Physics Letters B 547 (2002) 291–296
www.elsevier.com/locate/npe
De Sitter space as an arena for doubly special relativity
Jerzy Kowalski-Glikman1
Institute for Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, Pl-50-204 Wrocław, Poland
Received 1 August 2002; received in revised form 22 September 2002; accepted 30 September 2002
Editor: P.V. Landshoff
Abstract
We show that Doubly Special Relativity (DSR) can be viewed as a theory with energy–momentum space being the four-dimensional de Sitter space. Different formulations (bases) of the DSR theory considered so far can be therefore understood asdifferent coordinate systems on this space. The emerging geometrical picture makes it possible to understand the universalityof the non-commutative structure of space–time of doubly special relativity. Moreover, it suggests how to construct the mostnatural DSR basis, which turns out to be the bicrossproduct basis. 2002 Published by Elsevier Science B.V.
1. The DSR theory
Doubly special relativity theory is a new attemptto approach the problem of quantum gravity. Thistheory was proposed about a year ago by Amelino-Camelia [1] and is based on two fundamental assump-tions: the principle of relativity and the postulate ofexistence of two observer-independentscales, of speedidentified with the speed of lightc,2 and of massκ (orlength� = 1/κ) identified with the Planck mass. Thereare several theoretical indications that such a theorymay replace Special Relativity as a theory of relativis-tic kinematics of probes whose energies are close tothe Planck scale. First of all both loop quantum grav-
E-mail address:[email protected](J. Kowalski-Glikman).
1 Research partially supported by the KBN grant 5PO3B05620.2 In what follows we setc = 1.
ity and string theory indicate appearance of the min-imal length scale. It is therefore not impossible thatthis scale would be present in description of ultra highenergy kinematics even in the regime, in which gravi-tational effects are negligible. Secondly, in both infla-tionary cosmology [2] and in black hole physics [3]one faces the conceptual “trans-Planckian puzzle” ofordinary physical quanta being blue shifted up to thePlanck energies, which as advocated by many can besolved by assuming deviation from the standard dis-persion relation at high energies, and thus deviationfrom the standard relativistic kinematics. It should bealso stressed that some DSR models might providea resolution of observed anomalies in astrophysicaldata [4]. Moreover, predictions of the DSR scenariomight be testable in forthcoming quantum gravity ex-periments [5].
Soon after appearance of the papers [1] it wasrealized [6,7] that the so-calledκ-Poincaré algebra in
0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)02762-4
292 J. Kowalski-Glikman / Physics Letters B 547 (2002) 291–296
the bicrossproduct basis [8] provides an example ofthe energy–momentum sector of DSR theory.3 Thisalgebra consists of undeformed Lorentz generators
[Mi,Mj ] = iεijkMk, [Mi,Nj ] = iεijkNk,
(1)[Ni,Nj ] = −iεijkMk,
the standard action of rotations on momenta
(2)[Mi,pj ] = iεijkpk, [Mi,p0] = 0,
along with the deformed action of boosts on momenta
[Ni,pj ] = i δij
(κ
2
(1− e−2p0/κ
) + 1
2κ�p2
)
(3)− i1
κpipj
governed by the observer-independent mass scaleκ .The algebra (1)–(3) is, of course, not unique. The
presence of the second observer-independent scaleκ makes it possible to consider transformations toanother DSR basis, in which (1) holds, and thusthe Lorentz subalgebra is left unchanged, but oneintroduces new momentum variables
(4)p′0 = f
(p0, �p 2; κ
), p′
i = g(p0, �p 2; κ
)pi.
By constructionp′0 andp′
i transform under rotationsas scalar and vector, respectively. The functionsf andg are assumed to be analytical in the variablesp0 and�p 2, and in order to guarantee the correct low energybehavior one assumes that forκ → ∞f
(p0, �p 2) ≈ p0 + O(1/κ),
(5)g(p0, �p 2) ≈ 1+ O(1/κ).
It can be shown [12] that also vice versa, any deformedPoincaré algebra with undeformed Lorentz sector andstandard action of rotations, which has the standardPoincaré algebra as itsκ → ∞ limit can be related tothe algebra (1)–(3) by transformation of the form (4).One should note in passing that this means, in partic-ular, that any modified dispersion relation consideredin the context “trans Planckian problem” can be ex-tended to a DSR theory, and thus does not need to leadto breaking of Lorentz symmetry.
3 More recently another form of the DSR theory was presentedby [9]. Relations between different forms of DSR were discussedin [10,11].
The algebra (1)–(3) does not furnish the wholephysical picture of the DSR theory. To describephysical processes we need also a space–time sectorof this theory. The question arises as to if it is possibleto construct this sector from the energy–momentumsector. The answer turns out to be affirmative if oneextends the energy–momentum DSR algebra to thequantum (Hopf) algebra. It was shown in [12] thatsuch an extension is possible in the case of any DSRalgebra, in particular, for the algebra (1)–(3) one getsthe following expressions for the co-product
(6)�(pi) = pi ⊗ 1 + e−p0/κ ⊗ pi,
(7)�(p0) = p0 ⊗ 1 + 1 ⊗ p0,
(8)�(Ni) = Ni ⊗ 1 + e−p0/κ ⊗ Ni + 1
κεijkpj ⊗ Mk
(the co-product for rotations is trivial). Then onemakes use of the so-called “Heisenberg double” pre-scription4 [13] in order to get the following commuta-tors
[p0, x0] = i, [pi, xj ] = −iδij ,
(9)[pi, x0] = − i
κpi .
By using the same method one finds also that thespace–time of DSR theory is non-commuting
(10)[x0, xi] = − i
κxi
and that position operators transform under boosts inthe following way [12,13]
[Ni, xj ] = iδij x0 − i
κεijkMk,
(11)[Ni, x0] = ixi − i
κNi
(x0 and xi transform as scalar and vector underrotations).
It is important to note that as proved in [12] ifthe Heisenberg double method is used to derive thespace–time sector of the DSR theory, both the space–time non-commutativity (10) and the form of theboost action on position operators (11) is universalfor all the DSR theories, i.e., it is invariant of the
4 It should be stressed that the Heisenberg double method is nota unique way of deriving the space–time structure of the DSR theory(though appealing by its mathematical naturalness).
J. Kowalski-Glikman / Physics Letters B 547 (2002) 291–296 293
energy–momentum transformations (4), (5). As wewill see this observation finds its natural explanationin the complementary geometrical picture of DSR, tobe developed below, and this is, of course, a strongargument in favor of the Heisenberg double method.
2. DSR algebra and de Sitter space
Since the space–time algebra of Lorentz generatorsand positions given by (1), (10) and (11) is universal,it is worth to investigate it a bit closer. The first thingto note is that this algebra is theSO(4,1) Lie algebrawith Lorentz generators belonging to itsSO(3,1) Liesubalgebra (recall that in special relativity we have todo with the semidirect sum ofSO(3,1) and R4, in-stead). Let us recall now that both Lorentz generatorsand positions can be interpreted as symmetry genera-tors, acting on the space of momenta as “rotations” and“translations”, respectively. But then it follows that thespace of momenta can be identified with (a subspaceof) the group quotient spaceso(4,1)/so(3,1) which isnothing but the de Sitter space.
To see this explicitly, let us note that amonginfinitely many DSR bases, related to each other bytransformation (4), (5) one finds the basis, in whichthe action of Lorentz algebra on energy–momentumsector is classical, i.e.,
(12)[Ni,Pj ] = iδij P0, [Ni,P0] = iPi ,
while for positions we have the universal algebra (11).Moreover, one finds that in this basis
[Pi,Xj ] = −iδij
(1+ 1
κP0
)− i
κ2PiPj ,
[Xi,P0] = i
κPi + i
κ2PiP0,
[Xi,P0] = i
κPi + i
κ2PiP0,
(13)[P0,X0] = i
(1− 1
κ2P 20
),
and again the commutator of space and time is givenby (10).5
5 The energy–momentum sector of this basis is identical with theenergy–momentum sector of Snyder’s theory of non-commutativespace–time [14]. The relation between this basis and Snyder’s theory
As it stands, the algebra (10)–(13) looks like aparticular DSR basis. The important observation isthat, in agreement with general argument given at thebeginning of this section, the momentaP0 andPi canbe viewed as coordinates on de Sitter space. Indeed,let de Sitter space be defined by equation
(14)−η20 + η2
1 + η22 + η2
3 + η24 = κ2,
and let us define the coordinates
(15)Pµ = κηµ
η4, µ = (0, . . . ,3).
It is clear that the coordinatesPµ cover only halfof the whole de Sitter space (the points(ηµ,η4) and(−ηµ,−η4) are identified in these coordinates). Ifone now derives the form of generators ofSO(4,1)
symmetry of de Sitter space in these coordinates,such thatMi , Ni belong to itsSO(3,1) subalgebra,while Xµ are the remaining four generators belongingto the quotient of two algebrasSO(4,1)/SO(3,1), onefinds that they satisfy theSO(4,1) relations (1), (10),(11) as well as the cross relations (13).
This simple observation clarifies the universal sta-tus of the algebra satisfied by positions and boost androtation generators in any DSR basis (10), (11). Tounderstand this let us look at the DSR theory fromgeometric perspective suggested by the constructionabove. From this viewpoint the space of momentais not a flat space, as in Special Relativity, but acurved, maximally symmetric space of constant cur-vature. The fact that we need a maximally symmetricspace is related, of course to the fact that only suchspace has the required number of symmetry genera-tors, namely, six “rotations” identified with Lorentztransformations and four “translations” in the energy–momentum space, which can be identified with (non-commutative) positions. It is well known that there areonly three families of maximally symmetric spaces:de Sitter, anti-de Sitter and the flat space of con-stant positive, negative, and zero curvature, respec-tively. Next, it is clear that even though on the (mo-mentum) de Sitter space one can introduce arbitrarycoordinates (each corresponding to a particular DSR
was discussed in [12]. Note that I take liberty to call this basis aDSR theory even though the commutators (12) are undeformed. Thereason is that the deformation is still present in the action of boostson space–time variables (cf. (11)).
294 J. Kowalski-Glikman / Physics Letters B 547 (2002) 291–296
basis), the form of symmetries of this space does not,of course depend on the form of the coordinate sys-tem (recall that de Sitter space is the symmetric spaceso(4,1)/so(3,1)). In other words the momentum sec-tors of various DSR theories are in one-to-one corre-spondence with differential structures that can be builton de Sitter space, while the structure of the posi-tions/boosts/rotations, being related to the symmetriesof this space is, clearly, diffeomorphism-invariant. Letus note again that the fact that Heisenberg double con-struction leads to algebraic structure consistent withthe geometric picture of the DSR theory strongly indi-cates that it might be the right way of construction ofthe space–time sector of this theory.
At this point a question arises, namely if the coordi-nates (15) are the most natural ones from the geometricperspective. Indeed in these coordinates the physicalmeaning of positions as generators of translations inenergy–momentum space is far from being manifest.It is therefore useful to try to construct a coordinatesystem in which the physical role played by positionsexhibits itself in a more clear way. This can be done asfollows. Consider the pointO in de Sitter momentumspace with coordinates(ηµ,η4) = (0, κ). This pointcorresponds to the zero momentum in the coordinatesystem (15) and we assume that it corresponds to zeromomentum state in any coordinates as well. Geometri-cally this assumption corresponds to defining the pre-ferred point in de Sitter space, but, of course it iswell motivated physically. Since de Sitter space equalsso(4,1)/so(3,1), the stability group of this point isjustso(3,1) of Mi andNi , and the remaining four gen-erators ofso(4,1), Xµ can be used to define points onde Sitter space as follows. One observes that the groupelements exp(iP0X0), exp(iPiXi) (for fixed i) havenatural interpretation of “large translations” in the mo-mentum de Sitter space (since this space is curved thetranslations cannot commute, of course). Indeed
exp(iP (1)
0 X0)exp
(iP (2)
0 X0)
= exp(i(P (1)
0 +P (2)0
)X0
),
exp(iP (1)
i Xi
)exp
(iP (2)
i Xi
) = exp(i(P (1)
i +P (2)i
)Xi
)while
exp(iP0X0)exp(iPiXi)exp(−iP0X0)
(16)= exp(ie−P0/κPiXi
).
Now one can define the natural coordinates on themomentum de Sitter space by labelling the point
℘± ≡ G±(P0,Pi )O(17)= exp(±iPiXi)exp(±iP0X0)O
with coordinates (momenta)Pµ.With the help of explicit form ofXµ generators in
matrix representation
X0 = i
κ
0 0 1
0 0 01 0 0
,
(18)�X = i
κ
0 �ε T 0
�ε 0 �ε0 −�ε T 0
,
where�ε is a three-vector with one non-vanishing unitcomponent and�ε T the associated transposed vector,one finds
exp(±iP0X0) = coshP0
κ0 ∓sinhP0
κ
0 1 0∓sinhP0
κ0 coshP0
κ
,
(19)exp(±iPiXi) =
1+ �P 2
2κ2 ∓ �P T
κ
�P 2
2κ2
∓ �Pκ
1 ∓ �Pκ
− �P 2
2κ2 ± �P T
κ1− �P 2
2κ2
,
and thus the coordinates(Pµ) label the point℘± =(η0, . . . , η4) with
η0 = −κ sinhP0
κ±
�P 2
2κe∓P0
κ , ηi = −Pie∓P0
κ ,
(20)η4 = ±κ coshP0
κ∓
�P 2
2κe∓P0
κ .
Let us now observe that position and boost operatorsXµ andNi form the basis of theSO(4,1) algebra, andso their commutators withηA, A = 0, . . . ,4, are givenby
[X0, η4] = i
κη0, [X0, η0] = i
κη4,
[X0, ηi ] = 0, [Xi,η4] = [Xi,η0] = i
κηi,
[Xi,ηj ] = i
κδij (η0 − η4), [Ni,η0] = iηi,
[Ni,ηj ] = iδij η0, [Ni,η4] = 0.
J. Kowalski-Glikman / Physics Letters B 547 (2002) 291–296 295
Using now Leibnitz identity one can read off fromthese equations the form of non-vanishing cross com-mutators
[P0,X0] = i, [Pi ,X0] = ± i
κPi ,
[Pi ,Xj ] = −iδij e±2P0/κ + i
κ2
( �P 2δij − 2PiPj
),
(21)[P0,Xi] = ∓2i
κPi .
Remarkably, the action of boosts on momenta has theform
[Ni,Pj ] = iδij
(±κ
2
(e±2P0/κ − 1
) ∓ 1
2κ�P 2
)
± i1
κPiPj ,
(22)[Ni,P0] = iPi ,
which is nothing but the boost action in the bi-crossproduct basis (in the “−” case; in the “+” caseone has to do with the bicrossproduct basis withκ re-placed by−κ). Let us observe now that sinceη4 com-mutes with boosts (and, of course, rotations as well) itmust be related to the quadratic Casimir of the alge-bra (22). Indeed we find
η4 = ± 1
2κ
((2κ sinh
P0
2κ
)2
− �P 2e∓P0/κ
)± κ
(23)= ± 1
2κ
(C± + 2κ2),
where C± is the quadratic Casimir of DSR in bi-crossproduct basis.
Of course, one could consider another bases, de-fined by the prescription
℘̃± ≡ G̃±(P0,Pi )O= exp(±iP0X0)exp(±iPiXi)O
which can be easily found with the help of Eq. (16)and would lead to another DSR theory. Specificallyone gets (in the “+” case; the “−” one can be obtainedby changing the sign ofκ)
η0 = −κ sinhP0
κ+
�P 2
2κeP0/κ , ηi = −Pi,
(24)η4 = κ coshP0
κ−
�P 2
2κeP0/κ
which leads to
[P0,X0] = i, [Pi ,Xj ] = −iδij
(1−
�P 2
κ2
)eP0/κ ,
(25)[P0,Xi ] = −2i
κPie
P0/κ ,
[Ni,Pj ] = iδij
(κ sinh
P0
κ− �P 2
2κeP0/κ
),
(26)[Ni,P0] = iPieP0/κ .
The quadratic Casimir of the algebra (24) is againrelated toη4 and has the form
(27)C = −(
2κ sinhP0
2κ
)2
+ �P 2eP0/κ .
3. Conclusions
The main result of this Letter is that any DSRtheory can be regarded as a particular coordinatesystem on de Sitter space of momenta. In addition,the Lorentz transformations have interpretations ofstabilizers of the zero-momentum point in the deSitter space, while positions are identified with theremaining four generators ofSO(4,1). Moreover, onecan single out the basis, which is most natural from thegeometric point of view, and this basis turns out to bethe bicrossproduct one.
The observation that different DSR bases are re-lated by diffeomorphisms in the momentum de Sitterspace leads naturally to the question if this relation be-tween different DSR theories is of any physical rel-evance, i.e., if the “general coordinate invariance inmomentum space” can be somehow given a statusof physical symmetry. To answer this question oneshould check if it is possible to construct a (de Sitter)momentum space theory in a manifestly coordinate in-dependent way (which would, of course include themetric tensor on this space). This problem is currentlyunder investigation.
Acknowledgements
The idea to investigate de Sitter structure of themomentum space of DSR theories arose in the course
296 J. Kowalski-Glikman / Physics Letters B 547 (2002) 291–296
of many discussions with Giovanni Amelino-Cameliaduring my visit to Rome. I would like to thank him aswell as the Department of Physics of the University ofRome “La Sapienza” for their worm hospitality duringmy visit.
References
[1] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 35, gr-qc/0012051;G. Amelino-Camelia, Phys. Lett. B 510 (2001) 255, hep-th/0012238;G. Amelino-Camelia, Nature 418 (2002) 34, gr-qc/0207049.
[2] J. Martin, R.H. Brandenberger, Phys. Rev. D 63 (2001)123501, hep-th/0005209;R.H. Brandenberger, J. Martin, Mod. Phys. Lett. A 16 (2001)999, astro-ph/0005432;J.C. Niemeyer, Phys. Rev. D 63 (2001) 123502, astro-ph/0005533;J. Kowalski-Glikman, Phys. Lett. B 499 (2001) 1, astro-ph/0006250;A.A. Starobinsky, JETP Lett. 73 (2001) 371, astro-ph/0104043.
[3] W.G. Unruh, Phys. Rev. D 51 (1995) 2827;S. Corley, T. Jacobson, Phys. Rev. D 54 (1996) 1568;S. Corley, Phys. Rev. D 57 (1998) 6280;S. Corley, T. Jacobson, Phys. Rev. D 59 (1999) 124011;A. Blaut, J. Kowalski-Glikman, D. Nowak-Szczepaniak, Phys.Lett. B 521 (2001) 364.
[4] G. Amelino-Camelia, T. Piran, Phys. Rev. D 64 (2001) 036005,astro-ph/0008107;S. Sarkar, Mod. Phys. Lett. A 17 (2002) 1025, gr-qc/0204092;R. Aloisio, P. Blasi, A. Galante, P.L. Ghia, A.F. Grillo, astro-ph/0205271.
[5] J.P. Norris, J.T. Bonell, G.F. Marani, J.D. Scargle, astro-ph/9912136;A. de Angelis, astro-ph/0009271.
[6] J. Kowalski-Glikman, Phys. Lett. A 286 (2001) 391, hep-th/0102098.
[7] N.R. Bruno, G. Amelino-Camelia, J. Kowalski-Glikman, Phys.Lett. B 522 (2001) 133, hep-th/0107039.
[8] J. Lukierski, A. Nowicki, H. Ruegg, V.N. Tolstoy, Phys. Lett.B 264 (1991) 331;S. Majid, H. Ruegg, Phys. Lett. B 334 (1994) 348;J. Lukierski, H. Ruegg, W.J. Zakrzewski, Ann. Phys. 243(1995) 90.
[9] J. Magueijo, L. Smolin, Phys. Rev. Lett. 88 (2002) 190403,hep-th/0112090.
[10] J. Kowalski-Glikman, S. Nowak, Phys. Lett. B 539 (2002) 126,hep-th/0203040.
[11] J. Lukierski, A. Nowicki, hep-th/0203065.[12] J. Kowalski-Glikman, S. Nowak, hep-th/0204245.[13] J. Lukierski, A. Nowicki, in: H.-D. Doebner, V.K. Dobrev
(Eds.), Proceedings of Quantum Group Symposium at Group21, July 1996, Goslar, Germany, Heron Press, Sofia, 1997,p. 186;A. Nowicki, math.QA/9803064;J. Lukierski, hep-th/9812063.
[14] H.S. Snyder, Phys. Rev. 71 (1947) 38.
