Brick walls and AdS/CFT

Abstract

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 brickwall modification of a Klein Gordon field in the Hartle-Hawking-Israel state on 1+2 dimensional Schwarzschild AdS (BTZ) 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.

The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes

“…This was clarified by the works of Rehren [Re00a,Re00b] (who proposed what is now known as Rehren duality, cf. [Ri07a,Ri07b] for generalizations to asymptotically AdS spacetimes) and Dütsch and Rehren [DR02,DR03,DR11], and was further studied by Kay and Larkin [KL08] and Kay and Ortíz [KO14]. We also refer to the recent work of Zahn [Za15] for a holographic prescription with features similar to the field-theoretical AdS/CFT, though in a different setting.…”

Entropy and Quantum Gravity

“…Our results so far [15,16,27] show some promising indications that not only can they indeed be reconciled, but that a clearer understanding of black hole entropy and a clearer resolution of the information loss puzzle emerges once the string theory results are reworked and reinterpreted so as to be compatible with our proposal. Further results [29,30,31] suggest a different from usual interpretation of the AdS/CFT correspondence according to which it is a bijection between the boundary CFT and a subtheory of the bulk theory consisting of just its matter degrees of freedom. To end this essay, we outline the main ideas and results of each of these pieces of work.…”

“…Our argument for the AdS/CFT correspondence being a bijection between the boundary CFT and just the matter degrees of freedom of the bulk theory is that this would naturally fit with the equality of the latter von Neumann entropy of the bulk matter with the von Neumann entropy of the thermal state of the boundary CFT. As a reasonableness check on this, we verified, in [29] in a simple linear scalar field model (with vanishing boundary conditions on the conformal boundary) on 1+1 and 1+2 dimensional analogues to the geometry of the Schwarzschild-AdS right wedge that (when both are suitably regularized by regularizing the bulk entropy with the brick-wall model of [25,26]) the entropy of the boundary CFT according to fixed background holography is the same as the entropy of the bulk scalar field when the latter is in the Hartle-Hawking-Israel state. It remains for us to explain why we believe the horizons of Schwarzschild-AdS to be unsta-ble.…”

“…Our second set of results concerns the AdS/CFT correspondence [33] which is usually thought to be a full equivalence between a quantum gravity theory in the bulk of Anti de Sitter space (AdS) and a conformal field theory (CFT) on the AdS conformal boundary. By considering states of quantum gravity which contain black holes and which are modelled classically by the Schwarzschild-Anti de Sitter (Schwarzschild-AdS) spacetime, and by arguing that it is correct to describe these states as in our above discussed description of black hole equilibrium states in terms of a pure total state, we have argued in [29] and [30] that the AdS/CFT correspondence is, instead, a bijection between the boundary CFT and just the matter degrees of freedom of the bulk AdS quantum gravity theory. As explained in those papers, this seems to offer a resolution to a puzzle raised [1] by Arnsdorf and Smolin: The puzzle arises, if one adopts the usual view of full equivalence, from the fact that Rehren has shown in [34,35] that any CFT on the conformal boundary of AdS is also equivalent, under a natural form of fixed-background holography which he introduced in these papers, to a quantum field theory on the AdS bulk (satisfying vanishing boundary conditions at the conformal boundary and) obeying an appropriate version of commutativity at spacelike separation.…”

“…A key part of our discussion in [29] centers around the question: What becomes of a (non-gravitational) quantum field theory on a fixed Schwarzschild-AdS spacetime background when one switches on gravity? As is well-known, the maximally extended such classical Schwarzschild-AdS spacetime has a quadruple-wedge structure similar to that of the Kruskal spacetime.…”