algebra, albeit (at non-zero temperature) the (GNS) Hilbert spaces of bulk and boundary theories are still the ‘same’—the total bulk state being pure, while the boundary state is mixed (thermal). We also argue from the finiteness of its boundary (and hence, on our assumptions, also bulk) entropy at finite temperature, that the Rehren dual of the Maldacena boundary CFT cannot itself be a QFT and must, instead, presumably be something like a string theory.
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Notes
The small Roman numerals in angle-brackets refer to the end section, Sect. 7, entitled ‘Notes’.
References
Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, J., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rept. 323, 183–386 (2000). arXiv:hep-th/9905111
Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [also published in 2000 as, Int. J. Theor. Phys. 38, 1113 (1999)]. arXiv:hep-th/9711200
Rehren, K.H.: Algebraic holography. Ann. Henri Poincaré 1, 607–623 (2000). arXiv:hep-th/9905179
Rehren, K.H.: Local quantum observables in the anti-de-Sitter-conformal QFT correspondence. Phys. Lett. B 493, 383–388 (2000). arXiv:hep-th/9905179
Haag, R.: Local Quantum Physics, 2nd edn. Springer, Berlin (1996)
Kay, B.S., Larkin, P.: Pre-Holography. Phys. Rev. D 77, 121501R (2008). arXiv:0708.1283
Greenberg, O.W.: Generalized free fields and models of local field theory. Ann. Phys. 16, 158–176 (1961)
Duetsch, M., Rehren, K.H.: Generalized free fields and the AdS-CFT correspondence. Ann. Henri Poincaré 4, 613–635 (2003). arXiv:math-ph/0209035
Bertola, M., Bros, J., Moschella, U., Schaeffer, R.: A general construction of conformal field theories from scalar anti-de Sitter quantum field theories. Nucl. Phys. B 587, 619 (2000). arXiv:hep-th/9908140
Arnsdorf, M., Smolin, L.: The Maldacena conjecture and Rehren duality (2001). arXiv:hep-th/0106073
Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998). arXiv:hep-th/9802150
Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105 (1998). arXiv:hep-th/9802109
Rehren, K.H.: QFT lectures on AdS-CFT (2004). arXiv:hep-th/0411086
Duetsch, M., Rehren, K.H.: A comment on the dual field in the AdS-CFT correspondence. Lett. Math. Phys. 62, 171 (2002). arXiv:hep-th/0204123
Buchholz, D., Junglas, P.: Local properties of equilibrium states and the particle spectrum in quantum field theory. Lett. Math. Phys. 11, 51 (1986)
Bañados, M., Henneaux, M., Teitelboim, C., Zanelli, J.: Geometry of the (2+1) black hole. Phys. Rev. D 48, 1506 (1993). arXiv:gr-qc/9302012
Lifschytz, G., Ortiz, M.: Scalar field quantization on the (2+1)-dimensional black hole background. Phys. Rev. D 49, 1929 (1994). arXiv:gr-qc/9310008
Avis, S.J., Isham, C.J., Storey, D.: Quantum field theory in anti-de Sitter space–time. Phys. Rev. D 18, 3565 (1978)
Hartle, J.B., Hawking, S.W.: Path integral derivation of black hole radiance. Phys. Rev. D 13, 2188 (1976)
Israel, W.: Thermo field dynamics of black holes. Phys. Lett. A 57, 107 (1976)
Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate Killing horizon. Phys. Rept. 207, 49–136 (1991)
Kay, B.S.: Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum fields on space-times with horizons. J. Math. Phys. 34(1993), 4519 (1993)
Unruh, W.G.: Notes on black hole evaporation. Phys. Rev. D 14, 870 (1976)
Maldacena, J.M.: Eternal black holes in anti-de Sitter. JHEP 0304, 021 (2003). arXiv:hep-th/0106112
Kay, B.S.: Instability of enclosed horizons (2013). arXiv:1310.7395
Witten, E.: Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math. Phys. 2, 505 (1998). arXiv:hep-th/9803131
Ross, S.F.: Black hole thermodynamics (2005). arXiv:hep-th/0502195
Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752 (1977)
Hawking, S.W.: The path-integral approach to quantum gravity. In: Hawking, S.W., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey. Cambridge University Press, Cambridge (1979)
Hawking, S.W., Page, D.N.: Thermodynamics of black holes in anti-de Sitter space. Commun. Math. Phys. 87, 577 (1983)
Ortíz, L.: Hawking effect in the eternal BTZ black hole: an example of Holography in AdS spacetime. Gen. Relativ. Gravit. 45, 427 (2013). arXiv:1110.4451
Ortíz, L.: Quantum fields on BTZ black holes. Ph.D. Thesis. University of York (2011)
Keski-Vakkuri, E.: Bulk and boundary dynamics in BTZ black holes. Phys. Rev. D 59, 104001 (1999). arXiv:hep-th/9808037
Kay, B.S.: The double wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes. Commun. Math. Phys. 100, 57–81 (1985)
Boulware, D.G.: Quantum field theory in Schwarzschild and Rindler spaces. Phys. Rev. D 11, 1404 (1975)
’t Hooft G.: On the quantum structure of a black hole. Nucl. Phys. B 256, 727 (1985)
Mukohyama, S., Israel, W.: Black holes, brick walls and the Boulware state. Phys. Rev. D 58, 104005 (1998). arXiv:gr-qc/9806012
Winstanley, E.: Renormalized black hole entropy in anti de Sitter space via the ‘brick wall’ method. Phys. Rev. D 63, 084013 (2001). arXiv:hep-th/0011176
Kim, S.-W., Kim, W.T., Park, Y.-J., Shin, H.: Entropy of the BTZ black hole in 2+1 dimensions. Phys. Lett. B 392, 311–318 (1997). arXiv:hep-th/9603043
Kay, B.S.: Entropy defined, entropy increase and decoherence understood, and some black hole puzzles solved (1998). arXiv:hep-th/9802172
Kay, B.S.: Decoherence of macroscopic closed systems within Newtonian quantum gravity. Class. Quantum Gravity 15, L89–L98 (1998). arXiv:hep-th/9810077
Abyaneh, V., Kay, B.S.: The robustness of a many-body decoherence formula of Kay under changes in graininess and shape of the bodies (2007). arXiv:gr-qc/0506039
Kay, B.S., Abyaneh, V.: Expectation values, experimental predictions, events and entropy in quantum gravitationally decohered quantum mechanics (2007). arXiv:0710.0992
Haag, R., Hugenholtz, N.M., Winnink, M.: On the equilibrium states in quantum statistical mechanics. Commun. Math. Phys. 5, 215–236 (1967)
Kay, B.S.: Quantum field theory in curved spacetime. In: J.-P., Naber, G., Tsou, S.T. (eds.) Encyclopedia of Mathematical Physics edited by Françoise, vol. 4, p. 202. Academic [Elsevier], Amsterdam, New York (2006). arXiv:gr-qc/0601008
Fulling, S.A., Ruijsenaars, S.N.M.: Temperature, periodicity and horizons. Phys. Rep. 152, 135–176 (1987)
Kay, B.S.: Purification of KMS states. Helv. Phys. Acta 58, 1030–1040 (1985)
Takahashi, Y., Umezawa, H.: Thermo field dynamics. Collect. Phenom. 2, 55–80 (1975)
Takahashi, Y., Umezawa, H.: Thermo field dynamics. Int. J. Mod. Phys. B10, 1755–1805 (1996)
Larkin, P.: Pre-Holography. Ph.D. Thesis, University of York (2007)
Spradlin, M., Strominger, A.: Vacuum states for AdS(2) black holes. JHEP 9911, 021 (1999). arXiv:hep-th/9904143
Reed, M., Simon, B.: Fourier Analysis, Self-adjointness, Methods of Modern Mathematical Physics, vol. 2. Academic, New York (1975)
Kenmoku, M., Kuwata, M., Shigemoto, K.: Normal modes and no zero mode theorem of scalar fields in BTZ black hole spacetime. Class. Quantum Gravity 25, 145016 (2008). arXiv:0801.2044
Ichinose, I., Satoh, Y.: Entropies of scalar fields on three-dimensional black holes. Nucl. Phys. B 447, 340–370 (1995). arXiv:hep-th/9412144
Hawking, S.W.: Breakdown of predictability in gravitational collapse. Phys. Rev. D 14, 2460–2751 (1976)
Giddings, S.B.: Is string theory a theory of quantum gravity? To appear. In: G ’t Hooft, Verlinde, E., Dieks, D., de Haro, S. (eds.) 2012 Forty Years of String Theory: Reflecting on the Foundations Found. Phys. special issue to appear (2011). arXiv:1105.6359
Susskind, L.: The world as a hologram. J. Math. Phys. 36, 6377 (1995). arXiv:hep-th/9409089
Kay, B.S.: On the origin of thermality (2012). arXiv:1209.5215
Kay, B.S.: Modern foundations for thermodynamics and the stringy limit of black hole equilibria (2012). arXiv:1209.5085
Kay, B.S.: More about the stringy limit of black hole equilibria (2012). arXiv:1209.5110
Park, I.Y.: Fundamental versus solitonic description of D3-branes. Phys. Lett. B468, 213–218 (1999). arXiv:hep-th/9907142
Kay, B.S.: A uniqueness result for quasifree KMS states. Helv. Phys. Acta 58, 1017–1029 (1985)
Dimock, J.: Locality in free string field theory. J. Math. Phys. 41, 40–61 (2000)
Dimock, J.: Locality in free string field theory. Ann. Henri Poincare 3, 613–634 (2002). arXiv:math-ph/0102027
Gubser, S.S., Klebanov, I.R., Peet, A.W.: Entropy and temperature of black 3-branes. Phys. Rev. D54, 3915–3919 (1996). arXiv:hep-th/9602135
Klebanov, I.R.: TASI lectures: Introduction to the AdS/CFT correspondence (2000). arXiv:hep-th/0009139
Ortín, T.: Gravity and Strings. Cambridge University Press, Cambridge (2004)
Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 [Erratum-ibid. 1976 46 206] (1975)
Susskind, L., Uglum, J.: Black hole entropy in canonical quantum gravity and superstring theory. Phys. Rev. D 50, 2700–2711 (1994). arXiv:hep-th/9401070
Barbón, J.L.F., Emparan, R.: On quantum black hole entropy and Newton constant renormalization. Phys. Rev. D 52, 4527–4539 (1995). arXiv:hep-th/9502155
Wald, R.M.: General Relativity. University of Chicago Press, Chicago and London (1984)
Francesco, P.D., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, New York (1997)
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series and products. Academic, New York (1980)
Fewster, C.J., Verch, R.: Dynamical locality and covariance: what makes a physical theory the same in all spacetimes? (2011). arXiv:1106.4785
Fewster, C.J., Verch, R.: Dynamical locality of the free scalar field (2011). arXiv:1109.6732
Louko, J., Marolf, D.: Single exterior black holes and the AdS/CFT conjecture. Phys. Rev. D59, 066002 (1999). arXiv:hep-th/9808081
Yang, D.: A simple proof of monogamy of entanglement. Phys. Lett. A 360, 249–250 (2006). arXiv:quant-ph/0604168
Acknowledgments
LO thanks the Mexican National Council for Science and Technology (CONACYT) for funding his research studentship in York. BSK is grateful to Michael Kay for helpful comments and suggestions.
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Kay, B.S., Ortíz, L. Brick walls and AdS/CFT. Gen Relativ Gravit 46, 1727 (2014). https://doi.org/10.1007/s10714-014-1727-x
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DOI: https://doi.org/10.1007/s10714-014-1727-x