PHYSICAL REVIEW D 69, 044001 ~2004!

2¿1 gravity and doubly special relativity

Laurent Freidel*Perimeter Institute for Theoretical Physics, Waterloo, Canada

and ENS-Lyon, Lyon, France

Jerzy Kowalski-Glikman†

Institute for Theoretical Physics, University of Wrocław, Poland

Lee Smolin‡

Perimeter Institute for Theoretical Physics, Waterloo, Canadaand Department of Physics, University of Waterloo, Waterloo, Canada

~Received 21 July 2003; published 6 February 2004!

It is shown that gravity in 211 dimensions coupled to point particles provides a nontrivial example ofdoubly special relativity~DSR!. This result is obtained by interpretation of previous results in the field and byexhibiting an explicit transformation between the phase space algebra for one particle in 211 gravity found byMatschull and Welling and the corresponding DSR algebra. The identification of 211 gravity as a DSR systemanswers a number of questions concerning the latter, and resolves the ambiguity of the basis of the algebra ofobservables. Based on this observation a heuristic argument is made that the algebra of symmetries of ultrahigh energy particle kinematics in 311 dimensions is described by some DSR theory.

DOI: 10.1103/PhysRevD.69.044001 PACS number~s!: 04.90.1e, 04.60.2m, 11.30.Cp

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I. INTRODUCTION

Recently a proposal has been much discussed concehow quantum theories of gravity may be tested experimtally. The doubly or deformed special relativityproposal~DSR!1 is that quantum gravity effects may lead in the limof weak fields to modifications in the kinematics of elemetary particles characterized by@5–9#

~1! Preservation of the relativity of inertial frames.~2! Nonlinear modifications of the action of Lorentz boos

on energy-momentum vectors, preserving a preferredergy scale, which is naturally taken to be the Planenergy,Ep . In some casesEp is a maximum mass and/omomentum that a single elementary particle can atta

~3! Nonlinear modifications of the energy-momentum retions, because the function ofE and pW that is preservedunder the exact action of the Lorentz group is no lonquadratic. This could result in Planck scale effects suas an energy-dependent speed of light and modificatof thresholds for scattering, that may be observablepresent and near future experiments.

*Electronic address: lfreidel@perimeterinstitute.ca†Electronic address: jurekk@ift.uni.wroc.pl‡Electronic address: lsmolin@perimeterinstitute.ca1Aspects of DSR theories have been proposed or studied m

than once in the past, only to be forgotten and then rediscovagain. Early formulations were by Snyder@1# and Fock@2#. Duringthe 1990s the mathematical side of the subject was developed uthe name ofk-Poincare´ symmetry@3,4#. The recent interest is duto the proposal that the effects of such theories may be both tesand derivable from some versions of quantum gravity, see forample@5–9#.

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~4! Modifications in the commutators of coordinates amomentum and/or non-commutativity of space-time cordinates.

Theories with these characteristics are invariant unmodifications of the Poincare´ algebra, called genericallyk-Poincare´ algebras, wherek is a dimensional parametethat measures the deformations, usually taken to be protional to the Planck mass.

In a recent paper@10#, it was argued that quantum gravitin 211 dimensions@11,12# with vanishing cosmological constant must be invariant under some version of ak-Poincare´symmetry. The argument there depends only on the assution that quantum gravity in 211 dimensions with the cosmological constantL50 must be derivable from theL→0limit of 211 quantum gravity with nonzero cosmologicconstant. The argument is simple and algebraic, the pointhat the symmetry which characterizes quantum gravity211 dimensions withL.0 is actually quantum deformed dSitter toSOq(3,1), with the quantum deformation parametq given by @14,15,17#

z5 ln~q!' l PlanckAL. ~1!

The limit L→0 then affects both the scaling of the transtion generators as the de Sitter group is contracted toPoincare´ group, and the limit ofq→1. It is easy to see thabecause the ratiok5\AL/z5G211

21 , whereG211 is New-ton’s constant in 211 dimensions, is held fixed, the limigives thek-deformed symmetry group in 211 dimensions.The conclusion is that the symmetry algebra of~211!-dimensional quantum gravity withL50 is not Poincare´, it isa k-deformed Poincare´ algebra. This means that the theomust be a DSR theory.

Quantum gravity in 211 dimensions has been the subjeof much study in both the classical and quantum doma

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FREIDEL, KOWALSKI-GLIKMAN, AND SMOLIN PHYSICAL REVIEW D 69, 044001 ~2004!

beginning with the work of Deser, Jackiw and ’t Hooft@12–23#. If that theory is a DSR theory than the features julisted above must be present, and this could not have beasily missed by investigators.

Indeed,all of the listed features have been seen inliterature on 211 gravity. In the next section we reviewsome of the long standing results in 211 gravity and showhow they may be understood using the language of Dtheories. To clinch the relation, in section III we exhibit aexplicit mapping between the phase space of quantum gity in 211 dimensions coupled to a single point particstudied in@21#, and the algebra of symmetry generators oDSR theory.

The observation that 211 gravity provides examples oDSR theories can help the study of both sides of the relatThe language of DSR theories and their foundations in teof general principles can unify and explain some resultsthe literature of~211!-dimensional gravity that, when firsdiscovered, seemed strange and unintuitive. We can nowthat some of the features of 211 gravity are neither strangnor necessarily unique to 211 dimensions, because they folow only from the general requirement that the transformtions between different inertial frames preserve an enescale.

Furthermore, what one has in the 211 gravity models,such as those with gravity coupled toN point particles, is aclass of nontrivial DSR theories that are completely expland solvable, both classically and quantum mechanicaThe existence of these examples answers a number of qtions and challenges that have been raised concerningtheories. Some authors have argued@24# that DSR theoriesare just ordinary special relativistic theories rewrittenterms of some nonlinear combinations of energy and mmentum, while, conversely, others have argued that tmust be trivial because interactions cannot be consisteincluded. Both criticisms are shown wrong by the existenof an explicit and solvable class of DSR theories, with intactions, given by quantum gravity in 211 dimensionscoupled to point particles and fields.

Furthermore, we see that in 211 dimensions the apparenproblem of the freedom to choose the basis of the symmalgebra of a DSR theory is resolved by the fact thatchoice of the coupling of matter to the gravitational fiepicks out the physical energy and momentum. We see inIII below that for the case of minimal coupling of gravity ta single point particle the basis picked out is the classbasis.

Finally, one can ask whether the fact that 211 gravity is aDSR theory has any implications for real physics in 311dimensions. In the final section of the paper we presenheuristic argument that it may.

II. SIGNS OF DSR IN 2¿1 GRAVITY

In this section we point out where effects characteristicDSR have been discovered already in the literature on~211!-dimensional gravity. We consider only the caseL50.

It is important first to note that Newton’s constant in 211dimensions, denoted here byG, has dimensions of invers

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mass ~with only c51 and no\ involved!.2 Thus, if theasymptotic symmetry group knows about gravity, it will hato preserve the scaleG21. Of course, in theories with sufficiently short range interactions the asymptotic symmegroup does not depend on the coupling constants. But in 211gravity the presence of matter causes the geometry of sptime to become conical and this deforms the asymptotic cditions in a way that depends onG. Further, since\ is notinvolved in the definition of the mass scale,G21, the defor-mation affects also the algebra of the classical phase spThis is the main reason why 211 gravity is a DSR theory.

In 211 gravity coupled to point particles, the Hamitonian,H, whose value is equal to the ADM mass, and henis measured by a surface term, is bounded from both aband below@21,23#,

0<H<1

4G. ~2!

This can be understood in the following way. In 211 dimen-sions the spacetime is flat, except where matter is presenparticle, or in fact any compactly supported distributionmatter, is surrounded by an asymptotic region, which iscally flat, and whose geometry is thus characterized bdeficit anglea. A standard result is that@12,16–18,21,23#,

a58pGH. ~3!

But a deficit anglea must be less than or equal to 2p. Hencethere is an upper limit on the mass of any system, as msured by the Hamiltonian. The upper limit holds for all sytems, regardless of how many particles there are and wtheir relative positions or motions are.

This upper mass limit must be preserved by tasymptotic symmetry group. Hence the asymptotic symmtry group cannot be the ordinary Poincare´ group, if it includeboosts it must be a DSR theory with a maximum energy.3

It has further been shown that the spatial componentsmomentum of a particle in 211 gravity are unbounded@21#.This, together, with a bounded energy, implies a modifienergy-momentum relation.

The phase space of a single point particle in 211 gravitywas constructed by Matschull and Welling in@21# and it wasfound that a classical solution is labeled by a three dimsional positionYm , m50,1,2 and momentumpm . They findexplicitly that the energy momentum relations and the actof the Poincare´ symmetry are deformed, in a way that pr

2G is identified with inverse of thek deformation parameter ok-Poincare´ algebra.

3Note that in the approach of Matschull and Louko@22,23# whichanchor the reference frame to the conical infinity, the asymptsymmetry group is the two dimensional group of isometry ofconical space time and it does not contain boosts. Howeverresults concerning the phase space structure of the relative mof particles, like noncommutativity of positions and curved aunbounded space of momenta still hold in this approach.

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211 GRAVITY AND DOUBLY SPECIAL RELATIVITY PHYSICAL REVIEW D 69, 044001 ~2004!

serves a fixed energy scale. They indeed make explicit reence to the work of Snyder@1#, which was an early proposafor DSR.

Furthermore, Matschull and Welling find that the spactime coordinatesYm of a particle are noncommutative undthe classical Poisson brackets,

@Ym ,Yn#522GemnrYr . ~4!

This property was found in@23# to extend to systems ofNparticles.

Matschull and Welling also find that the componentsthe energy-momentum vector for a point particle in 211gravity live on a curved manifold, which is~211!-dimensional anti–de Sitter spacetime. This was shown@29,31# to be a feature of DSR theories.4

Ashtekar and Varadarajan@16# found a relationship between two definitions of energy relevant for 211 gravity,which is reminiscent of nonlinear redefinitions of the enerused in changing bases between different realizations of Dtheories. The case they studied has to do with 311 gravity,with two Killing fields, one rotational and one axial. Onfirst dimensionally reduces to 211 dimensions, in whichcase the dynamics of GR in 311 is expressed as a scalar fiecoupled to~211!-dimensional GR. The ADM HamiltonianHstill exists and still is bounded from above as in Eq.~2!. Butin the presence of the additional, rotational Killing field, ttheory can be represented by a scalar field evolving in areference Minkowski spacetime, with the ordinary Hamtonian

H f lat51

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H f lat is of course unbounded above. They find the retionship between them is

H51

4G~12e24GHf lat!. ~6!

This exact relation is in fact present in the literature on D@29#. It holds in the a presentation of thek-Poincare´ algebraknown as the ‘‘bicrossproduct’’ basis. In that caseH f lat5Eis, as in the present case, the zeroth component of an enmomentum vector andH is the ‘‘physical rest mass,m0 de-fined by

1

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p

dE

dpUE5p0

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It is intriguing that this is the inertial mass, while, for thsolutions with rotational symmetry, the ADM energy is thactive gravitational mass. Since they are both expresse

4Although in Refs.@29,31# the momentum space for a classDSR theories was shown to be de Sitter spacetime. We disbelow the difference between positively and negatively curved mmentum spaces.

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terms of the zeroth component of the energy-momentum vtor by the same equation, they coincide on the subsesolutions on which they are both defined, which are thetationally invariant solutions. This appears to be a dirdemonstration of the equality of inertial and gravitationmass, within this context.

Indeed, this observation suggests that the AshteVaradarajan form of the ADM mass is more general ththeir calculation shows. Indeed it is not hard to see thatis the case. Let us study the free scalar field in~211!-dimensional Minkowski spacetime, with no condition of rtational symmetry. This system isnot a dimensional reduc-tion of general relativity, only a subspace of solutions, thowith rotational symmetry, are related to general relativiBut it still may serve as a useful example of a DSR theoOf course the theory has full Poincare´ invariance, with mo-mentum generatorsPi and boost generatorsKi satisfying theusual Poincare´ algebra. But Eq.~6! implies that they formwith H a DSR algebra

$Ki ,H%5~124GH!Pi

$Ki ,Pj%521

4Gd i j ln@124GH# ~8!

with the other commutators undeformed. The physicalergy momentum relations are deformed to

Pi21m25

1

16G2@ ln~124GH!#2. ~9!

Recent calculations@26# indicate that quantum deformations of symmetries play a role in gravitational scatteringparticles in 211 dimensions.

All of these pieces of evidence show that 211 gravitycoupled to matter can be understood as a DSR system.

Of course, the~211!-dimensional model system is nocompletely analogous to real physics in 311 dimensions.But this result answers cleanly several queries and criticisthat have been levied against the DSR proposal.

First, some authors have suggested that DSR theoriesphysically indistinguishable from ordinary special relativi@24#. They argue that in some cases, one can arrive at a Dsystem from a nonlinear mapping of energy-momentspace to itself. These results show that argument fails,there is no doubt that the model system of point particles211 gravity is physically distinguishable from the modsystem of free particles in flat~211!-dimensional spacetimeThis is here a clean result, with no quantization ambiguitibecause the deformation parameterk51/4G is entirely clas-sical and the modification is of the structure of the classiphase space. The two phase spaces are not isomorphic,gravity is turned on, the phase space is curved and the mas a maximum, but whenG50 the phase space is flat anthe mass has no bound.

This is clear also for the multiparticle system, where theare nontrivial interactions, depending onG, which make thesystem measurably distinct from the free particle case wG50.

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The multiparticle system in 211 gravity also serves as aexample of a counterintuitive property of some DSR modin 311 gravity. This is that the upper mass limitMupper51/4G is independent of the number of particles in the stem. This of course cannot be the case in the real world, sis good to know that there are implementations of DSR311 dimensions that do not have an upper mass limitsystems of many particles, or where the upper mass lgrows with the number of particles or the mass of the tosystem, in such a way as to not violate experience@25#.

However, it is also good to know that there is a modsystem, which is sensible physically, in which this noninttive feature is completely realized. Moreover it suggestsstart of a physical answer to one of the puzzling questiabout DSR models. This is that the addition of energy amomentum in DSR theories is nonlinear. This can be undstood as a consequence of the nonlinear action of the Lorgroup, for example it follows from the fact that the energmomentum space has nonzero curvature. It appears to reeven in realizations of DSR that remove the mass limitcomposite systems.

Some physicists have criticized the DSR proposalpointing out that the nonlinear corrections to additionenergy-momentum vectors for a system of two particlesbe interpreted by saying that there is a binding energytween pairs of particles that does not depend on the distabetween them, but depends only on the individual energand momenta.

This may be counter-intuitive, but it is precisely the whhappens in 211 dimensions. Because spacetime is locaflat, each particle contributes a deficit angle to the ovegeometry that affects all the other particles’ motions, no mter how far away. The result is that there is a binding enethat is independent of distance.

This suggests a speculative remark: might there be ein 311 dimensions a small component of the binding eneof pairs of particles, of orderl pM1M2, which is independenof distance? Might this be interpreted as a kind of quantgravity effect?

In the last section we make some speculative remaconcerning the question of whether these results havebearing on real physics in 311 dimensions.

III. PHASE SPACE OF DSR IN 2¿1 DIMENSIONS

In this section we will compare the phase space of~211!-dimensional DSR with that of~211!-dimensional gravitywith one particle. Let us start with the former.

A. Phase spaces of DSR

As in 311 dimensions, the starting point to find the phaspace of DSR theory in the~211!-dimensional case is th~211!-dimensionalk-Poincare´ algebra@4#, the quantum al-gebra whose generators are momenta5 pm5(p0 ,pi) and Lor-

5The Greek indices run from 1 to 3, the Latin ones from 1 towhile the capital ones from 0 to 3.

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entz algebra generatorsJm5(M ,Ni) boosts. Taking the co-algebra of thek-Poincare´ quantum algebra and using thso-called ‘‘Heisenberg-double construction’’@27–29# it ispossible to derive the position variables, conjugate to mmenta,xm , as well as the brackets between them andk-Poincare´ algebra generators.

This ~quantum! algebraic construction has a geometriccounterpart, described in@30,31#. Here the manifold onwhich momenta live is de Sitter space~in the case at handthe 3 dimensional one!. The positions and the Lorentz tranformations are symmetries acting on the space of momeThus they form the three dimensional de Sitter algeSO(3,1). It is convenient to define the de Sitter spacemomenta as a three dimensional surface

2h021h1

21h221h3

25k2 ~10!

in the four dimensional Minkowski space with coordinat(h0 , . . .h3). The physical momentapm are then the coordi-nates on the surface~10!. This means that we can think ohA5hA(pm) as of the given functions of momenta, fowhich Eq.~10! is identically satisfied. In the DSR terminoogy, the choice of a particular coordinate system on de Sspace corresponds to a choice of the so called DSR basis~see@29,31#!. It turns out that in order to relate DSR to the 211gravity one has to choose the so called classical basis, cacterized byhm5pm . This choice will be implicit below,however we find it more convenient to write down the fomulas below in terms of the variableshA .

The algebra of symmetries of the de Sitter space of mmenta~10! can be most easily read off by writing down thaction of these symmetries on the four-dimensioMinkowski space with coordinateshA and then pulling themdown to the surface~10!. Let us note however that while it iseasy to identify the Lorentz generatorsJm5(M ,Ni) as theelements of theSO(2,1) subalgebra of theSO(3,1), it is amatter of convenience which linearly independent combition of generators is to be identified with positions~i.e. thegenerators of translation in momentum space!. Technicallyspeaking we are free to choose the decompositionSO(3,1) into the sum ofSO(2,1) and its remainder.

In the case of the DSR phase space, the action ofsymmetries is given by

@M ,h i #5e i j h j , @Ni ,h j #5d i j h0 , @Ni ,h0#5h i ,~11!

@Jm ,h3#50, ~12!

with Jm satisfying the algebra

@M ,Ni #5e i j Nj , @Ni ,Nj #52e i j M ~13!

@x0 ,h3#51

kh0 , @x0 ,h0#5

1

kh3 , @x0 ,h i #50,

~14!

@xi ,h3#5@xi ,h0#51

kh i , @xi ,h j #5

1

kd i j ~h02h3!.

~15!,

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211 GRAVITY AND DOUBLY SPECIAL RELATIVITY PHYSICAL REVIEW D 69, 044001 ~2004!

Note that it follows from these equations that

@x0 ,xi #521

kxi , @xi ,xj #50. ~16!

It is worth mentioning also that such a decompositionpossible in any dimension. In particular in the 311 case thebracket ~16! describes the so-calledk-Minkowski type ofnon-commutativity.

One can repeat this geometric construction in the cwhen the momenta manifold is the anti–de Sitter space

2h021h1

21h222h3

25k2. ~17!

Now the symmetry algebra isSO(2,2), having again thethree dimensional Lorentz algebraSO(2,1) described byEqs. ~11!, ~12! as its subalgebra. The algebra of positiowhich we denoteym ~i.e. translations of momenta! changesonly slightly and now reads

@y0 ,h3#521

kh0 , @y0 ,h0#5

1

kh3 , @y0 ,h i #50,

~18!

@yi ,h3#521

kh i , @yi ,h0#5

1

kh i ,

@yi ,h j #51

kd i j ~h02h3!, ~19!

From Eqs.~18!, ~19! it follows that

@y0 ,yi #521

kyi1

1

k2Ni , @yi ,yj #52

2

k2e i j M . ~20!

We see that the bracket~20! does not describe thk-Minkowski type of noncommutativity. Since the noncommutativity type is related to the co-algebra structure ofquantum Poincare´ algebra, this result indicates that alonwith the k-Poincare´ algebra there exists another quantuPoincare´ algebra with the same algebra, but different calgebra, which we expect to be related to the former btwist.6

B. Phase space of 2¿1 gravity

The phase space algebra of one particle in~211!-dimensional gravity is the algebra of asymptotic chargThis algebra has been carefully analyzed by MatschullWelling in @21#. They find that the physical momentummanifold is anti–de Sitter space and thathm5pm , as statedabove. This means that 211 gravity seems to pick the classical basis of DSR as the one having physical relevanFurther, Matschull and Welling employ a particular deco

6This expectation is based on the classification of Poisson sttures on Poincare´ group presented in@32#.

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position of theSO(3,1) algebra, in which the positionsYm

act on momenta as right multiplication and have the folloing brackets withhA :

@Y0 ,h3#521

kh0 , @Y0 ,h0#5

1

kh3 ,

@Y0 ,h i #521

ke i j h j , ~21!

@Yi ,h3#521

kh i , @Yi ,h0#5

1

ke i j h j ,

@Yi ,h j #51

k~e i j h02d i j h3!. ~22!

Comparing the expressions~18!, ~19! with Eqs.~21!, ~22!we easily find that these decompositions are related by

Y05y021

kM , Yi5yi2

1

k~Ni2e i j Nj !. ~23!

It can be also easily checked that

@Ym ,Yn#522

kemnrY r. ~24!

Thus the DSR anti–de Sitter phase space is~up to a trivialreshuffling of the generators! equivalent to the phase spacof a single particle in 211 gravity.

It is an open problem whether one can get de Sitter spas a manifold of momenta from 211 quantum gravity. Itwould be interesting to see if this is the case. If so, there etwo kinds of phase spaces of a particle in a~211!-gravitational field corresponding to two DSR phase spalgebras presented above.

IV. IMPLICATIONS FOR PHYSICS IN 3 ¿1 DIMENSIONS

We present here an argument that suggests that the reof this paper, and of those we reference, concerning~211!-dimensional quantum gravity coupled to point particles mhave implications for real physics in 311 dimensions.

The main idea is to construct an experimental situatthat forces a dimensional reduction to the~211!-dimensionaltheory. It is interesting that this can be done in quanttheory, using the uncertainty principle as an essential elemof the argument.

Let us consider a system of two relativistic interactielementary particles in 311 dimensions, whose masses aless thanG21. In the center of mass frame the motion wbe planar. Let us consider the system as described byinertial observer who travels perpendicular to the planethe system’s motion, which we will call thez direction. Fromthe point of view of that observer, the system is in an eigstate of total longitudinal momentum,Pz

total , with some ei-

genvaluePz . Since the system is in an eigenstate ofPztotal

the wavefunction of the center of mass will be uniform inz.c-

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Further, since there was initially zero relative momentubetween the particles in thez direction it is also true in theobservers frame that

Pzrel5Pz

12Pz250. ~25!

This implies of coursePz15Pz

25Pztotal/2. Then the above

applies as well to each particle, i.e. their wave functionsuniform in thez direction as their wave functions have wavlength 2L where

L5\

Pztotal

. ~26!

At the same time, we assume that the uncertainties intransverse positions are bounded a scaler, such that r!2L.

Then the wave functions for the two particles have sport on narrow cylinders of radiusr which extend uniformlyin the z direction.

Finally, we assume that the state of the gravitational fiis semiclassical, so that to a good approximation, withinCthe semiclassical Einstein equations hold

Gab58pG^Tab&. ~27!

Note that we do not have to assume that the semiclasapproximation holds for all states. We assume somethmuch weaker, which is that there are subspaces of statewhich it holds.

Since the wave functions are uniform inz, and since weare interested in the particle kinematics in flat space wesume that the dynamical degrees of freedom of the gravtional field are switched off, this implies that the gravittional field seen by our observer will have a spacelike Killifield ka5(]/]z)a.

Thus, if there are no forces other than the gravitatiofield, the scattering of the two particles described semicsically by Eq.~27! must be the same as that of two paralcosmic strings. This is known to be described by an equlent ~211!-dimensional problem in which the gravitationfield is dimensionally reduced along thez direction so thatthe two ‘‘cosmic strings’’ which are the sources of the gratational field, are replaced by two punctures.

The dimensional reduction is governed by a lengthd,which is the extent inz that the system extends. We canntake d,L without violating the uncertainty principle. It isthen convenient to taked5L. Further, since the system consists of elementary particles, they have no intrinsic extentthere is no other scale associated with their extent in thzdirection. We can then identifyz50 andz5L to make anequivalent toroidal system, and then dimensionally redalongz. The relationship between the four dimensional Neton’s constantG311 and the three-dimensional NewtonconstantG2115G, which played a role so far in this papergiven by

G2115G311

2L5

G311Pztot

2\. ~28!

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Thus, in the analogous~211!-dimensional system, whichis equivalent to the original system as seen from the poinview of the boosted observer, the Newton’s constant depeon the longitudinal momenta.

Of course, in general there will be an additional scafield, corresponding to the dynamical degrees of freedomthe gravitational field. We will for the moment assume ththese are unexcited, but exciting them will not affect tanalysis so long as the gravitational excitations are invaralso under the killing field and are of compact support.

Now we note that, if there are no other particles or excidegrees of freedom, the energy of the system can to a gapproximation be described by the HamiltonianH of the twodimensional dimensionally reduced system. This is describy a boundary integral, which may be taken over any cirthat encloses the two particles. But this is bounded frabove, by Eq.~2!. This may seem strange, but it is easysee that it has a natural four-dimensional interpretation.

The bound is given by

M,1

4G2115

2L

4G311~29!

whereM is the value of the ADM Hamiltonian,H. But thisjust implies that

L.2G311M5RSch ~30!

i.e. this has to be true, otherwise the dynamics of the gratational field in 311 dimensions would have collapsed thsystem to a black hole. Thus, we see that the total bofrom above of the energy in 211 dimensions is necessary sthat one cannot violate the condition in 311 dimensions thata system be larger than its Schwarzschild radius.

Note that we also must have

M.Pztot5

\

L. ~31!

Together with Eq.~30! this impliesL. l Planck, which is ofcourse necessary if the semiclassical argument we are giis to hold.

Now, we have put no restriction on any componentsmomentum or position in the transverse directions. Sosystem still has symmetries in the transverse directions.thermore, the argument extends to any number of particso long as their relative momenta are coplanar. Thus,learn the following.

Let H QG be the full Hilbert space of the quantum theoof gravity, coupled to some appropriate matter fields, wL50. Let us consider a subspace of statesH weak which arerelevant in the low energy limit in which all energies asmall in Planck units. We expect that this will have a symetry algebra which is related to the Poincare´ algebraP 311

in 311 dimensions, by some possible small deformatioparameterized byG311 and \. Let us call this low energysymmetry groupP G

311 .Let us now consider the subspace ofH weak which is de-

scribed by the system we have just constructed. It conta

1-6

b

e

he

atsac

s,etry

dd,b-

-

er-ted

dduby

211 GRAVITY AND DOUBLY SPECIAL RELATIVITY PHYSICAL REVIEW D 69, 044001 ~2004!

two particles, and is an eigenstate ofPztot with largePz

tot andvanishing relative longitudinal momenta. Let us call this suspace of Hilbert spaceHPz

.The conditions that define this subspace break the gen

tors of the~possibly modified! Poincare´ algebra that involvethez direction. But they leave unbroken the symmetry in t~211!-dimensional transverse space. Thus, a subgroupP G

311 acts on this space, which we will callP G211,P G

311 .We have argued that the physics inHPz

is to good ap-proximation described by an analogue system in of two pticles in 211 gravity. However, we know from the resulcited in the previous sections that the symmetry algebraing there is not by the ordinary~211!-dimensional Poincare´algebra, but by thek-Poincare´ algebra in 211 dimensions,with

k2154G311Pz

tot

\. ~32!

In particular, there is a maximum energy given by

Mmax~Pztot!5k5

M Planck2

4Pztotal

. ~33!

This gives us a last condition,

s.

n,

v,

04400

-

ra-

of

r-

t-

M Pztotal,

M Planck2

4~34!

which is compatible with the previous conditions. Thuwhen all the conditions are satisfied, the deformed symmalgebra must be identified withP G

211 .Now we can note the following. WhateverP G

311 is, itmust have the following properties:

It depends onG311 and \, so that its action oneachsubspaceHPz

, for each choice ofPz , is thek deformed 211

Poincare´ algebra, withk as above.It does not satisfy the rule that momenta and energy a

on all states inH, since they are not satisfied in these suspaces.

Therefore, whateverP G311 is, it is not the classical Poin

caregroup.Thus the theory of particle kinematics at ultra high en

gies is not special relativity, and the arguments presenabove suggest that it might be doubly special relativity.

ACKNOWLEDGMENTS

We would like to thank Giovanni Amelino-Camelia anArtem Staroduptsev for discussions. ENS is UMR 5672CNRS. The Research of J.K-G. was partially supportedKBN grant 5PO3B05620.

el,

v.

lal

ial

-

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