Planck-scale phenomenology with anti-de Sitter momentum space
Michele Arzano, Giulia Gubitosi, Joao Magueijo, Giovanni Amelino-Camelia
PPlanck-scale phenomenology with anti-de Sitter momentum space
Michele Arzano, Giulia Gubitosi, Jo˜ao Magueijo, and Giovanni Amelino-Camelia Dipartimento di Fisica, Universit`a La Sapienza and Sez. Roma1 INFN, P.le A. Moro 2, 00185 Roma, Italia Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom (Dated: October 10, 2018)We investigate the anti-de Sitter (AdS) counterpart to the well studied de Sitter (dS) model forenergy-momentum space, viz “ κ -momentum space” space (with a structure based on the propertiesof the κ -Poincar´e Hopf algebra). On the basis of previous preliminary results one might expect thetwo models to be “dual”: dS exhibiting an invariant maximal spatial momentum but unboundedenergy, AdS a maximal energy but unbounded momentum. If that were the case AdS momentumspace could be used to implement a principle of maximal Planck-scale energy, just as several studiesuse dS momentum space to postulate of maximal Planck-scale spatial momentum. However severalunexpected features are uncovered in this paper, which limit the scope of the expected duality,and interestingly they take different forms in different coordinatizations of AdS momentum space.“Cosmological” AdS coordinates mimic the dS construction used for κ -momentum space, and pro-duce a Carrol limit in the ultraviolet. However, unlike the κ -momentum space, the boundary of thecovered patch breaks Lorentz invariance, thereby introducing a preferred frame. In “horospherical”coordinates we achieve full consistency with frame independence as far as boost transformationsare concerned, but find that rotational symmetry is broken, leading to an anisotropic model for thespeed of light. Finally, in “static” coordinates we find a way of deforming relativistic transforma-tions that successfully enforces frame invariance and isotropy, and produces a Carrol limit in theultraviolet. However, the phenomenological implications appear to be too weak for any realisticchance of detection. Our results are also relevant for a long-standing debate on whether or not coor-dinate redefinitions in momentum space lead to physically equivalent theories: our three proposalsare evidently physically inequivalent, leading to alternative models of Planck-scale effects. As acorollary we study the UV running of the Hausdorff dimension of momentum space in the first andthird model, obtaining different results. I. INTRODUCTION
In recent years findings in several areas of quantum-gravity research (see, e.g. , Refs. [1, 2] and referencestherein) have motivated the investigation of Planck-scalemodified dispersion relations (MDRs), and this has at-tracted interest in MDRs as a possible avenue for Planck-scale phenomenology associated with astrophysical andcosmological observations [3–7]. It has become clearthat some of the key predictions arising from MDRs de-pend crucially on whether the relevant framework breaks or merely deforms relativistic symmetries. A preferred-frame scenario is inevitable if the transformation lawsbetween inertial observers remain the standard special-relativistic ones, since they only leave invariant the usualEinsteinian dispersion relation E − p = m . How-ever, it is possible to introduce a deformation of rela-tivistic symmetries preserving the equivalence of refer-ence frames and leaving the MDR observer-independent[8–10]. Notable examples of such “DSR” (doubly-special,or deformed-special relativity) scenarios include theo-ries based on a maximally-symmetric curved momentumspace. This has been investigated in great detail if mo-mentum space has de Sitter geometry. Here we seek toinvestigate momentum space with anti-de Sitter geome-try, a possibility which has so far received very little at-tention in the literature (see, however, [11, 12]). In doingso we will uncover several significant differences betweendS and AdS models of momentum space. If DSR-relativistic scenarios arise from maximally-symmetric momentum space it is easy to see how onecan achieve compatibility between some MDRs and thelaws of transformation between inertial observers. Oneusually introduces ordinary special relativity by takingas the starting point the isometries of Minkowski space-time, but one could equally well start from the isometriesof Minkowski momentum space. In either case one canderive the transformation laws of momenta and space-time coordinates by consistency [13]. Since the isome-tries of de Sitter (or anti-de Sitter) space can be seenas a deformation of the isometries of Minkowski space,any set of transformation laws derived from the isome-tries of de Sitter (or anti de Sitter) momentum space isas “relativistic” as special relativity (i.e. it abides by theprinciple of the relativity of inertial frames). However,such a construction entails a deformations of the trans-formation laws between inertial observers, and these willleave invariant a modified (deformed) dispersion relation.Constructions based on de Sitter momentum spacehave been extensively studied in the literature, withmany authors registering the expectation that the coun-terpart AdS model would have properties easily obtain-able from those of dS momentum space. Several argu-ments suggest that the two models should be “dual”, withdS exhibiting an invariant maximal spatial momentumbut unbounded energy, and AdS a maximal energy butunbounded spatial momentum. However, as we will showin this paper, many crucial novelties arise in AdS curvedmomentum space that are not captured by this expected a r X i v : . [ g r- q c ] D ec duality. Whereas previous arguments focused exclusivelyon local properties of the two momentum spaces, one ofthe key ingredients of our analysis is the realization ofthe fact that different coordinates cover different patchesof the manifold, and that this leads to different physi-cal statements on what is the free theory. A number ofoptions appear, mimicking—or not—constructions previ-ously considered for dS. We will find that in all of themAdS momentum space is qualitatively very different fromdS, the main point made in this paper.An important reference for us is the so-called “ κ -momentum space”, a coordinatization of a certain patchof de Sitter momentum space which has been found tohave remarkably good relativistic properties, and can beinspired by the formal structure of the κ -Poincar´e Hopfalgebra [14–16]. As we observed in Ref.[17], κ -momentumspace can be viewed as the momentum space equivalent ofthe “cosmological” representation of dS spacetime. Afterreviewing this construction of κ -momentum space (Sec-tion II) in Section III A we find the corresponding con-struction for AdS. In such “cosmological” coordinatesa simple representation for the Casimir invariant, mo-mentum space metric and integration measure is found.However, unlike with dS, the boundary of the coveredpatch breaks Lorentz invariance. If analyzed only atthe level of infinitesimal transformations the model isDSR-relativistic, but a breakdown of Lorentz invarianceis noticed when considering finite Lorentz transforma-tions. We know of no previous examples in the literatureof such subtle breakdown of relativistic symmetries (see,however [18]), and we speculate that such a possibilitycould play an important role in the phenomenology ofdepartures from ordinary special relativity, a case some-where in between the one of full breakdown of relativisticsymmetries (breakdown appreciable already for infinitesi-mal transformations) and the DSR-relativistic case (fullyrelativistic picture).Another possible approach mimicking the κ -momentum space consists of using “horospherical”coordinates, which cover a patch of AdS. We do thisin Section III B, only to encounter a similar problemto that found for cosmological coordinates, but thistime regarding the rotations. Similarly to what happensfor boosts in “cosmological” coordinates, the boundaryof the patch covered by horospherical coordinatesbreaks invariance under rotations, and so the theoryis anisotropic. The ensuing formalism is somewhatawkward, and the expression for the Casimir is far morecomplex. However, we argue that this could be a goodmodel for encoding anisotropic MDRs and speed oflight. We should however bear in mind some potentialpathologies: the model does not allow one spatialmomentum to take arbitrary negative values if we wantto preserve invariance under finite boosts.In view of the symmetry breaking properties of thesetwo models, in Section III C we investigate an alterna-tive construction which does not purport to mimic the κ -momentum space. We propose a set of coordinates analogous to “static” coordinates in the spacetime pic-ture. They cover the whole of AdS and do not breakLorentz invariance in any way. They lead to simple ex-pressions for the metric, Casimir and integration mea-sure. As with the first model, we find a Carroll limit inthe UV, i.e.: the speed of light goes to zero in the UV.As an application, in Section IV we briefly investigatethe issue of running of dimensionality for the first andthird model (the matter is far less obvious for the sec-ond model, due to its anisotropy). We do this by choos-ing linearizing coordinates and evaluating the measure ofintegration on momentum space (a procedure describedin [17, 19], known to match the spectral dimension inthe UV limit in all cases studied so far). We find thatthe two models exhibit running to different dimensions, aparticularly transparent indication of the fact that theyare physically distinct models, though both based on AdSmomentum space.Given that static coordinates have not been consideredfor dS, for completeness in Section V we present them.We find that they break Lorentz invariance in a fashionsimilar to that found for AdS in cosmological coordinates.We also examine running of dimensionality in the corre-sponding model, finding a very suggestive result. In aconcluding Section we collect the main results of this pa-per and discuss their implications. II. DE SITTER MOMENTUM SPACE
As mentioned in the Introduction the action of rel-ativistic symmetries on momenta can be deformed ifone considers a maximally symmetric curved momentumspace. A widely studied example of deformed Poincar´esymmetries reflecting such non-trivial geometry of mo-mentum space is the so-called κ -Poincar´e algebra [14–16]. Indeed, as first shown in [20], in a κ -deformed frame-work momenta can be seen as coordinates on a portion ofde Sitter momentum space defined as a four-dimensionalhyper-surface: − P + P + P + P + P = 1 (cid:96) . (1)embedded in five dimensional Minkowski space, with lineelement: ds = − dP + dP + dP + dP + dP , (2)selected by the inequality P − P > , (3)where the “cosmological constant” is the inverse of theparameter which governs the deformation of the alge-braic structures in κ -Poincar´e, κ = 1 /(cid:96) . The naturalparameterization of this submanifold, inherited by thebi-crossproduct basis of the κ -Poincar´e algebra [21], isgiven by bi-crossproduct coordinates , which correspondin position space to the “cosmological” or “flat slicing” FIG. 1: The portion of (2-dimensional) de Sitter momentumspace known as the “2D κ -mometum space”. The blue planeis defined by the condition P − P = 0, so the κ -mometumspace is on the upper-left side of the plane. The red lines rep-resent the mass-shells, defined by the condition P = const .In order to have the mass-shells completely within the allowedregion one has to further restrict to P > P < rendition of de Sitter space. They are related to the em-bedding coordinates via: P ( E, (cid:126)p ) = sinh( (cid:96)E ) (cid:96) + (cid:96)p e (cid:96)E ,P i ( E, (cid:126)p ) = − p i e (cid:96)E ,P ( E, (cid:126)p ) = − cosh( (cid:96)E ) (cid:96) + (cid:96)p e (cid:96)E , (4)where p ≡ | (cid:126)p | . With these coordinates the line elementtakes the familiar “cosmological” de Sitter metric form: ds = − dE + e (cid:96)E (cid:88) j =1 dp j (5)from which it is easy to infer the invariant integrationmeasure in momentum space: dµ ( E, (cid:126)p ) = e (cid:96)E p dEdp . (6)The deformed mass-shell is given by the intersection ofa plane P = const. with the momentum manifold: − P + (cid:126)P = 1 (cid:96) − P = m . (7)In Figure 1 we show the κ -momentum space and themass-shells given by the above constraint. In the mass-less case, using the relations above, it can be shown thatthe mass-shell condition reads C (cid:18) (cid:96) C (cid:19) = 0 , (8) where C is the Casimir invariant of the κ -Poincar´e algebrain bi-crossproduct coordinates: C = − (cid:96) sinh ( (cid:96)E/
2) + e (cid:96)E p . (9)Looking at the mass-shell, it is clear that one has toperform a further restriction on the allowed range forthe embedding coordinates in order for the theory to berelativistic. In fact, a crucial request is that any mass-shell is completely within the allowed portion of de Sittermomentum space. Failure to meet this condition wouldresult in the possibility for a finite boosts to bring out-side the allowed region some points that were originallywithin it. The restriction one has to enforce is given bythe conditions P > , P < . (10)However let us mention that at a field theoretic level theHopf algebraic structures of the κ -Poincar´e algebra en-sure that the model is fully consistent without the furtherrestriction above [22]. A. Running of Hausdorff dimension of momentumspace for the κ -momentum space scenario In [17] we showed that the κ -momentum space is char-acterized by a running of its Hausdorff dimension whengoing from the IR regime to the UV. We considered ageneral D + 1 de Sitter manifold, and we allowed forthe mass-shell to be a generic function of the κ -Poincar´eCasimir, parameterized as m = C (cid:0) (cid:96) γ C γ (cid:1) . Here wereview the argument found in [17] for UV dimensionalrunning, specializing to the D = 3, γ = 1 case, which isthe one discussed in the previous subsection (the exactcoefficient of the UV-dominant term in the mass-shellrelation does not affect the UV value of the Haussdorfdimension). In doing so, we will add some remarks thatwill facilitate comparison with the AdS constructions.The phenomenon of dimensional running can be char-acterized by choosing a set of “linearizing coordinates”,rendering the dispersion relations trivial in the UV, andexamining the dimensionality associated with the inte-gration measure in such coordinates. The linearizing co-ordinates for C , are [17]:˜ E = 2 sinh( (cid:96)E/ (cid:96) ˜ p i = p i e (cid:96)E/ . (11)which in the UV limit (defined as E → ∞ and p → /(cid:96) )become: ˜ E ≈ e (cid:96)E/ (cid:96) ˜ p i ≈ e (cid:96)E/ (cid:96) (12)(we note that in the UV limit ˜ E ≈ ˜ p , even off-shell). Interms of the new coordinates the measure (6) is given by: dµ = ˜ E ˜ p d ˜ E d ˜ p . (13)As explained in [17], for on shell relations which in theUV limit take the form C γ , one finds d H = 61 + γ , (14)so in the case of interest here ( γ = 1) one finds that theHausdorff dimension runs to 3 in the UV.Notice that we could obtain the same result by lineariz-ing directly the on-shell relation that comes out of the κ -momentum space construction, Eq. (7). This amountsto choosing the embedding coordinates themselves as lin-earizing coordinates. In the UV, their relation to thebi-crossproduct coordinates is:˜ E = P ≈ e (cid:96)E (cid:96) (1 + (cid:96) p )˜ p i = P i ≈ p i e (cid:96)E (15)where the approximate signs refer to the UV limit ap-proximation. We note also here that in the UV limit˜ E ≈ ˜ p , even off-shell. The measure (6) in the new coor-dinates now reads: dµ = ˜ p ˜ E d ˜ p d ˜ E (16)from which we can directly read d H = 3 . The last de-scription will be useful in establishing a comparison withAdS constructions. It implies that if we take the MDRthat comes most naturally out of dS (i.e., Eq. (7)) thenwe would observe dimensional reduction from D + 1 to D . This can be equivalently obtained from C with γ = 1. III. ADS MOMENTUM SPACE
As with dS space, AdS momentum space can be de-scribed as a four-dimensional hyper-surface embedded ina five-dimensional flat space, this time with signature − , − , + , + , +. The sub-manifold is now defined by: − P + P + P + P − P = − (cid:96) (18) In the more general D + 1-dimensional case, and allowing forredefinitions of the mass-shell with UV limit m = (cid:96) γ ( P − (cid:126)P ) γ one would get d H = D γ . (17)Note that this is another example of correspondence between theUV Hausdorff dimension of momentum space and the UV limit ofthe spectral dimension. The second one was computed in [23] forthe 4- and 3- dimensional cases, and the results are in agreementwith formula (17). and the corresponding line element is: ds = − dP + dP + dP + dP − dP . (19) A. Cosmological coordinates for AdS
In analogy with the κ -momentum-space constructionover dS we seek coordinates casting a portion of AdSin the form of a cosmological metric (which is no longera “flat slicing”, as it was for dS). It can be shown (seeAppendix A) that the cosmological AdS coordinates aredefined by the following relation with the embedding co-ordinates: P ( E, (cid:126)p ) = 1 (cid:96) sin (cid:96)EP ( E, (cid:126)p ) = p cos( (cid:96)E ) P ( E, (cid:126)p ) = p cos( (cid:96)E ) P ( E, (cid:126)p ) = p cos( (cid:96)E ) P ( E, (cid:126)p ) = 1 (cid:96) (cid:112) (cid:96)p ) cos (cid:96)E (20)In these coordinates the metric reads: ds = − dE + cos ( (cid:96)E ) (cid:18) dp (cid:96) p + p d Ω (cid:19) . (21)The sub-manifold covered by these coordinates is definedby the constraint − /(cid:96) ≤ P ≤ /(cid:96) , or, if we require theenergy to be positive, 0 ≤ P ≤ /(cid:96) . Using the lineelement (21) we easily deduce the invariant integrationmeasure for AdS momentum space in these coordinates: dµ ( E, p ) = cos ( (cid:96)E ) (cid:112) (cid:96) p p dEdp. (22)In analogy with dS, the mass-shell relation can be in-ferred by imposing P = const upon the surface condi-tion: − P + (cid:126)P = − (cid:96) + P = − m (23)From this we see that we must require that m ≤ /(cid:96) . Interms of the cosmological AdS cordinates the mass-shellcondition takes the form: − (cid:96) sin (cid:96)E + p cos ( (cid:96)E ) = − m . (24)In Figure 2 we plot the sub-manifold of adS covered bycosmological coordinates as well as the mass-shells givenby the constraint (23).
1. Maximal energy and speed of light in the UV limit
The mass-shell condition given by Eq. (24) implies thepresence of a maximal energy in the theory (just likeon the κ -momentum space there is a maximal spatial FIG. 2: Portion of AdS momentum space covered by cos-mological coordinates. The condition for the allowed regionis − /(cid:96) < P < /(cid:96) , which is the portion of the adS mani-fold between the two blue planes. The red lines represent themass-shells, defined by the condition P = const . momentum). Let us consider a massless particle, in thiscase the dispersion relation is given by1 (cid:96) tan( (cid:96)E ) = p (25)and it is evident that E ≤ E max = π (cid:96) , (26)whereas there is no maximal spatial momentum. In addi-tion, we see that the speed of light goes to zero as p → ∞ c = dEdp → . (27)This is nothing but the Carroll limit [24]. For massiveparticles the mass shell can be written as:cos (cid:96)E = 1 − (cid:96) m (cid:96) p . (28)(remember that the constraint m < /(cid:96) must be satis-fied). As p → ∞ again we get E → E max . Notice thatif m = 1 /(cid:96) then the MDR does not fix the momentum,and the energy saturates. A plot of the behaviour of theMDR is shown in Fig. 3.
2. Violation of Lorentz invariance
Despite the care taken not to introduce a preferredframe, this has in fact sneaked in by virtue of the fact (cid:45) (cid:45) (cid:45)
FIG. 3: Dispersion realation for a massive particle (28) in AdSmomentum space with cosmological coordinates, with (cid:96) = 1and m = 0 . that the boundary of the sub-manifold is not invariantunder the action of the Lorentz group. This is a crucialdifference between the analogous constructions for dS (forwhich the boundary is P = P ) and AdS (where theboundary is | P | = 1 /(cid:96) ). It might seem a subtle point,but the implications are obvious if we write down theLorentz transformations in momentum space.These can be described as a non-linear representationof the Lorentz group, inferred from the standard ones asapplied to the embedding coordinates through the rela-tions (20).For a finite boost in the ˆ1 direction, the explicit trans-formation rules are: E (cid:48) = 1 (cid:96) arcsin (cid:18) γ (cid:96) (cid:18) (cid:96) sin( (cid:96)E ) − vp cos( (cid:96)E ) (cid:19)(cid:19) (29) p (cid:48) = γ (cid:0) p cos( (cid:96)E ) − v (cid:96) sin( (cid:96)E ) (cid:1)(cid:113) − (cid:96) γ (cid:0) (cid:96) sin( (cid:96)E ) − vp cos( (cid:96)E ) (cid:1) (30) p (cid:48) = p cos( (cid:96)E ) (cid:113) − (cid:96) γ (cid:0) (cid:96) sin( (cid:96)E ) − vp cos( (cid:96)E ) (cid:1) (31) p (cid:48) = p cos( (cid:96)E ) (cid:113) − (cid:96) γ (cid:0) (cid:96) sin( (cid:96)E ) − vp cos( (cid:96)E ) (cid:1) (32)These can be shown to be generated by: L = p cos( (cid:96)E ) ∂ E + 1 (cid:96) tan( (cid:96)E ) ∂ p . (33)It is obvious that there is something pathological withthe finite transformations: as the transformation for theenergy shows, there is clearly a maximal boost parame-ter, such that any larger boost would bring the value ofenergy outside the allowed range.Another way to see that this framework is not invariantunder finite boosts, is by noticing that any mass shellthat goes through the allowed region of the adS manifoldis not completely included within that region (see Fig.2:), and there is no way to further restrict the allowedregion so to solve the problem. This means that for anyvalue of the mass and energy, there will always be a finiteboost pushing the particle outside the allowed range ofparameters.Yet another indicator of the breakdown of relativis-tic invariance is the fact that not only is the energy E bounded, but also the embedding one: P ≤ P = 1 (cid:96) (34)and this is clearly incompatible with its standard trans-formation rules under boosts. This is in sharp contrastwith the corresponding situation in de Sitter space, where p is bounded, but not its embedding counterpart. B. Horospherical coordinates
An AdS coordinate system which mimics more closelythe dS properties of the κ -momentum space is given bythe so-called horospherical coordinates [25, 26]. In termsof the embedding coordinates they read P = 1 (cid:96) cosh ( (cid:96)k ) + (cid:96) e (cid:96)k k i k i ,P = e (cid:96)k k ,P = e (cid:96)k k ,P = e (cid:96)k k ,P = 1 (cid:96) sinh ( (cid:96)k ) − (cid:96) e (cid:96)k k i k i , (35)where now k i k i = − k + k + k . It is easy to verify thatthey satisfy constraint (18) but only cover the P + P > spurious embeddingcoordinate has to be time-like in this case and indeed itis easily verified that this must be P since it diverges for (cid:96) →
0, the flat momentum space limit.Let us note that k is now one of the components of thespatial momentum, and k is the energy. It is physicallymore transparent to write the new coordinates as: P = 1 (cid:96) cosh ( (cid:96)p ) + (cid:96) e (cid:96)p ( − E + p + p ) ,P = e (cid:96)p E ,P = e (cid:96)p p ,P = e (cid:96)p p ,P = 1 (cid:96) sinh ( (cid:96)p ) − (cid:96) e (cid:96)p ( − E + p + p ) . (36)In such coordinates the line element is given by ds = e (cid:96)p ( − dE + dp + dp ) + dp , (37)and the associated integration measure is: dµ = e (cid:96)p dE dp . (38)Analogously to what we have done in the previous sub-sections, we find the mass-shell condition by requiringthat the spurious time-like coordinate is constant, whichin this case amounts to asking P = const : P − (cid:96) = − (cid:126)P + P = − m . (39) FIG. 4: Portion of AdS momentum space covered by horo-spherical coordinates. The condition for the allowed regionis P + P >
0, which is the portion of the AdS manifold onthe lower-right side of the blue plane. The mass-shell is givenby P = const. and is in red. Remember that now the P coordinate is the one related to energy and P is a spatialcoordinate. Also here this implies that the mass can not be arbitrarilylarge, m ≤ (cid:96) . In terms of the embedding coordinates themass-shell condition reads: − m = − e (cid:96)p E + e (cid:96)p ( p + p )+ (cid:18) (cid:96) sinh( (cid:96)p ) − (cid:96) e (cid:96)p ( − E + p + p ) (cid:19) (40)In Figure 4 we plot the sub-manifold of AdS coveredby horospherical coordinates, as well as the mass-shells.Despite the obvious anisotropy introduced by these co-ordinates, a deformed description of rotations exists, adcan be derived in analogy of what was done for boostsin cosmological coordinates. This means that the sub-manifold is invariant under infinitesimal transformations.However, one can see that the further restriction of themanifold to the P > P > P >
1. Anisotropic speed of light
It is interesting to look at the behaviour of the speed oflight in this model. For notational simplicity we restrict Θ (cid:72) Θ (cid:76) FIG. 5: Speed of light in horospherical coordinates as a func-tion of θ , for (cid:96) = 1 , p = 0 . to the case of 2 + 1-dimensional AdS momentum spaceand we write the spatial momenta in polar coordinates p = p cos θ, p = p sin θ . The dispersion relation for amassless particle reads E = 12 (cid:96) (cid:32) e − (cid:96)p cos θ + (2 + (cid:96) p sin θ ) − e − (cid:96)p cos θ ·· (cid:114) − p (cid:96) sin θ (cid:16) e (cid:96)p cos θ (4 + 3 (cid:96) p sin θ ) (cid:17)(cid:33) . (41)The deformation of rotations which guarantee local in-variance of the manifold leads to a direction-dependentdispersion relation and as a consequence to a direction-dependent speed of light.The general expression is quite complicated (a plot ofits angular dependence can be seen in Fig. 5), so herewe only write down the two special cases θ = 0 (speed oflight along the p direction) and θ = π/ p direction) c ( p, θ = 0) = e − (cid:96)p (42) c ( p, θ = π/
2) = (cid:96)p (cid:16) (cid:96) p + A (cid:17) A (cid:112) (cid:96) p − A (43)where A = (cid:112) − (cid:96) p − (cid:96) p . Note that the speed oflight becomes imaginary whenever p and θ are such thatthe condition P > C. “Static” coordinates
Static coordinates cover the full AdS manifold, and soclearly they do not break Lorentz invariance, but ratherdeform it. They can be defined from the embedding co-ordinates via: P = 1 (cid:96) sin( (cid:96)E ) cosh( (cid:96)p (cid:48) ) P r = 1 (cid:96) sinh( (cid:96)p (cid:48) ) P = 1 (cid:96) cos( (cid:96)E ) cosh( (cid:96)p (cid:48) ) (44)where P r ≡ (cid:112) P + P + P . The line element in thesecoordinates is: ds = − cosh ( (cid:96)p (cid:48) ) dE + dp (cid:48) + 1 (cid:96) sinh ( (cid:96)p (cid:48) ) d Ω . (45)We can also use an areal coordinate: p = sinh( (cid:96)p (cid:48) ) (cid:96) (46)resulting in: P = 1 (cid:96) sin( (cid:96)E ) (cid:112) (cid:96)p ) P r = pP = 1 (cid:96) cos( (cid:96)E ) (cid:112) (cid:96)p ) (47)for which the line element is: ds = − (1 + ( (cid:96)p ) ) dE + dp (cid:96)p ) + p d Ω . (48)Notice that invariance under (deformed) Lorentz trans-formations is not spoiled if we ask the energy to be pos-itive, i.e. this requirement is compatible with the de-formed transformation rules. This can be seen from (47):enforcing the positivity of E is equivalent to enforcingthe positivity of P . But the embedding coordinate P transforms with the standard Lorentz transformations, soasking it to be positive works in the same way as in theusual special relativistic case. Another interesting fea-ture of such coordinates is that the integration measureis undeformed: dµ = p dp dE. (49)We can find the mass-shell relation again by requiringthat P = const : − P + (cid:126)P = − (cid:96) + P = − m , (50) In this work we always assume (cid:96) > which in static coordinates becomes: − sin ( (cid:96)E ) (cid:96) [1 + ( (cid:96)p ) ] + p = − m . (51)In order to explore the physics and UV limit we considermassless particles. We see that their spatial momentumis unbounded, but their energy tends to a maximum: p → ∞ (52) E → E max = π (cid:96) . (53)We take these limiting values as the UV limit of themodel. The speed of light is given by: c = dEdp = 11 + ( (cid:96)p ) (54)and in the UV limit this goes to zero, what is known inthe literature as the Carroll limit [24].
IV. UV DIMENSIONAL REDUCTION IN ADSMOMENTUM SPACE
In a series of recent papers [17, 19] we have shown thatit is possible to characterize the phenomenon of dimen-sional reduction in the UV dispensing with the concept ofspectral dimension altogether. This is beneficial, as thelatter appeals to a fictitious time parameter, requires theEuclideanization of the space, and is not always basedon a properly defined probability distribution. Insteadwe showed that we could transfer all the non-trivial ef-fects of the MDRs into the measure, adopting linearizingvariables and then study the Hausdorff dimension of theenergy-momentum space in these variables. This is aphysically clearer procedure, and asymptotically (i.e. inthe deep UV limit) it produces results coinciding withthose using the spectral dimension in all known cases.Given the difficulties in defining asymptotic spectral di-mension for AdS momentum space, we favour our proce-dure here.As with [17, 19] we shall be concerned with MDRswhich in the UV limit have the form:Ω = f ( C ) ≈ C γ (55)where C is the Casimir invariant of the theory. We willexamine the UV running of the Hausdorff dimension forcosmological and static coordinates which have a clearUV limit leaving aside the case of horospherical coordi-nates whose anisotropic nature renders the notion of UVlimit ambigous. A. Cosmological coordinates
Following [17, 19], we find linearizing variables in 2steps: first by assuming γ = 0, then generalizing to γ (cid:54) = 0. When γ = 0 (i.e. when the MDRs are just the Casimir)the linearizing coordinates are just the embedding coor-dinates, as in (20): ˜ E ≡ P , ˜ p ≡ P . In D + 1 space-timedimensions the momentum space integration measure insuch variables is: d ˜ µ = ˜ p D − (cid:113) (cid:96) (˜ p − ˜ E ) d ˜ Ed ˜ p, (56)which, in the UV limit (as defined above), becomes: d ˜ µ ≈ ˜ p D − d ˜ Ed ˜ p. (57)Therefore the Hausdorff dimension is reduced by 1. Wenote that this is just the general measure studied in [17]: dµ ( ˜ E, ˜ p ) ∝ ˜ p D x − ˜ E D t − d ˜ Ed ˜ p (58)with values D t = 1, and D x = D − γ (cid:54) = 0 the linearizing coordinates can be found byfollowing the “step 2” described in [17], but we stressthat the procedure here does not rely on Euclideaniza-tion. In [17] we were dealing with an Euclideanized ver-sion of momentum space (because we wanted to study the asymptotic coincidence of spectral and Hausdorff dimen-sions), but the procedure carries through with a minimaladaptation if we remain Lorentzian. All we need do isintroduce hyperbolic (instead of spherical) polar coordi-nates: ˜ E = r cosh θ (59)˜ p = r sinh θ (60)so that the MDRs becomeΩ = r γ ) . (61)We can then define a linearizing variableˆ r = r γ (62)such that: dµ ∝ ˆ r Dt + Dx γ − (cos θ ) D t − (sin θ ) D x − d ˆ r dθ (63)leading to the conclusion that in the UV: d H = D t + D x γ . (64)For the AdS model we are considering this therefore be-comes: d H = D γ . (65) B. Static coordinates
Similarly to what happens for the cosmological coor-dinates, the linearising coordinates for the model are theembedding coordinates found in (47). The integrationmeasure is the same as (56). However the UV limit nowentails ˜ E ≈ ˜ p leading to an undeformed measure. Thismodel therefore has non-trivial physical effects (e.g. ithas a Carroll limit) but it does not present running ofthe dimensionality, if γ = 0. If γ (cid:54) = 0 one can straight-forwardly calculate d H = 1 + D γ , (66)and we therefore have a non-trivial running of the dimen-sionality. V. DE SITTER MOMENTUM SPACE INSTATIC COORDINATES
The first two models above arise from attempts toconstruct four-momenta defined on AdS space using a duality approach to dS space of momenta associated to κ -Poincar´e. The third model based on “static coordi-nates”, however, was proposed without reference to a dSconstruction, so one might wonder what the equivalentdS model would be. As we shall see, whilst static coor-dinates lead to an AdS momentum space model whichdoes not break any symmetry, its dS counterpart breaksLorentz invariance.Static coordinates for dS may be built from: P = 1 (cid:96) sinh( (cid:96)E ) (cid:112) − ( (cid:96)p ) P r = pP = 1 (cid:96) cosh( (cid:96)E ) (cid:112) − ( (cid:96)p ) (67)leading to metric: ds = − (1 − ( (cid:96)p ) ) dE + dp − ( (cid:96)p ) + p d Ω . (68)and an undeformed integration measure. The Casimir is: C = − sinh ( (cid:96)E ) (cid:96) [1 − ( (cid:96)p ) ] + p = m . (69)and we see that the theory has a maximum spatial mo-mentum, p max = 1 /(cid:96) , but unbounded energy, just likethe κ -Poincar´e case. (This maximal momentum coincideswith the location of de Sitter’s horizon, in the counter-part position space version of the space.)The UV limitmay be accordingly defined by: p → p max = 1 (cid:96) (70) E → ∞ . (71)and in such limit the speed of light c = dEdp = 11 − ( (cid:96)p ) (72) goes to infinity in the UV.All of these features are very similar to what is foundin κ -Poincar´e in the bicrossproduct basis . However thismodel breaks Lorentz symmetry in a way that mimicsclosely what happens for cosmological AdS coordinates.Indeed looking at the embedding coordinates (67), it isclear that the maximum value of momentum is reflectedinto a maximum value of the embedding coordinate P r .Since the embedding coordinates transform according tostandard Lorentz transformations, this is inconsistentwith the relativity of inertial frames.We conclude by noticing that this model also exhibitsrunning of dimensionality. Working out the integrationmeasure in linearizing coordinates, ˜ E ≡ P , ˜ p ≡ P r ,leads to: d ˜ µ = ˜ p D − (cid:113) − (cid:96) (˜ p − ˜ E ) d ˜ Ed ˜ p , (73)that in the UV limit becomes d ˜ µ ≈ ˜ p D − ˜ E d ˜ Ed ˜ p . (74)Following our standard calculation, we find for the UVHausdorff dimension of momentum space d H = D γ . (75)This is suggestively similar to what we found for AdSin cosmological coordinates. It also matches the resultobtained for dS linearising directly from the embeddingcoordinates, as discussed in Section II. VI. CONCLUSIONS
In this paper we took a first stab at defining a curvedmomentum space based on AdS geometry, in analogywith previous work for dS space. A number of signifi-cant novelties were uncovered in the process.The equivalent of the bicrossproduct basis was soughtin two ways. Firstly, we noted that we can regard the bi-crossproduct basis as the momentum space counterpartof the “cosmological” covering of dS, and sought similarcoordinates for AdS. We found that the equivalent con-struction for momentum space AdS, while simpler thandS and superficially more elegant, in fact breaks Lorentzinvariance instead of deforming it. The model must intro-duce a preferred frame because the boundary of the cor-responding sub-manifold is no longer invariant under theaction of the Lorentz group. The ensuing model may thusbe useful as a way of encoding subtle frame-dependencedue to the boundary effects: the frame dependence is onlyobvious with sufficiently large Lorentz transformations.One can also look at the bicrossproduct momenta asso-ciated to κ -Poincar´e as horospherical coordinates on dSmomentum space. Such coordinates can be introduced0also for AdS momentum space and we defined the as-sociated energy and momentum. We find that the cor-responding construction introduces spatial anisotropy inmomentum space and thus one must deform not onlyLorentz symmetry but also spatial rotations. However,similarly to what happens in the “cosmological” coordi-nates setting, the boundary is not invariant under rota-tions, breaking isotropy. The result is awkward in sev-eral other ways, including the fact that the speed of lightand the MDRs are anisotropic. Again this may serveas a useful way of encoding phenomenology, in this caseanisotropic dispersion relations and anisotropic speed oflight.A third construction, based on “static coordinates”,whilst not mimicking the usual set up for κ -Poincar´espace, proves to be the best one conceptually and interms of simplicity. It leads to an undeformed integrationmeasure and a very simple Casimir invariant. It models amaximal energy and unbounded spatial momentum with-out introducing a preferred frame. The speed of light goesto zero in the UV limit, and this is achieved isotropically.We advocate this construction as the most conservativemodel for AdS momentum space. For completeness, inthis paper we have also considered a dS model based onstatic coordinates, the counterpart to the last AdS modelproposed in this paper. Curiously the dS static modelbreaks Lorentz invariance in a way similar to what hap-pens to the AdS model in cosmological coordinates.As a first application of these models we investigatedthe phenomenon of running of the dimensionality. We didthis by considering “linearizing” coordinates (i.e. coor-dinates which render the dispersion relations trivial) andevaluating the integration measure in terms of them, tofind the associated Hausdorff dimension. This procedurewas considered in the past [17, 19], and found to matchthe spectral dimension in the UV limit in all cases stud-ied. In this paper we found that the (Lorentz breaking)AdS model based on cosmological coordinates runs to: d H = D γ (76)in the UV limit, whereas the (non-Lorentz breaking)model based on static coordinates runs to: d H = 1 + D γ (77)showing further that the two models are physically dis-tinct models. It is curious that the equivalent result fordS in static coordinates matches the result found for AdSin cosmological coordinates (cf. Eq.(75) and Eq.(65)).This also matches the result for the κ -Poincar´e space [23]if we linearize the Casimir coming directly from the em-bedding variables, as explained in the discussion leadingto Eq. (17). Could this be pointing us to an interestingduality? VII. ACKNOWLEDGMENTS
We were all supported by the John Templeton Foun-dation. The work of MA was also supported by a MarieCurie Career Integration Grant within the 7th EuropeanCommunity Framework Programme. JM was also fundedby a STFC consolidated grant and the Leverhulme Trust.
Appendix A: Derivation of cosmological coordinatesfor AdS
One starts by setting [28]: P = 1 (cid:96) sin (cid:96)EP i = ˆ P i cos (cid:96)EP = ˆ P cos (cid:96)E (A1)where i = 1 , , P µ are the embedding coordinates.In this way Eq.(18) becomes a condition requiring thespatial homogeneous leaves to be hyperboloids:ˆ P i − ˆ P = − (cid:96) . (A2)In terms of these coordinates the metric induced onthe 4-surface is the cosmological rendition of (a portionof) AdS: ds = − dE + cos ( (cid:96)E ) dσ (A3)where the spatial metric is dσ = d ˆ P + d ˆ P + d ˆ P − d ˆ P (A4)subject to (A2). Introducing polar coordinates in the { ˆ P i } space: ˆ P = p cos θ ˆ P = p sin θ cos φ ˆ P = p sin θ sin φ (A5)ensures that p will be a comoving areal coordinate. In-deed, then dσ = dp + p d Ω − dP , and dP can at mostcorrect the dp component of the metric. Specifically wecan solve (A2) as: ˆ P = (cid:112) (cid:96) p (cid:96) (A6)and by differentiating and inserting in dσ we get thecosmological form of the AdS metric: ds = − dE + cos ( (cid:96)E ) (cid:18) dp (cid:96) p + p d Ω (cid:19) . (A7)1The explicit expression relating the two sets of coordi-nates is therefore: P ( E, (cid:126)p ) = 1 (cid:96) sin (cid:96)EP ( E, (cid:126)p ) = p cos( (cid:96)E ) cos θP ( E, (cid:126)p ) = p cos( (cid:96)E ) sin θ cos φP ( E, (cid:126)p ) = p cos( (cid:96)E ) sin θ sin φP ( E, (cid:126)p ) = 1 (cid:96) (cid:112) (cid:96)p ) cos (cid:96)E (A8)This transformation can be abbreviated using notation: P ( E, (cid:126)p ) = 1 (cid:96) sin (cid:96)EP r ( E, (cid:126)p ) = p cos (cid:96)EP ( E, (cid:126)p ) = 1 (cid:96) (cid:112) (cid:96)p ) cos (cid:96)E (A9) where the { P i } are to be obtained from P r via the usualpolar coordinate formulae. Then Eq.(A9) is valid in anynumber D of spatial dimensions, as long as we employthe standard polar coordinate D − ds = − dE + cos ( (cid:96)E ) (cid:18) dp (cid:96) p + p d Ω D − (cid:19) . (A10) [1] G. Amelino-Camelia, Living Rev. Rel. (2013) 5[arXiv:0806.0339 [gr-qc]].[2] D. Mattingly, Living Rev. Rel. (2005) 5 [gr-qc/0502097].[3] G. Amelino-Camelia, J. R. Ellis, N. E. Mavromatos,D. V. Nanopoulos and S. Sarkar, Nature (1998) 763[astro-ph/9712103].[4] U. Jacob and T. Piran, Nature Phys. (2007) 87 [hep-ph/0607145].[5] S. Alexander, R. Brandenberger and J. Magueijo, Phys.Rev. D (2003) 081301 [hep-th/0108190].[6] S. Alexander and J. Magueijo, Proceedings of the XIIIrdRencontres de Blois ’Frontiers of the Universe’, pp281,The Gioi Publishers, 2004 [hep-th/0104093].[7] G. Gubitosi, L. Pagano, G. Amelino-Camelia, A. Mel-chiorri and A. Cooray, JCAP (2009) 021[arXiv:0904.3201 [astro-ph.CO]].[8] G. Amelino-Camelia, Int. J. Mod. Phys. D (2005)2167 [gr-qc/0506117].[9] J. Magueijo and L. Smolin, Phys. Rev. Lett. , 190403(2002) [hep-th/0112090].[10] J. Kowalski-Glikman and S. Nowak, Int. J. Mod. Phys.D (2003) 299 [hep-th/0204245].[11] G. Amelino-Camelia, M. Arzano, S. Bianco andR. J. Buonocore, Class. Quant. Grav. (2013) 065012[arXiv:1210.7834 [hep-th]].[12] M. Arzano, D. Latini and M. Lotito, SIGMA , 079(2014) [arXiv:1403.3038 [gr-qc]].[13] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikmanand L. Smolin, Phys. Rev. D (2011) 084010[arXiv:1101.0931 [hep-th]].[14] J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B (1992) 344.[15] J. Lukierski and H. Ruegg, Phys. Lett. B (1994) 189[hep-th/9310117].[16] J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoi,Phys. Lett. B (1991) 331.[17] G. Amelino-Camelia, M. Arzano, G. Gubitosiand J. Magueijo, Phys. Lett. B (2014) 317[arXiv:1311.3135 [gr-qc]].[18] J. Magueijo and L. Smolin, Phys. Rev. D (2003)044017 [gr-qc/0207085].[19] G. Amelino-Camelia, M. Arzano, G. Gubitosi andJ. Magueijo, Phys. Rev. D , no. 10, 103524 (2013).[20] J. Kowalski-Glikman and S. Nowak, hep-th/0411154.[21] S. Majid and H. Ruegg, Phys. Lett. B (1994) 348[hep-th/9405107].[22] M. Arzano, J. Kowalski-Glikman and A. Walkus, Class.Quant. Grav. , 025012 (2010) [arXiv:0908.1974 [hep-th]].[23] M. Arzano and T. Trzesniewski, Phys. Rev. D (2014)124024 [arXiv:1404.4762 [hep-th]].[24] J.M. L´evy-Leblond, Ann. Inst. H. Poincar´e (1965) 1.[25] N.J. Vilenkin and A.U. Klimyk, “Representation of LieGroups and Special Functions,” 3 volumes Kluwer, 1991,1993.[26] H. Lu, C. N. Pope and P. K. Townsend, Phys. Lett. B , 39 (1997) [hep-th/9607164].[27] K. Land and J. Magueijo, Phys. Rev. Lett. , 071301(2005) [astro-ph/0502237].[28] I. Bengtsson,, 071301(2005) [astro-ph/0502237].[28] I. Bengtsson,