Holography without holography: How to turn inter-representational into intra-theoretical relations in AdS/CFT
HHolography without holography: How to turninter-representational into intra-theoretical relations inAdS/CFT
Rasmus Jaksland ∗ Niels S. Linnemann † For the published version (open access), please see https://doi.org/10.1016/j.shpsb.2020.04.007 ∗ [email protected], Department of Philosophy and Religious Studies, NTNU Norwegian University ofScience and Technology, NTNU Dragvoll, 7491 Trondheim, Norway † [email protected], Institute of Philosophy, University of Bremen, 28359 Bremen, Germany a r X i v : . [ phy s i c s . h i s t - ph ] A ug bstract We show by means of the AdS/CFT correspondence in the context of quantum gravity howinter-representational relations—loosely speaking relations among different equivalent represen-tations of one and the same physics—can play out as a tool for intra-theoretical developmentsand thus boost theory development in the context of discovery. More precisely, we first showthat, as a duality, the AdS/CFT correspondence cannot in itself testify to the quantum ori-gin of gravity (though it may be utilized for this purpose). We then establish through twocase studies from emergent gravity (Jacobson (2016), Verlinde (2017)) that the holographicAdS/CFT correspondence can, however, still excel as a guiding principle towards the quantumorigin of gravity (similar in nature to quantisation).
Keywords: quantum gravity, dualities, context of discovery, entanglement, holography, guidingprinciples
Contents dual = CFT and the quantum origin of gravity 10 dual = CFT as a guiding principle to quantum gravity 27 Introduction
Ever since its discovery, the AdS/CFT correspondence has intrigued researchers in many areas ofphysics. The correspondence conjectures a duality between D = 10 type IIB superstring theory on AdS × S and D = 4, N = 4 Super Yang-Mills theory defined on a fixed spacetime backgroundconformal to the asymptotic boundary of AdS ; here D denotes spacetime dimensions, and N number of supersymmetries. The AdS/CFT correspondence is an example of a gauge/gravityduality: a duality between a D -dimensional gauge theory and a D +1-dimensional theory of gravity. As such, gauge/gravity dualities and therefore the AdS/CFT correspondence realise a holographicsetting: the physics of the system may be represented both by a theory defined in a volume enclosedby a surface and by another theory defined on the surface enclosing the volume (the physics of thesystem can be seen as being projected from the boundary of the volume).From the perspective of quantum gravity, the AdS/CFT correspondence is enticing: it promisesa relation between a quantum field theory and a theory of gravity beyond semi-classical gravityand effective field theories, i.e. an apparent breakthrough in the attempt to unify quantum degreesof freedom with gravitational ones. In the first part of this paper we argue that the AdS/CFTcorrespondence, qua duality, cannot immediately fulfill this promise to be a general guide to thequantum origin of gravity, by which we mean a general prescription of how classical gravity emergesfrom an underlying quantum level of description. We argue that the only theory of quantum gravity directly advanced by the AdS/CFT correspondence is the type of string theory found on the AdSside; typically type IIB superstring theory. In involving a duality, any entry of the AdS/CFT dictio-nary is a statement about the relation among quantities in two different equivalent representationsof the same underlying bare theory. Thus, when the AdS/CFT correspondence relates gravita-tional degrees of freedom to quantum degrees of freedom on the boundary, this is not implying thatgravity emerges from boundary degrees of freedom, just as the validity of Fourier transformation isnot demonstrating that momentum degrees of freedom emerges from positional ones, or vice versa .What we seek in a theory of quantum gravity is not a re-representation of a theory in new guises(in principle, if you know one representation of the bare theory, you do not learn anything extraabout it through another representation). Rather, what we are after are the underlying degrees offreedom from which general relativity and the standard model are expected to emerge. This is notwhat the AdS/CFT correspondence provides although it does serve to improve our understandingof the theory of quantum gravity we find on the AdS side namely string theory (more on this issection 3.1).The AdS/CFT correspondence, however, can nevertheless be of general utility in the search fortheories of quantum theory: In the second (constructive) part of this paper, we will defend the viewthat the discoveries of the AdS/CFT correspondence—in order to allow for general advances towards AdS stands for Anti-de-Sitter,
CFT for conformal field theory. There are other examples of AdS/CFT correspondences, e.g.
AdS /CF T (involving type IIA superstring theory)(Aharony et al., 2008) and AdS /CF T (involving type IIB superstring theory) (Maldacena, 1999, section 5). Ourfocus here shall be on AdS /CF T , but all said applies equally well to other AdS/CFT correspondences. The samegoes for other examples of gauge/gravity dualities, for instance the one from heterotic string theory (Chen et al.,2013) While the AdS/CFT correspondence is a relation between a four-dimensional and a ten-dimensional theory, it isstill a gauge/gravity duality. This is so since the ten dimensional spacetime includes five compact dimensions—thoseof the S —such that the boundary of the spacetime is the four-dimensional boundary of AdS . See the next section for a precise clarification. Rather, the duality relations of interest have to be changed individually from inter-representational relations into intra-theoretical ones by embedding the dual elements into the samedescription of reality. The ambition for the second part of the paper is to provide a more detailedprocedure for how to achieve this—the methodology we denote ‘holography without holography’—than those already found in the literature.In a nutshell, this methodology amounts to a procedure whereby symmetric duality relationsof the AdS/CFT correspondence serve as guiding principles in the search for the relation betweenunderlying quantum degrees of freedom and gravity without , however, realising the AdS/CFT corre-spondence and holography in particular. The methodology has already been implicitly implementedin works by Jacobson (2016) and Verlinde (2017) who in their respective ways use insights from the(holographic) AdS/CFT correspondence in an exploration of the emergence of gravity within a non-holographic context: The work of Jacobson (2016) suggests taking the CFT degrees of freedom asan inspiration (but nothing more!) for the degrees of freedom from which gravity arises. Relationsbetween these underlying degrees of freedom and gravity that resemble those of the AdS/CFT corre-spondence are then proposed to hold in virtue of thermodynamic effects rather than as holographicduality relations. In this way, what used to be different representations of the same bare theoryare—modulo proper modifications—turned into thermodynamic relations of coarse-graining withinthe same representation. Verlinde’s emergent gravity program (2017) also uses holography as aninspirational tool for an actual intra-theoretical statement from an inter-representational statementbut does not completely leave it to this: In his account of de Sitter space as an excited state ofAnti-de Sitter space (thus the ground state), holography still genuinely features at the ground statelevel.Before we proceed a brief terminological remark is in order. We will refer to relations thatobtain between different dual representations as inter-representational relations. Equalities betweenquantities of different dual representations, i.e. quantities standing in an inter-representationalrelation, will be marked with the superscript ‘dual’: ‘ dual = ’. In contrast, the mere equality sign ‘=’will be used exclusively for statements of equality between properties of the same representation,thus intra-theoretical , as opposed to inter-representational relations. In this terminology, most Following De Haro (2018), dualities are inexact if “they are not instantiated by the models in an exact manner.”(footnote 19) We stick to the standard nomenclature here according to which dualities are relations between theories; that is,each representation is a theory. This way of talking, however, stands in mild terminological tension with thinking ofa supposed common core underlying the two representations as the actual theory at play.
In this section, we give a short introduction to the AdS/CFT correspondence and point out itsholographic nature.Naively, two theories are often said to be dual if they are equivalent with respect to all theirphysically/empirically significant elements. Nevertheless, as for instance argued by De Haro andButterfield (2018), theories can be dual even if they are not about the same physics/the sameempirical content: in fact, two dual theories each of which are seen as embedded within different(external) physical contexts are generally not at all empirically equivalent. So, rather than linkingdualities to a notion of physical or empirical equivalence, dualities should strictly speaking beconceived of as formal relations holding between uninterpreted theories, i.e. theories not linked tothe world yet. We will thus follow the so-called ’Schema’ for dualities as explicated in De Haro(2017), De Haro and Butterfield (2018), and De Haro (2018): two uninterpreted theories (whichwe shall denote ‘representations’) are dual iff they are isomorphic representations of one and thesame common core theory (called bare theory). Each representation consists of a copy of thebare theory and some specific structure. This schema is straightforwardly illustrated through ananalogy with group theory: an abstract group (the bare theory) can be represented by concretegroup representations which can be thought of as a pair containing the structure linked to the baretheory and specific structure linked to its concrete, individual nature as a specific representation.The abstract group SU (2) is for instance represented by various concrete matrix groups. Theseconcrete group representations are of course not generally isomorphic to one another; only if theyare isomorphic do they count as dual. Cf. Rickles (2017, 62): “a pair of theories is said to be dual when they generate the same physics.” De Haro and Butterfield (2018) give the example of the Kramers-Wannier duality between the high and lowtemperature regimes of the statistical mechanics of a lattice. For the present purposes, we will disregard any subtleties arising from the multiple interpretive stances one maytake towards the metaphysics of dualities (see Read (2016) and Le Bihan and Read (2018) for an overview of these). G = ( R , +) that is isomorphic to the positive real numbers with multiplication G = ( R + , · )under the group isomorphism f : G → G , x exp { x } : if x + y = z then exp { x } · exp { y } = exp { z } for all x, y, z ∈ R . They are two different representations of the same abstract group. If onewants to add real numbers or multiply exponentials, then one can use either representation. As anillustration, consider Alice, a stubborn and mathematically gifted kid with a single shortcoming: shestruggles with addition. Consequently, she has simply given up on addition, and instead adopted (1)the exponential sequence exp { } , exp { } , ... for counting, and (2) multiplication for adding them.To make things easier, she has memorized a type of dictionary for moving back and forth betweenher numbers, the exponentials, and the ordinary numbers used by everyone else where the entriesof this dictionary are given by the isomorphic function f ( x ) = exp { x } . With this system, Alice cando what everyone else does with ordinary numbers: she can make sure that she has coin enough tobuy her three chosen pieces of candy, she can follow math classes, she can assess which lake in thepark holds more swans and how many there are in total. All cases where we use real numbers andaddition, Alice can use exponentials and multiplication, irrespective of what the numbers represent!As such, the story of Alice makes vivid the formal character of isomorphisms and thus dualities:they are independent of interpretation; an aspect that will become important in our criticism ofthe role of AdS/CFT correspondence in the search for the quantum origin of gravity. If somesituation can be represented in terms of ( R , +), then it can be represented by ( R + , · ). There isno additional question whether the isomorphism is realised; Alice does not have to check in everycase where real numbers are added, whether she can represent the situation using exponentialsand multiplication. The formal relation, the isomorphism, guarantees that whenever people add(or subtract) real numbers, Alice can multiply exponentials instead. But this also entails thatAlice’s knowledge of the isomorphism is not an insight about that which is counted and added,but rather an insight about the abstract mathematical group instantiated by counting and adding.Generally, isomorphisms do not give us degrees of freedom that were not already there. Knowingtwo representations is helpful if you, like Alice, struggle with one of them, but this does not entailthat one representation can disclose something absent in the other (though different aspects mightbe manifest). This is so even if we want to insist that there are two swans in the lake, and notexp { } as Alice claims. In a sense, only ( R , +) represents when it comes to swans in lakes. Buteven this makes no difference for Alice! Even if ( R + , · ) somehow misrepresent the ontology of thesituation, Alice can still use it to find out how many swans the park’s lakes have in total. Again, thisis so because the isomorphism is a purely formal relation that is unaffected by the interpretation ofeither representation.While such dualities are well known in mathematics, they are more surprising in physics. Oneof the most profound examples known from physics—due to pioneering work by Juan Maldacena(1999)—is the conjectured duality between D = 10 type IIB superstring theory on an asymptot-ically AdS × S background (the AdS side) and D = 4, N = 4 Super Yang-Mills theory (SYM)defined on a background identical to the conformal boundary of AdS —known as the AdS/CFTcorrespondence. As a duality, the AdS/CFT correspondence conjectures an isomorphism be- Thanks to an anonymous reviewer for pressing us here. The AdS/CFT correspondence is strictly speaking just a conjecture, that is a so-far unproven theorem. For anoverview on reasons why to accept the AdS/CFT correspondence nevertheless, see Wallace (2017a, section 5.4) andAmmon and Erdmenger (2015, chapter 6-8). For the Poincar´e patch of AdS spacetime, the conformal boundary is Minkowski spacetime, and for (the universalcovering of) global AdS (in D + 1 dimensions), the conformal boundary is R × S D − . R , +) and ( R + , · ), a dictionary exists that translates between the stringtheory with a dynamical spacetime on the AdS side to the quantum field theory without gravityon the CFT side, however remarkable this may seem. Any expression and operation on the AdSside can be translated to an equivalent expression and operation on the CFT side, and vice versa ,modulo technical complications analogous to Alice’s potential problems relating to the evaluationand multiplication of exponentials.While the AdS/CFT correspondence in its general form involves a duality between string theoryand conformal field theory, in the low energy, weak coupling limit the AdS side can be approxi-mated by type IIB supergravity from which one can derive the Einstein-Hilbert action with negativecosmological constant (together with some additional matter fields following from the limit fromsuperstring theory ). Taking the corresponding limit on the CFT side, one arrives at a dual-ity between strongly coupled N = 4 SYM and semi-classical gravity. Thus, it promises to relatesemi-classical gravity to quantum field theory; semi-classical gravitational degrees of freedom toquantum ones.As its name indicates, the AdS/CFT correspondence is a correspondence and not merely aduality, though it is a duality nonetheless. Being a correspondence, this duality comes with theadditional interpretative commitment that the two sides of the duality are indeed representations ofthe same physics, and not just an abstract structural similarity between two distinct physical sys-tems (see De Haro and Butterfield (2018) for more on this difference). This suggests an immediateutility of the AdS/CFT correspondence in overcoming difficulties in one representation by trans-lating to the other; again Fourier transformations and their use in moving between position andmomentum representation is a good illustration. We take no issue with this use of the AdS/CFTcorrespondence as a transformation. What we want to argue is that this use in itself will not attestto the quantum origin of gravity though it might be of service in the exploration of theories, such astype IIB string theory, that do so. The AdS/CFT correspondence has also been employed to exploreissues of more conceptual nature relevant for quantum gravity research such as the informationparadox (see Harlow (2016) for a review), cosmic censorship and cosmological bounce (Engelhardtand Horowitz, 2016), locality (Hamilton et al., 2006), and the relation between entanglement andgeometry that we will discuss in more detail below. These, however, are still just exploiting theresources of re-representation. They translate from one side of the duality—typically the AdS sidesince it features gravity—to the other where different technical capacities can be used to explore thesame (qua correspondence and not only duality) physics. From this procedure we can, for instance,learn about the quantum nature of AdS black holes. These lessons then promise to generalize toother black holes and thus quantum gravity in general under the additional assumption that theydo not rely on elements peculiar to this holographic setting. This additional assumption is not andcannot be sanctioned by the AdS/CFT correspondence, so even these conceptual insights, in prin-ciple, go beyond the direct import of the AdS/CFT correspondence. In fact, they are one typicalexample of the heuristic use of the AdS/CFT correspondence that by their straightforward use of More precisely, this is the limit where the string length, l s , is much smaller than the characteristic length scaleof the spacetime background and the string coupling, g s , is much smaller than one. One essentially obtains supergravity upon inserting additional fields. On the CFT side, this is the limit where the rank of the gauge group goes to infinity and the ’t Hooft couplingis large but finite. For further details, see Ammon and Erdmenger (2015, chapter 5). Thank you to an anonymous reviewer of this journal for pressing us on this issue. A number of these were raised to our attention by an anonymous reviewer of this journal.
Soon after its discovery, the AdS/CFT correspondence was linked to the notion of holography(Witten, 1998). Holography was first conceived by ’t Hooft (1994) as the conjecture thatgiven any closed surface, we can represent all that happens inside it by degrees of freedomon this surface itself. This, one may argue, suggests that quantum gravity should bedescribed entirely by a topological quantum field theory, in which all physical degreesof freedom can be projected onto the boundary (’t Hooft, 1994) .In other words, a system is holographic if and only if the physics of the system can be representedboth by a theory defined in the volume enclosed by the surface—commonly referred to as thebulk—and by another theory (with different degrees of freedom) defined on a surface enclosing thevolume—commonly referred to as the boundary.Thus, holography as formulated by ’t Hooft—a holographic duality—obtains if a system ad-mits both a D -dimensional and a ( D + 1)-dimensional representation such that the D -dimensionalrepresentation is defined on a background identical to the boundary of the ( D + 1)-dimensionalrepresentation.It may not be immediately clear how holography is realised by the AdS/CFT correspondence.After all, the AdS/CFT correspondence involves a duality between a four-dimensional gauge theoryand a ten-dimensional theory of gravity. However, only five of these dimensions—those of AdS —are extended dimensions and therefore relevant from the point of view of holography. Hence, thebulk degrees of freedom can be encoded by boundary degrees of freedom as required by holography.According to the AdS/CFT correspondence, the five-dimensional theory of gravity has an alternativerepresentation in terms of a four dimensional theory “living” on its boundary. The understanding of the AdS/CFT correspondence has recently advanced with the discovery ofa relation between the entanglement entropies for subsystems on the CFT side and the area ofextremal co-dimension two-surfaces on the AdS side (Ryu and Takayanagi, 2006). To introducethis, some set-up is required. Since this relation proves important in the discussion of Jacobson(2016), we will present this setup more carefully than what might seem at first necessary: Considera CFT state | Ψ i with an AdS dual that features a classical spacetime M Ψ (see the illustrationon figure 1). By holography, | Ψ i is a state in the Hilbert space for a CFT which itself is definedon a spacetime identical to the asymptotic boundary of M Ψ (denote it as ∂M Ψ ). To constructthe Hilbert space of the CFT, define a spatial slice of ∂M Ψ (denote it as Σ ∂M Ψ ). We then have | Ψ i ∈ H Σ ∂M Ψ . Now, divide Σ ∂M Ψ into two regions B and B , such that B ∪ B = Σ ∂M Ψ . We canregard the full quantum system as composed of two subsystems, Q B and Q B , associated with the See also (Susskind, 1995). Σ 𝜕𝑀 𝜓 = 𝐵 ∪ 𝐵 𝐵 𝐵𝑢𝑙𝑘
𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦, 𝜕𝑀 𝜓 𝑡 𝑀 𝜓 𝐵෨ Figure 1: For the purpose of illustration, a spacetime is depicted here with boundary S D − × R .With the AdS metric in the interior of the cylinder, this is a depiction of the universal covering ofglobal AdS spacetime. Figure is taken from Jaksland (2020).two spatially separated regions B and B . As a consequence, the Hilbert space of the full systemcan be decomposed as a tensor product of the Hilbert spaces of Q B and Q B .On the CFT side, we define the entanglement entropy S B = − tr( ρ B log( ρ B )) with the densitymatrix ρ B = tr B ( | Ψ i h Ψ | ). S B is the entanglement (von Neumann) entropy associated with entan-glement between the quantum systems Q B and Q B over an entangling surface that coincides withthe boundary of B , that is ∂B .On the AdS side, we define ˜ B ; the co-dimension two-surface of minimal area whose boundary(“endpoints”) is such that it separates B from B , i.e. ∂ ˜ B ≡ ˜ B | ∂M Ψ = ∂B (see figure 1). We are nowin the position to formulate the mentioned relation between entanglement entropy for subsystemson the CFT side, and area of extremal co-dimension two-surfaces on the AdS side: the so-calledRyu-Takayanagi formula conjectures that S B dual = A ( ˜ B )4 G N (cid:126) (1)where A ( ˜ B ) is the area of ˜ B .The Ryu-Takayanagi formula relates entanglement on the CFT side to a spacetime surfaceproperty—its area—on the AdS side. Thereby, the Ryu-Takayagani formula cashes out the AdS/CFT9orrespondence in a vivid manner by relating entanglement to spacetime. It also seems to shed newlight on the Bekenstein-Hawking formula that relates the horizon area, A BH , to the entropy, S BH ,of stationary black hole. Now, AdS/CFT correspondence or not, the Bekenstein-Hawking formulais still expected to be satisfied in the bulk. So if A ( ˜ B ) = A BH then S BH dual = S B where B is thefull Cauchy surface on which the thermal CFT state is defined, i.e. the complement of B is empty.Again, this is a relation among quantities of different representations. However, it cannot but makeone speculate whether the origin of these degrees of freedom are related. Since the origin of S B is known to be the entanglement entropy of some CFT state—it might be tempting to envisagethat the black hole degrees of freedom simply are these CFT degrees of freedom. This idea is byno means foreign in the physics literature: “one can regard the origin of the black hole entropyas the entropy of the corresponding gauge theory, namely the number of microscopic states of thegauge theory” (Natsuume, 2015, 34-35). It is even often conjectured for general spacetimes in thecontext of the AdS/CFT correspondence: a “spacetime in which gravity operates is emergent fromthe collective dynamics of the quantum fields; the latter by themselves reside on a rigid spacetimesans gravity” (Rangamani and Takayanagi, 2017, 3). So does this not suggest that AdS/CFTcorrespondence might provide a theory for the quantum origin of gravity? It is after all one of thecentral expectations of such a theory that it will feature the degrees of freedom that give rise tothe Bekenstein-Hawking formula. dual = CFT and the quantum origin of gravity
In this section, we argue that the AdS/CFT correspondence qua involving a duality can merely offerre-representations. It can, in other words, not directly show how gravitational degrees of freedomemerge from quantum ones despite its enticing relations between these and the (apparent) hopes tothe contrary. The AdS/CFT correspondence cannot be a direct guide for this central componentof a theory of quantum gravity.Broadly speaking, we take the problem of quantum gravity to consist in improving on the lackof predictivity perturbative quantum gravity suffers from due to its non-renormalizability. Themeans of improvement here standardly include attempting non-perturbative quantisation instead(such as canonical quantum gravity, and canonical loop quantum gravity), taking the renormaliza-tion group flow seriously (asymptotic safety), and reproducing perturbative quantum gravity onlyin some lower energetic limit (such as done by string theory). Following Crowther and Linnemann(2017), we can conceive of the problem of quantum gravity as finding a theory incorporating per-turbative quantum gravity in some way or another which is UV-better, i.e. more predictive athigh energies than perturbative quantum gravity; the problem is not necessarily to arrive at an‘UV-complete’ theory, i.e. a theory that stays formally predictive up to all energies.The theory on the AdS side—superstring theory—is now one of the main contending approaches Describing this as holography already alludes to this ontological prioritising of boundary since in a hologram theinformation to create a three dimensional image is stored in a two dimensional surface. It therefore appears threedimensional under certain circumstances, but it is really two-dimensional. Other examples are Horowitz and Polchinski (2009, 178), Faulkner et al. (2014, 3), and Hubeny (2015, 2). Perturbative quantum gravity can be treated as an effective field theory and thus as effectively renormalizableonly far below the Planck energy scale — at higher energy scales, measurements need to be taken again and againon the coupling constants to make statements about that energy scale in question. but standardly only admits a perturba-tive formulation. The AdS/CFT correspondence, however, promises a non-perturbative, and thusUV-better formulation in terms of the CFT side: the AdS/CFT correspondence “is itself our mostprecise definition of string theory, giving an exact construction of the theory with AdS × S bound-ary conditions” (Horowitz and Polchinski, 2009, 182). Thus, there is a minimal sense in which atheory of quantum gravity can feature through the AdS/CFT correspondence. This type of roleof the AdS/CFT correspondence in the development of string theory is what De Haro (2018) de-notes the theoretical function of dualities. On this view, the dualities are assumed to be exact, butthey can still be employed in “extracting the content of a theory ‘that is somehow already there’,even if only implicitly, using a set of rules” (De Haro, 2018, 10). The isomorphism entailed by theconjecture that the AdS/CFT correspondence is an exact duality can be exploited to better ourunderstanding of one side of the duality using the other; an effect that is only further amplified bythe AdS/CFT correspondence being a weak/strong duality (De Haro, 2018, 18).However, the scope of the AdS/CFT correspondence seems to extend even further since itrelates a theory of gravity to a gauge theory. This appears to open new avenues from the per-spective of quantum gravity for understanding the origin of gravitational degrees of freedom innon-gravitational quantum degrees of freedom; as already suggested by the remarks of Natsu-ume (2015, 34-35) and Rangamani and Takayanagi (2017, 3). One might therefore hope that theAdS/CFT correspondence suggests a more general prescription of how gravity emerges from non-gravitational quantum degrees of freedom. Hubeny (2015, 2) for instance writes in a recent reviewof the AdS/CFT correspondence:This new type of holographic duality not only provided a more complete formulation ofthe theory, but also profoundly altered our view of the nature of spacetime: the gravita-tional degrees of freedom emerge as effective classical fields from highly quantum gaugetheory degrees of freedom. This harks back to earlier expectations motivated by blackhole thermodynamics, that spacetime arises as a coarse-grained effective description ofsome underlying microscopic theory, but with a new twist: the relevant description islower-dimensional.The claim we want to defend in this section is that the AdS/CFT correspondence does not withoutalteration provide such a new prescription of quantum origin of gravity. The AdS/CFT correspon-dence thus only offers itself as a theory of quantum gravity in the minimal sense of one side of theduality being a theory of quantum gravity; for the variant of AdS/CFT correspondence consideredthis is a type IIB string theory on asymptotically
AdS × S , i.e. as potentially providing a “morecomplete formulation” of this type of string theory. The problem lies in that the AdS/CFT correspondence involving a duality, albeit with additionalinterpretive commitments, entails an isomorphism between the AdS side and the CFT side suchthat they are different representations of the same bare theory. That this and other isomorphisms Just as loop quantum gravity for instance must aim at being (at least) a UV-better theory. In particular cases this is a relation between semi-classical gravity and N = 4 SYM, though the more generalformulation of the gravity side is as type IIB string theory. R , +) and ( R + , · ). But for the same reason, it cannot be a deep insight about the way the world isthat Alice can count swans like this; particular expressions about the relation between Alice’s andordinary peoples’ ways of counting—such as 4 swans dual = exp { } swans—are (conjectured) analytictruths. This also applies to the duality relations coming out the AdS/CFT correspondence: Theyare not candidates for deep insights about the way the world is; though they might be very profoundinsights about the relations among different representations of reality.In particular, in a world where the outcome of any experiment can be described by the AdSside there is also always (in principle) an alternative account of these experiments in terms of theCFT side and vice versa , assuming their conjectured duality; though this alternative representationmight not be known. No additional or external assumptions about the realisation of the AdS/CFTcorrespondence go into this, since the duality aspect of their relation is a purely formal relation.As such, there is not an additional fact about the realisation of the duality aspect of the AdS/CFTcorrespondence that might or might not obtain in the world beyond the question whether the CFTor AdS description obtains (as was the case for the isomorphism between ( R , +) and ( R + , · )). Thesame goes in general for any gauge/gravity duality.A similar point is made by Dean Rickles (2013, 317), who observesthat the duality relation is formally symmetric, so it would apparently make just asmuch sense to say that the gauge theory emerges from gravitational theory as the otherway around. But rather than expressing this is terms of the symmetry between the two sides of the duality, we willemphasise that the absence of direct import from the AdS/CFT correspondence for the quantumorigin of gravity derives from its inter-representational nature. This becomes clear through contrast-ing an (inter-representational) area law from the AdS/CFT correspondence—the Ryu-Takayanagiformula presented in the previous section—to the Bekenstein-Hawking formula (Bekenstein, 1973)(which is an intra-theoretical relation). The Bekenstein-Hawking formula relates the horizon area, A BH , of a stationary black hole to its entropy, S BH : S BH = A BH G N (cid:126) (2)Comparing to the Ryu-Takayangi formula (see equation 1), the similarity is striking. Both relateentropies to areas in terms of the same proportionality constant. But while the Bekenstein-Hawkingformula applies only to certain horizon structures (in particular that of black holes), the Ryu-Takayanagi formula is a generic relation between areas on the AdS side and (entanglement) entropieson the CFT side. Still, it might be alluring to conceive of the Ryu-Takayanagi formula as ageneralisation of the Bekenstein-Hawking formula. The Ryu-Takayanagi formula might even be Assuming that relations among mathematical representations are analytic. We will leave the precise terminolog-ical issue to the philosopher of mathematics. Being a weak/strong coupling, only one side of the duality is actually tractable in most cases. This is only furtheramplified by the absense of a non-perturbative formulation of the AdS side. Similar observations are made by Teh (2013) and de Haro (2017). all accounts locate the degrees of freedom associated with the entropy in the samespacetime as that of the black hole horizon. The Bekenstein-Hawking formula, in other words,states a relation among quantities in the same representation in which the black hole finds itself,and is thus an intra-theoretical relation. As such, it signifies an intriguing discovery in the context ofblack holes about the relation between a thermodynamic property related to the number of degreesof freedom, entropy, and spacetime, in the form of the horizon area.This is in stark contrast to the Ryu-Takayanagi formula. Here the entropy is associated withsome CFT degrees of freedom. But in the CFT representation where these degrees of freedomlive, the co-dimension two-surface, ˜ B , whose area they relate to is nowhere to be found. Instead,this is a surface in the alternative representation of the same bare theory in terms of the AdS side.Whereas the Bekenstein-Hawking formula is the surprising discovery of a relation between verydistinct quantities in the same representation (an intra-theoretical relation), the Ryu-Takayanagiformula relates two different representations of the same bare theory to one another (an inter-representational relations). The Ryu-Takayanagi formula is not a relation between the entropy ofa volume and the surface area of this volume but rather a relation between the entropy in onerepresentation and a surface in another. It is the equivalent of discovering that you can count andadd both like your 9-year old alter ego and like mathematically gifted Alice—using an exponentialsequence, and multiplication instead! This might be (very) useful, but it is not in itself a profoundinsight about reality. And this generalises to all other relations coming out of the AdS/CFT corre-spondence; duality relations cannot directly attest to the origin of back hole degrees of freedom—asalluded to by Natsuume (2015, 34-35) above—nor the origin of any other gravitational degrees offreedom. They simply attest to the discovery that the very same bare theory—and the same physicsunder the interpretation as a correspondence—has two alternative representations, and how theserepresentations relate to each other.To drive our point home: Any relation coming out of the AdS/CFT correspondence is—modulosignificant differences in complexity and transparency—analogous to the truism ‘all bachelors areunmarried men’. The relation between bachelors and unmarried men holds irrespective of the waythe world is, but this also means that it is not an insight about reality: while this relation is helpfulto anyone not knowing what a bachelor is it cannot directly do more than lifting such kind of Cf. for instanceit seems far from clear as to whether we should think of these degrees of freedom as residing outsideof the black hole (e.g., in the thermal atmosphere), on the horizon (e.g., in Chern-Simons states), orinside the black hole (e.g., in degrees of freedom associated with what classically corresponds to thesingularity deep within the black hole) (Wald, 2001, 31).It is beyond the scope of this paper to go into the details of these different accounts of black hole entropy. We referthe reader to Wald (2001) and Carlip (2014); and for a more philosophical discussion on the status of black holethermodynamics as thermodynamics proper to Dougherty and Callender (2016) and Wallace (2017b). It happens to be entanglement entropy but this is not the important contrast with the Bekenstein-Hawkingformula. If we established in some way or another that there are bachelors, then there is noadditional question as to whether all these bachelors are unmarried men. Similarly, it is nonsense toask whether the existence of unmarried men causes the existence of bachelors or whether bachelorsemerge as a consequence of unmarried men. Unmarried men are neither more nor less real orfundamental than bachelors. It is a relation among representations like an exact duality.So, while the AdS/CFT correspondence can serve to improve our understanding of either side ofthe duality using the other (for purposes of quantum gravity most likely using the CFT side to learnabout the string theory on the AdS side), nothing else can be achieved directly by the AdS/CFTcorrespondence qua duality; particularly not concerning the emergence of gravity from quantumdegrees of freedom in any other theory. Rather, it must (and, as we will show, does) result fromhighly non-trivial heuristics if the AdS/CFT correspondence—as Hubeny (2015, 2) claims—showshow “spacetime arises” from a “lower dimensional” quantum theory.
To activate the discoveries of the AdS/CFT correspondence for general advances in the search for thequantum origin of gravity, one would have to seek means for embedding its inter-representationalrelations into one and the same theory, that is to promote inter-representational relations intointra-theoretical ones. As alluded to above, this goes beyond rendering the duality in-exact; themethodology of holography without holography, developed below, does this by keeping aspectsof the formal framework of the AdS/CFT correspondence while leaving holography, and thus theinter-representational relations, behind. Holography without holography follows the spirit of theheuristic function of dualities as explicated by De Haro (2018) where dualities are used to buildnew theories.The methodology of holography without holography amounts to a particularization of thisheuristic function of dualities where the symmetric duality relations of the AdS/CFT correspon-dence serve as guiding principles in the search for the relation between underlying quantum degreesof freedom and gravity without , however, realising the AdS/CFT correspondence. We demonstratein this section how the works of Jacobson and Verlinde respectively manage to do this and therebyimplement holography without holography, though only implicitly so. These case studies then nicelyillustrate the point already made above that this heuristic use of the AdS/CFT correspondence can-not be achieved by simply stipulating that the relations of the AdS/CFT correspondence are notexact. As, for instance, Jacobson’s implementation of the Ryu-Takayanagi formula demonstrates,the problem is not that the relation—and the duality as a whole—is exact (and thus symmetric), butrather how to embed the entropy and the area in the same theory. The methodology of holographywithout holography provides for a procedure towards this end. Having only a perturbative formulation of the AdS side is analogous to not knowing well what a bachelor is;each of these cases of ignorance can be shed light on by exploring the arguably easier-to-deal-with CFT side andconceptualization in terms of unmarried men, respectively. It is still a profound insight but one about the surprising relation between two theoretical frameworks that werehitherto conceived to be distinct. .1 Jacobson 2016: Entanglement Equilibrium and the Einstein fieldequation Ted Jacobson (2016) promises thatThe present work combines the local spacetime setting of the equation of state approach[as in (Jacobson, 1995)], with the statistical, compact-region setting of the holographicanalysis [as in Faulkner et al. (2014)], but it proceeds directly in spacetime, making nouse of holography (Jacobson, 2016, 2).The mentioned “holographic analysis” is the derivation of linearised Einstein field equations fromthe first law of entanglement entropy in the context of the AdS/CFT correspondence by van Raams-donk and collaborators (Van Raamsdonk, 2010; Lashkari et al., 2014; Faulkner et al., 2014)). Thementioned “equation of state approach” is Jacobson’s 1995 own thermodynamic take on gravityin which the Einstein field equations are derived from a thermodynamic-like equation of state.Motivated by these two programs, Jacobson manages to derive the Einstein field equations froman entanglement equilibrium condition. In the following, we will first depict the derivation of vanRaamsdonk and collaborators in the holographic setting, and then show how Jacobson manages toinvoke the spirit of this derivation in a completely non-holographic setting.
In this section, we review the equivalence of constraints on the entanglement entropy on the CFTside to the validity of the linearised Einstein field equations on the AdS side as considered byLashkari et al. (2014); Faulkner et al. (2014); Swingle and Van Raamsdonk (2014). We in particularemphasise those aspects of the derivation of this equality that are paralleled by Jacobson’s non-holographic derivation of the field equations.In any QFT, entanglement entropy features in a first-law-like expression (‘first law of entangle-ment entropy’): δS B = δ h H B i (3)where B is some (sub)region of the domain of the QFT, δS B is the first order variation of the entan-glement entropy of this region away from the vacuum state of the QFT , and δ h H B i is the first ordervariation of the expectation value of the modular Hamiltonian H B ≡ − log (cid:0) ρ V acB (cid:1) ( ρ V acB denotesthe vacuum state linked to region B ). The modular Hamiltonian is the Hamiltonian with respectto which the state ρ V acB can be expressed as a thermal state, that is ρ V acB = exp( − H B ). Though thefirst law of entanglement entropy looks like the Clausius relation for thermal systems (‘ T dS = dE ’),it is a general result for any QFT—in particular, it does not presuppose an equilibrium(-like state).By the Ryu-Takayanagi formula, δS B can be related to the variation of the area of the co-dimension two surface ˜ B in the dual representation (see figure 1) such that δA ( ˜ B )4 G N (cid:126) dual = δ h H B i . (4)15his relation, however, is in general not of much use since the modular Hamiltonian, H B , is generallynot a local operator. But if B is a ball-shaped region in Minkowski spacetime, then thereexists a conformal mapping between the state ρ V acB and the corresponding state ρ η as seen by aconstantly accelerating observer (called the ‘Rindler observer’) (Casini et al., 2011). In a conformalfield theory, the operators are invariant under conformal transformation, and we may therefore usethis mapping to express the modular Hamiltonian in terms of the boost Hamiltonian associatedwith the accelerating observer. In doing so, we find a local expression for h H B i as an integralof the energy-momentum tensor over the ball-shaped region B , which we may identify as somehyperbolic energy, E HypB , of the quantum state on B . But to re-emphasise, this expression of themodular Hamiltonian as a local hyperbolic energy only holds in a conformal field theory. Insertingthis into eq. (4), we obtain a relation between an area variation on the AdS side and an energyvariation on the CFT side: δA ( ˜ B )4 G N (cid:126) dual = δE HypB . (7)It is a general result of the AdS/CFT correspondence that energies of the CFT state correspondto energies of the dual spacetime (Balasubramanian and Kraus, 1999). We can therefore obtainan interpretation of the CFT energy E HypB on the AdS side, and thus relate the AdS area variationto an AdS energy variation. In a final step, this can then be shown (as done by Lashkari et al.(2014) and Faulkner et al. (2014)) to be equivalent to the linearised vacuum Einstein field equationswith negative cosmological constant (that are solved by AdS spacetime). Moreover, adding firstorder corrections to Ryu-Takanagi formula due to bulk entanglement over ˜ B will give the linearised,semi-classical Einstein field equations with negative cosmological constant; it is speculated that theinclusion of all orders in δS and δE HypB will give the full semi-classical Einstein field equations(Swingle and Van Raamsdonk, 2014).In sum, starting from the formal result δS B = δ h H B i , the left hand side, δS B , receives itsholographic interpretation as an area on the AdS side using the Ryu-Takayanagi formula. Theright hand side, h H B i , is first recognised as a local hyperbolic energy using a conformal mappingand then translated to the AdS using the general relation between CFT and AdS energy. The A local operator is an operator that can be expressed in terms of an integral of the quantum fields and theirderivatives. More precisely, the ball-shaped region is given by what is usually called a (spacelike) geodesic ball of radius l with center p . See for instance Jacobson (2016, appendix A). Also known as the Rindler Hamiltonian. For completeness the full expression is: h H B i = 2 π Z B d D − x R − ( ~x − ~x ) R (cid:10) T tt ( x ) (cid:11) ≡ E HypB (5)where T tt ( x ) is the energy density of the CFT and R is the radius of the ball shaped region, B . For later purposes, observe that for an infinitesimally small ball-shaped region δ (cid:10) T tt ( x ) (cid:11) may be approximatedby a constant throughout the ball which allows the evaluation of the integral (5) for the first order variation: δ (cid:10) H InfB (cid:11) = 2 π Ω D − R D D − δ (cid:10) T tt ( x ) (cid:11) (6)where H InfB is the modular Hamiltonian of the infinitesimal ball and Ω D − is the surface area of a sphere of dimension D − Here the energy of a spacetime is interpreted as some quasi-local energy momentum tensor, since any localoperator depending only on the metric and its first order derivatives must vanish in a generally covariant theory.
The outset for Jacobson’s derivation of Einstein’s field equation is a quantum state | Ψ i defined ona maximally symmetric spacetime, ( M , g µν ). One then considers the domain of dependence—the causal diamond—of a ball shaped region, B , (‘geodesic ball’) within this spacetime. Thequantum degrees of freedom within the ball will generally be entangled with the degrees of freedomoutside the ball. Thus, the ball is associated with a non-zero entanglement entropy, S tot B , i.e. anentanglement across the boundary surface of B , ∂ B .In the holographic version, the first step of the derivation involves finding a relation betweenan energy and an entanglement entropy (component) that in turn can be related to the area ofa co-dimension two surface of the bulk spacetime. The first step in Jacobson’s derivation lies inestablishing an analogous relation while however sticking to the theory defined in ( M , g µν ), thatis without using a holographic relation. This entails that the featured co-dimension two-surfacecannot go into some kind of bulk (relative to which the theory ( M, g ) ‘sits’ on the boundary) butmust be identified with a co-dimension two-surface in M itself. And similarly, the entanglemententropy cannot be associated with that of a different representation (living on the boundary) ofthe system. In contrast to the holographic version, this setting is intra-theoretical rather thaninter-representational; the inter-representational derivation is only used as a guiding principle inthe generation of an analogous intra-theoretical derivation that does not assume holography. Thisis the methodology of holography without holography and we shall return to this role of it as aguiding principle in section 5.In the non-holographic setting, the most obvious candidate for a co-dimension two-surface isthe boundary of B , ∂ B . Indeed, it is a generic result of quantum many body systems that theentanglement over the boundary of a subsystem (such as ∂ B ) scales (to leading order) with the areaof the boundary (in this case, the area of ∂ B ) (Eisert et al., 2010). This leading order behavior isprimarily due to vacuum fluctuations in the near vicinity of ∂ B governed by the UV physics andnot the state of the subsystem. We can therefore conceive of this as a UV contribution to the totalentanglement entropy that scales with the area of the boundary and is independent (to leadingorder) of the state of the subsystem: S UV B = ηA ( ∂ B ), where η is some proportionality constant.Any other contributions to the total entanglement across ∂ B is due to IR (long range) physics andis determined by the state of the subsystem. This is, in other words, entanglement across ∂ B thatis not originating in the entanglement between degrees of freedom immediately in the vicinity of ∂ B . Thus, the total entanglement entropy relative to the vacuum between the quantum subsystemon B and the rest of the system can be expressed as: S tot B = S UV B + S IR B = ηA ( ∂ B ) + S IR B . (8)Consider a simultaneous variation of the metric and the quantum state ( δg µν , δ | Ψ i ) away from the The maximally symmetric spacetimes are Minkowski spacetime and (Anti-)de-Sitter spacetime. Whereas the holographic version considered a ball-shaped region on the boundary, denoted B , this non-holographic version considers a ball-shaped region in the bulk, which we denote B to avoid confusion. If η is a constant (as is assumed), and if it holds that S UV B = ηA ( ∂ B ) then thevariation of S UV B only depends on δg µν since A ( ∂ B ) is independent of | Ψ i to leading order. Thevariation of S IR B , on the other hand, depends only on δ | Ψ i since it is, to leading order, independentof the exact location of the boundary. We therefore get: δS tot B = ηδA ( ∂ B ) + δS IR B . (9)As stated above, it is a general result of any QFT that the variation of the entanglemententropy can be related to the variation of the expectation value of the modular Hamiltonian whenthe variation is taken with respect to the vacuum state. However, since S UV B is independent of avariation of the state, it follows that only the variation of the IR component of the entanglemententropy contributes to the variation of the modular Hamiltonian, i.e. δS IR B = h H B i . We thereforehave: δS tot B = ηδA ( ∂ B ) + δ h H B i . (10)Finally, Jacobson assumes what he calls the maximal vacuum entanglement hypothesis :When the geometry and quantum fields are simultaneously varied from maximal sym-metry, the entanglement entropy in a small geodesic ball is maximal at fixed volume(Jacobson, 2016, 1).The first order variation of entanglement entropy away from the vacuum state vanishes for fixedvolume, i.e. δS tot B = 0 when the variation of the geometry and quantum state is such that thevolume of the subsystem is unchanged. From this, we obtain: 0 = ηδA ( ∂ B ) + δ h H B i closelyresembling eq. (4) from the holographic version. However, this time the area variation relates toa modular Hamiltonian in the same representation: this is an intra-theoretical rather than inter-representational relation.The issue, however, is the same: H B is generally not a local operator. Had this been a conformalfield theory, we could directly proceed by using the conformal mapping of this state from a ballshaped region to the state as seen from the Rindler observer system and express h H B i in terms of theRindler Hamiltonian. As in the holographic version above, we could thereby find a local expressionfor h H B i as an integral of the energy-momentum tensor over the ball-shaped region B , which we mayidentify as some hyperbolic energy, E Hyp B , of the quantum state on B . Although we cannot generallyexpect to have conformal invariance, Jacobson (2016, 4) speculates that the matter considered hereis suitably described by a theory with a UV fixed point, and, furthermore, that the energy regimesof interest are sufficiently close to that of the fixed point such that it can in fact be treated asapproximately conformal for small length scales compared to the characteristic length scale of the Notice that we do two simultaneous variations here, whereas in holographic version we only varied the CFT statewhich entailed a consequent variation of the AdS spacetime metric. To leading order, we can disregard backreaction from the matter fields on the metric. For fixed volume, a thermodynamic system at equilibrium has minimal free energy, that is ∂F = ∂E − T ∂S = 0.Since the energy of a maximally symmetric spacetime is zero, minimisation of free energy corresponds to maximisationof entropy. Note that the variation of area is at fixed volume as opposed to fixed radius. Comparing to the holographicanalysis of Van Raamsdonk and others (see the previous section), this is a special case where the state of the systemis assumed to be such that the energy of the ball vanishes (as the variation happens around a maximally symmetricspacetime for which in fact E = 0). However, the derivation still runs in parallel. Jacobson therefore assumes that the modular Hamiltonian canbe expressed in terms of a hyperbolic energy of B , E Hyp B , with some spacetime scalar correction, X , that must be assumed to be small. The IR entanglement entropy can therefore be expressedas: δS IR B = δ h H B i = δE Hyp B + δX . Thus, assuming that the fundamental quantum degrees offreedom are sufficiently similar to CFT degrees of freedom, we can obtain a relation similar up toa sign and small scalar to that obtained in the successful derivations of Einstein field equations inthe AdS/CFT correspondence: 0 = ηδA ( ∂ B ) + δE Hyp B + δX. (12)From this constraint, Jacobson then derives the full Einstein field equations by a proceduresimilar to that of Faulkner et al. (2014). Jacobson can derive the full Einstein equation and notonly the linearised equation due to differences in the expressions of the area variation. In building on the Ryu-Takayanagi formula and thus holography, the derivation of van Raams-donk and collaborators merely show that the Einstein equations on the AdS side are equivalent to anentanglement constraint on the CFT side. Despite its promise to relate gravity and entanglement,the derivation is simply a formal result within the AdS/CFT framework; it signifies how to translatethe Einstein field equations when moving from the gravity representation on the AdS side to the non-gravitational representation on the CFT side. Jacobson reproduces—with some deviations—thisderivation, but replaces any components coming from the AdS/CFT correspondence with plausiblenon-holographic assumptions. First, the inter-representational Ryu-Takayanagi formula—needed tohave a relation to spacetime in the first place—is replaced by a conjectured leading order propor-tionality between the area and entanglement entropy of the same ball; an intra-theoretical relation.Second, since it is only part of the entanglement that is related to area, Jacobson has to help himselfto the additional assumption that the total entanglement entropy vanishes, but together these twoassumption reproduce, up to a sign, eq. (4) from the holographic version in a non-holographic set-ting. Third, assuming that the microscopic degrees of freedom are close to conformal—a propertythat followed directly in the holographic version, the CFT being conformally invariant—Jacobsonreproduces, up to a sign and a small scalar, eq. (7) from which the Einstein field equations can bederived. Whereas the holographic version shows that the Einstein field equations on the AdS sideare equivalent to an entanglement constraint on the CFT side, an inter-representational relation,Jacobson promotes this into a relation between the microscopic quantum degrees of freedom andan emergent leading order gravitational dynamics that accords with GR. More precisely, the (ap-parent) gravitational dynamics is collective behavior resulting from entanglement thermodynamics See Cao and Carroll (2018) for another non-holographic derivation of the Einstein field equations that is in-spired by the holographic derivation of section 4.1.1 but which trades the requirement of a UV-fixed point for otherassumptions. As it turns out, X is related to the curvature scale of the spacetime such that the derivation as a whole is onlyvalid if X vanishes or is small everywhere compared to E Hyp B which signifies the importance of the assumption thatthe degrees of freedom are close to conformal. For completeness, we have: δS IR B = δ h H B i = 2 π (cid:126) Ω D − R D D − δ h T tt ( x ) i + δX ) ≡ E Hyp B (11)Compare with eq. (6). In essence, Jacobson can help himself to more assumptions since he is not constrained by the framework of theAdS/CFT correspondence. B of the boundary, ∂M Ψ quantum system defined on a re-gion B in the bulkBalance equation δS B = δ h H B i δS tot B = δS UV B + δ h H B i = 0Area law S B dual = A ( ˜ B )4 G N (cid:126) S UV B = ηA ( ∂ B )Table 1: Comparison of basic elements in the derivation of the Einstein field equations in theholographic (van Raamsdonk) and non-holographic (Jacobson) version.in the form of the hypothesis that the vacuum entanglement is maximal ( δS tot B = 0), i.e. at an equi-librium. Indeed, this result that a small ball—a small local system—should be at an entanglementequilibrium after some time is supported by quantum thermalisation in generic many-body systems(Kaufman et al., 2016). The duality between the Einstein field equations and an entanglementconstraint derived in the holographic version informs a thermodynamic coarse-grained relation inthe non-holographic setting where entanglement equilibrium, assumed UV-completion, and leadingorder area scaling of the entanglement entropy replace the inter-representational relations of theholographic version. Erik Verlinde (2017) also explicitly draws on the relations between spacetime and entanglementcoming out of the AdS/CFT correspondence. His ambition is, in accordance with the methodologyof holography without holography, to let these relations inspire prospective insights about theemergence of spacetime and gravity with the goal to “apply them to a universe closer to our own,namely de Sitter space” (Verlinde, 2017, 3). Verlinde’s argument builds on a remarkable relation between the AdS/CFT correspondence andthe multi-scale entanglement renormalization ansatz (MERA) known from quantum many-bodysystems. In its original context, MERA can be regarded as a coarse-graining schema: If weconsider a chain of n spins, MERA involves joining p neighbouring spins into one block such thatthe number of spins at this first level of coarse-graining, (call it u = 1), is n/p . Notably, thisscheme can be applied iteratively until only one block remains such that after u = 2 , , , ... stepsof coarse-graining, the number of spins is n/p u . An associated mapping from the coarse-grained to the fine-grained Hilbert space is to be described by an isometric tensor, i.e. a tensor that As such, Jacobson’s derivation might be conceived of as realizing the anticipated role of the AdS/CFT correspon-dence as a correspondence principle `a la Bohr that van Dongen et al. (2019) trace back to the inception of AdS/CFTby Maldacena (1999). Similarities and dissimilarities between this case and that of Bohr and his contemporariesmight be an interesting venue for further research. Verlinde does at places speak as if AdS spacetime and gravity are emergent from the CFT degrees of freedomand thus disregards (or overlooks) the inter-representational character of the AdS/CFT correspondence. However,the reconstruction of his argument below avoids this misunderstanding. The presentation here largely follows Rangamani and Takayanagi (2017, ch. 14). p spins into one. For each additional level of coarse-graining, u = 2 , , , ... , another coarse-graining tensor is required. Together, these coarse-graining tensorscompose a map from a fine-grained state to a respectively more coarse-grained level u = 2 , , , ... .(Or seen conversely again, the corresponding isometric tensors compose a map from a more-coarse-grained level to a more fine-grained state.) Such a map can be depicted graphically as a networkwhere each isometric/coarse-graining tensor (the nodes) takes one input (referred to as the ‘incominglegs’) and delivers p outputs (called the ‘outgoing legs’) each of which in turn serves as input forother tensors (see figure 2). The legs are in other words the edges connecting the nodes. In thistensor network picture, the boundary of the network corresponds to the map between the original(fine-grained) state, u = 0, and the first coarse-grained level, u = 1. As one moves into the bulk,one goes towards higher u and towards maps between ever more coarse-grained levels.In order for the coarse-grained states to be renormalizable, all entanglement must be removed This is basically the block spin renormalisation group as championed by Kadanoff (1966). More generally, a tensor T with indices αβ...ab... can be used to express quantum states such as | ψ i = P Dα,β,...,a,b,... =1 T αβ...ab... | α i | β i ... | a i | b i ... — but also to depict maps from one Hilbert space H A into anotherHilbert space H B with dim( H A ) ≤ dim( H B ) such as | α i | β i ... P Da,b,... =1 T αβ...ab... | a i | b i ... . It is the latter usewe have in mind here. In a graphical representation of such a tensor-induced map, one would call the indices α, β, ... incoming legs, and the indices a, b, ... outgoing legs (see Pastawski et al. (2015)). Tensor networks are then graphstructures where each node is associated with such a tensor (and only one tensor), and each edge between nodes witha summation of a joint index of two tensors. Note that norm-preserving tensors can have at most as many incominglegs as outgoing legs; a norm-preserving tensor can also be (non-uniquely) pseudo-inverted and then thought of as acoarse-graining operation — as we are doing here partly. An inverted norm-preserving tensor has at least as manyincoming legs as outgoing legs. In the formalism, each edge corresponds to a summation of joint indices between the tensors connected by theedge. removed by the coarse-graining. More concretely: Movingoutward towards the boundary of the tensor network (from the coarse-grained to the fine-graineddescription of the system), the tensors linked to the map from the level u to u − u into their constituents at the level of description of u − u − u renormalizable.So much for the account of MERA as a coarse-graining scheme. What is interesting from theperspective of the AdS/CFT correspondence is that the tensor network has an intriguing similarityto a discretised AdS spacetime when the original (boundary) state is a CFT state (Swingle, 2012).A first indication of this is gathered from the alluring similarity between the depiction of a tensornetwork of figure 2, and a graphical depiction of hyperbolic 2-space geometry (that constant timesslices of AdS spacetimes belong to) on a flat projection plane. Instead of being a coarse-grainingscale, u can be reinterpreted as the radial bulk coordinate of AdS spacetime and each tensor isconceived as encoding the local geometrical neighbouring relations. The evidence for this geometricinterpretation of the MERA tensor network is still tentative (see Bao et al. (2015) for a review)but includes: the matching of the length of trajectories between AdS spacetime and the tensornetwork (as captured by the number of links crossed), the reproduction of a discretised versionof the Ryu-Takayanagi formula, and the occurrence of error-correcting features known from theactual AdS/CFT correspondence. In summary, the geometric interpretation of the tensor networkis corroborated by elements of the AdS/CFT correspondence and the similarity between the radialAdS coordinate and the coarse-graining scale u .This all provides for a discretised version of the AdS/CFT correspondence. The tensor networkis interpreted geometrically while it — due to its origin as a coarse-graining procedure — stillcarries the full CFT state on its boundary. In particular, any operator acting on an incoming ‘leg’of a tensor in the network can be represented — ‘pushed through’ — to act as an equal normoperator on the outgoing leg. In this way, every operator can be pushed to the boundary suchthat the operator acting in the bulk can be re-represented as an operator acting on the boundary.Thus, holography obtains. (As basically already said above, we can regard the tensor networkfrom the outside towards the inside — then local tensors of the network — should be understoodas coarse-graining operations; or we can regard the tensor network from the inside towards theoutside.)Verlinde now aims to use this tensor network version of the AdS/CFT correspondence in anexploration of de Sitter spacetime. Essentially, his proposal consists in considering what happenswhen one changes the tensor network so it is no longer holographic by connecting it differently, i.e.changing what tensors are joined by each edge. From the perspective of the coarse-graining scheme,this would be nonsensical: The tensor network depends on and is tailored to the original state andshould be subject to change only when the state is changed. An arbitrary change in the bulk of thenetwork would most likely entail that the approximation breaks down in the sense that the networkwould no longer map a coarse-grained description of some system to a fine-grained one. Once thetensor network receives its geometric interpretation, changes in the bulk of the network can be See Pastawski et al. (2015), section 2. u and the radial coordinate of the AdS spacetime and the support comingfrom the AdS/CFT correspondence. Neither are satisfied for arbitrary tensor networks, but ratheronly obtains for holographic tensor networks.Verlinde’s proposal, therefore, is to take as the vacuum state a holographic tensor network whichdoes admit a geometric interpretation as an AdS spacetime. From the coarse-graining perspective,the tensors of the network codified the short range entanglement that were removed between eachcoarse-grained level. With the geometric interpretation, the tensors no longer belong to differentlevels of description, but rather different depths in the bulk space. Verlinde therefore conjecturesthat the tensors are not transformations that reinstate entanglement, but rather entities that cap-ture the entanglement structure underlying space(time). The network is in other words regardedas a network of entanglement (quantum information) where the links between the tensors in thenetwork encode their mutual entanglement. This entanglement is then what gives rise to space-time: “spacetime geometry is viewed as representing the entanglement structure of the microscopicquantum state” (Verlinde, 2017, 3). Verlinde explicitly acknowledges that this idea comes from therelation between spacetime and entanglement in the AdS/CFT correspondence. His account, how-ever, is different in an important respect (though he does not emphasize this himself): Spacetimeis not dual to an entanglement structure (inter-representational), rather spacetime is conjecturedto emerge from entanglement (intra-theoretical). As argued in section 3.1, this cannot simply beachieved by stipulation. Crucial for Verlinde’s account therefore is the insight that the tensors canbe elevated to comprise real networks of entanglement rather than transformations. Thus the dualCFT state serves to inspire, but only inspire, the quantum degrees of freedom which spacetimeoriginates in. Re-representation in terms of a CFT state in still possible, but only when the tensornetwork finds itself in the configuration where it is equivalent to a MERA construction based onthat CFT state; this is the holographic state that Verlinde proposes to be the ground state andwhich is indicated to give rise to an AdS spacetime.For Verlinde, therefore, the network is not dependent on the CFT state (for which the MERA ismeant as a coarse-graining scheme). The holographic state is simply one configuration of the tensornetwork. As it turns out, the holographic state is one where each tensor is maximally entangledwith its nearest neighbours; this is what allows one to push through every bulk operator all the wayto the boundary. Verlinde writes: “we take an alternative point of view by regarding all these bulktensors as physical qubits, and interpreting the short distance entanglement imposed by the networkas being due to stabilizer conditions” (Verlinde, 2017, 7). The short range entanglement entailsthat the Bekenstein-Hawking formula, the intra-theoretical proportionality between entanglemententropy and area, holds in the bulk when the stabilizer conditions are satisfied. This allows forthe derivation of Einstein field equations which further corroborates the geometrical interpretationof the holographic tensor network state—the conjectured ground state—as a spacetime: “Ourinterpretation of general relativity and the Einstein equations is that it describes the response ofthe area law entanglement of the vacuum spacetime to matter” (Verlinde, 2017, 15).Replacing some of the short range entanglement with long range entanglement in the tensornetwork——that is, connecting non-neighbouring tensors in the network—will have two conse-quences: First, the tensor network will no longer be holographic and second, the entanglement See Pastawski et al. (2015), Yang et al. (2016), and Hayden et al. (2016). n indices are placed (here, n = 6); contraction of tensor indicesgives rise to a tensor network. Read from the inside to the outside to the outside, there are alwaysmore outgoing tensor indices than incoming ones. In case of figure 2 (a), each of the six indices of atensor x is contracted with that of neighbouring tensors y such that mutual entanglement betweeneach index pair is maximised (stabiliser state); schematically, this can be written as T αα | α i x | α i y = P Dα =1 1 √ D | α i x | α i y where x and y denote the neighbouring tensors and D = 2 , , ... for qbit/tribit/...states. In case of figure 2 (b), tensor indices are contracted in the same fashion except for thatone index per tensor is left open as a “free slot”. The tensor network in (b) serves as a mapwhich transforms quantum states associated with the free slots in the bulk (red dots) into physicalquantum states associated with the boundary (white dots). Thereby, the network provides a mapfrom ‘red dot’ inputs to ‘white dot’ outputs. The tensor network in figure (a) is obtained fromthat in figure (b) by successively contracting those pairs of free bulk indices (red dots) which areneighbouring across a tiling vertex. Thereby, hexagon tilings are turned into pentagon tilings (seefigure 2 (b)). (Note that the graphical depictions are slightly misleading, in particular by putativelysingling out the tensor in the center of the figure as special. Rather, there are graph isomorphismsfor moving any specific tensor into the center without changing the local tiling structure of thegraph.) Now, when long-range entanglement enters the system through thermal excitation, itdissolves the maximal entanglement structure as displayed in figure 2 (a); contractions on the levelof neighbouring tensors are then partly broken up (as the case in (b)) and replaced by contractionsbetween non-neighbouring tensors. By this, the relationship between entropy and area fadesaway (area scaling of entropy is linked to short-range entanglement only — as it is the case forAdS-spacetime) and volume-scaling contributions to the entropy ultimately begin to dominate — Thus, Verlinde as well as Jacobson assumes that the ground state is at an entanglement equilibrium. This mightprove to be another generic feature of quantum gravity theories.
24s it is the case for dS-spacetime for large-scales. Rather than contracting all free indices amongneighbouring indices then (which gets one from (b) to (a)), the free indices should be thought ofas being connected among all kind of tensors across the bulk. In particular, a re-representation ofbulk states via boundary states is not possible anymore.A second way to support the picture of dS as the excitation of AdS is based on energy-entropyconsiderations (see section 2.3, Verlinde (2017)). First, note that the energy for a ball-shaped regionof radius r in AdS/dS spacetime (with AdS/dS curvature radius L ) is given by E ( A ) dS ( r ) = ± ( d − d − πGL V ( r )where V ( r ) is the volume of the ball-shaped region considered. Then, using the AdS/CFT cor-respondence, one can partly re-express this energy expression in terms of the number of possibleconfigurations C realising the corresponding CFT state. For r = L , the energy is provided by E AdS ( L ) = − (cid:126) d − L C ( L ) . Similarly, one can reformulate the corresponding energy expression for dS-spacetime such that itlooks—from an usual quantum mechanical perspective—like an energy state corresponding to anexcitation of AdS-spacetime : E dS ( L ) = − (cid:126) d − L ( N ( L ) − C ( L )) , where N ( L ) = 2 C ( L ) tracks the additional number of configurations relative to the ground state.As a consistency check, use the energy state expression for dS qua excitation of the AdS state toderive the entropy formula for dS. This gives the same entropy as usually attributed to a region ofdS-spacetime. The claim that dS is an excitation of AdS is thus in the end established as plausibleby showing that a state counting method available from the AdS/CFT correspondence provides theusually accepted entropy formula for a dS spacetime region at its curvature radius L .A decisive element of establishing the dS-state as an excitation of the AdS-state in the energy-based argument strikingly consists of turning the inter-representational relation holding at theground level between the AdS spacetime and a CFT into an intra-theoretical one, that is reinterpretthe number of configurations on the CFT side as in fact the number of configurations for realisingthe AdS spacetime (the AdS-state) simpliciter . Thereby, a similar formula for the energy in termsof configurations as in the case of a CFT can be used at the excited level of the AdS state (where thecorrespondence to a CFT as such is normally not available anymore ). But also the entropy-based Thanks a lot to X for clarifications on this matter. The AdS and the dS metric for a static coordinate patch is given by ds = − f ( r ) dt + dr f ( r ) + r d Ω where f ( r ) = 1 + r L for AdS, and f ( r ) = 1 − r L for dS. Roughly, the entropy formula is obtained by counting the ways in which the excitations N ( L ) can be distributedover the number of degrees of freedoms C ( L ). The derivation is unfortunately only alluded to in Verlinde (2017).See however van Leuven et al. (2018, section 4.4) for a detailed, alternative derivation of the same formula. Or at least not straightforwardly. See van Leuven et al. (2018) for an attempt towards correspondence relationsbetween non-AdS-spacetimes (such as de Sitter- and Minkowski-spacetime) and CFTs more generally. These specificcorrespondence relations are however based on connecting the non-AdS-spacetimes to AdS-spacetime, and thus againon the AdS/CFT correspondence in one way or the other.
We now argue that both Verlinde (2017) and Jacobson (2016) in their respective ways employinsights gained from the AdS/CFT correspondence to advance non-holographic theories of quantumgravity; in accordance with the general tenet of the heuristic function of dualities as identified byDe Haro (2018). We call this particular methodology ‘holography without holography’.In Jacobson’s case, an entire derivation in the context of the AdS/CFT correspondence servedas a guide towards an analogous intra-theoretical, and thus more promising, derivation in a non-holographic setting made possible by means of additional (plausible) assumptions. Verlinde usesthe tensor network analogue of the AdS/CFT correspondence to model the degrees of freedom inthe AdS vacuum (exploiting the holographic stabiliser condition implied by the AdS/CFT corre-spondence), and then speculates that this generalises even when the tensor network is no longerholographic (due to long range entanglement). This generalisation is then also supported by inde-pendent consistency checks against known results for de Sitter spacetime entropy scaling. In bothcases, crucial aspects of the formal framework surrounding the relevant parts of the AdS/CFT cor-respondence are kept in place to secure the expedience in the non-holographic setting: In Verlinde’scase the tensor network structure and in Jacobson’s case non-holographic equivalents of crucialrelations such as eq. 1 and eq. 7. The work already done in the context of the AdS/CFT corre-spondence ensures that these are controlled environments which enable the type of manipulationnecessary for promoting the inter-representational relations of the AdS/CFT correspondence intointra-theoretical relations.We conjecture that the promises of this methodology is not idiosyncratic to the two cases studiedbut can be generalised to generic inter-representational relations (and structures of relations) inthe AdS/CFT correspondence and possibly beyond. The overall heuristic of holography withoutholography can be decomposed into the following steps:1. Identify features of the AdS/CFT correspondence that are desirable, if they can be conceivedas intra-theoretical relations. Jacobson:
The relation between EFEs and entanglement (entropy).
Verlinde:
Holographic stabiliser condition for entanglement networks.2. Identify the specific formal (sub)framework of the AdS/CFT correspondence that sustainsthese features.
Jacobson:
The Ryu-Takayanagi formula, the relation between entanglement entropy andenergy, and the relation between area and energy.
Verlinde:
Tensor network formalism.3. Seek to embed this formal (sub)framework into the intra-theoretical context of interest mod-ifying it adequately, while preserving enough of its structure such that it can still sustain thefeatures of interest. 26 acobson:
Entanglement equilibrium, (leading order) area scaling of entanglement entropy,and (approximate) conformal invariance.
Verlinde:
Identify the holographic stabiliser condition for entanglement network as a groundstate, thus allowing for (and requiring at the same time) its violation at the level ofexcited states.4. Exploit the setup to derive intra-theoretical results analogous to the inter-representational onesof the AdS/CFT correspondence and/or utilize the now intra-theoretic formal (sub)frameworkto inform the physics of an otherwise intractable context.
Jacobson:
Derives the Einstein Field Equations from entanglement constraints.
Verlinde:
Obtains excitation states (in particular, dS comes out as an excitation of AdS).Given that holography without holography requires non-trivial innovative creativity in its ap-plication — as signified in both case studies —, it is by no means an automaton for discovery. Butit suggests a way to activate the AdS/CFT correspondence in the context of discovery for generalpurposes of quantum gravity beyond the string theory on the AdS side. dual = CFT as a guiding principle to quantum gravity
So far, we first defended the view that the AdS/CFT correspondence in itself does not breaknew grounds towards the quantum origin of gravity. We then went on to argue that it can stillexcel in a methodology referred to as ‘holography without holography’. Reconsidering the roleof the AdS/CFT correspondence as an inspirational template for intra-theoretical relations frominter-theoretical relations, the AdS/CFT correspondence got to be understood as the key guidingprinciple for applying the methodology of holography without holography. In this section, we makeour conception of the AdS/CFT correspondence as a guiding principle more precise. In particular,we explain that the AdS/CFT correspondence in the context of holography without holography isbest conceived of as what we suggest calling an analytic guiding principle. It is not a secret that the search of a theory of quantum gravity suffers from the unavailability ofempirical data. Therefore, in addition to efforts of extending quantum phenomenology—the searchfor new means of experimentally probing the relevant regimes in which we expect quantum gravi-tational effects to kick in—insights into a theory of quantum gravity can otherwise only come fromthe imposition of principles and the implementation of these principles within specific theoreticalprograms such as string theory, loop quantum gravity, or asymptotic safety (to mention a few). To-wards a theory of quantum gravity, such principles provide guidance, motivation of the problem ofquantum gravity in the first place and sorts of non-empirical justification . Examples of guiding This is a specific way in which the AdS/CFT correpondence can be said to have a ‘heuristic function’ (cf. De Haro(2018)). Without much empirical data at all, there is hardly any empirical data in need of explanation, either. Thus,the motivation for quantum gravity is largely theoretical; as such, it typically rests on the demand for realisationof certain principles such as UV completion (the idea that the theory holds formally up to all high energies) orunification. Dawid (2013) for instance suggests adopting means of non-empirical theory confirmation , at least in the contextof string theory. In any case, it remains uncontroversial that, in the ongoing context of discovery, principles help inthe preliminary appraisal of hypotheses and theory proposals at the level of plausibility arguments (see Schickore(2018), section 9.2).
UV-completion , and quantisation ; examples of (weakly) justificatory principlesinclude minimal length and quantisation ; and examples of motivation include unification , to namesome. Notably, such principles are intentionally not strictly formalised in order to keep them asframework-independent as possible (for instance, the idea of UV completion, that a theory holds upto arbitrarily high energies, can upon its general conception still be fleshed out in different specificscenarios). Now, we see it as a task for the philosopher to work out the principles used and alludedto in different approaches, and to make their mutual dependence relationships transparent (to givean example: in many frameworks, the principle of minimal length implies UV completion, raisingthe question whether minimal length is the intended principle one wants to commit to — or ratherUV completion). Our discussion of the AdS/CFT correspondence as a sort of guiding principle canbe seen as a contribution to this more general project. Now, the duality aspect of the AdS/CFT correspondence entails a (conjectured) mathematicalisomorphism whereas most guiding principles are in comparison (also) desiderata for instantiationsof certain physical or metaphysical properties (call these physical and metaphysical guiding princi-ples respectively). This becomes clearer via examples: unification is usually understood (or desired)to express not just unity of representation but unity of nature ; UV completion not just (predictive)completeness of our description (our theory) but as predictive completeness in our picture of na-ture; etc. In light of this contrast, we refer to the AdS/CFT correspondence as a (merely) analyticguiding principle as it is not intended to apply to nature as such, that is to have any significanceother than that of formal relationships.To clarify the notion of an analytic guiding principle and the role of the AdS/CFT corre-spondence qua analytic guiding principle, it is instructive to make a comparison to other guidingprinciples which can be conceived of as analytic, such as quantisation and minimal coupling.
67, 68
Quantisation is a prescription for going over from the classical to the quantum theory. In somesense, it amounts to a (not always unambiguous) prescription taking classical observables to quan-tum observables as well as classical evolution to quantum evolution. There are different attemptsto make this notion technically rigorous (as for instance geometric quantisation, and deformationquantisation); none of these can however provide a satisfactory account of quantisation in everydesirable scenario (see for instance Landsman (2006)). Minimal coupling in the context of GR isa specific prescription for associating matter equations in flat spacetime to corresponding matter Roughly, the idea that spatiotemporal structure consists of discrete chunks. See Hossenfelder (2013) for more. See Crowther and Linnemann (2017) for a detailed discussion. At the end of the day, the goal would be to create a hierarchy network of principles which provide a powerfultool for comparing the commitments in different approaches to one another, and allow for exploring further optionsupon dropping or adding the commitment to certain principles in specific approaches. Unification as a guiding principle for instance has both physical and metaphysical strands. For a discussion, seeSalimkhani (2018). Both are applicable in contexts well beyond quantum gravity. One might object to the classification of quantisation as an analytic guiding principle given that physical con-siderations are taken into account in quantisation (for instance, in order to avoid ambiguities, one can demand thatcertain classical symmetries must have corresponding quantum symmetries, that is quantum anomaly avoidance);however, quantisation theory as such covers a vast space of theories for which a clear physical interpretation is notreadily given. It is for this reason that quantisation in our view first of all amounts to a mathematical (that is,formal) theory. Consider for instance the normal ordering issue. See for instance Landsman (2006). In other words, the general principle of quantisation can only be cashed out (so far) through a cluster of various(not completely mutually compatible) technical renderings, which is however the case for many other principles (suchas unification, or UV completion) as well. This is not a problem: the notion of principle is after all supposed tocapture a general idea at least prima facie independent of framework-specific theoretical renderings. As a prescription, it suffers from ordering ambiguities in mappinghigher derivative expressions in flat spacetime to that in curved spacetime, analogous to those oc-curring in quantisation (see Misner et al. (2017, chapter § varying ingredientfor fleshing out the recipe of holography without holography (see section 4.3), and not a recipe fortheory change itself: • in the context of Jacobson’s work, the inter-representational AdS/CFT correspondence servesas a formal template towards corresponding intra-theoretical relations without being realised(not even approximately) at any point itself. The AdS/CFT correspondence is used as aformalistic (thus analytic) guide towards inter-theoretical relations in a specific frameworkchosen by Jaocobson while holography as a physical feature falls out of the picture. • in the context of Verlinde’s work, the inter-representational AdS/CFT correspondence inits tensor network formulation is used to define the ground state of the system based onthe stabiliser condition for holography; thus again embedding parts of its formal frameworkin an intra-theoretical context. Relying on established techniques from the tensor networkformulation of the AdS/CFT correspondence, the nearest neighbour entanglement due to thestabilizer condition can be systematically broken to suggest possible excitations of the system.Holography without holography, in other words, forms a general strategy in the context of theAdS/CFT correspondence that can be carried out in different ways. The unifying idea is, as ex-plicated in section 4.3, the identification of those inter-representational features of the AdS/CFTcorrespondence that are desirable if they can be made intra-theoretical, and the subsequent embed-ding of the formal framework sustaining these features into the intra-theoretical context of interest.What are desirable features will depend on one’s purposes, and how to succeed with the embeddingwill depend on the specific context. A substantial point of working out the methodology behind holography without holographyis to acknowledge an even more high-level tranformational methodology of changing an inter-representational into an intra-theoretical relation . After all, a methodology analogous to that ofholography without holography has for instance been tacitly assumed in discussions on the empiricalrelevance of global symmetries for longer now: global symmetries of say Newtonian theories (such asglobal translation/rotation/... invariance) are — as such —, pace orthodoxy, inter-representationalrelations. They tell us how two different Newtonian representations (two different Newtonian“worlds”) can be mapped into another. They however obtain empirical relevance (say become ob-servable — either directly, or indirectly) via associated conserved quantities — as usually done — At the level of dynamical equations, any instance of the flat metric is substituted by the corresponding generalrelativistic metric; and any instance of the flat spacetime covariant derivative by the covariant derivative associatedwith the general relativistic metric. Again, as emphasized in section 4, this embedding of originally inter-representational elements into an intra-theoretical context cannot be achieved by simply stipulating that the relations of interest are not exact. Global symmetries are arguably analogous to dualities. To use the slogan mentioned by De Haro and Butterfield(2018), “a duality is like a symmetry, but at the level of a theory” (p. 6). See paragraph (2) in section 1.1 thereinand Read and Møller-Nielsen (2018) for a more detailed discussion of the analogy. See for instance Greaves and Wallace (2013). one
Newtonian representation, that is as intra-theoretical relations. So, an overarching method-ology behind holography without holography includes changes from inter-representational dualityrelations into intra-theoretical relations (with adequate new theoretical embedding) like those byJacobson and Verlinde; but also, for instance, (2) the change from an inter-model symmetry toan intra-model (physical) symmetry. Note that the overarching methodology suggested throughholography without holography is hereby still more specific than the generic heuristics of “changeone of your guiding principles” or “change your guiding principle of holography”.
Nevertheless, what is the status of the specific methodology of holography without holography em-ployed in this paper, especially given that we are willing to build such bold (general) methodologicalclaims on it? Two core objections suggest themselves: (1) The approaches considered are highlyspeculative and only followed by a small group of researchers; in other words, they are neither em-pirically nor sociologically well supported. Why then think that anything of general value can beconcluded from their consideration? (2) Even in the extremely lucky scenario that the approachesof Jacobson or Verlinde turned out to culminate in anything close to empirically adequate theories,why should it matter how we got to them (if this methodology was used at all)?Let us start by addressing the second objection. First, it is not important whether Jacobsonor Verlinde actually used our proposed methodology extracted from their approaches but that itcould have been used. That a rationale is not used, does not rob the rationale its relevance. Then, turning the objection around, one way to understand our work is in fact as (arguably another) demonstration — contra long-lived Popperian biases — that there is a rationality of discovery possible in the context of emergent gravity and — just as already hinted at in various other specialcases — a fortiori in science.Concerning the first objection we cannot do much more than stay defiant: In the end, it isup to us philosophers of physics whether we stick to grand plans of helping out physicists in themoment of crisis (as for instance called for by the Huggett and W¨uthrich (2013)) or back off inthe very first moment that we realise that ongoing scientific research can simply turn out to beill-directed (as any fallible enterprise)). To engage in current research, whether as a physicist or That Jacobson (2016, 1) has used a methodology resembling holography without holography in the context ofdiscovery is suggested by his clear reference in the paper to the work by van Raamsdonk and collaborators as asource of inspiration quoted above. Compare this for instance to the distinction between ‘discoverability’—something could have been discovered ata certain point of time—and ‘discovery’—something was actually discovered at a certain point of time—by Nickles(1988). Arguably, discoverability is more interesting than actual discovery: working with the notion of discoverabilityis more likely to reveal stable conceptual relations while decreasing the risk of taking historical contingencies asrelevant. See Nickles (2012). Dividing up the context of discovery into a context of generation, and a context of pursuit (following Laudan(1981)), the statement is really that there is (some) rationality (not necessarily logic though) for generating scientifichypotheses and perhaps even theories. That there is a rationality in the context of pursuit (say in the form ofappraising plausibility arguments to decide which research direction to tackle next), is much less debated (see Nickles(1988)). Consider for instance the work of Darden (2002) on how specific biological mechanisms can be systematicallyarrived at through instantiating abstract mechanism schemes.
30s a philosopher (and at least scientists know this), also amounts to accepting the possible butunknown opportunity cost. To address the sociological aspect of this objection specifically: not thenumber of individuals working in an approach but rather the groundedness in accepted principlesand spontaneous reproduction of approaches by more or less independent researchers—as it is thecase for the works of Jacobson and Verlinde—strike us as sensible criteria for which proposals toconsider. Note also that in a sense it is not correct that Verlinde and Jacobson’s approaches arejust pursued by a few individuals; their programs are part of a general trend towards considerationsof horison and entanglement entropy in order to make a step forward towards a theory of quantumgravity . Working out a (joint) rationality of discovery behind their approaches, should thusbenefit a whole field. In this paper we uncovered an enticing methodology where inter-representational relations are,at least for heuristic purposes, turned into intra-theoretical ones. Both of our two case studiesfocused on how in particular the AdS/CFT correspondence can be put to use in non-holographicsettings when re-interpreted intra-theoretically. It was argued that this is of particular relevancein the context of the AdS/CFT correspondence: after all, although the AdS/CFT correspondencepromises a relation between gravitational and quantum degrees of freedom, its nature as a dualityprohibits it from being a theory of the quantum origin of gravity. While this complication has beenalluded to by a number of authors, this paper offered more concrete details how to activate theAdS/CFT correspondence for general purposes in quantum gravity: the methodology of holographywithout holography. This methodology was concretely exemplified in the two case studies of worksby Jacobson (2016) and Verlinde (2017), who in their respective ways draw inspiration from the(inter-representational) AdS/CFT correspondence for making an intra-theoretical claim. Thus, weargued that the AdS/CFT correspondence here serves as an analytic guiding principle—consistingof purely mathematical relations among representations of the bare theory—that can inform intra-theoretical relations for the purpose of developing prospective theories for the quantum origin ofgravity.In so far as holography without holography proves to be successful beyond these two case studies,it serves as an apology for the extensive research on the AdS/CFT correspondence. Critics arguethat such research is misguided since the actual world is not AdS; that research on the AdS/CFTcorrespondence is thus merely esoteric mathematics re-expressing a wrong-headed theory and, to putit boldly, the equivalent of studying the relation between bachelors and unmarried men in a worlddevoid of men. Holography without holography offers a response to such criticism: the AdS/CFTcorrespondence can serve as an important guiding principle towards a theory of quantum gravity forthe actual world despite the fact that it is not realised (probably not even approximately). Giventhe current status of quantum gravity research we should focus more on the indirect contributionof the AdS/CFT correspondence as it transpires through the methodology of holography withoutholography. See for instance works by Maldacena and Susskind (2013), Chirco et al. (2014), Han and Hung (2017), Bayta¸set al. (2018), Cao and Carroll (2018), and Chirco et al. (2018). eferences Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. N = 6 superconfor-mal Chern-Simons-matter theories, M2-branes and their gravity duals.
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