CCHEP XXXXX
Bulk Locality and Asymptotic Causal Diamonds
Chethan KRISHNAN a ∗ a Center for High Energy Physics,Indian Institute of Science, Bangalore 560012, India
Abstract
In AdS/CFT, the non-uniqueness of the reconstructed bulk from boundary subregions hasmotivated the notion of code subspaces. We present some closely related structures thatarise in flat space. A useful organizing idea is that of an asymptotic causal diamond (ACD):a causal diamond attached to the conformal boundary of Minkowski space. The space ofACDs is defined by pairs of points, one each on the future and past null boundaries, I ± .We observe that for flat space with an IR cut-off, this space (a) encodes a preferred class ofboundary subregions, (b) is a plausible way to capture holographic data for local bulk recon-struction, (c) has a natural interpretation as the kinematic space for holography, (d) leadsto a holographic entanglement entropy in flat space that matches previous definitions andsatisfies strong sub-additivity, and, (e) has a bulk union/intersection structure isomorphic tothe one that motivated the introduction of quantum error correction in AdS/CFT. By slidingthe cut-off, we also note one substantive way in which flat space holography differs from thatin AdS. Even though our discussion is centered around flat space (and AdS), we note thatthere are notions of ACDs in other spacetimes as well. They could provide a covariant wayto abstractly characterize tensor sub-factors of Hilbert spaces of holographic theories. ∗ [email protected] a r X i v : . [ h e p - t h ] A ug Introduction
We will start by observing a few reasons to believe that holography is likely to be auniversal feature of quantum gravity .Firstly, observables in a diffeomorphism invariant (quantum) theory with a dynamicalmetric, are naturally integrals over all of spacetime . This means that bulk spacetime is adummy variable in quantum gravity and that observables are supported only at the boundary.Secondly, black holes seem to be the highest entropy objects in theories of gravity, and theirentropy scales with area and not volume. This hints at the fact that the Hilbert space size ofa region of space in quantum gravity does not scale with volume . Thirdly, the open-closedduality of string theory is a suggestion that strings that contain gravity are dual to stringsthat contain only non-gravitating fields. Since open strings are excitations of D-branes whichwrap submanifolds, it is perhaps not too far-fetched (at least not in hidsight!) to think thatthere should be a desription for gravity in terms of a lower dimensional non-gravitataionaltheory.In the above paragraph, we have not mentioned the word “AdS” even once. This is becausewe expect holography to hold more generally than in AdS – in fact, the idea of holographywas introduced by ’tHooft and Susskind [1, 2] long before the AdS/CFT correspondence[3, 4, 5]. However there are two features that make holography shockingly more impressivein AdS than elsewhere. One is that we know explicit examples of holographic duals (eg.[3] and many more) in some asymptotically AdS spaces thanks to string theory . Secondly,at least at the semi-classical level, in asymptotically AdS spaces we know how to formulatea correspondence between bulk calculations and boundary calculations [5]. This is in bigparts due to the fact that the holographic boundary in AdS/CFT is as close to a physical boundary as one can hope for.In some ways, these two aspects of AdS/CFT are quite independent. The former is a This is clearly a vague statement. The challenge is to clarify in what precise sense this is true in varioustheories/spacetimes/etc. The local metric acquires infrastructural meaning when G N → , where gravity is weak and close tonon-dynamical. The key point is that in a spacetime with a non-dynamical metric, diffeomorphism invarianceends up becoming a trivial gauge redundency: one can gauge-fix the metric to a form where its isometries(if any) are manifest, and then use coordinates in that gauge to label points in spacetime. This is what onedoes in Poincare invariant field theory, which has local observables. But when the metric is dynamical, it isnot clear how to solve away the gauge redundency, and we seem forced to go to the boundary of spacetimeto find observables. Though perhaps not as widely appreciated as they should be, these facts are known oneway or another since the days of Einstein’s “hole argument”. Dynamical diff invariance and black holes are two defining features of gravity, and the takeaway fromthe above discussion is that both of them ndicate a holographic definition of quantum gravity. See [6] for a non-AdS example for quantum gravty. . This suggests the possibility thatif we could guess the analogous structures in other spacetimes, we might be able to makeprogress with only minimal knowledge about the specific theories involved . In this paper,we hope to take a very small step in this rough direction.Our starting point is the observation that the natural “unit” of holography in a region ofspacetime is a causal diamond . The time evolution inside the causal diamond (at least atthe semi-classical level) is entirely determined by the data on the diamond. Versions of thisidea have been exploited in many interesting ways for almost two decades now, and manyinteresting papers have explored various aspects of it [8, 9, 10, 12, 13, 14, 15]. We will adda new twist to this ingredient, by introducing the notion of an asymptotic causal diamond .In most of our discussion, we will stick to flat space for concreteness, but in the final sectionwe will make some comments about other spacetimes.The motivation for introducing an asymptotic causal diamond is local bulk reconstruc-tion . In AdS, we know that local bulk reconstruction can be accomplished from boundary subregions through causal wedges and related ideas [16, 17, 18], and we wish to know whatare the structures required for accomplishing this more generally, in particular in flat space.It is easy enough to convince oneself that in flat space, the plausible answer is a causaldiamond attached to the conformal boundary. This is equivalent to choosing two points, oneeach on the two null boundaries. It naturally encodes a boundary subregion (in a manner wewill make precise), while at the same time we expect it to contain precisely enough data toreconstruct local bulk regions. The idea of taking asymptotic causal diaomnds seriously forholographic purposes beyond AdS/CFT is likely to be fruitful: we will present developments Note that the existence of a sparse spectrum and a large central charge [7] in the CFT are pre-conditions for a semi-classical weakly curved description, and so are not structural to the semi-classical correspondencein the sense that we use here. Of course there exists the possibility that this might give us enough of a hint to characterize (perhapseven fully explicitly) holographic theories. This idea has its origins (I think) in covariant versions of entropy bounds, made precise by using light-sheets as the holographic screens by Bousso [8, 9]. It turns out that this entropy is bounded by appropriatelydefined areas: this is the (Fischler-Susskind-)Bousso bound [10, 8, 11]. There is evidence [8] that the originof these bounds is statistical and not thermodynamical, which again is a strong suggestion that it is a counton the number of degrees of freedom of a holographic fundamental theory. These ideas are one of the majorsources for our work. See the appendix for a discussion on the precise role that an asymptotic causal diaomond plays in bulkreconstruction. kinematic space . In flat space,since the boundary has null components, a causal structure defined instrinsically on theboundary is ill-defined, so this idea does not work directly. But we will see that bulk causaldiamonds attached to the conformal boundary can still be used to capture precisely analogousinformation. In particular, the fact that the space of certain pairs of points on the boundaryremains of significance even in flat space, we find quite striking. Our work is also closelyrelated to the HRT construction [12]: the choice of a point each on the two null boundariesdefines a canonical class of HRT surfaces for flat space with a cut-off.
To set the stage, we will consider d + 1 dimensional Minkowski space, M d +1 , with d = 2 for concreteness and ease in drawing pictures – but we emphasize that the statements wemake here generalize quite readily to all d . In many places, all one has to do is replacestraight lines by hyperplanes and circles by (hyper)spheres.Let us write down some formulas for the conformal structure of flat space to set up ournotation: we largely follow the conventions of Hawking&Ellis. Flat space metric in polar4 urface (q = constant)Surface (p = constant)Timelike geodesicNull geodesic r=0,t=0 ii Figure 1: A plot of the ( t (cid:48) , r (cid:48) , φ ) coordinates of 3-dimensional flat space. The r (cid:48) coordinateis plotted radially, but it should be understood that the metric in the radial direction is notflat becaause of the conformal transformation. All points on the circle drawn at r (cid:48) = π areidentified, and becomes the point at infinity. Our figure should be compared to figure 15 (i)in Hawking&Ellis. But we believe that what is labeled as a spacelike geodesic in the figurethere is only a spacelike curve , at least for the 2+1 dimensional case that we consider here.We will have more to say about spacelike geodesics in this paper.coordinates ds = − dt + dr + r dφ is conformal to the Einstein static form ds = − dt + dr (cid:48) + sin r (cid:48) dφ (2.1)which is locally of the form R × S , but with a constrained range for t (cid:48) , r (cid:48) : − π < t (cid:48) + r (cid:48) < π, − π < t (cid:48) − r (cid:48) < π, r (cid:48) ≥ . (2.2)This metric is conformally flat, and upto the conformal factor (which will not be importantfor us) it turns into flat space under the coordinate change t + r = tan (cid:18) t (cid:48) + r (cid:48) (cid:19) , t − r = tan (cid:18) t (cid:48) − r (cid:48) (cid:19) (2.3)It is also convenient to introduce the null coordinates p, q via p + q = t (cid:48) , p − q = r (cid:48) . (2.4)In figure 1 we have drawn the Einstein static universe after opening up the r (cid:48) coordinates intoa disc with an identified boundary circle. This is useful for maintaining the usual intuitionfor a radial coordinate, while working with the conformally compactified coordinates.5 Bulk Causal Diamonds Anchored to the Conformal Boundary
Asymptotic Causal Diamonds:
Our goal is to construct an analogue of Rindler-AdS/entanglement wedge reconstrcution, that is useful in flat space. For this we find ituseful to introduce the notion of an asymptotic causal diamond. The basic idea is that anasymptotic causal diamond in M is defined by two points: one on the future boundarynull cone I + which we call p F , and one on the past boundary null cone I − , which we call p P . Note that the null boundaries I ± are defined by p = π/ and q = − π/ respectively,and therefore these points are fixed uniquely by one null coordinate (and the angles): p F = ( π/ − (cid:15), q, φ ) , p P = ( p, − π/ (cid:15) (cid:48) , φ (cid:48) ) . (3.1)These points we will call the vertices of the diamond. We work with the conformal coordi-nates here to locate our asymptotic causal diamonds. We have exhibited the possibility ofregulating the points at the boundary by introducing an (cid:15), (cid:15) (cid:48) for convenience in some cal-culations when taking the asymptotic limit, but they can be set to zero . The intersectionof the past light cone of p F and the future light cone of p P defines an asymptotic causaldiamond. Spacelike Geodesics:
Note that if we consider the future and past points to be ordinarypoints (instead of points at the conformal boundary) these intersections are simply circles[15, 13]. The difference, when we take these points to the conformal boundary is that thesecircles end up having infinite radius, and end up becoming (spacelike) straight lines. It isstraightforward to show this systematically and we will present it soon, but the result isintuitive enough. This means that bulk regions that can be reconstructed from boundarydata on the asymptotic causal diamond, are regions anchored at the boundary and boundedby straight lines . The key point is that even though we have not specified an explicit bulkreconstruction map as in the AdS-Rindler wedge [17, 18, 21], we expect just from the factit is the interior of an asymptotic causal diamond, that this entire region is re-constructiblefrom the conformal boundary. This is the key requirement in building the connection withquantum error correction.To present some explicit formulas, we can work with the spatial slice of the bulk thatcorresponds to the t = 0 instant (which is identical to the t (cid:48) = 0 instant) as is usually done We view the space of asymptotic casual diamonds as part of the associated data of asymptotically flatspace. Later we will also introduce a radial cut-off for some purposes. It will be interesting to relate thisand the (cid:15), (cid:15) (cid:48) more concretely. Let us emphasize a trivial point: the straight lines that we talk about here are straight lines in theoriginal flat space. They can be shown to be circles on the S of the conformally flat patch of the Einsteinstatic universe corresponding to Minkowski space, see Appendix. Indeed, the nature of the precise holographic data and the form of the bulk reconstruction map in flatspace must differe from the familiar ones in AdS/CFT. We will comment about this briefly in an Appendix. x -coordinate as a radial coordinate togetherwith a Z freedom (the angle degree of freedom in 1+1 d), this will turn into a more conven-tional Penrose diagram. The local bulk aspects we discuss are invisible in this simple 1+1dimensional situation. We show only ACDs attached to the right boundary here.in the AdS/CFT case in discussions of bulk reconstruction and code subspaces. This slicehas a simple description in terms of asymptotic causal diamonds: simply make sure that weonly consider those ACDs in (3.1) with p = − q ≡ q , φ = φ (cid:48) ≡ φ . (3.2)where q , φ are arbitrary (within their allowed ranges). In other words, these are the sym-metric asymptotic causal diamonds. For this class, we can write down explicit formulaswhich are not at all ugly, and that is another motivation to make things explicit. By trans-lating these coordinates back to the standard flat space coordinates, we can see that theyare the limiting cases of the classes of casual diamonds defined by the two points ( t, r, φ ) = ( ± ˜ T , ˜ R, φ ) (3.3)in the limit where ˜ T , ˜ R are both going to infinity, but ˜ T − ˜ R is held fixed (and is controlledby q ). Explicit formulas are simple to write down, we will present one: the circle that is atthe waist of the casual diamond turns into the promised straight line in this limit: r cos( φ − φ ) = ˜ R − ˜ T (3.4) We can write explicit finite expressions for ˜ T , ˜ R if we retain the (cid:15), (cid:15) (cid:48) .
7e will sometimes use k ≡ ( ˜ R − ˜ T ) / in what follows for brevity. The asymptotic causaldiamonds whose waists are defined by these straight lines are a natural generalization of theusual Rindler wedge. Metric on the Generalized Rindler Wedge:
We can define a Rindler-like metric onthese asymptotic causal diamonds. Specializing (without any real loss of generality) to thecase φ = 0 in (3.4) the following coordinate transformation brings the original flat spacecoordinates ( r, t, φ ) to a natural Rindler-like form in the variables ( R, T, Φ) : R = (cid:112) ( r − k/ cos φ ) − t , T = tanh − (cid:18) tr − k/ cos φ (cid:19) , Φ = φ (3.5)The generalized Rindler form of the metric follows from this – the explicit form is straight-forward, but of not much use here.To see that this is a natural generalization of Rindler, note that when q = 0 = k ,the asymptotic causal diamond has vertices that are “half-way” to time-like infinity, andcorresponds to the usual Rindler wedge. Then the “waist” of the causal diamond is a straightline that passes through the origin of the Minkowski space: r cos φ = 0 . (We have againtaken φ = 0 without loss of generality.) It is immediate to check in this case that thecoordinate transformation (3.5) above reduces to the simple metric ds = − R dT + dR + R cosh T d Φ (3.6)when k = 0 . This metric is the standard (spherical) Rindler metric. See [24] to see a recentdiscussion of a related but different generalization of the spherical Rindler metric.From the perspective of causal structure and bulk reconstruction, what we have ob-tained via the ACD coordinates ( R, T, Φ) above is a coordinate system that is similar to theMinkowski coordinates ( r, t, φ ) . The former covers all of the (asymptotic) causal diamondin one chart, while the latter covers all of Minkowski space. In the Penrose diagram, theconstant R and T slices have a structure in the ACD that has some similarities to the r and t coordinates in the full Minkowski space. A field theory at R = R cut on the ACD could be auseful way to describe holographic data at the boundary of the ACD for bulk recnstruction.This is analogous to how r = r cut in standard Minkowski space could be a useful way tocapture holographic data in Minkowski space. Note in particular that the R cut → ∞ limittakes one to the conformal boundary of the ACD . This is a field theory living on de Sitter space, which is believed to be well-defined if one takes theBunch-Davies vacuum to define correlators [25]. Note that the spherical Rindler metric is a time-dependentbackground in T . This is a field theory living on the Einstein static universe. Such theories are well-defined, the scalarcase is standard [26], fermionic and superysmmetric examples go back to the work of D. Sen [27, 28]. In a previous version of this paper, there were statements in this paragraph to the effect that the
8t is worth mentioning however that despite some parallels with AdS-Rindler and despiteits apparent simplicity, the ( R, T, Φ) coordinates have one important difference when itcomes to bulk reconstruction: in higher than 1+1 dimensions, the metric on the ACD is time-dependent. This means that the problem of solving the wave equations and identifyingappropriate spacelike Green functions could be harder in an ACD than it was in Rindler-AdS. Nonetheless, as far as making our points are considered, this is merely a technicality:the causal structure is our key concern here. Let us also note that it is perhaps significantthat at the cut-off, the geometry is de Sitter as we mentioned in a footnote. ACD Reconstruction:
Even thought we will not write an explicit ACD reconstructionmap in this paper, the above discussions lead to a direct parallel with the causal wedgereconstruction picture in AdS, because we expect from the causal structure that the data atthe boundary of the causal diamond (ie., the intersection of the boundary of ACD and theconformal boundary of Minkowski space) is enough to fix bulk data in the interior of theACD.From this point of view, the Rindler-AdS/causal wedge reconstruction and ACD recon-struction are parallel structurally as ingredients for local bulk reconstruction. We will usethis fact to make the connection with code subspaces momentarily.Before proceeding, let us also note that for any bulk point in Minkowski space, we candefine the regions on the boundary that are spacelike separated from it. This set of points isa natural candidate for the region in the boundary from which we expect to have an analogueof global bulk reconstruction. Thanks to the previously noted fact that a radial cut-off inMinkowski space is natrually comaptible asymptotically with the conformal boundary, wehope to report on some progress on an explicit construction of this type in the near future[19].
Geometry of Asymptotic Causal Diamonds in the Conformal Diagram:
It isuseful for the ensuing discussion to have an understanding of the detailed geometry of ACDs.In the conformal coordinates on the Einstein static sphere, the straight lines (3.4) take theform r (cid:48) /
2) cos( φ − φ ) = tan q ≡ k. (3.7) generalized Rindler wedge coordinates do not reach the boundary in a suitable way, and therefore cannotbe used for reconstruction. This made it seem that one needed alternate coordinates on the ACD for bulkreconstructiom (and it was not clear what was the natural choice). This problem goes away now, becausethe premise was in error. I thank Vyshnav Mohan for bringing this to my attention. Note that since this is just flat space in some other coordinates, the isometries still contain Minkowskitime translation. The point however is that in the natural coordinates here, the time direction T is not anisometry. Z Figure 3: The y = 0 slice of the Einstein static sphere (which corresponds to φ = 0 ) where x, y, z are auxiliary embedding coordinates. The poles are at r (cid:48) = 0 (North) and π (South).South pole is the point at infinity of r in M . Changing φ corresponds to rotating thisconfiguration around the z -axis.Note that tan r (cid:48) cos( φ − φ ) = const . corresponds to great circles (geodesics) on the sphere ,and so these curves are not geodesics (naturally). But noting the similarity of the two formssuggests that the easiest way to have an intuition for these curves is to go to the Cartesiancoordinates in which the sphere is embedded. When φ = 0 , we find that these curves arethe intersections of the planes z = xk , (3.8)with the sphere x + y + z = 1 . These are circles, but not great circles. For different valuesof k , the projections of these circles on the x - z plane are plotted in Figure 3.In Figure 1, the Einstein static sphere is portrayed as a disc with its boundary pointsidentified, and on the t = 0 = t (cid:48) plane, the above circles will be represented by the plots of(3.7). These curves connect diametrically opposite points: this is natural because a straightline in space provides two opposite ways to reach the point at infinity, and the fact that thisline is (a finite distance) away from the origin becomes irrelevant at infinity. The schematiccross section on the equatorial plane, of three different causal diamonds is presented in Figure4. We also present the picture of the ACD on a ( t (cid:48) , r (cid:48) , φ ) diagram where r (cid:48) is drawn as aradial coordinate, in Figure 5. Finite IR Cut-off and Code Subspace Structure:
With these, we have almost Remember that r (cid:48) is the polar angle on the Einstein static sphere. BCCOOAA
Figure 4: The “width” of the asymptoctic causal diamond is determined by how close tothe equator its vertex is. A vertex at timelike infinity covers the entire Minkowski diamond,while a vertex “half-way” leads to the usual spherical Rindler wedge. The end points aredetermined by the anglular coordinates of the vertex and are independent of the vertexheight of the ACD. They are diametrically opposite to each other, when there is no radialcut off. The curves are schematic.everything we need to show that the bulk local structure that arises out of reconstructionfrom boundary subregions in flat space is parallel to that in AdS. The key observation of [21]is that the bulk regions that are re-constructed from boundary subregions satisfy certainunion and intersection rules, corresponding to the fact that they arise from unions andintersections at the boundary.It can be seen that an identical structure arises here as well, if we use asymptotic causaldiamonds to define bulk reconstruction together with a fixed (but arbitrary) radial cut-off.We will take this radial cut-off to be r = r cutoff or equivalently r (cid:48) = r (cid:48) cutoff , but we do notexpect the picture to change substantively for large classes of cut-offs. As is evident fromthe figures, the shape of the bulk reconstruction region has changed in AdS compared to flatspace. In flat space they are straight lines as we emphasized before. But as long as the cut-offis finite, the union/intersection structure remains intact. This is entirely analogous to thesituation in AdS. Note that the unions and intersections are trivial to construct. Examplesof the various kinds of configurations discussed in [21] are shown in figures. See our Figure 6which should be compared to figure 3 in [21]. The causal reconstructibility of the bulk pointfrom the conformal boundary is identical in both cases.The discussion of the union/intersection structure is clear in standard polar coordinateswith a cut-off, because we have regions bounded by straight lines and their unions andintersections are intuitive. But it is instructive to also consider the analogous picture in the11igure 5: The shaded region is the “back side” of the asymptotic causal diamond, the regionon the Minkowski diamond bounded by the boundary of this region is the “front" whichwe have not marked as to not clutter the figure. The inner curve connecting diametricallyopposite points is a straight line (spacelike geodesic) in Minkoswki space. The region betweenthis curve and the “lower”semi-circle is the interior of the ACD. As the vertices of the ACDmove closer to the equator of the conformal diagram, the inner curve gets closer to theboundary. The curves in the figure are schematic, their connectivity is what we wish toemphasize. A B A BC x O(Y) (x)
Figure 6: This figure should be compared to figure 3 in [21].12igure 7: Comparison of the conformal coordinates and the standard Minkowski polar co-ordinates. The r = ∞ of polar coordinates gets mapped to r (cid:48) = π . The union/intersectionstructures within the cut-off are isomorphic to that in AdS.conformal Einstein static coordinates: after all, we found in a previous discussion that theACDs there span the same boundary half-circle irrespecive of how deep into the bulk theygo, as long as the angular coordinates of their vertices are the same. This is clearly distinctfrom the AdS situation.But again when we have an IR cut-off, the situation changes dramatically, and is illus-trated in Figure 7. We find that indeed the union/intersection structure is identical to thatwhich was found in [21]. Note also that as the tips of the causal diamonds move up(down)towards the future(past) timelike infinity, the causal diamond sweeps out the bulk regionprecisely once. In particular, this means that the spherical Rindler wedge – the case whenthe vertex is half-way up(down) in the conformal diagram – covers half of the boundary.This is consistent with the requirements of [21] (see eg., eqn. (4.28) in [29]) which requireshalf of the boundary for reconstructing the center of the bulk. Together, these facts showthat the protection against boundary erasures follows a structure isomorphic to that in AdS.One can also think about the above construction in terms of the causal structure ofthe cut-off geometry. For a radial cut-off, this is an Einstein static universe in one lowerdimension ( R × S in the present case). We expect the HRT construction of the entanglementwedge of subregions on this boundary to lead to the same conclusions. What ACDs do is toprovide a class of sub-regions that are natural from the perspective of the underlying casual13igure 8: Sliding the cut-off outward demonstrates that scrambling never stops in flat space.structure. Scrambled Subregions and What Makes Flat Space Different:
There is onesubstantive way in which the above picture differs from AdS, however. The key point isthat unlike in AdS, the sub-regions we have identified do not have a simple limit when thecut-off is taken to infinity. This should be clear from Figure 8. In AdS, the subregions of theconformal boundary when the cut-off has been taken to infinity give us a God-given set ofboundary subregions. This is related to the fact that gravity has decoupled at the boundaryand therefore there is a canonical notion of sub-regions/tensor factors. Here we do not havesuch a simple limit. This is related to the fact we previously noted, that the ACDs (andtherefore holographic data) spreads out all over half of the celestial sphere when the cut-offis taken to infinity. We expect that this is related to the non-locality of the dual theory andto the fact that the holographic entanglement entropy in flat space scales with the volume.
Integral Geometry and Kinematic Space:
Let us take a moment here to notethat the spacelike straight lines that bound our ACDs should be compared to the space ofspacelike bulk geodesics that were used to define the CFT kinematic space in [14]. The worksof [14, 15, 13] and various follow-ups, developed this notion of a CFT kinematic space, whichin the AdS/CFT context can be viewed as the space of causal diamonds in the Minkowskispace where the CFT lives. In flat space, since the conformal boundary does not have a well-14efined causal structure, this perspective needs change. Interestingly, our proposal gives anatural bulk construction of the boundary kinematic space, for holography in flat space. Itis noteworthy that the space of (suitably chosen) pairs of points seems to have a natural rolein flat space as well.Adapting results of [15, 13], one can write down the metric on the space of our asymptoticcausal diamonds, and potentially connect the wave equation on it, with the Casimirs of thePoincare group. This is clearly a direction that has substantial potential for development,especially in light of the fact that Poincare is a contraction of the conformal group. We willexplore this in more detail in [20]. It will be interesting to see if we can define some versionof a Poincare OPE block in a way analogous to the (conformal) OPE block defined in [15].We expect also that the Kirillov-Kostant form on the co-adjoint orbit of the Poincare groupshould be related to the Crofton form on our kinematic space, in analogy with the AdSresults of [30].
Holographic Entanglement Entropy:
Let us conclude this section by noting thatagain at finite cut-off the spacelike lines (or hyperplanes in higher dimensions) match thepreviously noted Ryu-Takayanagi surfaces of flat space: see [32] for a Euclidean discussionand section 7 of [33] for a Lorentzian discussion which is closer in spirit to ours. Since theseare minimal surfaces, the proofs of strong sub-additivity [34, 35] goes through as it does inAdS [36]. In fact in our 2+1 dimensional bulk, both the strong subadditivity statementsregarding boundary sub-regions S A + B + S B + C ≥ S A + B + C + S B and S A + B + S B + C ≥ S A + S C turn into the statement that the sum of diagonals of a cyclic quadrilateral is greater than thesum of any two opposite sides: an immediate consequence of triangle inequality. It might beinteresting to investigate other entangelement entropy inequalities in this setting. We willnot pursue this line further since we do not have a separate definition of the dual theory,and have nothing to compare with. Note also that the holographic entanglement entropyhere scales with the volume (of the cut-off) as it should, since the holographic dual of flatspace is expected to have some non-local features. We will briefly comment on this again inthe next section. This paper is fairly broad in scope, and many directions of exploration automaticallypresent themselves. We will only comment on a few limited aspects in this section.In some of our discussions above, an IR cut off played an interesting role. The precisenature of this cut-off was not too important for us (in the sense that we expect large classes of Let us emphasize that it is the bulk IR cut off that we are referring to. .We feel this perspective can be instructive, even if only quantities which have well-definedlimits as the cut-off is tken to infintiy are ultimately what we are after.To calculate holographic correlators in AdS, a finite IR cut-off has been quite useful fromthe beginning days of AdS/CFT (see eg., [42]). This is in spite of the fact that it is only in thestrict z → limit of the AdS metric, that the AdS isometries reduce to conformal isometriesat the boundary. At finite values of the cut-off, the bulk isometries move the cut-off , justas they do in flat space. An IR box for flat space has many parallels also to the discussionsthat arise in the context of AdS scattering amplitudes [45, 46]. There it is known thatcertain boundary correlators in the AdS box have a natural flat space S-matrix like limit.Even more strikingly, there is evidence there for the emergence of analogues of soft theorems[47]. We will view this as a suggestion that the soft-theorem/BMS/null-infinity aspects offlat space should be viewed as a sub-sector of the full set of holographic correlators that onecan define in flat spacetime with an IR cut-off. Yet another indication that an IR cut offis worth exploring comes from [49], where fairly detailed evidence was presented that thethermal aspects of AdS black holes have natural analogues for flat space with box boundaryconditions. Our discussions in this paper extend these previous considerations and presentsevidence that parallel structures exist also for causal/local aspects of the reconstructed bulkin flat space.To summarize, even if one’s ultimate goal is the limit where the “cut off has gone toinfinity”, it might be useful to study boundary correlation functions for flat space with afinite cut-off. We will view this as a way to describe an isolated gravitating region in aholography-compatible way.We will have more to say about the radial cut-off in future work [19], but for our presentpurposes, we have used r = R cut in standard spherical polar coordinates . Note in particular Most investigations in flat space (eg., on soft theorems and BMS invariance) restrict their attention evenfurther, to massless particles and the null boundary. See discussions and references in [37, 38, 39, 40, 41] fora point of entry into various perspectives. There are many interesting and clearly important ideas here, butwe do not think a framework based entirely on the null boundary is a candidate for a complete descriptionof holography in flat space, which should include massive particles. This is perhaps best viewed as a manifestation of the inexact decoupling between closed and open stringmodes as one moves away from the strict z → limit. A related observation is that a fully satisfactoryholographic renormalization group has been difficult to formulate in AdS/CFT, see [43, 44] for interestingattempts to make the scale-radius duality precise. Another natural candidate is to use an Ashtekar-Hansen [48] type radial coordinate. r -coordinate in the ( t, r ) coordinate grid hasthe nice property that the r → ∞ limit takes us to the entire conformal boundary of thefull Minkowski causal diamond. This suggests that a natural box where the holographicdual of flat space can live is on the Einstein static universe at r = R cut with coordinates ( t, Ω d − ) where Ω d − stands for the angular coordinates of d + 1 dimensional asymptoticallyflat space. It will be interesting to calculate the boundary correlators on this space fromthe bulk at finite cut-off following adaptations of the standard GKP-W prescription [4, 5],and then see what interesting information can be extracted when taking suitable R cut → ∞ limits. We expect that a suitably defined set of boundary correlators of this type (or perhapssuitably identified S-matrix elements) will exhibit the full Poincare invariance of the bulk .Note also that the intersection of the asymptotic causal diamond with the cut-off surfaceoffers a natural notion of causal structure on the cut-off surface .One potential upshot of this discussion is to study field theory on the Einstein staticuniverse [26, 27, 28] that gets suitably “frozen" as the radius of the sphere becomes infinite,and whose appropriate correlators attain an enhanced bulk Poincare invariance in this limit.S-matrix elements might be related to such correlators. It will be interesting to connect thiswith the discussions in [50, 51] which note that there is entanglement on the celestial spherein flat space. It has been suggested that these theories have non-propagating local degreesof freedom, but non-local constraints and non-trivial entanglement. See, [52, 53]. This isalso related to our previous discussions of holographic entanglement entropy in flat space.In flat space, with only marginal extra baggage we were able to reproduce many of thestructures that arose in AdS. Even though the ACD perspective has not been emphasized inAdS (because the kinematic space has many equivalent definitions there), one takeaway ofour work is that it works equally well in both flat space and AdS. Let us make one commentabout what is it that asymptotic causal diamonds capture. In AdS they directly capturea boundary subregion, which one can think of as a tensor factor of the Hilbert space ofquantum gravity. A causal diamond encodes this tensor factor in a covariant way. In flatspace, similar statements might hold: an ACD gives us an abstract and covariant way tocapture the degrees of freedom of quantum gravity. This begs the question: what aboutother spacetimes? See [54, 55, 56, 57, 58] for some recent discussions about various aspectsof boundaries in holographic settings.Interestingly, there exists a perfectly well-defined notion of an asymptotic causal diamondin de Sitter space as well: they are defined by a point each, on the future and past boundaries. One of the problems with the null boundary is that it has no natural causal structure. Note however thetrivial (but possibly useful) fact that not all curves on a null hypersurface are null. Eg: take a light cone,cut it by a t = constant surface. The resulting curve on the light cone is a spacelike circle. As the reader will notice, the rub lies in the word “capture”. To make it precise will require new ideas. r and t coordinates inthe Penrose diagram. It will be very interesting to develop this fully. Another direction thatnaturally presents itself, since pairs of points play a distinguished role in our construction,is that of bi-local holography of Das and Jevicki [64, 65]. One thing we have not mentionedat all in this paper is the connection between causal diamonds and tensor networks. This isclearly an idea worth exploring. See eg. [68, 69]. Acknowledgments
I thank Budhaditya Bhattacharjee, (especially) K. V. Pavan Kumar, Alok Laddha, RaghuMahajan, Vyshnav Mohan, Aninda Sinha, Ronak Soni and Amandeep Singh for discussions.I am particularly indebted to Aman for creating the pdf versions of my hand-drawn figures.I also thank the usual suspects at TIFR for stimulating questions and comments during atalk based on this material. 18
Non-Standard PDE Data and Holography
Let us comment on a few points regarding bulk reconstruction from the ACD . We willonly discuss the scalar field in a fixed background below, which is the usual context of theHKLL like bulk reconstruction. Dynamical gravity introduces extra subtleties, which arenot crucial for our purposes here.It is best to not mix up the following two questions: • Is it possible to determine the scalar field in (some region of) the bulk if we are giventhe field and its normal derivative (as required for a second order PDE), on some time -like surface (say, r = R cut ) near the boundary ? A closely related question is, if thereis such a region, what characterizes it? • If the answer to the above question is yes, what are the constraints on such “Cauchy”data for it to be suitable for describing holography?In AdS, that the answer is affirmative to the first question is an implicit (and often un-emphasized) message of HKLL . The explicit message of HKLL is the answer to the secondquestion: the two pieces of “Cauchy” data in AdS are to be taken as the non-normalizablemode which we are instructed to set to zero, and the normalizable mode which we are freeto specify. Given this “Cauchy” data, HKLL gives us an explicit construction of the bulkscalar field in terms of this “Cauchy” data.Our claim is that in both AdS and flat space, for data provided on (an appropriatetimelike cut-off of) an asymptotic causal diamond, the answer is "yes" to the first question.We will not try to answer the second question in this paper. All we are concerned with is thequestion of reconstructibility, given two appropriate pieces of “Cauchy” data. What furtherconstraints should those two pieces satisfy for capturing various aspects of holography, is aquestion we will come back to in future work.Now, let us provide some circumstantial evidence for the above claim for reconstructibilityfrom a timelike surface. This is a somewhat non-standard type of data for second orderhyperbolic PDEs, and even though the problem feels fairly basic, we have not been able tofind references that deal precisely with the type of problem we are after . The usual Cauchyboundary conditions involve data on a spacelike slice. A comment from Sandip Trivedi has influenced some of my thoughts in this section. I thank GautamMandal, Shiraz Minwalla and Sandip Trivedi for discussions. We will call such data, “Cauchy” data with quotes, to differentiate it from the unquoted Cauchy data,which is given on spacelike slices. Note that a Rindler-AdS wedge associated to a (spherical) boundary subregion in which we can do thisreconstruction is the natural notion of an asymptotic causal diamond of AdS. This does not necessarily mean that they do not exist. The trouble is partly that it is hard to find . The intersection of this surface with the interior of the causal diamond, wewill call a timelike cross-section of the causal diamond . We claim that the "Cauchy" data,aka. the value of the field and its derivative on a timelike cross-section, is precisely enoughdata for reconstructing the field everywhere inside the causal diamond. In 1+1 dimensionsthis is straightforward to see, for massless scalar waves. We can simply use the fact thatflipping t ↔ r only results in an overall sign change of the wave equation and the fact thattimelike slices are Cauchy surfaces for evolution along r . Another (more explicit) way tocome to the same conclusion is to use d’Alembert’s solution to the 1+1 dimensional waveequation (see for example eq. (6.41) in [66]) and express the "Cauchy" data in terms of theCauchy data (ie., the functions φ and v in [66]). It can be seen that this is a bijection.This is also related to the uniqueness properties of solutions of the wave equation in theirdomains. Generalizations of d’Alembert’s solution exist in all dimensions. One can alsorelate the spacelike and timelike data by using Fourier series to deompose the solution.We expect that versions of these arguments should hold also in higher dimenions, andalso when there is mass [67]. One reason to suspect this is that the standard obstructionto development of a hyperbolic PDE from a (non-null) surface for "Cauchy" data is theexistence of "charatecteristic surfaces". These are basically places where the normal to thesurface becomes zero norm, aka these are just null surfaces.Based on these, we conjecture that in both AdS and flat space, the (suitably regu-lated) outer boundary of an asymptotic causal diamond (ACD) is a region where if you give"Cauchy" data, you can do evolution into the spacelike bulk, and the region in which youcan do this reconstruction, is precisely what defines an ACD. This is of course, demonstrablytrue in AdS thanks to the Ridler-wedge version of the HKLL construction. It will be veryinteresting to do something similar explicitly in flat space [67]. References [1] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C , 284(1993) [gr-qc/9310026]. suitable keywords for google, that zero in precisely on what we are after, especially on a topic as old andvast as PDEs. Letting the slice pass through the center is merely a matter of convenience: we expect data on anytimelike slice to work. Note the crucial fact that when a causal diamond goes to infintiy and becomes an ACD, the center ofthe diamond tends to the boundary! So if we can re-construct the diamond from its central time-like slice,it is natural to expect that ACDs can re-construct the bulk. , 6377 (1995)doi:10.1063/1.531249 [hep-th/9409089].[3] J. M. Maldacena, “The Large N limit of superconformal field theories and supergrav-ity,” Int. J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)]doi:10.1023/A:1026654312961, 10.4310/ATMP.1998.v2.n2.a1 [hep-th/9711200].[4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B , 105 (1998) doi:10.1016/S0370-2693(98)00377-3[hep-th/9802109].[5] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253(1998) doi:10.4310/ATMP.1998.v2.n2.a2 [hep-th/9802150].[6] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model:A Conjecture,” Phys. Rev. D , 5112 (1997) doi:10.1103/PhysRevD.55.5112 [hep-th/9610043].[7] S. El-Showk and K. Papadodimas, “Emergent Spacetime and Holographic CFTs,” JHEP , 106 (2012) doi:10.1007/JHEP10(2012)106 [arXiv:1101.4163 [hep-th]].[8] R. Bousso, “A Covariant entropy conjecture,” JHEP , 004 (1999) doi:10.1088/1126-6708/1999/07/004 [hep-th/9905177].[9] R. Bousso, “Holography in general space-times,” JHEP , 028 (1999)doi:10.1088/1126-6708/1999/06/028 [hep-th/9906022].[10] W. Fischler and L. Susskind, “Holography and cosmology,” hep-th/9806039.[11] T. Banks and W. Fischler, “An Holographic cosmology,” hep-th/0111142.[12] V. E. Hubeny, M. Rangamani and T. Takayanagi, “A Covariant holographic entangle-ment entropy proposal,” JHEP , 062 (2007) doi:10.1088/1126-6708/2007/07/062[arXiv:0705.0016 [hep-th]].[13] J. de Boer, F. M. Haehl, M. P. Heller and R. C. Myers, “Entanglement, hologra-phy and causal diamonds,” JHEP , 162 (2016) doi:10.1007/JHEP08(2016)162[arXiv:1606.03307 [hep-th]].[14] B. Czech, L. Lamprou, S. McCandlish and J. Sully, “Integral Geometry and Holography,”JHEP , 175 (2015) doi:10.1007/JHEP10(2015)175 [arXiv:1505.05515 [hep-th]].2115] B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, “A Stereoscopic Look intothe Bulk,” JHEP , 129 (2016) doi:10.1007/JHEP07(2016)129 [arXiv:1604.03110[hep-th]].[16] A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Local bulk operators inAdS/CFT: A Boundary view of horizons and locality,” Phys. Rev. D , 086003 (2006)doi:10.1103/PhysRevD.73.086003 [hep-th/0506118].[17] A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Holographic representation oflocal bulk operators,” Phys. Rev. D , 066009 (2006) doi:10.1103/PhysRevD.74.066009[hep-th/0606141].[18] I. A. Morrison, “Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schliederproperty in holography,” JHEP , 053 (2014) doi:10.1007/JHEP05(2014)053[arXiv:1403.3426 [hep-th]].[19] B. Bhattacharjee and C. Krishnan, “Holography in Cut-Off Flat Space", in progress.[20] C. Krishnan, K. V. P. Kumar, V. Mohan and A. Singh, “The Kinematics of Holographyin Flat space", in progress.[21] A. Almheiri, X. Dong and D. Harlow, “Bulk Locality and Quantum Error Correctionin AdS/CFT,” JHEP , 163 (2015) doi:10.1007/JHEP04(2015)163 [arXiv:1411.7041[hep-th]].[22] F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, “Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence,” JHEP ,149 (2015) doi:10.1007/JHEP06(2015)149 [arXiv:1503.06237 [hep-th]].[23] T. Kohler and T. Cubitt, “Complete Toy Models of Holographic Duality,”arXiv:1810.08992 [hep-th].[24] V. Balasubramanian, B. Czech, B. D. Chowdhury and J. de Boer, “The entropy of a holein spacetime,” JHEP , 220 (2013) doi:10.1007/JHEP10(2013)220 [arXiv:1305.0856[hep-th]].[25] J. Maldacena, “Vacuum decay into Anti de Sitter space,” arXiv:1012.0274 [hep-th].[26] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,”doi:10.1017/CBO9780511622632[27] D. Sen, “Fermions in the Space-time R X S**3,” J. Math. Phys. , 472 (1986).doi:10.1063/1.527246 2228] D. Sen, “Supersymmetry in the Space-time R X S**3,” Nucl. Phys. B , 201 (1987).doi:10.1016/0550-3213(87)90033-2[29] D. Harlow, “TASI Lectures on the Emergence of Bulk Physics in AdS/CFT,” PoS TASI , 002 (2018) doi:10.22323/1.305.0002 [arXiv:1802.01040 [hep-th]].[30] R. F. Penna and C. Zukowski, “Kinematic space and the orbit method,”arXiv:1812.02176 [hep-th].[31] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy fromAdS/CFT,” Phys. Rev. Lett. , 181602 (2006) doi:10.1103/PhysRevLett.96.181602[hep-th/0603001].[32] W. Li and T. Takayanagi, “Holography and Entanglement in Flat Spacetime,” Phys.Rev. Lett. , 141301 (2011) doi:10.1103/PhysRevLett.106.141301 [arXiv:1010.3700[hep-th]].[33] X. L. Qi and Z. Yang, “Butterfly velocity and bulk causal structure,” arXiv:1705.01728[hep-th].[34] E. H. Lieb and M. B. Ruskai, “A Fundamental Property of Quantum-Mechanical En-tropy,” Phys. Rev. Lett. , 434 (1973). doi:10.1103/PhysRevLett.30.434[35] E. H. Lieb and M. B. Ruskai, “Proof of the strong subadditivity of quantum-mechanicalentropy,” J. Math. Phys. , 1938 (1973). doi:10.1063/1.1666274[36] M. Headrick and T. Takayanagi, “A Holographic proof of the strong subadditivity of en-tanglement entropy,” Phys. Rev. D , 106013 (2007) doi:10.1103/PhysRevD.76.106013[arXiv:0704.3719 [hep-th]].[37] G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP ,062 (2010) doi:10.1007/JHEP05(2010)062 [arXiv:1001.1541 [hep-th]].[38] B. Oblak, “BMS Particles in Three Dimensions,” doi:10.1007/978-3-319-61878-4arXiv:1610.08526 [hep-th].[39] A. Bagchi, R. Basu, A. Kakkar and A. Mehra, “Flat Holography: Aspects of the dualfield theory,” JHEP , 147 (2016) doi:10.1007/JHEP12(2016)147 [arXiv:1609.06203[hep-th]].[40] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,”arXiv:1703.05448 [hep-th]. 2341] A. Laddha and A. Sen, “Sub-subleading Soft Graviton Theorem in Generic The-ories of Quantum Gravity,” JHEP , 065 (2017) doi:10.1007/JHEP10(2017)065[arXiv:1706.00759 [hep-th]].[42] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Correlation functions in theCFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B , 96 (1999) doi:10.1016/S0550-3213(99)00053-X [hep-th/9804058].[43] I. Heemskerk and J. Polchinski, “Holographic and Wilsonian Renormalization Groups,”JHEP , 031 (2011) doi:10.1007/JHEP06(2011)031 [arXiv:1010.1264 [hep-th]].[44] T. Faulkner, H. Liu and M. Rangamani, “Integrating out geometry: Holo-graphic Wilsonian RG and the membrane paradigm,” JHEP , 051 (2011)doi:10.1007/JHEP08(2011)051 [arXiv:1010.4036 [hep-th]].[45] J. Polchinski, “S matrices from AdS space-time,” hep-th/9901076.[46] J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,” JHEP , 025 (2011) doi:10.1007/JHEP03(2011)025 [arXiv:1011.1485 [hep-th]].[47] A. L. Fitzpatrick and J. Kaplan, “AdS Field Theory from Conformal Field Theory,”JHEP , 054 (2013) doi:10.1007/JHEP02(2013)054 [arXiv:1208.0337 [hep-th]].[48] A. Ashtekar and R. O. Hansen, “A unified treatment of null and spatial infinity in generalrelativity. I - Universal structure, asymptotic symmetries, and conserved quantities atspatial infinity,” J. Math. Phys. , 1542 (1978). doi:10.1063/1.523863[49] P. Basu, C. Krishnan and P. N. B. Subramanian, “Hairy Black Holes in a Box,” JHEP , 041 (2016) doi:10.1007/JHEP11(2016)041 [arXiv:1609.01208 [hep-th]].[50] H. Jiang, W. Song and Q. Wen, “Entanglement Entropy in Flat Holography,” JHEP , 142 (2017) doi:10.1007/JHEP07(2017)142 [arXiv:1706.07552 [hep-th]].[51] A. Bagchi, R. Basu, D. Grumiller and M. Riegler, “Entanglement entropy in Galileanconformal field theories and flat holography,” Phys. Rev. Lett. , no. 11, 111602(2015) doi:10.1103/PhysRevLett.114.111602 [arXiv:1410.4089 [hep-th]].[52] E. Hijano and C. Rabideau, “Holographic entanglement and PoincarÃľ blocks inthree-dimensional flat space,” JHEP , 068 (2018) doi:10.1007/JHEP05(2018)068[arXiv:1712.07131 [hep-th]].[53] E. Hijano, “Semi-classical BMS blocks and flat holography,” JHEP , 044 (2018)doi:10.1007/JHEP10(2018)044 [arXiv:1805.00949 [hep-th]].2454] C. Krishnan and A. Raju, “A Neumann Boundary Term for Gravity,” Mod. Phys. Lett.A , no. 14, 1750077 (2017) doi:10.1142/S0217732317500778 [arXiv:1605.01603 [hep-th]].[55] C. Krishnan, K. V. P. Kumar and A. Raju, “An alternative path integral for quan-tum gravity,” JHEP , 043 (2016) doi:10.1007/JHEP10(2016)043 [arXiv:1609.04719[hep-th]].[56] C. Krishnan, A. Raju and P. N. Bala Subramanian, “Dynamical boundary for anti deSitter space,” Phys. Rev. D , no. 12, 126011 (2016) doi:10.1103/PhysRevD.94.126011[arXiv:1609.06300 [hep-th]].[57] C. Krishnan, S. Maheshwari and P. N. Bala Subramanian, “Robin Gravity,” J.Phys. Conf. Ser. , no. 1, 012011 (2017) doi:10.1088/1742-6596/883/1/012011[arXiv:1702.01429 [gr-qc]].[58] C. Krishnan, R. Shekhar and P. N. Bala Subramanian, “A Hairy Box in Three Dimen-sions,” arXiv:1905.11265 [gr-qc].[59] “Bulk Reconstruction from the Black Hole horizon", in progress.[60] S. Raju, “Smooth Causal Patches for AdS Black Holes,” Phys. Rev. D , no. 12, 126002(2017) doi:10.1103/PhysRevD.95.126002 [arXiv:1604.03095 [hep-th]].[61] V. Balasubramanian, P. Kraus and A. E. Lawrence, “Bulk versus bound-ary dynamics in anti-de Sitter space-time,” Phys. Rev. D , 046003 (1999)doi:10.1103/PhysRevD.59.046003 [hep-th/9805171].[62] V. Balasubramanian, P. Kraus, A. E. Lawrence and S. P. Trivedi, “Holo-graphic probes of anti-de Sitter space-times,” Phys. Rev. D , 104021 (1999)doi:10.1103/PhysRevD.59.104021 [hep-th/9808017].[63] D. Marolf, “States and boundary terms: Subtleties of Lorentzian AdS / CFT,” JHEP , 042 (2005) doi:10.1088/1126-6708/2005/05/042 [hep-th/0412032].[64] S. R. Das and A. Jevicki, “Large N collective fields and holography,” Phys. Rev. D68