Gravitational Lorentz violation and superluminality via AdS/CFT duality

Gravitational Lorentz violation and superluminality via AdS/CFT duality

Raman Sundrum

Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles St. Baltimore, Maryland 21218, USA(Received 3 January 2008; published 25 April 2008)

Aweak quantum mechanical coupling is constructed permitting superluminal communication within a

preferred region of a gravitating AdS5 spacetime. This is achieved by adding a spatially nonlocal

perturbation of a special kind to the Hamiltonian of a four-dimensional conformal field theory with a

weakly coupled AdS5 dual, such as maximally supersymmetric Yang-Mills theory. In particular, two

issues are given careful treatment: (1) the UV-completeness of our deformed conformal field theory

(CFT), guaranteeing the existence of a ‘‘deformed string theory’’ AdS dual and (2) the demonstration that

superluminal effects can take place in AdS, both on its boundary as well as in the bulk. Exotic Lorentz-

violating properties such as these may have implications for tests of general relativity, addressing the

cosmological constant problem, or probing behind horizons. Our construction may give insight into the

interpretation of wormhole solutions in Euclidean AdS gravity.

DOI: 10.1103/PhysRevD.77.086002 PACS numbers: 11.25.Tq, 11.10.Lm, 11.30.Cp

I. INTRODUCTION

Relativistic invariance is a pillar of the fundamental lawsof physics. It is worth questioning whether this structure isexact or just a (very good) approximation. The issue issubtle in the context of general relativity which promotesPoincare invariance to a local symmetry, whose breakingtherefore requires some sort of Higgs mechanism. Whilelow-energy effective field theories with partial Higgsing ofgeneral relativity [1,2], consistent with observation, havebeen constructed, their incorporation into UV-completetheories of quantum gravity such as string theory has notbeen demonstrated.

In the description in terms of a Higgs mechanism,relativistic invariance is respected by the dynamics andbroken only by the state of some ‘‘Higgs’’ degrees offreedom. However, such a broad categorization encom-passes some rather familiar and unremarkable cases. Forexample, the preferred frame occupied by the cosmicmicrowave background can formally be thought of asspontaneously breaking Lorentz invariance, and by goingto comoving coordinates general coordinate invariance iseffectively ‘‘Higgsed.’’ But there may also be exotic Higgsphases, breaking relativistic invariance with much moredramatic implications. There is of course the possiblephenomenology of measurable quantitative deviationsfrom standard expectations of general relativity. SeeRefs. [3] for examples. But there may be important quali-tative effects as well. In Lorentz invariant theories, super-luminal propagation and interactions in one referenceframe would imply acausal effects in other frames. Butthis need not be the case for Lorentz-violating interactions,which may have a preferred frame in which causal unitaryevolution is defined. Superluminal interactions would beliberating in our vast universe, and might also allow us toprobe behind black hole [4] and cosmological horizons,normally off limits by relativistic causality. If Lorentz

violation is significant, it can go beyond being merely aprobe of horizons, it can modify their character [5]. Theobservation that some apparently ‘‘innocent’’ effectivefield theories display superluminal behavior [6] would nolonger immediately be a disqualification. Lorentz violationin general relativity may also help us understand some ofgravity’s other mysteries. For example, in Ref. [7] it wasshown that large standard model quantum contributions todark energy can be canceled by a symmetry, ‘‘energy-parity,’’ but the longevity of flat empty space then requiresa Lorentz-violating short-distance breakdown of generalrelativity. Finally, if relativistic invariance is an approxi-mation, it may well be an emergent (accidental) symmetry,simple examples of which occur in the long-wavelengthapproximation of some condensed matter systems. Thequestion then arises, what more fundamental structure orsymmetry underlies relativity.In this paper, we exploit the powerful approach to quan-

tum gravity offered by the AdS/CFT correspondence [8](reviewed in Ref. [9]), to engineer UV-complete gravita-tional dynamics exhibiting weak breaking of (local)Poincare invariance and superluminal action-at-a-distance.The construction is made on the CFT side of the corre-spondence, specifically by perturbing strongly coupledlarge-Ncolor N ¼ 4 supersymmetric Yang-Mills (SYM)theory by suitably chosen spatially nonlocal operators.The advantage of working in terms of these holographicdegrees of freedom is that it finesses the tricky issues ofbreaking gauge symmetries, such as general coordinateinvariance, that appear in the dual description of AdSgravity. In particular, the correspondence relates the break-ing of gauge symmetry in AdS to the breaking of globalsymmetry in the CFT, which is much easier to understand.This feature is illustrated by two well-known examples (inwhich, however, the relevant Higgs dynamics in AdS takesquite familiar and unexotic forms). The first example is

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given by simply adding a spacetime-dependent mass termfor some SYM scalars, m2ðxÞ TrA2ðxÞ, to the CFTLagrangian, thereby explicitly breaking the globalPoincare invariance (or conformal invariance for that mat-ter). In the dual AdS picture the perturbation is reflected bygravitating spacetime-dependent fields turned on in thebulk, sourced from the AdS boundary. But even awayfrom the boundary, these fields break AdS isometries.This is a physical effect, apparent to a bulk observer.Theoretically, one can phrase this as the turned-on fields‘‘Higgsing’’ the bulk general coordinate invariance,although we usually do not adopt this language. The sec-ond example is provided in Ref. [10] which studied asupersymmetric field theory with a global Uð1ÞR symme-try, explicitly broken by an anomaly. The field theory has asupergravity dual in which the global Uð1ÞR is mapped toan AdS gauge field. The explicit breaking in the fieldtheory must map to a Higgsing of the AdS gauge field.Therefore we know on general grounds that the requisiteHiggs condensate must appear on the AdS side of the dual,and Ref. [10] shows in more detail that this is the case.

Our final construction has the following properties:(i) The deformation of SYM takes the form of a man-

ifestly Hermitian perturbation to the SYMHamiltonian, thereby guaranteeing unitary andtime-local quantum mechanical evolution.

(ii) The deformation explicitly violates the Poincare in-variance of SYM. It mediates superluminal processesat first order.

(iii) To all orders in perturbation theory, the deformedtheory is UV finite (no new divergences beyond therenormalized SYM CFT). The finiteness propertiesare related to the spatial nonlocality of our perturba-tion Hamiltonian. In particular, the deformedHamiltonian is well defined.

(iv) The deformed SYM theory is indeed a weak pertur-bation of all SYM processes, viewed for a finiteperiod of time. That is, perturbation theory can betrusted. This implies that there must be some gravitydual of our deformed theory, including the dual of(ii).

(v) Some degree of superluminality in the gravity dual istaking place in the AdS bulk, not just at the AdSboundary. Since the bulk spacetime gravitates, thenecessary Higgs setup must appear, but we are un-able to give its explicit form.

(vi) Our deformed Hamiltonian is a sum of squares ofHermitian operators and therefore has an energybounded from below. But we cannot prove that theSYM vacuum is a ground state of the deformedtheory. For instance, Ref. [11] proves a positiveenergy theorem in the gravitational dual based onthe assumption of relativistic causality of the CFT,but our deformation violates this assumption.Instead, it is likely that the SYM vacuum corre-sponds to an excited state (not even an energy eigen-

state), and therefore can decay into the true groundstate of the full Hamiltonian. At first sight this ap-pears problematic since we wish to consider propa-gation of simple objects in the ‘‘recognizable’’ AdSvacuum, with weak superluminal corrections. Thedecay of the SYM vacuum implies the decay of theAdS vacuum into an unknown state. In such a statewe would not necessarily know what spacetime met-ric to use to even define superluminality.

(vii) Fortunately, the vacuum decay rate can be controlledby the weakness of our perturbation, and we canallow the perturbation to be turned on for only afinite duration, thereby ensuring that most regionsof space do not experience vacuum decay. This stillleaves a small but nonzero amplitude for superlumi-nal propagation and interaction of bulk quanta in this(approximate) AdS(CFT) vacuum. In this way, weengineer rare breakdowns of the general relativisticapproximation, with long-range superluminalconsequences.

The simultaneously weak and long-range character ofthe superluminal Lorentz-violating interactions distin-guishes our construction from earlier Lorentz-violatingdeformations of SYM and their gravitational duals [12],and points to how such striking effects might be compatiblewith real world gravity. It is possible that the puzzling CFTinterpretation of wormhole solutions in Euclidean AdSgravity [13–15] is related to constructions similar in spiritto ours. However, in this paper we work in Lorentziansignature spacetime. Several Lorentzian aspects of theAdS/CFT correspondence are discussed in Ref. [16].We work up to our construction in the following stages.

In Sec. II, we illustrate the long-wavelength emergence ofrelativistic invariance, in the absence of gravity, in a simplequantum lattice model which fundamentally has a pre-ferred reference frame. We then add a perturbation thatleads to instantaneous action-at-a-distance in what wouldotherwise have been the continuum relativistic regime. InSec. III, we generalize this action-at-a-distance within along-wavelength effective field theory containing gravity.The notion of ‘‘instantaneous’’ is ill-defined in a generalrelativistic context, but is replaced by superluminality. Therequisite combination of Higgs effects is described. InSec. IV, we review the emergent nature of (higher-dimensional) quantum general relativity, via the AdS/CFT correspondence, from N ¼ 4 SYM. We then de-scribe the generalization of Sec. II to SYM. This sectionis primarily intended for contrast with Sec. V, where ourmain results appear. The AdS dual setup contains super-luminality in a general relativistic context, but localized tothe AdS5 boundary. The deformation also leads to UVdivergences, that can, however, be treated by renormaliza-tion. We explain how AdS/SYM vacuum decay can besuppressed by making the SYM deformation act for a finiteduration. In Sec. V, we describe a perturbation of the SYM

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Hamiltonian whose AdS dual contains superluminality inthe AdS ‘‘bulk.’’ We show UV-finiteness and perturbativityof our deformation to SYM, and again indicate how SYMvacuum decay can be suppressed by making the deforma-tion act temporarily. Section VI provides our conclusions.

II. EMERGENT SPECIAL RELATIVITYANDACTION-AT-A-DISTANCE

Consider a very simple example. We start with an under-lying Hamiltonian for a theory without Poincare invarianceliving on a spatial cubic lattice (continuous time),

H ¼ 1

2

X~n

��2

~n þX3

i¼1

ð�~nþi ��~nÞ2�; (1)

where i are spatial unit vectors, and ~n are lattice points. Infamiliar fashion, for long-wavelength modes of this systemwe arrive at the approximately relativistic theory of amassless free scalar field,

H � 1

2

Zd3 ~xf�2ð ~xÞ þ ð@i�ð ~xÞÞ2g: (2)

Quantization of both Hamiltonians, written above in termsof Schrodinger picture operators, is of course straightfor-ward. Even in this simplest of examples, the underlyingtheory contains couplings, �~nþi� ~n, which instantaneouslyconnect two points at finite spatial separations.

But we can arrange for a more drastic breakdown ofPoincare invariance right in the midst of the relativisticregime, for example,

H ¼ 1

2

X~n

��2

~n þX3

i¼1

ð�~nþi ��~nÞ2�þ �

�X~n

J ~n�~n

�2; (3)

where J � 0 is a lattice function with finite support, andnormalized to

X~n

J ~n ¼ 1: (4)

The entire perturbation to the Hamiltonian is obviouslyalso � 0 and minimized at the vacuum h�i ¼ 0. If thesupport of J has a typical size L� lattice-spacing� 1 andJ is smooth on that scale, then the perturbation can be madeweak by taking �� L. In the support region of J, a latticequantum can be absorbed by the perturbation in one loca-tion ~n1 and be instantaneously emitted at a distant location~n2.Note that because energy is conserved and because J is

smooth on the lattice scale, soft incoming quanta neces-sarily scatter (nonlocally in space) to soft quanta, so thatthe continuum long-wavelength approximation is not bro-ken by the perturbation. There is therefore a good contin-uum approximation to this model,

H � 1

2

Zd3 ~xf�2ð ~xÞ þ ð@i�ð ~xÞÞ2g þ �

�Zd3 ~xJð ~xÞ�ð ~xÞ

�2;

(5)

where J is normalized as

Zd3 ~xJð ~xÞ ¼ 1: (6)

Again, the weak � coupling can absorb relativistic quantaand instantaneously emits such quanta far away. Sucheffects can appear acausal in another relativistic referenceframe, but that frame is not coequal to the defining oneabove. We see that the lattice structure is irrelevant to thequestion of this type of long-range interaction. Once onedeclares there to be a preferred frame, all that is required isa causal unitary theory in that frame. It may have a rela-tivistic approximation when some (in this case, long-range)interactions are neglected, but this relativity is not an exactprinciple that disqualifies nonrelativistic perturbations.While the original lattice model, being just a discrete set

of quantum mechanical degrees of freedom is manifestlyUV-finite, we should check that this is the case for thecontinuum approximation (that is check that we can trulydecouple the lattice structure and have a continuum limit).Thinking of � as a perturbation quadratic in fields, we seethat the only (potentially divergent) loop diagrams are tovacuum energy. Since there is no gravity in this model,vacuum energy is physically irrelevant and we can ignorethese diagrams.

III. EFFECTIVE GRAVITYANDSUPERLUMINALITY

Are superluminal long-range effects consistent withgeneral relativity? Let us try to construct a long-wavelength continuum description of such a combination,generalizing (5). In this section we will not worry about theissue of UV completeness. To simplify our task a little, letus first aim for the limit in which J is supported on just twopoints between which we want to arrange for superluminalinteractions, Jð ~xÞ ¼ 1

2�3ð ~x� ~x1Þ þ 1

2�3ð ~x� ~x2Þ, tempo-

rarily turning a blind eye to the loss of smoothness andthe product of coincident �-functions. We will addsmoother sources at the end of this section. Since space-time symmetries are gauged by general relativity, we willrealize the breaking of Poincare invariance as a Higgseffect. Really, two Higgs effects are required: one to pickout the two special locations, ~x1, ~x2, and one to define‘‘simultaneous’’ times on these two locations at which thelong-range interactions occur.We take the first Higgs effect to be of a familiar type: we

add to our theory a new species of particle, , with massmmuch greater than the UV cutoff of our effective descrip-tion, but smaller than the Planck scale. We also assign it aZ2 charge so that it can only be destroyed or created inpairs. Heavy pairs cannot be created within the effective

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description from light gravitational and � quanta muchsofter than m, but the light particles can interact with apreexisting pair. We take this pair of heavy particles to beso distantly separated at some initial time, that they cannotannihilate for a very long time to come. Since light and softquanta cannot appreciably accelerate the massive , wecan take the pair to be approximately at rest with respectto an asymptotic Minkowski frame. Because of their largeinertia and small Compton wavelength with respect to theUV cutoff of the light quanta, the preexisting particleswill act as effectively fixed pointlike locations. By thismeans, the ‘‘preferred’’ locations of the -pair effectivelyspontaneously breaks (local) Poincare invariance. Theirtwo locations will generalize the fixed locations ~x1;2 of

our long-range interaction.We want an interaction between and � so that a �

quantum that propagates to the location of one of thepreexisting ’s can ‘‘instantly’’ jump to the location ofthe other . But this requires an identification of time onthe worldline of one of the ’s with time on the worldlineof the other distant . Such a preferred pairing of timesfurther breaks general coordinate invariance and necessi-tates the second Higgs effect. Minimally, this Higgs effectcan be localized to the worldlines, defining the preferredtimes as the proper time along each worldline since theirpair-creation in the distant past. However, it is convenientto use a Higgs effect already in the literature that defines apreferred time everywhere in space, namely, the ‘‘Ghost-condensate’’ [2]. In a generally covariant and consistent,but unusual, effective field theory a new scalar field, �, iscoupled to gravity so as to admit a nontrivial stable solu-tion, namely, Minkowski spacetime metric with

�ðxÞ ¼ kx0; (7)

where k is a fixed constant parameter from the � action.This time dependence arises spontaneously and partiallyHiggses general coordinate invariance. Small � flucuationsabout (7) are ‘‘eaten’’ by the metric fields. Therefore inunitary gauge (7) is exact while the gravitational action ismodified. Nevertheless, over a large regime this effectivefield theory reproduces standard general relativity. Thefield (7) then gives us a global ‘‘clock.’’

Putting together the ingredients, we take our model ofsuperluminality to be given by

S¼ SEinstein½g�� þ Sghost½g��;��þ 1

2

Zd4x

ffiffiffiffiffiffiffi�gp fg��@��@��þ g��@� @� �m2 2g

� �

4

Zd4y

ffiffiffiffiffiffiffiffiffiffiffiffiffi�gðyÞ

q 2ðyÞ�ðyÞ

�Zd4z

ffiffiffiffiffiffiffiffiffiffiffiffiffi�gðzÞ

q 2ðzÞ�ðzÞk�ð�ðyÞ ��ðzÞÞ: (8)

First note that this action is generally coordinate invariant.Somewhat similar nonlocal operators were discussed in

Ref. [17] as coordinate invariant observables in ordinaryeffective general relativity. Here, the nonlocal operatorrepresents a true modification of the dynamics, not just aprobe of standard gravity.After passing to the ghost-condensate unitary gauge the

nonlocal term above becomes

Ssuperluminal ¼ � �

4

Zdt

Zd3 ~y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�gðt; ~yÞ

q 2ðt; ~yÞ�ðt; ~yÞ

�Zd3 ~z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�gðt; ~zÞ

q 2ðt; ~zÞ�ðt; ~zÞ; (9)

which is nonlocal in space, but local in time. The leadingbehavior of this system can be seen in the limit that the UVcutoff is � m� MPl. In the limit, with k held fixed,gravity decouples from the dynamics, but � continues toprovide a global time. The pair of distant heavy particlesbecome infinitely massive and pointlike, with static loca-tions ~x1;2. Therefore � effectively has an action in this

limit,

Seff ¼ 1

2

Zd4xð@��Þ2 � �

4

Zdtð�ðt; ~x1Þ þ�ðt; ~x2ÞÞ2;

(10)

which is equivalent to (5) with J ¼ 12�

3ð ~x� ~x1Þ þ12�

3ð ~x� ~x2Þ.With MPl, m large but finite, the � interaction is instan-

taneous in the unitary gauge, but not in a fully generalcoordinate invariant sense. Rather, the general statement isthat the � interaction is superluminal with respect to themetric g��.

Finally, let us discuss how to generalize this constructionto allow a smooth Jð ~xÞ. The simplest way is to replace thepair of heavy elementary particles with a smooth solitonand an antisoliton. For example suppose a is an isotripletHiggs field for an SOð3Þ gauge theory, which Higgses thesymmetry down to SOð2Þ (Georgi-Glashow model [18]).This theory supports smooth magnetic monopole solitons.We take our heavy preexisting particles to be a distantlyseparated monopole þ antimonopole pair. Let us general-ize our 2� couplings above to ð a a � v2Þ�, where v isthe magnitude of the a vacuum expectation value (VEV).Therefore the � interaction turns on smoothly as one entersthe cores of the monopoles where a deviates appreciablyfrom its VEV. The J we have engineered has negligiblesupport except in the two widely separated soliton cores.It is not known if this relatively simple effective gravi-

tational dynamics exhibiting superluminality can be UVcompleted, but it is a sensible low-energy effective fieldtheory (at least over a finite but long time interval to avoidany gravitationally induced collapse) and illustrates theprinciples we will pursue, indirectly, in the context ofAdS/CFT.


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IV. ADS/CFT AND BOUNDARYSUPERLUMINALITY

The CFT ! AdS correspondence is in a very real sensea case of emergent gravity and relativity. The UV com-pleteness of the CFT transfers to the AdS quantum gravity.Let us specialize to the CFT given by strongly coupledlarge-Ncolor N ¼ 4 SYM. There are good arguments [19–21] to suggest that this theory might itself be realizable asthe IR limit of a lattice theory (continuous time) with apreferred frame for unitary quantum evolution. Such alattice system would violate all of Poincare invarianceexcept for time translation invariance. Poincare and con-formal invariance would emerge in the continuum long-wavelength limit. The gravity dual of such a lattice systemwould have a ‘‘UV’’ boundary at which Poincare invari-ance is badly broken, reflecting the YM lattice structure,with IIB superstring field profiles emanating from the UVboundary, perturbing the usual AdS5 � S5 background.But the dual of the statement that the far IR of the latticetheory is successfully approximated by continuum SYMtranslates to saying that, far away from the UV boundary inthe IR of the bulk, the AdS5 � S5 background and fluctua-tions are gradually restored (the deviating profiles dampout). In this sense, (higher-dimensional) general relativitycan emerge from a quantum theory which fundamentallydoes not enjoy (even special) relativistic structure. Theabove features follow on general grounds, but details ofthe AdS dual of such a lattice gauge theory are not known.However, a provocative related example, with a singlelattice dimension, has been studied in Ref. [20]. A generalmoral to keep in mind is this: if a UV-complete quantumtheory has a regime or approximation in which it matches aCFTwhich has an AdS gravity (string) dual, then the entirequantum theory must have a dual description which has agravitational regime or approximation. This latter gravita-tional (string) dual must also reflect the deviations fromCFT behavior, and must possess the objects and defectsnecessary to do so.

For most of this section we work directly in the contin-uum (only briefly invoking a possible lattice realization insubsection IVD). We generalize (5) by perturbing theSYM CFT by a bi-local interaction, but now each localfactor must be a SUðNcolorÞ gauge-invariant compositeoperator. SYM has six ‘‘flavors’’ of real color-adjointscalar fields, AI¼1;...6. Flavor-adjoint color-singlet scalar

bilinears, TrAIAJ � �IJ6 TrAK AK, are primary operators

of the SYM CFT of dimension 2 (related by extendedsupersymmetry to conserved currents). We will pick anyof them, say OðxÞ � TrA1A2, to build a bi-local perturba-tion to SYM:

H ¼ HCFT þ �

�Zd3 ~xJð ~xÞOð ~xÞ

�2: (11)

Here the operator O is in Schrodinger picture. We havechosen a very low-dimension operator so as to minimize

the issues of UV divergences, studied in subsection IVB.Local double-trace operator deformations were studied inRefs. [22–26].We can also pass to the action formulation and path

integral quantization:

S ¼ SCFT � �Zdt

�Zd3 ~xJð ~xÞOðt; ~xÞ

�2

¼ SCFT � �Zd4xJð ~xÞOðxÞ

Zd4yJð ~yÞOðyÞ�ðx0 � y0Þ:

(12)

A. Superluminality

We consider the reference frame of H as the preferredone in which quantum time-evolution is defined. As inSec. II, the � perturbation can absorb CFT excitationsand instantly reemit them far away in the support of J,consistent with causality in the defining frame. This effectis reflected as superluminality in the gravity dual. Without�, the dual vacuum configuration is of course the well-knownAdS5 � S5. For the point we want to make, the S5 isjust a detail. Wewill not bother keeping track of locality onthe S5, just Kaluza-Klein reducing from 10 dimensionsdown to 5. Choose Poincare coordinates in AdS, in thesame preferred frame as the perturbed CFT,

ds2AdS ¼ ��dx�dx� � dz2

z2: (13)

Let us focus on the propagation of the AdS scalar,�ðx; zÞ, dual to the operator O. It is a ‘‘good’’ tachyonwith 5D mass-squared of�4, saturating the Breitenlohner-Freedman stability bound [27]. Consider two spacelike-

separated events in the AdS bulk spacetime, ð0; ~0; zÞ andðt > 0; ~x; zÞ, with 2z < t� j ~xj, so that causal communica-tion between them is ordinarily (� ¼ 0) impossible.

However, let us now suppose that ~0 and ~x are both withinthe support of J. To first order in �, perturbation theorypulls down from the action �

Rdt0ðRd3 ~x0Jð ~x0ÞOðt0; ~x0ÞÞ2.

The AdS dual of this perturbation at leading order in largeNcolor is that each Oðt0; ~x0Þ maps to a bulk-boundary free-field AdS propagation of the � scalar, with the boundarypoint being ðt0; ~x0; 0Þ. Denoting the free-field bulk-boundary propagator between ðx; zÞ and x0 by Kðx�x0; zÞ, we see that our leading correction to the bulk-to-

bulk propagator between ð0; ~0; zÞ and (t > 0, ~x, z) is

�Zdt0

Zd3 ~x0d3 ~y0Jð ~x0ÞJð ~y0ÞKðt0; ~x0; zÞKðt� t0; ~x� ~y0; zÞ:

(14)

Now,, for example, boundary points such as (t0 � t=2, ~x0 �~0) and (t0 � t=2, ~y0 � ~x) contribute to this integral. In AdS,such boundary points are causally connected to our bulk

points ð0; ~0; zÞ and (t > 0, ~x, z), respectively. That is eachK


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factor allows causal communication (they have imaginaryparts) and hence so does the entire bulk-to-bulk correction.This result is also a limiting case of that of Sec. V, whichprovides a more formal derivation.

(This leading order superluminality is UV finite in thecontinuum. Therefore it is also a good approximation to alattice realization of the deformed SYM theory, as long asall the relevant length scales are much larger than thelattice spacing, in particular t, j ~xj, z above and the domi-nant wavelengths of J.)

A quantum gravitational theory thereby admits super-luminal propagation, although in this case the ‘‘magic’’ islocalized to the AdS boundary. (A brief comment to similareffect is made in the discussion of Ref. [15].) Still, the AdSstring theory must possess the necessary boundary defectsthat allow this to occur, as long as our CFT deformation isUV complete. But we cannot argue that this takes the formof a Higgs mechanism on the AdS boundary since in asense gravity and general coordinate invariance end there.

B. Renormalizability

We must take care to understand what divergencesemerge due to the multiple operator products of O appear-ing in � perturbation theory. The reader may wish to followthis section by using the free SYM field theory as a simpleexample. Although we are primarily interested in stronglycoupled SYM so that the AdS dual is weakly coupled, theoperator O has very similar divergence properties at arbi-trary coupling because of its supersymmetry-protecteddimension.

We will power-count � perturbation theory to identifythe superficial divergences. The bilocal nature of our per-turbation makes this somewhat unfamiliar. We can mas-sage it a little to make it more amenable to standard power-counting. We attribute our perturbation to one that is linearin O,

�S ¼Zd4x�1=2Jð ~xÞOðxÞ�ðxÞ; (15)

where � is an auxiliary field with a ‘‘propagator,’’ for � ¼0,

G0ðx; yÞ � �2i�ðx0 � y0Þ: (16)

We can use this propagator to ‘‘integrate out’’ � and returnto the original perturbation. This is a formal device in thatG0 does not follow from some quadratic � action, but it isuseful for power-counting purposes. Since O has scaledimension 2, for power-counting purposes the background

field �1=2Jð ~xÞ has dimension 3=2, and from its propagator itis clear that �ðxÞ has power-counting dimension 1=2.

Let us consider what superficial divergences there can bein the � perturbation expansion. These must be local prod-ucts of CFT operators multiplied by powers of �ðxÞ and�1=2J (and derivatives), with total dimension 4. Since asingle operator insertion of O in the CFT is not divergent,

the divergences can only begin at quadratic order in �1=2J,already ‘‘costing’’ dimension 3. At most this could bemultiplied by powers of �, since all CFT gauge-invariantoperators have scaling dimension >1. By �! ��, J !�J symmetry, the only divergent structures can be �J2ð ~xÞand �J2ð ~xÞ�2ðxÞ. The first of these divergent structures canonly arise from the leading order VEVof the perturbationHamiltonian, a physically irrelevant real c-number con-stant, that can be renormalized away by simply subtractingit from our Hamiltonian (or action).We are thus only left to contend with �J2ð ~xÞ�2ðxÞ, that

is, a divergence in the� self-energy. Indeed there really is alogarithmic divergence of this form, and it is coupled to therest of the CFT because each �ðxÞ field in it can becontracted with �’s elsewhere in the perturbative expan-sion, so we do have to address this divergence. The leadingcorrection to the � self-energy takes the form

�Zd4xJð ~xÞ�ðxÞ

Zd4yJð ~yÞ�ðyÞh0jTfOðxÞOðyÞgj0i

¼ �c lnð�aÞZd4xJ2ð ~xÞ�2ðxÞ+finite; (17)

where a is a short-distance cutoff, � is an arbitrary renor-malization scale put in to separate out the UV divergence,and c is an unimportant constant. We have used thatO is adimension 2 primary operator in constraining the form ofits correlator in the usual way.This divergence is removed by renormalization of �. To

see this let us resum the � self-energy contributions to itspropagator arising from integrating out the CFT. Define theself-energy correction

��ðx; yÞ ¼ �Jð ~xÞJð ~yÞh0jTfOðxÞOðyÞgj0i: (18)

Log divergences are removed by the subtraction

��subðx; yÞ ¼ ��ðx; yÞ � ��divðx; yÞ; (19)

where by (17)

��divðx; yÞ � �c lnð�aÞJ2ð ~xÞ�4ðx� yÞ: (20)

Resumming this self-energy in the � propagator gives,in an obvious matrix notation,

�G ¼ �G0ðI � ��G0Þ�1

¼ �G0ðI � ��subG0 � ��divG0Þ�1: (21)

This is the only combination in which � and G0 appear inthe perturbative expansion, and all powers of � are explic-itly shown. Note that in the expansion of this expression�div always appears in the sandwich G0�divG0, for whichit is easy to prove,

G0�divG0 ¼ lnð�aÞJG0; (22)

where J is just the finite constant


086002-6

J � cZd3 ~xJ2ð ~xÞ: (23)

Using this relation, we can rewrite the resummed � propa-gator as

�G ¼ �G0ðI � ��subG0 � �J lnð�aÞIÞ�1

¼ �Rð�ÞG0ðI � �Rð�Þ�subG0Þ�1; (24)

where we define a renormalized coupling at �,

�Rð�Þ � �

1� �J lnð�aÞ : (25)

Since there are no more superficial divergences otherthan the real divergence subtracted by this renormalization,the perturbative expansion is now finite in terms of �R asthe short-distance cutoff a! 0.

C. Suppressed vacuum decay

As discussed in the introduction, there is no guaranteethat the SYM vacuum remains the true ground state of thedeformed theory. However, whatever the true ground state,the amplitudes for the decay of the SYM vacuum followfrom our renormalizable theory, giving some finite decayrates (per unit volume) perturbatively in �R. Above, we didsubtract a single infinite (order �J2) correction to vacuumenergy as part of renormalization, but this is irrelevant forvacuum decay since this divergence is real while it is theimaginary parts of ‘‘vacuum’’ energy that encode vacuumdecays via the optical theorem. For sufficiently small �R(renormalized at the scale typical of J) these decay rates,and Lorentz-violating processes in general, are suppressed,but a sufficiently long time will always overcome the weakcoupling and lead to complete SYM(AdS) decay. To pre-vent this from happening we will consider �ðtÞ to besmoothly time dependent with finite support. This makesthe Hamiltonian time dependent. Now we can choose �R soweak as to not lead to catastrophic vacuum decay. This alsosuppresses the probability of superluminal propagation,but it does not vanish and we are only seeking this quali-tative fact. Our previous analysis of superluminality insubsection IVA is only altered in that the �! �ðt0Þ nowsits inside the time integral. Superluminality continues tohold as long as the duration of nonvanishing � is taken tocover the events being discussed there.

D. Comments

We have seen that the deformed CFT is renormalizablein the same sense that QED is, logarithmic UV divergencesbeing eliminated by � renormalization, but the catch is that� runs in the UV to strong coupling where our power-counting breaks down. Thus, we have not yet demonstratedtrue UV completeness, but we are close. Of course, onepossibility is that the short-distance cutoff a may really befinite, if, for example, it is a lattice spacing for SYM on aspatial lattice. A spatial lattice theory would regulate the

divergences of the continuum field theory by converting itinto quantum mechanics of discrete lattice degrees of free-dom. In that case, the renormalizability usefully translatesinto insensitivity of the long-wavelength theory to detailsof the lattice structure.Another possibility is not to regulate the CFT itself, but

rather to replace each OðxÞ with a ‘‘slightly’’ nonlocaloperator so as to regulate the operator product divergencesappearing in � perturbation theory. A seemingly separatequestion is whether superluminality can be realized in thegravitating bulk of AdS, rather than the AdS boundarywhere the gravitational dynamics ends. We really wouldlike to test if superluminality can appear right in the midstof quantum general relativity. Presently, bulk quanta mustpropagate relativistically to the AdS boundary before theysee superluminal effects. As it turns out, we can indeedengineer bulk superluminality, and the trick is to replacethe pair of local operators appearing in the CFTHamiltonian by a pair of nonlocal operators. As an addedbenefit, this replacement renders the deformed theory UVfinite, that is, it regulates the UV divergences discussedabove. We pursue this approach next.

V. SUPERLUMINALITY IN THE BULK

In this section, O denotes an arbitrary local scalar pri-mary operator of the SYM CFT, of scaling dimension d,which has the single-trace limit as Ncolor ! 1. Insubsection VD we will restrict d to ensure perturbativityof our deformation under all circumstances. Until then wesimply assume perturbation theory in the CFT deformationis to be trusted.

A. Warm-up on nonlocal operators and UV finiteness

Consider the simple, but time-dependent, deformationof the CFT Hamiltonian of the form

HðÞ � HCFT þ�HðÞ � HCFT þZd3 ~xJð; ~xÞOð ~xÞ:

(26)

All the operators appearing here are Schrodinger operators,time dependence appearing only in the source, Jð; ~xÞ.Since time-ordering subtleties will be important in this

section, we use Hamiltonian and operator methodsthroughout. The master formula for time-ordered perturba-tion theory is

Te�iRt2t1dHðÞ ¼ e�iHCFTðt2�t1Þ

X1

n¼0

ð�iÞnn!

Z t2

t1

d1

�Z t2

1

d2 . . .Z t2

n�1

dn�HðnÞ . . .�Hð1Þ:

(27)

We uniformly use hats to distinguish Heisenberg operators,


086002-7

�HðÞ � eiHCFT�HðÞe�iHCFT;

Oð; ~xÞ � eiHCFTOð ~xÞe�iHCFT:(28)

Consider the example of the CFT-vacuum persistenceamplitude at order J2,

Zd4xd4yJðxÞJðyÞh0jTfOðxÞOðyÞgj0i

¼Zd4xd4yJðxÞJðyÞ

Zdm2

Zd3 ~p

jh0jOð0Þjm; ~pij22

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

p

� ððx0 � y0Þe�ip:ðx�yÞ þ ðy0 � x0Þeip:ðx�yÞÞ/Zd4xd4yJðxÞJðyÞ

Zdm2m2d�4Gðx� y;mÞ

¼Zdm2m2d�4

Zd4q

j~JðqÞj2q2 �m2 þ i�

¼ 1: (29)

The first equality follows in passing to the spectral repre-sentation by inserting a complete set of states, integratingover their invariant mass-squared and momentum. Theproportionality follows from the fact that by Lorentz in-variance the matrix element of the scalar operator O canonly depend on the mass m, not the momentum ~p, and themass-dependence follows by dimensional analysis in theCFT. Gðx� y;mÞ denotes a free-field scalar Feynmanpropagator of mass m. The final equality arises because,whereas the smoothing due to J cuts off the q-integral, them integral remains divergent.

This divergence is very closely related to the one of theprevious section, and is rather standard when perturbing by(superpositions of) local operators. In perturbation theory,

Oð; ~xÞ creates a pointlike disturbance at time , whichthen begins to spread out. However, at second order a

second Oð0; ~x0Þ can sample the disturbance created bythe first, and this creates our divergence when the twopoints coincide. Such divergences are avoided if the point-like disturbances are ‘‘thickened’’ to finite size. One con-venient way of doing this is to use a fake time evolution tospread out the pointlike disturbance created byO. A simpleillustration is provided by the Hamiltonian,

H ¼ HCFT þZd4xJðxÞeiHCFTx0Oð ~xÞe�iHCFTx0 : (30)

All the operators appearing are in Schrodinger picture. Oneproduct of these operators happens to be a Heisenbergoperator in form, but the whole effect of the x0 ‘‘timeevolution’’ on O is to turn this spatially localSchrodinger operator into a spatially nonlocal one, byevolving the disturbance it creates for a finite time whichcauses the disturbance to spread over a finite spatial region.Note that since x0 is integrated over, the SchrodingerH and�H are time independent. It is important to note that byspecifying a Hamiltonian, the resulting dynamics is auto-matically local in time. But the perturbation is spatiallynonlocal because it cannot be written as a superposition of

local Schrodinger operators. If this looks unfamiliar it isbecause it is inconsistent with Lorentz invariance, fromwhich we are deviating in this paper.To contrast this with our earlier example, let us again

apply (27) to calculate the CFT-vacuum persistence ampli-tude at order J2,

/Z 1

�1d

Zd4xd4yJðxÞJðyÞh0jTfOðx0 þ ; ~xÞOðyÞgj0i

¼Zd

Zdm2

Z d3 ~p

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

p fðÞe�iffiffiffiffiffiffiffiffiffiffiffi~p2þm2

p

þ ð�Þeiffiffiffiffiffiffiffiffiffiffiffi~p2þm2

pgj~Jð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

q; ~pÞh0jOð0Þjm; ~pij2

/Zd

Zdm2m2d�4

Z d3 ~p

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

p j~Jðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~p2 þm2

q; ~pÞj2

� fðÞe�iffiffiffiffiffiffiffiffiffiffiffi~p2þm2

p þ ð�Þei

ffiffiffiffiffiffiffiffiffiffiffi~p2þm2

pg<1: (31)

The time ordering is with respect to only. Here, we seethat the smoothness of J does cut off them and ~p integrals,and the integral also converges as ! 0. This merelyreflects the thickened, as opposed to completely local,operator that perturbs the CFT in the second example.The comparison of the relatively simple examples in this

subsection should orient the reader in the full constructionof the next.

B. Superluminality

Consider the Hamiltonian,

H ¼ HCFT þ�H

� HCFT þ �

�Zd4xJðxÞeiHCFTx0Oð ~xÞe�iHCFTx0

�2; (32)

where everything has been written in terms of Schrodingeroperators, and where J is a smooth spacetime-dependentsource of compact support. Note that x0 is a dummyintegration variable and that H is in fact time independent.(We will, however, introduce time dependence insubsection VD.)To demonstrate superluminality, we consider properties

of the bulk-to-bulk propagator for the scalar �, dual to theprimary operator O, between the two spacetime points

ðy�; zÞ and ð0; zÞ, namely h0jTf�ðy; zÞ�ð0; zÞgj0i. (This awell-defined object if we imagine having gauge-fixed gen-eral coordinate invariance.) Our approach (but not ourresult) shares some similarities with that of Ref. [26] study-ing AdS implications of double-trace but local deforma-tions of SYM. We will take

0< y0 < j ~yj; (33)

so that the two bulk points are ordinarily (� ¼ 0) causallydisconnected (recalling that our AdS metric is (13)), im-plying the vanishing of the commutator,


086002-8

h0j½�ðy; zÞ; �ð0; zÞ�j0i ¼ 2i Imh0jTf�ðy; zÞ�ð0; zÞgj0i� sgnðy0Þ: (34)

The right-hand form shows that this information on cau-sality is contained in the propagator.

We will further consider that

y0 < z; (35)

so that if superluminality were concentrated on the AdSboundary, there is simply no time to causally propagatethrough the bulk to get to the boundary to take advantage ofit. Therefore we are guaranteed that if �H gives a nonzerocorrection to (34), then it must be due to superluminaleffects in the gravitating AdS bulk spacetime. In particularalso, the spatially nonlocal operator �H must have anaction at the instant it acts which is dual to disturbing theAdS bulk, rather than just a modification of AdS boundaryconditions as is the case for local CFT operators.

Let us check that there is indeed a nonvanishing correc-tion to the bulk commutator VEV. First let us formalize thequestion in the gravity description. To do this let us adopt aHamiltonian approach to the AdS gravitational theory,using the AdS asymptotics to define energy and hencethe Hamiltonian, HAdS. The AdS/CFT dictionary tells ushow insertions of the local operator O map to the AdS

boundary, so that we are able to map the effects of our CFTdeformation, �H, and regard it as a deformation added toHAdS (at least to any fixed order in � perturbation theory).Below we will work in the gravity description, and theappearance ofO or�H will always denote the correspond-ing objects mapped to the gravity side. So, for example, thetotal gravity-side Hamiltonian is

Hgravity ¼ HAdS þ �H: (36)

In this canonical approach let �ð ~x; zÞ denote the

Schrodinger operator dual to O, with �ðx; zÞ �eiHgravityx0�ð ~x; zÞe�iHgravityx0 being the usual Heisenberg op-erator construction. Using the very general result of (27) itis then straightforward to prove that to first order in �H,

½�ðy; zÞ; �ð0; zÞ� ¼ iZ y0

0d½½�ðy; zÞ;�HðÞ�; �ð0; zÞ�;

(37)

where we have used that at zeroth-order (relativistic) cau-

sality implies ½�ðy; zÞ; �ð0; zÞ� ¼ 0.We can then calculate the VEVof this commutator in the

large-Ncolor limit, where the resulting four-point VEVs

factorize as h0j� � O O j0i � h0j� O j0ih0j� O j0i, etc.The result is

h0j½�ðy; zÞ; �ð0; zÞ�j0i � 2iZ y0

0d

Zd4xd4x0JðxÞJðx0Þh0j½Oðx0 þ ; ~xÞ; �ðy; zÞ�j0ih0j½Oðx00 þ ; ~x0Þ; �ð0; zÞ�j0i

¼ �8iZ y0

0d

Zd4xd4x0JðxÞJðx0Þ Imh0jTfOðx0 þ ; ~xÞ�ðy; zÞ�gj0isgnðx0 þ � y0Þ

� Imh0jTf½Oðx00 þ ; ~x0Þ�ð0; zÞ�gj0isgnðx00 þ Þ� �8i

Z y0

0d

Zd4xd4x0JðxÞJðx0Þ ImKðy0 � x0 � ; ~y� ~x; zÞ

� ImKð�x00 � ;� ~x0; zÞsgnðx0 þ � y0Þsgnðx00 þ Þ; (38)

finally arriving at an expression in terms of bulk-boundarypropagators of the undeformed theory.

Choose some intermediate � y0=2 as an example. Wesee that if the support of J has sufficiently positive x00 so

that ðx00 þ Þ2 > ð ~x0Þ2 þ z2 and has sufficiently negative x0so that ðy0 � � x0Þ2 > ð ~y� ~xÞ2 þ z2, then causal com-munication between the bulk point ð0; zÞ and boundarypoint ðx00 þ ; ~x0Þ is possible and causal communication

between the bulk point ðy; zÞ and boundary point (x0 þ ,~x) is also possible. Therefore by the relation betweencommutator VEVs and propagators (the analog of (34))each ImK factor can be nonzero and so we have demon-strated causal communication between ð0; zÞ and ðy; zÞ.

It is important to stress that although we have expressedthis communication mathematically as a product of com-munication from bulk to boundary and then back in theundeformed theory, there is in fact not enough time avail-

able in our setup for bulk to boundary communication toproceed unless there is superluminality in the bulk. Thisbulk superluminal communication must therefore be tak-ing place. It is a mere convenience that we are parametriz-ing the requisite bulk disturbances in terms of boundarysources that could produce them given enough time.In this admittedly indirect manner we have shown that

superluminality is taking place in the AdS bulk, and there-fore the exotic Higgs effect necessary to make this possiblein a gravitating spacetime must be present. But we muststill ask if the gravitating theory is UV complete. This isguaranteed if the deformed CFT is UV complete. We nowshow this.

C. UV finiteness

Consider the amplitude to evolve from a state jAi to astate jBi over a time interval t. We consider the two states


086002-9

to be energy-momentum eigenstates of the undeformed CFT. For clarity, we begin with the example of the order �2

contribution to this amplitude following from (27),

/Z t

0d0

Z t

0d

Zd4xd4yd4x0d4y0JðxÞJðyÞJðx0ÞJðy0ÞhBjOðx0 þ ; ~xÞOðy0 þ ; ~yÞOðx00 þ 0; ~x0ÞOðy00 þ 0; ~y0ÞjAi

/Zd0

Z0d

Zd4xd4yd4x0d4y0JðxÞJðyÞJðx0ÞJðy0Þ

Zþd4p

Zþd4q

�Zþd4khBjOðx0 þ ; ~xÞjpihpjOðy0 þ ; ~yÞjqihqjOðx00 þ 0; ~x0ÞjkihkjOðy00 þ 0; ~y0ÞjAi

¼Zd0

Z0d

Zþd4p

Zþd4q

Zþd4k~JðpB � pÞ~Jðp� qÞ~Jðq� kÞ~Jðk� pAÞeiðpB0�q0Þeiðq0�pA0Þ0

� hBjOð0ÞjpihpjOð0ÞjqihqjOð0ÞjkihkjOð0ÞjAi: (39)

We have inserted complete sets of states between opera-tors, explicitly summing over their possible momenta, withpositive mass-squared and energy (the ‘‘ þ’’ subscript onthe momentum integrals), and implicitly over any otherlabels. Such integrals represent sums over nonvacuumstates related by Poincare symmetry and scale symmetry.One can also insert the vacuum state, in which case therelevant momentum integral drops out in the obvious way.We have not written these terms because they are lessdangerous to finiteness.

UV divergences can only arise in the expression whenthe momentum integrals or integrals diverge. There areno UV divergences in the above expression, however,because the smoothness of J translates into the rapid damp-ing of ~J for large momenta, and because the , 0 integralshave finite range with only well-behaved phase factorintegrands. This generalizes the finiteness we saw in thesecond example of the last subsection. The reader caneasily extend this check of finiteness to arbitrary order in� perturbation theory by repeated insertion of a completeset of states between operators. The smooth J factorsalways make each momentum integral converge.

Thus our deformed CFT, and its deformed AdS dual, areUV complete. We can easily compare this with Sec. IV bynoting that we revert to that case in the limit

JðxÞ ! Jð ~xÞ�ðx0Þ; ~JðqÞ ! ~Jð ~qÞ: (40)

Consider the simple case where the jpi; jki states arereplaced by the CFT vacuum state, so there is only onemomentum integral to worry about, and only one (relative)� � � 0 > 0 integral. The relevant term is

Z0d�

Zþd4qeiq0�jh0jOjqi~Jð ~qÞj2: (41)

By scale invariance, jh0jOjqij2 / ðq2Þd�2, and we see thatwithout ~J to help cut off q0, the integral diverges as � !0, q0 ! 1.

D. Perturbativity and vacuum decay

Schematically, each order in perturbation theory in �brings an expression,

�Zd

Zd4xJðxÞ

Zd4yJðyÞ

Zþd4p

Zþd4q . . .

� Oðx0 þ ; ~xÞjpihpjOðy0 þ ; ~yÞjqihqj . . . (42)

The coupling combination �JðxÞJðyÞ has dimension 9�2d which we will ascribe completely to �, taking J to be asmooth dimensionless function taking values of order unityin a spacetime volume of order L4. We have seen in the lastsubsection how the intermediate state momenta are tied tothe external momenta, which we characterize to be of orderE, with the J integrals providing momentum shifts of order1=L. We begin by considering E� 1=L. Each J integral

suppresses oneRþ d

4p integral, fixing p� E. Each Othen counts as Ed in its matrix elements and each jpihpjcounts as E�4, by dimensional analysis and the fact that thehard external E scale is dominant. Finally, the depen-dence in the operators turns into a phase factor ei�E,where �E is an energy change allowed by the ‘‘back-ground’’ J, of order 1=L. Therefore the

Rd integral

(with whatever time-ordered limits of integration) is atmost of order L.Putting these factors together, we find that every order in

perturbation theory counts as the dimensionless combina-tion, �LE2d�8, for E� 1=L. If d > 4 then, no matter howsmall we take �L, there will be processes where perturba-tion theory is breaking down (although we still do not findUV divergences perturbatively). We will avoid this byrestricting the primary operator O to have scaling dimen-sion d 4. Then the condition for perturbativity is�L9�2d � 1 for E> 1=L.We must still check perturbativity in the far IR, E�

1=L. Now all momentum scales above are dominated by1=L in all the integrals and matrix elements, and theperturbative strength is simply �L9�2d. So there is consis-tency in the perturbativity requirements for IR and UVprocesses, �L9�2d � 1.Our deformation of SYM is finite and perturbative. In

particular, the deformed Hamiltonian requires no subtrac-tions and is a sum of squares (HCFT is by supersymmetry),so that there is some well-defined ground state. The SYMvacuum is a finite and well-defined excitation above this


086002-10

ground state. We can make it arbitrarily long-lived bymaking � weaker and weaker. Or we can do what we didin subsection IVC, make �! �ðÞ and our Hamiltoniantime dependent so that the deformation is turned on for afinite duration. We then take � small enough that the SYM(AdS) vacuum in most regions of space survives the periodof deformation. This suppresses the amplitude for super-luminality in the bulk but it can still take place.

VI. CONCLUSIONS

Our central construction has been a weak Lorentz-violating deformation of the N ¼ 4 SYM CFTHamiltonian by a superposition of spatially nonlocal op-erators. We checked that, at leading order in the perturba-tion, the standard AdS/CFT map gives a nonvanishingpropagator between two spacetime points which are ordi-narily causally disconnected in the AdS bulk. The two bulkpoints are also ordinarily out of causal contact with theAdS boundary during the time interval separating them,implying that the superluminal behavior is taking place inthe gravitating bulk. Finally, we checked that our deforma-tion was UV complete, in that there were no new sources ofUV divergence outside the renormalized CFT. Therefore,there must be a complete deformed AdS gravity/string dualof the superluminal behavior.

However, given our indirect CFT approach to this con-clusion, the detailed AdS description of bulk superlumin-ality is not apparent. Indeed, the specific form of ourspatially nonlocal deformation was chosen to demonstratebulk superluminality in terms of the simplest object of theAdS/CFT dictionary, namely, the bulk-boundary propaga-tor. It is possible that a different nonlocal deformationmight yield a simpler AdS spacetime description, althoughlikely at the cost of a more complex translation from theCFT side.

We have presented a simple building block for super-luminality, but there are clearly other directions to pursue.It appears straightforward that a similar deformation couldbe used to couple two different, otherwise decoupled,CFTs, whose dual would describe bulk coupling of AdSdegrees of freedom from both CFTs. Deformations by localoperators (as opposed to nonlocal ones such as we aresuggesting) connecting otherwise disconnected CFTs

have been discussed in Refs. [28]. More ambitiously, wewould like to understand regimes in which the Lorentz-violating or superluminal effects become important, as, forexample, required in Ref. [7], in resolving the cosmologi-cal constant problem by energy-parity, or in [5] in modify-ing the character of black hole horizons. We would alsolike to see if superluminal effects can be engineered withinUV-complete quantum gravity theories as weak probes ofordinary horizons, as well as the interesting singularitiesthat they can hide.As discussed in the introduction, Lorentz violation in a

gravitational context must appear formally as a type ofHiggs effect. This suggests that even when the violation isexplicit on the CFT side as in our case, on the AdS side itshould appear as a property of a state or solution, not amodification of the gravitational dynamics itself. As re-marked above, our approach does not straightforwardlygive a detailed description of such states in AdS. It isintriguing, however, that, from the opposite direction,wormhole solutions in Euclidean AdS gravity pose apuzzle for CFT interpretation [13–15] precisely becausethey suggest nonlocal interactions on the CFT side.Perhaps the resolution of this puzzle lies in nonlocal de-formations of the CFT, at least similar in spirit to theexample of Sec. V of this paper.The example of superluminality and Lorentz violation

provided in this paper has a certain ‘‘premeditated’’ feel toit, and one naturally wonders whether it is too contrived tobe at work in nature. That would, however, be a prematureconclusion, because we have only given an existence proof.It is possible that real world gravity and relativity is a richemergent phenomenon with a more natural framework forthese exotic effects. Hopefully we can understand thetheoretical possibilities well enough to devise the rightexperimental tests to decide.

ACKNOWLEDGMENTS

The author is grateful to Nima Arkani-Hamed, JuanMaldacena, and Joe Polchinski for discussions. This re-search was supported by the National Science FoundationGrant No. NSF-PHY-0401513 and by the Johns HopkinsTheoretical Interdisciplinary Physics and AstrophysicsCenter.

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Gravitational Lorentz violation and superluminality via AdS/CFT duality