Wormhole Spaces: the Common Cause for the Black Hole Entropy-Area Law,the Holographic Principle and Quantum Entanglement
aa r X i v : . [ h e p - t h ] O c t Wormhole Spaces: the Common Causefor the Black Hole Entropy-Area Law,the Holographic Principle andQuantum Entanglement
Manfred Requardt
Institut f¨ur Theoretische PhysikUniversit¨at G¨ottingenFriedrich-Hund-Platz 137077 G¨ottingen Germany(E-mail: [email protected])
Abstract
We present strong arguments that the deep structure of the quantumvacuum contains a web of microscopic wormholes or short-cuts. We de-velop the concept of wormhole spaces and show that this web of wormholesgenerate a peculiar array of long-range correlations in the patterns of vac-uum fluctuations on the Planck scale. We conclude that this translocalstructure represents the common cause for both the BH-entropy-area law,the more general holographic principle and the entanglement phenomenain quantum theory. In so far our approach exhibits a common structurewhich underlies both gravity and quantum theory on a microscopic scale.A central place in our analysis is occupied by a quantitative derivationof the distribution laws of microscopic wormholes in the quantum vac-uum. This makes it possible to address a number of open questions andcontroversial topics in the field of quantum gravity.
Introduction
In the following we want to give a new explanation of the area law of blackhole (BH) entropy and the more general and stronger holographic principle .Furthermore, we provide (in our view) convincing arguments that an importantstructural ingredient of the deep structure of our quantum vacuum is a networkof microscopic wormholes. In contrast to e.g. string theory and loop quantumgravity (LQG), which both employ the quantum laws more or less unalteredall the way down to the remote
Planck scale , we regard this as an at leastdebatable assumption. We rather view the holographic hypothesis as a means tounderstand how both quantum theory and gravitation do emerge as derived andsecondary theories from a more fundamental theory living on a more microscopicscale. A central role in this enterprise is played by an analysis of the microscopicstructure of the quantum vacuum which leads to the key concept of wormholespaces .This important conceptual structure makes it possible to understand theholographic aspects of quantum gravity, on the one hand, and the (non-local)entanglement phenomena pervading ordinary quantum physics, on the otherhand, in a relatively natural way. Furthermore we think that there exist linksto the old ideas of e.g. Sakharov and Zeldovich, dubbed induced gravity (see forexample [1],[2],[3],[4]).Some words are in order regarding the relation of our investigation to theanalysis of BH entropy in, say, string theory. Three scenarios are in our viewin principle possible. Either, both approaches adress the same phenomena indifferent languages, or they deal with them on different scales of resolution ofspace-time. Be that as it may, we think that our observation that the trueground state of our quantum vacuum seems to be what we call a wormholespace (see section 3) is an aspect which is not apparent in the original stringtheory approach and may be helpful to fix the proper ground state in stringtheory.In [5] Bekenstein remarked that the deeper meaning of black-hole entropy(BH-entropy) remains mysterious. He asks, is it similar to that of ordinaryentropy, i.e. the log of a counting of internal BH-states, associated with asingle BH-exterior? ([6],[7] or [8]). Or, similarly, is it the log of the numberof ways, in which the BH might be formed. Or is it the log of the number ofhorizon quantum states? ([9],[10]). Does it stand for information, lost in thetranscendence of the hallowed principle of unitary evolution? ([11],[12]). Hethen claims that the usefulness of any proposed interpretation of BH-entropydepends on how well it relates to the original “statistical” aspect of entropy as ameasure of disorder, missing information, multiplicity of microstates compatiblewith a given macrostate, etc.Quite a few workers in the field argue that the peculiar dependence of BH-entropy on the area of the event horizon points to the fact that the degrees offreedom (DoF), responsible for BH-entropy, are situated near the event horizon.This seems to be further corroborated by the corresponding behavior of the so-called entanglement entropy, i.e. its (apparent) linear dependence on the area1f the dividing surface (cf., just to mention a few sources, [13],[14],[15] or thelively debate in [16], concerning entanglement entropy in a more general set-ting, [17],[18]). This linear dependence does however not generally hold withoutfurther qualifications. It does in particular not hold for excited states (see [19])!That is, while some particular sort of entanglement certainly plays an impor-tant role in this context, the real question is in our view the scale of resolutionof space-time where this entanglement becomes effective and the nature of thequantum vacuum on this level of resolution.Remark: We want to emphasize the in our view crucial (but frequently appar-ently not fully appreciated) point that the entropy content of a BH is maximal.We think, the usual version of entanglement, we observe on the scales of or-dinary quantum theory, is only an epiphenomenon , representing rather thecoarse-grained effect of a hidden structure which lives on a much more mi-croscopic scale. I.e., we are sceptical whether on such a microscopic scale thequantum vacuum can still be treated in the way of an ordinary quantum fieldtheory vacuum as suggested in some of the papers cited above. We think, the maximum-entropy property of the BH-interior suggests another interpretation.We will come back to this point in more detail in section 4 (cf. also the scep-tical remarks in some of the review papers by Wald, e.g. [20] (see in particularsect.6, Open Issues), [21] (see in particular sect.4, Some unresolved Issues andPuzzles],[22]).As BH-entropy is widely regarded as an observational window into the morehidden and primordial quantum underground of space-time, it should be ex-pected that it can be naturally explained within the frameworks of the leadingcandidates of such a theory, i.e., to mention the most prominent, string theoryor LQG. For certain extreme situations string theory manages to give an expla-nation of the BH-entropy-area law. Whether the explanation is really naturalis perhaps debatable (it relies in fact on a number of assumptions and corre-spondences as e.g. peculiar intersections of various classes of p-branes). In asense, it is rather a correspondence between BH-behavior and the configura-tional entropy of certain string states. To mention some representative papers,[23],[24],[25],[26],[27],[28]. In LQG, on the other hand, it is assumed from theoutset (at least as far as we can see) that the corresponding DoF are sittingat the BH-horizon. Therefore the observed area dependence of BH-entropy isperhaps not so surprising (cf. e.g. [29],[30]).In the enumeration of the most promising candidates for a theory of quan-tum gravity one approach is usually left out which, we nevertheless think, hasa certain potential. One may, for example, tentatively divide quantum grav-ity candidates into roughly three groups, the relativisation of quantum theory(with e.g. LQG and causal set theory as members), the quantisation of generalrelativity (string theory being a prominent candidate) or third, theories whichunderlie both general relativity and quantum theory but are in fact more fun-damental and structurally different from both and contain these two pillars ofmodern physics as derived and perhaps merely effective sub-theories, living oncoarser scales (cf. e.g. [31]). In the following we want to develop such a model2heory in more detail.As far as we can see, such a philosophy is also shared by ‘t Hooft whoemphasized this point in quite a few papers (see e.g. [32],[33],[34],[35]). Wequote from [33]: . . . it may still be possible that the quantum mechanical nature of thephenomenological laws of nature at the atomic scale can be attributedto an underlying law that is deterministic at the Planck scale but withchaotic effects at all larger scales. . . Since, according to our philosophy,quantum states are identified with equivalence classes. . .
Furthermore: . . . It is the author’s suspicion however, that these hidden variable theoriesfailed because they were based far too much upon notions from everydaylife and ‘ordinary physics’ and in particular because general relativisticeffects have not been taken into account properly.
While ’t Hooft usually chooses his model theories from the cellular automa-ton (CA) class, we are adopting a point of view which is on the one hand moregeneral and flexible but, on the other hand, technically more difficult and com-plex. Instead of a relatively rigid underlying geometric substratum in the case ofCA (typically some fixed regular lattice) on which the CA are evolving accord-ing to a given fixed (typically local) CA-law, we are employing quite irregular,dynamic geometric structures called by us cellular networks , the main pointbeing that connections ( edges or links ) between the respective nodes or cells can be created or annihilated according to a dynamical law which, in addition,determines the evolution of the local node- and edge-states.To put it briefly, the ‘matter distribution’ (i.e. the global pattern of node-states) acts on the geometry of the network (the global pattern of active edges)and vice versa. Thus, as in general relativity, the network is supposed to findboth its internal geometry and its matter-energy distribution with the help ofa generalized dynamical law which intertwines the two aspects (cf. e.g. [36] or[37] and further references given there). Technically, the geometric substructurecan be modelled by large, usually quite irregular ( random ) graphs .To make our point clear, this approach should not be confused with e.g.the spin network approach in LQG or various forms of (dynamical) triangula-tions. Our networks are usually extremely irregular and wildly fluctuating ona microscopic scale, resembling rather Wheeler’s space-time foam , and smoothgeometric structures (as e.g. dimensional notions) are hoped to emerge via somesort of a geometric renormalisation process (in fact a very particular organizedform of coarse-graining steps). Some of the interesting deeper mathematicalaspects can for example be looked up in [38].In our dynamical network approach to quantum space-time physics the nodesare assumed to represent cells of some microscopic size (presumably Planck size),the internal details of which cannot be further resolved in principle or are ig-nored and averaged over for convenience and will be represented instead by a3imple ansatz for a local (node) state. It can perhaps be compared with themany existing spin-models which are designed to implement certain characteris-tic features of complex solids. This is more or less the same philosophy as in theCA-framework. The elementary connections between the nodes (the edges ingraph theory) are assumed to represent elementary interactions or informationchannels among the cells and also carry simple edge-states . We made a detailednumerical analysis of the behavior of such networks in [39].Remark: We would like to emphasize however, that our approach does not re-ally rely on this particular framework. It rather serves as a means to illustratethe various steps in our analysis within a concrete model theory.The paper is organized as follows. In the next section we analyse the basicsubstratum, i.e. the microscopic patterns of vacuum fluctuations, in particularthe negative energy fluctuations. In section 3 we describe the three differentroads which lead (in our view: inevitably) to the concept of wormhole space .The preparatory sections 2 and 3 are then amalgamated in section 4 into a de-tailed analysis of the microscopic distribution pattern of short-cuts or wormholes and their consequences for the number of effective DoF in a volume of space.We introduce a new type of dimension, the so-called holographic dimension .Furthermore, we explain the microscopic basis of the holographic principle ingeneral and the bulk-boundary correspondence between the DoF in the interiorof e.g. a BH and the DoF on the boundary. Some apparent counter examplesconcerning the area-scaling property (see e.g. [92]; Wheeler’s ’bag of gold’-spacetimes ) are very briefly addressed. In the last section we briefly commenton a number of immediate applications of our microscopic holographic approachand (open) problems which can be settled with the help of our framework.
A characteristic feature of the dynamical network models we investigated istheir undulatory character. As a consequence of the feedback structure of thecoupling between node (cell) states and wiring diagram of edges (i.e. the pat-tern of momentary elementary connections or interactions) the network neversettles in a static, frozen final state. The network may of course end up in some attracting subset of phase space but typical are wild fluctuations on a small (mi-croscopic) scale with possibly some macroscopic patterns emerging on a coarserscale forming some kind of superstructure (see e.g. [39]).It is in our view not sufficiently appreciated that, in contrast to most of theother systems being studied in physics, the quantum vacuum is in a state of eter-nal unrest on a microscopic scale, with, for all we know, short-lived excitationsconstantly popping up and being reabsorbed by the seething sea.It therefore seems reasonable to regard our above network model (investi-gated in e.g. [36] to [39]) as a (toy) model of the quantum vacuum with the4nergy-momentum fluctuations on short scales being associated with the fluc-tuations of the local node and edge states.
Postulate 2.1
In the following we adopt the working hypothesis of a parallelismof network behavior and microscopic behavior of the quantum vacuum.
We now come to a detailed analysis of the microscopic pattern of vacuumfluctuations. In sect.4 of [40] we made a calculation which shows that, given thehuge number of roughly Planck-size grains in a macroscopic piece of space andassuming that the individual grains are allowed to fluctuate almost indepen-dently, more precisely, some grain variable like e.g. the local energy, the totalfluctuations in a macroscopic or mesoscopic piece of space of typical physicalquantities are still so large (i.e. macroscopic) that they should be observable.Note that with the number of nodes of roughly Planck-size, N P , in a macroscopicvolume, V , being gigantic, its square root is still very large (for the details of theargument see [40]). More precisely, with q i some physical quantity belongingto a microscopic grain (e.g. energy, momentum, some charge etc. and takingfor convenience h q i i =0) and Q V := P i q i being the observable belonging to thevolume V , the fluctuation of the latter behaves under the above assumption as h Q V Q V i / ∼ ( V /l p ) / (1)with N P ∼ V /l p the number of grains in V . This is a consequence of the central limit theorem . As such large integrated fluctuations in a macroscopicregion of the physical vacuum are not observed (they are in fact microscopic onmacroscopic scales), we conclude: Conclusion 2.2
The individual grains or supposed elementary DoF do not fluc-tuate approximately independently.
Remark: We note that this fact is also corroborated by other, independentobservations.We can refine the result further (cf. [40]) by assuming that the fluctuationsin the individual grains are in fact not independent but correlated over a certaindistance or, more precisely, are short-range correlated . In mathematical formthis is expressed as integrable correlations . This allows that “positive” and“negative” deviations from the mean value can compensate each other moreeffectively. Letting e.g. q ( x ) be the density of a certain physical observable and Q V := R V q ( x ) d n x the integral over V . In order that h Q V Q V i / ≪ V / (2)we proved in [40] that it is necessary that Z V d n y h q ( x ) q ( y ) i ≈ onclusion 2.3 Nearly vanishing fluctuations in a macroscopic volume, V ,together with short-range correlations imply that the fluctuations in the individ-ual grains are anticorrelated in a fine-tuned and non-trivial way, i.e. positiveand negative fluctuations strongly compensate each other which technically isexpressed by property (3). Remark: In [41] we extended such a vacuum fluctuation analysis and appliedit to measurement instruments, being designed to detect (possibly) microscopicfluctuations of distances due to passing gravitational waves.We hence infer that the fluctuation pattern of e.g. energy-momentum hasto be strongly anticorrelated . But under the above assumption it is possiblethat the underlying compensation mechanism which balances e.g. the positiveand negative energy fluctuations is of short-range type, viz., individual grain-energies may still fluctuate almost independently if their spatial distance issufficiently large. We show in the following that the true significance of theso-called holographic principle is it, to enforce a very rigid and long-ranged anticorrelated fluctuation pattern in the quantum vacuum. As we are at themoment only interested in matters of principle, we assume the simplest case toprevail, called the space-like holographic principle (holding in contexts like e.g.quasi-static backgrounds or asymptotic Minkowski-space; see e.g. the beautifulreview [42]).
Postulate 2.4
There exists a class of scenarios in which the maximal amountof information or entropy which can be stored in a spherical volume is propor-tional to the area of the bounding surface. The same holds then for the numberof available DoF in V . This is the spacelike holographic principle. In a series of papers Brustein et al. developed a point of view that relatestypical fluctuation results of quantum mechanical observables in quantum fieldtheory with the area-law-like behavior of entanglement entropy and BH-entropy(cf. e.g. [43]). We already made a brief remark to this approach in [40]. We notethat we arrived at related results using different methods in another context (seee.g. [44] and [45]). As a more detailed comment would lead us too far astray,we plan to discuss this subject matter elsewhere.What we are going to show in the following is that the mechanism leadingto the strange area-behavior of the entropy of an enclosed volume, V , is con-siderably subtler as usually envisaged. On the one hand, we will show thatthe number of elementary DoF contained in V is in principle proportional tothe volume. On the other hand we infer from observations on the macroscopicor mesoscopic scale that the fluctuations of e.g. the energy are strongly anti-correlated. However, as long as this compensation mechanism is short-ranged,we would still have a number of nearly independently fluctuating clusters ofelementary DoF which again happens to be proportional to the volume as thecluster size is roughly equal to the correlation length. So the conclusion seemsto be inescapable that the patterns of vacuum fluctuations must actually be long-range correlated on a microscopic scale.6ut we showed in [40] or [41] in quite some detail that even systems, display-ing long-range correlations, will usually have an entropy which is proportionalto the volume. A typical example is a (quantum) crystal ([40],[41]). It is cer-tainly correct that below a phase transition point a system of particles in thecrystal phase has a smaller entropy than in the liquid or gas phase, but still theentropy happens to be an extensive quantity. The reason is in our view thatthe system develops, as a result of the long-range correlations, new types ofcollective excitations (e.g. lattice phonons) which serve as new collective DoF.Approximately they may be treated as a gas of weakly interacting elementarymodes with the usual extensive entropic behavior.That is, the holographic principle entails that the elementary DoF have to belong-range anticorrelated (cf. also the remarks in [35] or sect.7 of [46]). But wesee that this is only a necessary but not a sufficient property for an entropy-arealaw to hold. We hence arrive at the preliminary conclusion: Conclusion 2.5
From our preceding arguments and observations we concludethat the holographic principle implies that the fluctuation patterns in V arelong-range anticorrelated in a fine-tuned way on a microscopic scale and areessentially fixed by the state of the fluctuations on the bounding surface. Thedynamical mechanism, which generates these long-range correlations must how-ever, by necessity, have quite unusual properties (cf. subsection 4.2). Before we derive the wormhole structure of the quantum vacuum on a primordialscale in the next sections, we continue with the general analysis of the patternof vacuum fluctuations and derive some useful properties of it.A particular role is usually played by the energy and its fluctuations. Fur-thermore, vacuum fluctuations are frequently discussed together with the so-called zero-point energies . While they are not exactly the same, they are closelyrelated. Both occur also in connection with the cosmological constant problem (to mention only a few sources see e.g. [47],[48],[49],[50],[51],[52],[53]).In the simplest examples like e.g. the quantized harmonic oscillator or theelectromagnetic field we have H = P / m + mω/ · Q (4)and with h P i = h Q i = 0 (5)in the groundstate, ψ , we have ~ · ω/ h H i = 1 / m · h ( P − h P i ) i + mω/ · h ( Q − h Q i ) i (6)with h ( P − h P i ) i · h ( Q − h Q i ) i ≥ ~ / P, Q ] = − i ~ . In the same way we have in (matter-free)QED: H = const · ( E + B ) (8)7ith h E i = h B i = 0 (9)so that again h H i is a sum over pure vacuum fluctuations of the non-commutingquantities E and B . One should however note that in the quantum field contextproducts of fields at the same space-time point have to be Wick-ordered (in orderto be well-defined). It is, on the other hand, frequently argued that with gravityentering the stage, these eliminated zero-point energy fluctuations have to betaken into account again. In our view, this problem is not really settled.We now come to an important point. It is our impression that in someheuristic discussions (vacuum fluctuations as virtual particle-antiparticle pairs)the consequences of the fact that the vacuum state is an exact eigenstate of theHamiltonian in a Hilbert space representation of some quantum field theory arenot fully taken into account. I.e., we have H Ω = 0 (10)(provided the ground state energy is for convenience normalized to zero; notehowever that this may be problematical in a theory containing gravity). Eigen-states, however, have the peculiar property that the standard deviation is nec-essarily zero, ∆ Ω H = h ( H − h H i Ω ) i / = h H i / = 0 (11)According to the standard interpretation of quantum theory combined withspectral theory, H ≥
0, this implies that in each individual observation processthe total energy of the vacuum which is, according to conventional wisdom, the(hypothetical) sum or superposition of local (small scale) fluctuations, happensto be exactly zero. In other words, the elementary fluctuations have to exactlycompensate each other in an apparently fine-tuned way. Put differently
Observation 2.6
If there are positive local energy fluctuations, there have to beat the same time by necessity negative energy fluctuations of exactly the sameorder. That is, at each moment, the global pattern of energy fluctuations inthe quantum vacuum is an array of rigidly correlated positive and negative localexcitations.
Remark: Note the similarity of this independent observation to what we havesaid above in connection with the holographic hypothesis.It should be mentioned that Hawking in [54] invoked exactly this picture of aparticle pair excitation near the event horizon with the virtual particle, havingnegative energy, falling into the BH while the one with positive energy escapesto infinity.It would be useful to get more quantitative information on the spectral prop-erties of the local observables, in particular estimates on negative fluctuations.One could try to make an explicit spectral resolution of these quantities, e.g.of the energy, contained in a finite volume, V , but this turns out to be diffi-cult in general, even if one has given an explicit model theory in some Hilbert8pace. As we prefer a more general, model independent approach (not necessar-ily based on Hilbert space mathematics), we proceed by using (similar to Bell inhis papers) a general probabilistic approach which rather exploits the statisticsof individual measurement results. Unfortunately, we found that the standardestimates, known to us in this context (e.g. the Markov-Chebyshev-inequality),always go in the wrong direction (see e.g. [55] or [56]). Therefore we present inthe following our own estimate.The strategy is the following. We take an observable, E V , localized in V with, for convenience, discrete spectral values, E i , and corresponding probabil-ities denoted by p i >
0. If we assume that the expectation of E V is zero (whichcan always be achieved by a simple shift) we have X p i = 1 , X p i · E i = 0 (12)Furthermore, we assume its standard deviation in e.g. the vacuum, Ω, to befinite (which is automatically the case for bounded operators, but we want toinclude also more general statistical variables) X p i · E i = (∆ Ω E ) < ∞ (13)In a first step we make the simplifying assumption (taking e.g. a boundedfunction of the energy) Assumption 2.7 | E i | ≤ Λ for all E i (14)We are interested in the amount of negative (e.g. energy) fluctuations we willobserve in measurements. A reasonable quantitative measure of it is X p − i · | E − i | (15)with E − i , p − i the negative spectral values and their corresponding probabilities.We then have (with | E − i | / Λ ≤ X p − i · | E − i | / Λ ≥ X p − i · | E − i | / Λ (16)For the lhs we have X p − i · | E − i | / Λ = X p + i · | E + i | / Λ (17)as the expectation of E was assumed to be zero.This yields X p − i · | E − i | / Λ = 1 / · (cid:16)X p − i · | E − i | / Λ + X p + i · | E + i | / Λ (cid:17) ≥ / · (cid:16)X p − i · | E − i | / Λ + X p + i · | E + i | / Λ (cid:17) (18)9.e. X p − i · | E − i | ≥ / · X p i · E i / Λ = 1 / · (∆ Ω E ) (19)On the other hand (Cauchy-Schwartz) (cid:16)X p − i · | E − i | (cid:17) = 1 / · (cid:16)X p − i · | E − i | + X p + i · | E + i | (cid:17) ≤ / · X p i · | E i | (20)We hence arrive at the result Conclusion 2.8
If the observable, E , is bounded, so that its spectral valuesfulfill | E i | ≤ Λ , we have the estimate / − (∆ Ω E ) ≤ (cid:16)X p − i · | E − i | (cid:17) ≤ / Ω E ) (21) with p i the probabilities that the negative spectral values E i occur in an obser-vation. That is, we manage to bound a quantity, which is difficult to measuredirectly, by quantities, which are usually more easily accessible. We can generalize this result to situations where the E i are not exactlybounded by some Λ but are bounded in at least an essential way. We assumethat there exists some Λ so that X | E i | > Λ p i · | E i | < ε Λ (22)We then have X p − i · | E − i | / Λ = 1 / (cid:16)X p − i · | E − i | / Λ + X p + j · | E + j | / Λ (cid:17) ≥ / X | E − i |≤ Λ p − i · | E − i | / Λ + X | E + j |≤ Λ p + j · | E + j | / Λ ≥ / X | E − i |≤ Λ p − i · | E − i | / Λ + X | E + j |≤ Λ p + j · | E + j | / Λ ≥ / (cid:16)X p i · | E i | / Λ − ε Λ / Λ (cid:17) (23) Corollary 2.9
Under the above assumption of an essentially bounded E wehave X p − i · | E − i | ≥ / (cid:0) (∆ Ω E ) − ε Λ (cid:1) (24)Another, rigorous, but not quantitative, argument can be derived from ax-iomatic quantum field theory (see e.g. [57]). It follows from the so-called Reeh-Schlieder theorem that there are no local observables or fields which can an-nihilate the vacuum (where by local we mean that the objects commute forspace-like separation). I.e., we have for any local A (with A = A ∗ ) A Ω = 0 ⇒ ( A Ω | A Ω) = (Ω | A Ω) = 0 (25)10e take now as local observable the energy density integrated over a certainspatial region, V , H V := Z V h ( x , d x (26)One usually normalizes h ( x ) to(Ω | h ( x ) Ω) = 0 ⇒ (Ω | Z V h ( x , d x Ω) = 0 (27)The classical expression of the energy density, being derived in Lagrangianfield theory, is positive. The corresponding quantized expression, after a neces-sary
Wick-ordering (see e.g. [58] or [59]) is however no longer positive definite asan operator (density). This can be seen as follows. If the quantized energy den-sity were still positive, one can take the square root (via the spectral theorem)of e.g. the positive operator H V and get:0 = (Ω | H V Ω) = ( H / V Ω | H / V Ω) (28)hence H / V Ω = 0 (29)As H / V is also a local observable this is a contradiction due to the Reeh-Schlieder theorem. Conclusion 2.10 H V is not a positive operator, hence its spectrum containsnegative spectral values. It is then easy to construct Hilbert-space vectors, ψ , sothat the measurement of H V in ψ yields negative values for the local energy. Remark: We recently learned that this argument is originally attributed toEpstein (unpublished;[60] or see [61]), while the derivation which can be foundin [62] is a completely different one.The important message (in our view) of all this is that, perhaps in contrastto naive expectation, the quantum vacuum contains a lot of negative energyexcitations which globally exactly balance the positive excitations. One maynow speculate about the possibility of making use of this observation.
In this section we want to describe (very) briefly and sketchily the three differentlines of reasoning which lead us to the concept of wormhole spaces . The first lineoriginated from our investigation of the structure and dynamical behavior of thenetworks we described above. In e.g. [36] we analyzed in some quantitative de-tail the unfolding of the network structure and the various network epochs underthe inscribed microscopic dynamical laws and developed the two-level conceptof the network structure (or, rather, a multi-scale structure), which, under theright conditions, is relatively smooth on a sufficiently coarse-grained level (level11) with, among other things, a distant measure (metric) of the more ordinarytype and (hopefully) an integer-valued geometric dimension, while on a moremicroscopic scale (level 1) the network structure is expected to be very erraticwith possibly a lot of links (elementary interactions or information channels)connecting regions which may be far apart with respect to the metric on level 2.The association of these links with microscopic wormholes thus suggests itself(cf. in particular observation 4.27 in [36]). Note furthermore that our networkdynamics implies that these translocal connections are dynamically switched onor off. Compare this observation with the point of view expounded in e.g. [63] . . . But if a wormhole can fluctuate out of existence when its entrancesare far apart . . . then, by the principle of microscopic reversibility, thefluctuation into existence of a wormhole having widely separated entrancesought to occur equally readily. This means that every region of spacemust, through the quantum principle, be potentially “close” to every otherregion, something that is certainly not obvious from the operator fieldequations which, like their classical counterparts, are strictly local.. . . It isdifficult to imagine any way in which widely separated regions of space canbe “potentially close” to each other unless space-time itself is embeddedin a convoluted way in a higher-dimensional manifold. Additionally, adynamical agency in that higher-dimensional manifold must exist whichcan transmit a sense of that closeness.
The quantitative network calculations in the mentioned papers have mainlybeen performed within the framework of random graphs . Important mathemat-ical tools for the network analysis in the transition from microscopic, stronglyfluctuating and geometrically irregular scales to coarse-grained and, by the sametoken, smoother scales have been the concepts of cliques of nodes, the clique-graph of a graph and an important network parameter which we dubbed in-trinsic scaling dimension (we later learned, [38], that this concept plays also animportant role in geometric group theory or Cayley-graphs where it is called the growth degree ). To give a better feeling what is actually implied, we give thedefinitions of clique, clique-graph and internal scaling dimension (more aboutgraph theory can e.g. be found in [64], [37], notions and properties of graphdimension were studied in e.g. [65]).
Definition 3.1
A simplex in a graph is a subset of vertices (nodes) with eachpair of nodes in this subset being connected by an edge. In graph theory it isalso called a complete subgraph. The maximal members in this class are calledcliques.
Definition 3.2
The clique graph, C ( G ) , of a graph, G , is built in the followingway. Its set of nodes is given by the cliques of G , an edge is drawn between tooof its nodes if the respective cliques have a non-empty overlap with respect totheir set of nodes. Graphs carry a natural neighborhood structure and notion of distance. Theneighborhood U n ( x ) of a node x is the set of nodes y which can be reached,12tarting at x in ≤ n consecutive steps, i.e. there exists a path of ≤ n consecutiveedges connecting the nodes x and y . Definition 3.3
The canonical network or graph metric is given by d ( x, y ) := min γ { l ( γ ) | γ a path connecting x and y } (30) Here l ( γ ) is the number of consecutive edges of the path. The above definitionfulfills all properties of a metric. Thus graphs and networks are examples ofmetric spaces. Definition 3.4 (Internal Scaling Dimension)
Let x be an arbitrary node of G . Let U n ( x )) denote the number of nodes in U n ( x ) .We consider the sequenceof real numbers D n ( x ) := ln( U n ( x ))ln( n ) . We say D S ( x ) := lim inf n →∞ D n ( x ) isthe lower and D S ( x ) := lim sup n →∞ D n ( x ) the upper internal scaling dimension of G starting from x . If D S ( x ) = D S ( x ) =: D S ( x ) we say G has internal scalingdimension D S ( x ) starting from x . Finally, if D S ( x ) = D S ∀ x , we simply say G has internal scaling dimension D S . Observation 3.5
We proved in [65] (among other things) that this quantitydoes not depend on the choice of the base point for most classes of graphs.
It turns out that this geometric notion is a very effective characteristic of thelarge-scale structure of graphs and networks. This topic was further studied ingreater generality in e.g. [38].In [37] we developed what we called the geometric renormalization group ,to extract important geometric coarse grained, that is, large scale informationfrom the microscopically quite chaotically looking network and its dynamics.The idea is, at least in principle, similar to the block spin transformation instatistical mechanics. That is, certain characteristic properties of the systemare distilled from the microscopically wildly fluctuating statistical system bymeans of a series of algorithmic renormalization steps (i.e. coarse-graining pluspurification). The central aim is it to arrive in the end at a system whichresembles, on the surface, a classical space-time, or, on the other hand, todescribe the criteria a network has to fulfill in order that it actually has such a classical fixed point .In the course of this analysis we observed (cf. section VIII of [37]) that theso-called critical network geometries , i.e. the microscopic network geometrieswhich are expected to play a relevant role in the analysis, are necessarily in avery specific way geometrically non-local , put differently, they have to contain avery peculiar structure of non-local links, or short-cuts , that is, in other words,the kind of wormhole structure , we already described above.Relations to non-commutative geometry were established and studied in [66].We mention in particular section 7.2 “Microscopic Wormholes and Wheeler’sSpace-Time Foam” and section 8 “Quantum Entanglement and Quantum Non-Locality”. The possible relevance for quantum theory is in fact quite apparent13as has also been emphasized in the papers by ’t Hooft), as these microscopicwormholes may be the origin of the ubiquitous entanglement phenomena inquantum theory. The following figures describe pictorially the nested structureof the cliques of nodes in consecutive renormalization steps and overlappingcliques of nodes, defining the local near-order of physical points together withshortcuts which connect distant parts of the coarse-grained surface structure.Figure 1: Nested Structure; the (overlapping) cliques of a given level are repre-sented as non-overlapping for reasons of pictorial clearnessFigure 2: Translocal links, connecting some local clusters of nodes (grains)A second complex of (related) phenomena emerges in the field of small worldnetworks . This is a particular class of networks of apparently quite a universalcharacter (described and reviewed in some detail, for the first time, in [67])with applications in many fields of modern science. They consist essentially ofan ordinary local network with its own local notion of distance superimposedby a typically very sparse network of so-called short-cuts living on the sameset of nodes and playing a structural role similar to the microscopic wormholesdescribed above. A typical example (with dimension of the underlying lattice k = 1) is given in the following figure. Some further (in fact very few) references,taken from quite diverse fields are e.g. [68],[69],[70]. Observation 3.6
Its, in our view, crucial characteristic is the existence of twometrics over the same network or graph. The first, d ( x, y ) , is defined (cf.definition 3.3) by taking into account the full set of edges (i.e., including the Figure 3: The smallworld model for k = 1. The number of nodes is N = 30.In this particular realisation we have inserted four additional shortcuts. Theunfilled nodes are the vertices which can be reached by for example ≤ x . The black nodes are the vertices not reached after threesteps. short-cuts) and a second (local) metric, d ( x, y ) , taking into account only theedges of the underlying local network. It hence holds d ( x, y ) ≤ d ( x, y ) (31)Remark: The metric d ( x, y ) may then be associated (after some renormalisa-tion or coarse-graining steps) with an ordinary macroscopic metric defined ona smooth space (without wormholes) like our classical space-time. d ( x, y ), onthe other hand, should be regarded as a microscopic distance concept whichemploys the existence of wormholes.While, on the surface, the origin of this concept of small world networksseems to be quite independent of the wormholes in general relativity, it is themore surprising that on a conceptual meta level various subtle ties do emerge.To mention only one (in our view) important observation. In [71] it is for exam-ple shown, that a sparse network of shortcuts superimposed upon an underly-ing local network, has the propensity to stabilize the overall frequency pattern( phase locking ) of so-called phase-oscillators which represent the nodes of thenetworks, the links representing the couplings. The oscillators are assumed tooscillate with (to a certain degree) independent frequencies. If we relate theselocal frequencies with some local notion of time (or clocks), we may infer that(microscopic) wormholes create or stabilize some global notion of time!We now come to the third strand, viz. the real wormholes of general rel-ativity or quantum gravity. We mainly concentrate on the wormholes in true,i.e. Lorentzian space-time. Euclidean wormholes also (may) play an importantrole and have been discussed extensively in the context of the (nearly) van-ishing value of the cosmological constant (see e.g. [72],[73],[74], [75],[76],[77]).Of particular relevance in the Lorentzian context are the so-called traversable wormholes. Their study started (as far as we know) with two seminal papers byThorne and coworkers (see [78]). The geometric construction of such solutions15s in fact not so difficult if performed by the so-called g-method . That is, oneconstructs a geometric wormhole, e.g. of the static type, and, in a second step,computes the energy-momentum tensor being consistent with this solution.Giving a rough outline, this can be done in following way. Two open ballsare removed from two different pieces of e.g. approximately flat 3-space. Theirboundaries are glued together with the junction being smoothed. As a conse-quence of the smoothing process a tube emerges interpolating between the twospheres (see e.g. [79]). It is a remarkable fact that in this process the weakenergy condition (WEC) has to be violated, the latter implying that T ≥ , T + T ii ≥ i = 1 , , Observation 3.7
In order to get a traversable wormhole, one has to violatethe WEC. The WEC is always satisfied by classical matter. Therefore quan-tum effects are needed. The kind of negative energy needed is also called exoticmatter.
We showed in quite some detail in the preceding section that the quantumvacuum abounds with negative energy fluctuations. Therefore the speculationin section H of the first paper in [78] does not seem to be too far-fetched. Ina next step one can study networks of such traversable wormholes. In [80] itis speculated that such a network, existing in the early universe, may solvethe horizon problem . The same situation was discussed from the point of viewof our network approach in section 4.1 (The Embryonic Epoch) of [36]. Allthis comes already quite near the general picture we envoked in the beginningof this section. Furthermore one can envisage solutions combining black andwhite holes. This corresponds to some of our networks where the orientation(direction) of the links connecting two nodes can change under the dynamics.A review of Lorentzian wormholes can be found in the book by Visser ([81]).Some other references are e.g. [82] and [83].The above picture of a hypothetical network of wormholes sitting in the deepstructure of the quantum vacuum is beautifully complemented by an approach(see e.g. [84],[85]) which investigates within a (semi)classical approximation theenergy of a quantum vacuum state containing such an array of wormholes (or,rather, a gas of such wormholes) and compare it with a vacuum state which inzeroth order is flat Minkowski space. It comes out (apparently being a kind ofCasimir effect) that the quantum vacuum containing the wormhole gas has inthis semiclassical approximation a lower energy compared to the state, beinga perturbation of Minkowski space. One should note however that this is afirst order quantum effect! Anyhow, this observation seems to corroborate thespace-time foam picture of e.g. Wheeler and we conclude this section with
Conclusion 3.8
From our analysis in this and the preceding section emerges amodel of the ground state of some preliminary version of quantum gravity which ontains as an essential ingredient a network of microscopic wormholes. Thesewormholes can be created and annihilated and are in our picture the carriers ofinformation between distant parts of classical space-time. Definition 3.9 (Wormhole Space)
We call such a physical structure a worm-hole space and regard our cellular or small world networks, discussed above, asmodels, encoding and representing the typical characteristics of such systems.The typical characteristic is the existence of two types of distance, a micro-scopic one and an ordinary local one, being similar to ordinary macroscopicmetrics on smooth spaces.
We learned in the preceding sections that two (presumably crucial) propertiesgovern the behavior of the quantum vacuum on a microscopic scale. First,the vacuum fluctuations are strongly long-range anticorrelated on a microscopicscale, i.e. there exists a fine-tuned pattern of positive and negative (energy)fluctuations. Second, a quantum mechanical stability analysis seems to showthat the quantum vacuum is pervaded by a network of microscopic wormholes.We argued above that these two features are not independent phenomena butrather are the two sides of the same medal. Furthermore, the presumed worm-hole structure has been supported by observations coming from other fields ofresearch like e.g. cellular or small-world networks.In this (central) section we will now combine these observations and showthat they underlie (among other things) the holographic principle and the entropy-area law of BH-thermodynamics. In the following we will use (for convenience)the language of our networks with the nodes of the network representing mi-croscopic grains of space (or space-time) of roughly Planck-size. Leaving outother details we treat our quantum vacuum as a wormhole space, i.e. as a(small world) network consisting of an ordinary local network structure beingsuperimposed by a (presumably) sparse random network with edges consistingof short-cuts, i.e. links, connecting regions of space or space-time, which may bequite a distance apart with respect to the metric, belonging to the underlyinglocal network. These short-cuts represent the wormholes of ordinary space-time.The crucial characteristic, from which everything is expected to follow, isthe pattern and distribution of these short-cuts being immersed in the under-lying local network. That is, we randomly select a node x in the network G ( G standing for graph) and study the distribution of short-cuts connecting x with nodes y on spheres of radius R around x (measured with respect to somemacroscopic metric or the natural metric of the underlying local network). Observation 4.1
We expect that the precise distribution law will depend onthe concrete type of space-time we are dealing with. This holds in particular if he space-time is not static. That is, our microscopic approach to holographymakes it possible to understand how holography may depend on the concretelygiven type of space-time (cf. e.g. the covariant entropy bound of Bousso, [42]). Remark: We emphasize that the network or the quantum vacuum it is represent-ing, is basically a statistical system with all local DoF fluctuating. That means,most of our statements in the following are about mean values or averages overfiner statistical details.
One can arrive at the law, describing the distribution of short-cuts or wormholesaround some arbitrary but fixed generic node (viz. some fixed place in space-time) in roughly two ways. One can e.g. motivate the distribution law byappealing to certain fundamental principles like e.g. scale-freeness or absenceof a particular and in some sense unnatural length scale on a fundamentallevel. Alternatively, one can show that a reasonable choice leads to far-reachingconsequences and corroborates the findings and observations made on a moremacroscopic level. To keep the discussion as briefly as possible we adopt in thissection the second point of view. In the following we want to concentrate, forthe sake of brevity, on a simple type of quantum vacuum, that is, the vacuumbelonging to ordinary Minkowski space or a space-time which is asymptoticallyflat (e.g. a Schwarzschild space-time). We postpone the analysis of more generalspace-times as they occur in general relativity.We make the following conjecture:
Conjecture 4.2
On the average the number of short-cuts from a central node x to nodes y , sitting on the sphere, S R ( x ) about x is independent of R . Denotingthis number by N S R ( x ) , we hence have N S R ( x ) = N (33)Remark: As this number is a statistical average, it need not be an integer.The situation is depicted in the following picture. Observation 4.3
We will show in subsection 4.2 in a detailed quantitativeanalysis that this result approximately holds as well for nodes, not sitting ex-actly in the center of the spheres S R (see the following picture). Definition 4.4
We denote the cluster of nodes in the ball B R being connectedto an x by short-cuts by C B R ( x ) . We previously introduced the internal scaling dimension of a network (seedefinition 3.4). It roughly describes how fast the network is growing with respectto some base node. As this growth degree is to a large degree independent of thebase node (see e.g. [65]) it is a global characteristic of a given network, in fact of awhole class of similar networks ([38]). It is well known that the generalization of18 hort−CutsSpheres
Figure 4: Short-Cuts from a central node to nodes lying on two different spheres.In this picture we assumed N = 2 Sphere of Radius RShort−Cuts
Figure 5: Short-Cuts from nodes not sitting in the center to nodes lying on afixed sphere S R .the concept of dimension away from smooth geometric structures is not unique.The above type of dimension has the tendency to grow if additional short-cuts are inserted into a given network geometry. We now introduce anotherdimensional concept which catches other important network properties beingmore closely related to the phenomena we want to analyze in this paper. It usesin an essential way the two metrics, d , d , introduced above. Observation 4.5
From the above we infer that the number of nodes in thecluster C B R ( x ) is approximately equal to N · R . Furthermore, if the networkof short-cuts is very sparse, the clusters C B R ( x i ) , C B R ( x j ) with x i = x j areessentially disjoint (the overlap is empty or very small). This is the phenomenoncalled spreading in the theory of random graphs. Hence, the following concept is reasonable.We define a holographic dimension , D H , of a network in the following way.We take some ball B R with macroscopic radius R around some fixed but arbi-trary node x with respect to the local metric d . We then form the U (1)1 ( y )-neighborhoods around the nodes y ∈ B R with respect to the microscopic metric d . We construct a minimal cover of B R by such U (1)1 ( y i ), i.e. a minimal19igure 6: Various clusters C B R ( x i ) with empty or marginal overlapselection of such y i s.t. [ i U (1)1 ( y i ) ⊃ B R (34)The cardinality of such a minimal set we denote by N C ( B R ). We take the limit R large or R → ∞ (in an infinite network) and define Definition 4.6
We call D H := lim R →∞ ln N C ( B R ) / ln R (35) the holographic dimension of the graph (network), provided the limit exists. Inthe more general situation we can, as in definition 3.4, define upper and lowerdimensions etc. Corollary 4.7
As for the previously defined graph dimension, the limit is in-dependent of the selected base point , x , if the network or graph is homogeneouson the average or in the large. Observation 4.8
Due to the sparseness of the embedded subgraph of short-cuts,which yields the spreading property mentioned above, the number N C ( B R ) scalesfor the wormhole spaces or small-world networks as N C ( B R ) ∼ R n − (36) with n the dimension of the local network or its coarse-grained continuum limitspace. Proof: The U (1)1 ( y )-neighborhoods consist of nodes lying in the neighborhoodswith respect to the local metric, d , U (2)1 ( y ), plus the vertices connected byshort-cuts with y . The cardinality of U (2)1 ( y ) is independent of R and typically(at least in our models) a small number. For R → ∞ U (1)1 ( y ) ∩ B R will thereforeconsist mainly of nodes connected to y by short-cuts. Sparseness of the short-cutgraph and spreading yield the result. ✷ onclusion 4.9 For the type of wormhole spaces or small-world networks, de-fined above, we then have D H = lim R →∞ ln( V ( B R ) /R ) / ln R = n − That is, in this case we have the important result D H = dim S R = n − B R , contains approximately | V ( B R ) | := V ( B R ) /l p (39)DoF or grains of Planck size. The typical cluster size is | C B R ( x i ) | ≈ N · R/l p (40)Due to the mentioned spreading property the number of (effectively) indepen-dent cluster in the above minimal cover is approximately N C ( B R ) ≈ ((4 / π · R /N · R ) · l − p = (3 N ) − · πR /l p =(3 N ) − · A ( S R ) /l p =: (3 N ) − · | A ( S R ) | (41)with A ( S R ) denoting the area of S R . Observation 4.10
The number of effectively independent clusters, C B R ( x i ) in B R is N C ( B R ) ≈ (3 N ) − · | A ( S R ) | = (3 N ) − · A ( S R ) /l p (42) with the typical cluster size | C B R ( x i ) | ≈ N · R/l p (43)To show now that the number of effective DoF in a generic volume (whereby generic we mean a region in space with the diameter in all directions be-ing roughly of the same order) is proportional to the surface area, A ( V ), of itsboundary, we employ a general observation, made e.g. in statistical mechan-ics. An important tool for the analysis of systems in statistical mechanics arecorrelation functions. Correlations decay usually for large separation of the re-spective DoF, but what is on the other hand certainly the case is, that nearestneighbors are strongly correlated (near order versus far order). Observation 4.11
We expect that the DoF in each of the U (1)1 ( x ) are stronglycorrelated. We hence take it for granted, that they act effectively as a singlecollective DoF. Conclusion 4.12 (Area Law)
Due to the existence of wormholes or short-cuts, distributed in space-time, the number of effective DoF (affiliated with therespective clusters C B R ( x i ) ) in e.g. a ball B R equals N C ( B R ) , that is DoF in B R ) ≈ (3 N ) − · | A ( S R ) | = (3 N ) − · A ( S R ) /l p (44)This is the area-law behavior of entropy or number of DoF in a volume of spacefound in e.g. BH-entropy. We note however, that this law, in our formulation, isessentially a statement about the collective behavior of the elementary DoF in(the interior of) a volume of space. I.e., the respective DoF are not really sittingon the boundary of V . As to the details of the bulk-boundary correspondence see the following subsection.If we adopt the entropy-area law of BH-thermodynamics, which is, expressedin Planck units, S = 1 / · | A | (45)we have the possibility to fix our parameter N , which gives the number ofwormholes connecting a central grain of space with the grains on a surroundingsphere S R for any R . However, entropy is not exactly identical to numberof DoF. To relate the two, we have to make a simple model assumption. Onefrequently makes the assumption of Boolean DoF , i.e. the DoF on an elementaryscale are two-valued . Observation 4.13
With this assumption we have the relation S = N · ln 2 i.e. N = | A | / · ln 2 (46) with S the entropy, N the number of DoF. Conclusion 4.14
With the help of this identification we get N = 4 / · ln 2 (47) which can in qualitative arguments be approximated by one! That is, in Planck units, there exists roughly one short-cut between a centralvertex and a surrounding sphere of radius R . This shows that on an extremelymicroscopic scale, the network of short-cuts is indeed very sparse. Howeverthe picture changes considerably if we go over to more accessible length scales.If we use, for example an atomic length-scale of e.g. l a := 10 − m , we haveapproximately (10 − ) / (10 − ) = 10 (48)grains of Planck-size in a volume element of diameter l a . If we then choose,instead of a sphere S R , a spherical shell of radius R and thickness l a we haveapproximately 22 bservation 4.15 The number of wormholes or short-cuts between a centralvolume element of size l a and a corresponding spherical shell of radius R isapproximately short-cuts ) ≈ · = 10 (49) which is quite a large number. If we choose for example R = 1 m , we see that roughly 10 grains in theshell are the endpoints of about 10 short-cuts coming from the central volumeelement of size l a . If we replace R by the approximate diameter of the universe,i.e. R ≈ ly, we get (with 1 ly ≈ m ): R ≈ m (50)and for the number of Planck-size grains in a spherical shell of this radius: R ) ≈ (51)with still 10 short-cuts ending there. That is, only one in 10 grains is theendpoint of a respective short-cut. But if we select a volume element of size l a in this shell, we have still Observation 4.16
The number of wormholes (short-cuts) between two volumeelements of size l a being a distance R apart, is still the large number short-cuts ) ≈ · − · = 10 − · = 10 (52) that is, even over such a large distance there exist still a substantial number ofwormholes connecting the two volume elements. But nevertheless, the networkis sparse, viewed at Planck-scale resolution. We now come to the last point of this section. From what we have learned above,it is intuitively clear, that the DoF sitting on the boundary S R of e.g. a ball B R should fix (or slave) the DoF in the interior. But we note that in order that thiscan hold, we have to verify our statement made in observation 4.3. Furthermore,it is of tantamount importance to understand in more quantitative detail theinfluence of different shapes of the region under discussion and the effect ofdifferent space-time geometries. The prerequisites for this enterprise will bederived in the following.As an example we employ, as we already did above, the simple geometry ofthe spacelike holographic bound. For reasons of simplicity we place the centerof the ball in the origin, i.e. x = . It is of great help if we can transformthe problem into a problem of ordinary continuous analysis. To this end weintroduce the probability that a node in the interior of B R and an arbitrarynode on the boundary S R are connected by a short-cut. With y ∈ S R and x ∈ B R there spatial euclidean distance in three dimensions is | y − x | = X i =1 ( y i − x i ) ! / (53)23 bservation 4.17 The edge probability is given by p ( | y − x | ) = N / | A ( S | y − x | ) | = ( N · l p / π ) · | y − x | − (54)Here | A ( S | y − x | | is the number of nodes (or Planck-scale grains) on the spherearound x with radius | y − x | .This follows directly from what we have learnedin the previous sections.What we are actually doing in the following is the calculation of the averagenumber of short-cuts between an arbitrary node x in the interior of B R and thenodes on the boundary S R . This will be done within the framework of randomgraphs . The above p is the so-called edge probability (for the technical detailssee [64] or [36],[37]). The sample space is the space of graphs with node set comprising the node in x and all the nodes sitting on the boundary S R and edgeset all possible different sets of short-cuts connecting x with the nodes on S R .The probability of each graph in the sample space is calculated with the help ofthe above elementary edge probability p and its dual q := 1 − p .We choose x arbitrary but fixed in B R ( ) and let y vary over the sphere S R ( ). The integral over S R ( ) will then give the mean number of short-cutsbetween x and the grains on S R ( ). The guiding idea is that the DoF in theinterior are fixed by the DoF on the boundary if this integral is essentially & d .To make the integration easier we choose, without loss of generality, x = z , z := k · R (55)with 0 ≤ k ≤
1. A straightforward calculation (using polar coordinates andappropriate variable transformations) yields for the average number of short-cuts, N S R ( x ), N S R ( x ) = ( N l p / π ) · l − p · Z S R | y − x | − do = (cid:0) N / π · R (cid:1) · π R − · Z +1 − du ((1 + k ) − ku ) − = N / · Z +1 − du ((1 + k ) − ku ) − (56) Observation 4.18
Note that the integrand ((1+ k ) − ku ) − is always positive.Furthermore, our choice of a Coulomb-like law (in three dimensions) for thedistribution of short-cuts in the previous subsection, i.e. p ∼ R − , makes theabove integral independent of R .
24e can find a closed expression for the definite integral, i.e. I := Z +1 − du ((1 + k ) − ku ) − = − / k · ln ((1 − k ) / (1 + k ) ) > x relative to the center and the boundarycan be regulated by the value of the parameter 0 ≤ k ≤
1. We have tabulatedthe integral for k from 0 to 0 . k I k B R apart from a thin shell near the boundary. But this is not reallysurprising because there the main contribution comes from the near side of theboundary and is no longer of a true short-cut character. Taking into accountthe additional prefactor, N /
2, in front of the integral which is ≈ / Conclusion 4.19
The number of short-cuts from an arbitrary node x in B R tothe boundary S R is approximately p ( x ) & for most of the nodes. Furthermore for our Coulomb-like distribution law itis independent of the radius of the sphere and is therefore consistent with theexpected holographic behavior for this geometry. It is instructive to evaluate the above formula for k >
1, i.e., the influencevia short-cuts of the sphere S R on a DoF in the exterior of S R . For k large, theintegral is dominated by the first term in the integrand, viz. for k large we have I ≈ Z − du (1 + k ) − ∼ k − (59) Conclusion 4.20
For nodes, x , lying outside of S R , the effect of the short-cutconnections between x and S R decays like a Coulomb-law. That is, the DoFin the exterior are no longer fixed by the DoF on S R . What remains insteadis a statistical influence in form of a correlation which decays with increasingdistance. By the same token, there cannot be an entropy-area law for the exteriorof the sphere relative to its internal boundary. Anyhow, this example does notreally contradict the correctness of the spatial holographic principle as beingpresented in this paper. It would be interesting to relate our findings to thecovariant holographic principle of e.g. Bousso, [42] This simple observation has an important consequence for arguments beingsometimes invoked against the general nature of the spatial holographic principle(cf. e.g. [42]). While we do not intend to discuss the holographic principle formore general space-times in this paper, we mention one counter-example whichone finds frequently in the literature, i.e. a universe containing a closed spatialslice, S with a small inner subregion, S (see the following picture).25 S1 S2
Figure 7: A closed spatial slice containing a small subregion
Observation 4.21
The area-law in the usual form applies for the subregion S relative to its boundary. However, according to our (microscopic) version ofspatial holography, the DoF on the inner boundary cannot slave the DoF in thelarge region S if the inner boundary becomes too small. They only establishsome kind of correlation in the exterior. The quantitative details are given byintegrating our Coulomb-like influence law over the inner surface. Another, related, class of interesting (but perhaps pathological) apparentcounter examples (which we plan to address in greater detail elsewhere) is dis-cussed in e.g. [92], i.e. spacetimes which are called by Marolf ’bag-of-goldspacetimes’. An essential ingredient is some FRW-spacetime hidden in the in-terior of a region which resembles an ordinary BH. The innner FRW-universehas of course an entropy which is proportional to its volume while from theoutside the whole configuration looks like a BH. This seeming contradiction canbe easily understood with the help of our microsopic holographic law as theFRW-spacetime is actually only weakly coupled with the exterior of the BH viawormholes. The technical arguments are the same as above.
In the preceding sections we developed only the groundwork of our approach.To keep the paper within reasonable size, we had to postpone a more detaileddiscussion of the many consequences and immediate applications. In this fi-nal section we at least undertake to briefly comment on a number of importantpoints. It is however obvious that a more detailed discussion of each point wouldrequire a paper of its own.i) The possible connections to the ubiquituous phenomenon of entanglement inordinary quantum theory are obvious. Interesting in this respect is e.g. the well-known tension in quantum theory between the locality and causality principleof special relativity and the instantaneous state reduction, accompanying themeasurement process (cf. the respective sections in e.g. [86]). We think, similarto e.g. ’t Hooft, that (the microscopic form of) holography (we developed in this26aper) is the common basis which may unite quantum theory and gravitation.ii) The consequences of the BH-entropy being maximal, which is quite unchar-acteristic for the ground state entanglement entropy in say ordinary quantumtheory, should be further analysed.iii) The ADS-CFT-correspondence is regarded in string theory as the paradigmfor bulk-boundary correspondence (we mention only the review [87] and thepopular account [88]). In it two, at first glance, fundamentally different theoriesare related to each other, the one living in the bulk, the other living on the boundary at infinity . We must however say that the concrete physical episte-mology of this latter notion is not entirely clear to us. The use of boundariesat infinity is wide spread in holography and is mathematically well-defined, inparticular for certain well-adapted coordinate systems being in use in hyperbolicgeometry . But in general it is rather an asymptotic property and not a concreteplace. Note that in our approach full information about the interior of a (spa-tial) region is distributed essentially everywhere in the exterior of the region viawormholes, but usually not in the form of another field theory!iv) A virulent problem (the unitarity problem ) in BH-thermodynamics is thequestion whether a pure state goes over into a mixed state or not, that is, if thelaws of ordinary quantum theory are possibly violated in BH-thermodynamics(instead of the many published papers we mention only the reviews by Wald,cited above). This is a quite intricate epistomological problem somewhat similarto the quantum measurement problem. We think, part of the problem is thatfrequently pure states and mixtures are regarded as complete opposites. Butthis is not really correct. It is here not the place to go into more details. But insome respect it lies rather in the eye of the beholder. That is, it is the problemof dealing with the complete microscopic information of a state, or rather withsome coarse-grained form. Note that in our approach microscopic informationis widely scattered via short-cuts or wormholes over essentially the whole space.I.e., it is not fully accessible to a local observer. We recommend the studyof some older classics on the ergodic theorem in quantum statistical mechanics([89],[90],[91]).v) Our analysis should be extended to more general space-times where possiblydifferent distribution laws may show up.
References [1] A.D.Sakharov: “Vacuum Theory in Curved Space-Time and the Theory of Grav-itation”, Sov.Phys.Dok. 12(1968)1040[2] A.D.Sakharov: “Spectral Density of Eigenvalues of the Wave Equation and Vac-uum Polarization”, Theor.Math.Phys. 23(176)435[3] Y.B.Zeldovich: “The Cosmological Constant and the Theory of Elementary Par-ticles”, Sov.Phys.Usp. 11(1968)381
4] T.Jacobson: “Black-Hole Entropy and Induced Gravity”, gr-qc/9404039[5] J.D.Bekenstein: “Do we understand Black-Hole Entropy?”, seventh Marcel-Grossmann Meeting, Stanford 1994, gr-qc/9409015[6] J.D.Bekenstein: “Black-Holes and Entropy”, Phys.Rev. D7(1973)2333[7] J.D.Bekenstein: “Statistical Black-Hole Thermodynamics” Phys.Rev.D12(1975)3077[8] S.W.Hawking: “Black Holes and Thermodynamics”, Phys.Rev. D13(1976)191[9] G.’t Hooft: “The Black-Hole Interpretation of String Theory”, Nucl.Phys.B335(1990)138[10] L.Susskind,L.Thorlacius,R.Uglum: “The stretched Horizon and Black-Hole Com-plementarity”, Phys.Rev. D48(1993)3743, hep-th/9306069[11] S.W.Hawking: “Break Down of Predictability in gravitational Collapse”,Phys.Rev. D14(1976)2460[12] S.Giddings: “Comments on Information Loss and Remnants”, Phys.Rev.D49(1994)4078[13] G.’t Hooft: “On the Quantum Structure of a Black-Hole”. Nucl.Phys.B256(1985)727[14] R.Sorkin: “Ten Theses on Black-Hole Entropy”, in Proc.Eur.Sci.Found.Conf. onPhilosphy and Found.Issues in Stat.Phys., Utrecht, Nov.2003, publ. Spec.Iss.ofStud.Hist.Phil.Mod.Phys. 36(2005)291, hep-th/0504037[15] R.Sorkin: “The Statistical Mechanics of Black-Hole Thermodynamics”, ChandraSymp. Chicago Dec.1996, p.177, ed. R.M..Wald, Univ.Chic.Pr. 1998)[16] T.Jacobson,D.Marolf,C.Rovelli: “Black Hole Entropy: inside or out?”,Int.Journ.Theor.Phys. 44(2005)1807, hep-th/0501103[17] L.Bombelli,K.Rabinder,K.Kaul,J.Lee,R.Sorkin: “Quantum Source of Entropy forBlack-Holes”, Phys.Rev. D34(1986)373[18] M.Srednicki: “Entropy and Area”, Phys.Rev.Lett. 71(1993)666[19] M.Requardt: “Entanglement-Entropy for Groundstates, Low-Lying and HighlyExcited Eigenstates of General (Lattice-)Hamiltonians”, hep-th/0605142[20] R.M.Wald: “The Thermodynamics of Black Holes”, Liv.Rev.Relativ. 4(2001), 6[online article][21] R.M.Wald: “Gravitation, Thermodynamics, and Quantum Theory”, CQG16(1999)A177, arXiv:gr-qc/9901033[22] R.M.Wald: “Black Holes and Thermodynamics”, Symposium on Black Holes andRelativistic Stars, Chicago, Dec. 1996, arXiv:gr-qc/9702022[23] A.Strominger,C.Vafa: “Microscopic Origin of the Bekenstein-Hawking Entropy”,PL B 379(1996)99, arXiv:hep-th/9601029
24] J.M.Maldacena: “Black Holes in Strin Theory”, Ph.D. thesis, hep-th/9607235[25] G.T.Horowitz: “The Origin of Black Hole Entropy in String Theory”, Talk givenat Pacific Conference of Gravitation and Cosmology, Seoul, South Korea, 1996,arXiv:gr-qc/9604051[26] G.T.Horowitz,J.Polchinski: “A Correspondence Principle for Black Holes andStrings”, PR D 55(1997)6189, arXiv:hep-th/9612146[27] A.W.Peet: “TASI Lectures on Black Holes in String Theory”, TASI 1999,arXiv:hep-th 0008[28] S.D.Mathur: “The Fuzzball Proposal for Black Holes”, Fortschr.Phys.53(2005)793, arXiv:hep-th/0502050[29] A.Ashtekar,J.Baez,A.Goricki,K.Krasnov: “Quantum Geometry and Black HoleEntropy”, PRL 80(1998)904[30] C.Rovelli: “Loop Quantum Gravity and Black Hole Physics”, Helv.Phys.Act.69(1996)582, arXiv:gr-qc/9608032[31] C.J.Isham: “Structural Issues in Quantum gravity”, Lecture given at the GR14-conference, Florence 1995, arXiv:gr-qc/9510063[32] G.’t Hooft: “How Does God Play Dice?, (Pre) Determinism at the Planck Scale”,An Essay in Honor of John S. Bell, arXiv:hep-th/0104219[33] G.’t Hooft: “Quantum Gravity as a Dissipative Deterministic System”, CQG16(1999)3263, arXiv:gr-qc/9903084[34] G.’t Hooft: “Quantum Information and Information Loss in general relativity”,Lectture held at the 5th Symposion on Foundations of Quantum Mechanics,Tokyo, Japan, August 1995, arXiv:gr-qc/9509050[35] G.’t Hooft: “Dimensional Reduction in Quantum Gravity”, Essay dedicated toAbdus Salam, arXiv:gr-qc/9310026[36] M.Requardt: “(Quantum) Space-Time as a Statistical Geometry of Lumps inRandom Networks”, CQG 17(2000)2029, arXiv:gr-qc/9912059[37] M.Requardt: “A Geometric Renormalisation Group in Discrete Quantum Space-Time”, Journ.Math.Phys. 43(2002)351, gr-qc/0110077[38] M.Requardt: “The Continuum Limit of Discrete Geometries”,Int.J.Geom.Meth.Mod.Phys. 3(2006)285, math-ph/0507017[39] T.Nowotny,M.Requardt: “Emergent Properties in Structurally Dynamic CellularAutomata”, J.Cell.Aut. 2(2007)273, cond-mat/0611427[40] M.Requardt: “Planck Fluctuations, Measurement Uncertainties and the Holo-graphic Principle”, Mod.Phys.Lett.A 22(2007)791, gr-qc/0505019[41] M.Requardt: “About the Minimal Resolution of Space-Time Grains in Experi-mental Quantum Gravity”, arXiv:0807.3619
42] R.Bousso: “The holographic principle”, Rev.Mod.Phys. 74(2002)825,hep-th/0203101[43] R.Brustein,A.Yarom: “Thermodynamics and area in Minkowski space”, P.R. D69(2004)064013, arXiv:hep-th/0311029, R.Brustein,A.Yarom,D.H.Oaknin: “Im-plications of area scaling of quantum fluctuations”, P.R. D 70(2004)044043,arXiv:hep-th/0310091, R.Brustein,A.Yarom: “Area-scaling of quantum fluctua-tions”, Nucl.Phys. b 709(2005)391, arXiv:hep-th/0401081, R.Brustein,A.Yarom:“Dimensional reduction from entanglement in Minkowski space”, JHEP 0501:046(2005), arXiv:hep-th/0302186[44] M.Requardt: “Symmetry Conservation and Integrals over Local Charge Densitiesin Quantum Field Theory”, Comm.Math.Phys. 50(1976)259[45] M.Requardt: “Fluctuation Operators and Spontaneous Symmetry Breaking”,J.Math.Phys. 43(2002)351, arXiv:math-ph/0003012[46] L.Susskind: “The world as a Hologram”, Journ.Math.Phys. 36(1995)6377,hep-th/9409089[47] S.E.Rugh,H.Zinkernagel,T.Y.Cao: “The Casimir Effect and the Interpretation ofthe Vacuum”, Stud.Hist.Phil.Mod.Phys. 30(1999)111[48] S.E.Rugh,H.Zinkernagel: “The Quantum Vacuum and the Cosmological Con-stant Problem”, Stud.Hist.Phil.Mod.Phys. 33(2002)663[49] W.Nernst: “Ueber einen Versuch, von quantenmechanischen Betra-chtungen zur Annahme stetiger Energieaenderungen zurueckzukehren”,Verh.deutsch.Phys.Ges. 18(1916)83[50] C.P.Enz,A.Thellung: “Nullpunktsenergie und Anordnung nichtvertauschbarerFaktoren im Hamiltonoperator”, Helv.Phys.Act. 33(1960)839[51] T.H.Boyer: “Quantum Zero-Point Energy and Long-Range Forces”, Ann.Phys.56(1970)474[52] S.Weinberg: “The cosmological constant problem”, Rev.Mod.Phys. 61(1989)1[53] N.Straumann: “On the mystery of the cosmic vacuum energy density”,Eur.Journ.Phys. 20(1999)419, astro-ph/0009386[54] S.W.Hawking: “Particle Creation by Black Holes”, Comm.Math.Phys.43(1975)199[55] H.Bauer: “Mass- und Integrationstheorie”, de Gruyter, Berlin 1990[56] W.Feller: “Probability Theory”, Wiley, N.Y. 1957[57] R.F.Streater,A.S.Wightman: “PCT, Spin & Statistics and All That”, Benjamin,N.Y. 1964[58] J.D.Bjorken,S.D.Drell: “Relativistic Quantum Fields”, McGraw-Hill, N.Y. 1965[59] C.Itzykson,J.-B.Zuber: “Quantum Field Theory”, McGraw-Hill, N.Y. 1985
60] H.Reeh: private communication[61] S.J.Summers: “Yet More Ado About Nothing, The Remarkable Relativistic Vac-uum State”, arXiv:0802.1854[62] H.Epstein,V.Glaser,A.Jaffe: “Nonpositivity of the Energy Density in QuantizedField Theories”, Nuov.Cim. XXXVI (1965)1016[63] A.Anderson,B.DeWitt: “Does the Topology of Space Fluctuate?”, Found.Phys.16(1986)91[64] B.Bollobas: “Modern Graph Theory”, Springer, N.Y. 1998, “Random Graphs”,Cambr.Univ.Pr., Cambridge 2001[65] T.Nowotny,M.Requardt: “Dimension Theory on Graphs and Networks”,J.Phys.A:Math.Gen. 31(1998)2447, arXiv:hep-th/9707082[66] M.Requardt: “Wormhole Spaces, Connes’
Points Speaking To Each Other , andthe Translocal Structure of Quantum Theory”, arXiv:hep-th/0205168[67] D.J.Watts: “Small Worlds, The Dynamics of Networks between Order and Ran-domness”, Princt.Univ.Pr., Princeton 1999[68] S.H.Strogatz: “Sync: The Emerging Science of Spontaneous Order”, Penguin,N.Y. 2003[69] M.S.Granovetter: “The Structure of Weak Ties”, Sociol.Th. 1(1983)203[70] A.Lochmann,M.Requardt: “An Analysis of the Transition Zone Between the Var-ious Scaling Regimes in the Small World Model”, J.Stat.Phys. 122(2006)255,arXiv:cond-mat/0409710[71] E.Niebuhr,H.G.Schuster,D.M.Kammen,C.Koch: “Oscillator-Phase Coupling forDifferent Two-Dimensional Network Connectivities”, PR A44(1991)6895[72] S.W.Hawking: “Wormholes in Spacetime”, PR D 37(1988)904[73] S.Coleman: “Why there is Nothing Rahter than Something”, Nucl.Phys. B310(1988)643[74] I.Klebanov,L.Susskind,T.Banks: “Wormholes and the Cosmological Constant”,Nucl.Phys. B 317(1989)665[75] J.Preskill: “Wormholes in Spacetime and the Constants of Nature”, Nucl.Phys.B 323(1989)141[76] W.G.Unruh: “Quantum Coherence, Wormholes and the Cosmological Constant”,PR D 40(1989)1053[77] S.W.Hawking: “Do Wormholes Fix the Constants of Nature”, Nucl.Phys. B335(1990)155[78] S.M.Morris,K.S.Thorne: “Wormholes in spacetime”, Am.J.Phys. 56(1988)395,S.M.Morris,K.S.Thorne,U.Yurtsever: “Wormholes, Time Machines, and theWeak Energy Condition”, PRL 61(1988)1446
79] S.V.Krasnikov: “Toward a Traversable Wormhole”, arXiv:gr-qc/0003092[80] D.Hochberg,T.W.Kephart: “Wormhole Cosmology and the Horizon problem”,PRL 70(1993)2665[81] M.Visser: “Lorentzian Wormholes”, Springer, Berlin 1996[82] I.H.Redmount,Wai-Mo Suen: “Is quantum spacetime foam unstable?”, PR D47(1993)R2163, “Quantum dynamics of Lorentzian spacetime foam”, PR D49(1994)5199[83] C.J.Fewster,Th.A.Roman: “On Wormholes with Arbitrarily Small Quantities ofExotic Matter”, arXiv:gr-qc/0507013[84] G.Preparata,S.Rovelli,S.-S.Xue: “Gas of wormholes: a possible ground state ofquantum gravity”, Gen.Rel.Grav. 32(2000)1859, arXiv:gr-qc/9806044[85] R.Garattini: “Large N-wormhole approach to space-time foam”, Phys.Lett. B446(1999)135, arXiv:hep-th/9811187, “Space-time foam and vacuum energy”,Talk given at the Spanish Relativity Meeting, Mao, Minorca, Spain Sept.2002,arXiv:gr-qc/0212013[86] Y.Aharonov,D.Rohrlich: “Quantum Paradoxes”, Wiley, N.Y. 2005[87] O.Aharony,S.S.Gubser,J.Maldacena,H.Oguri,Y.Oz: “ Large N Field Theories ,String Theory and Gravity”, Phys.Rep. 323(2000)183[88] J.Maldacena: “The Illusion of Gravity”, Sci.Am. Nov. 2005, 56[89] J.v.Neumann. “Beweis des Ergodensatzes und des H-Theorems in der neuenMechanik”, Zeitschr.Phys. 57(1929)30[90] W.Pauli,M.Fierz: “Ueber das H-Theorem in der Quanten Mechanik”,Zeitschr.Phys. 106(1937)572[91] N.G.vanKampen: “Grundlagen der Statistischen Mechanik der IrreversiblenProzesse”, Fortschr.Phys. 4(1956)405[92] D.Marolf: “Black Holes, ADS, and CFTs”, Gen.Rel.Grav. 41:903,2009,arXiv:0810.4886; S.D.H.Hsu,D.Reeb: “Unitarity and the Hilbert space of quan-tum gravity”, Class.Quant.Grav. 25:235007,2008, arXiv:0803.421279] S.V.Krasnikov: “Toward a Traversable Wormhole”, arXiv:gr-qc/0003092[80] D.Hochberg,T.W.Kephart: “Wormhole Cosmology and the Horizon problem”,PRL 70(1993)2665[81] M.Visser: “Lorentzian Wormholes”, Springer, Berlin 1996[82] I.H.Redmount,Wai-Mo Suen: “Is quantum spacetime foam unstable?”, PR D47(1993)R2163, “Quantum dynamics of Lorentzian spacetime foam”, PR D49(1994)5199[83] C.J.Fewster,Th.A.Roman: “On Wormholes with Arbitrarily Small Quantities ofExotic Matter”, arXiv:gr-qc/0507013[84] G.Preparata,S.Rovelli,S.-S.Xue: “Gas of wormholes: a possible ground state ofquantum gravity”, Gen.Rel.Grav. 32(2000)1859, arXiv:gr-qc/9806044[85] R.Garattini: “Large N-wormhole approach to space-time foam”, Phys.Lett. B446(1999)135, arXiv:hep-th/9811187, “Space-time foam and vacuum energy”,Talk given at the Spanish Relativity Meeting, Mao, Minorca, Spain Sept.2002,arXiv:gr-qc/0212013[86] Y.Aharonov,D.Rohrlich: “Quantum Paradoxes”, Wiley, N.Y. 2005[87] O.Aharony,S.S.Gubser,J.Maldacena,H.Oguri,Y.Oz: “ Large N Field Theories ,String Theory and Gravity”, Phys.Rep. 323(2000)183[88] J.Maldacena: “The Illusion of Gravity”, Sci.Am. Nov. 2005, 56[89] J.v.Neumann. “Beweis des Ergodensatzes und des H-Theorems in der neuenMechanik”, Zeitschr.Phys. 57(1929)30[90] W.Pauli,M.Fierz: “Ueber das H-Theorem in der Quanten Mechanik”,Zeitschr.Phys. 106(1937)572[91] N.G.vanKampen: “Grundlagen der Statistischen Mechanik der IrreversiblenProzesse”, Fortschr.Phys. 4(1956)405[92] D.Marolf: “Black Holes, ADS, and CFTs”, Gen.Rel.Grav. 41:903,2009,arXiv:0810.4886; S.D.H.Hsu,D.Reeb: “Unitarity and the Hilbert space of quan-tum gravity”, Class.Quant.Grav. 25:235007,2008, arXiv:0803.4212