De Sitter space as an arena for doubly special relativity

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  • Physics Letters B 547 (2002) 291296www.elsevier.com/locate/npe

    De Sitter space as an arena for doubly special relativity

    Jerzy Kowalski-Glikman 1

    Institute for Theoretical Physics, University of Wrocaw, Pl. Maxa Borna 9, Pl-50-204 Wrocaw, PolandReceived 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 energymomentum 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 spacetime 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-independent scales, of speedidentified with the speed of light c,2 and of mass (orlength = 1/) 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: jerzykg@poczta.wp.pl(J. Kowalski-Glikman).

    1 Research partially supported by the KBN grant 5PO3B05620.2 In what follows we set c= 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-called -Poincar algebra in

    0370-2693/02/$ see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)0 27 62 -4

  • 292 J. Kowalski-Glikman / Physics Letters B 547 (2002) 291296

    the bicrossproduct basis [8] provides an example ofthe energymomentum sector of DSR theory.3 Thisalgebra consists of undeformed Lorentz generators

    [Mi,Mj ] = iijkMk, [Mi,Nj ] = iijkNk,(1)[Ni,Nj ] = iijkMk,

    the standard action of rotations on momenta

    (2)[Mi,pj ] = iijkpk, [Mi,p0] = 0,along with the deformed action of boosts on momenta

    [Ni,pj ] = i ij(

    2(1 e2p0/)+ 1

    2p2)

    (3) i 1pipj

    governed by the observer-independent mass scale .The algebra (1)(3) is, of course, not unique. The

    presence of the second observer-independent scale 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)p0 = f(p0, p 2;

    ), pi = g

    (p0, p 2;

    )pi.

    By construction p0 and pi transform under rotationsas scalar and vector, respectively. The functions f andg are assumed to be analytical in the variables p0 andp 2, and in order to guarantee the correct low energybehavior one assumes that for f(p0, p 2

    ) p0 +O(1/),(5)g(p0, p 2) 1+O(1/).

    It can be shown [12] that also vice versa, any deformedPoincar algebra with undeformed Lorentz sector andstandard action of rotations, which has the standardPoincar algebra as its 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 spacetime sectorof this theory. The question arises as to if it is possibleto construct this sector from the energymomentumsector. The answer turns out to be affirmative if oneextends the energymomentum 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+ ep0/ pi,(7)(p0)= p0 1+ 1 p0,(8)(Ni)=Ni 1+ ep0/ Ni + 1

    ijkpj 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 ] = iij ,(9)[pi, x0] = i

    pi .

    By using the same method one finds also that thespacetime of DSR theory is non-commuting

    (10)[x0, xi] = ixi

    and that position operators transform under boosts inthe following way [12,13]

    [Ni, xj ] = iij x0 iijkMk,

    (11)[Ni, x0] = ixi iNi

    (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 thespacetime sector of the DSR theory, both the spacetime 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 spacetime structure of the DSR theory(though appealing by its mathematical naturalness).

  • J. Kowalski-Glikman / Physics Letters B 547 (2002) 291296 293

    energymomentum 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 spacetime 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 the SO(4,1) Lie algebrawith Lorentz generators belonging to its SO(3,1) Liesubalgebra (recall that in special relativity we have todo with the semidirect sum of SO(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 andtranslations, respectively. But then it follows that thespace of momenta can be identified with (a subspaceof) the group quotient space so(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 energymomentumsector is classical, i.e.,

    (12)[Ni,Pj ] = iij P0, [Ni,P0] = iPi ,while for positions we have the universal algebra (11).Moreover, one finds that in this basis

    [Pi,Xj ] = iij(

    1+ 1P0

    ) i2

    PiPj ,

    [Xi,P0] = iPi + i

    2PiP0,

    [Xi,P0] = iPi + i

    2PiP0,

    (13)[P0,X0] = i(

    1 12

    P 20

    ),

    and again the commutator of space and time is givenby (10).5

    5 The energymomentum sector of this basis is identical with theenergymomentum sector of Snyders theory of non-commutativespacetime [14]. The relation between this basis and Snyders 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 momenta P0 and Pi canbe viewed as coordinates on de Sitter space. Indeed,let de Sitter space be defined by equation

    (14)20 + 21 + 22 + 23 + 24 = 2,and let us define the coordinates

    (15)P = 4

    , = (0, . . . ,3).

    It is clear that the coordinates P cover only halfof the whole de Sitter space (the points (,4) and(,4) are identified in these coordinates). Ifone now derives the form of generators of SO(4,1)symmetry of de Sitter space in these coordinates,such that Mi , Ni belong to its SO(3,1) subalgebra,while X are the remaining four generators belongingto the quotient of two algebras SO(4,1)/SO(3,1), onefinds that they satisfy the SO(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 energymomentum 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 spacetime variables (cf. (11)).

  • 294 J. Kowalski-Glikman / Physics Letters B 547 (2002) 291296

    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 spacetime 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 inenergymomentum 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 point O in de Sitter momentumspace with coordinates (,4) = (0, ). 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 isjust so(3,1) ofMi andNi , and the remaining four gen-erators of so(4,1), X 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(ieP0/PiXi).

    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) P.With the help of explicit form of X generators in

    matrix representation

    X0 = i

    0 0 10 0 0

    1 0 0

    ,

    (18)X = i

    0

    T 0 0 0 T 0

    ,

    where is a three-vector with one non-vanishing unitcomponent and T the associated transposed vector,one finds

    exp(iP0X0)= cosh

    P0

    0 sinh P0

    0 1 0 sinh P0

    0 cosh P0

    ,

    (19)exp(iPiXi)=

    1+ P 222 P T

    P 222

    P

    1 P

    P 222 P T

    1 P 222

    ,

    and thus the coordinates (P) label the point =(0, . . . , 4) with

    0 = sinh P0

    P 22

    eP0 , i =Pie

    P0 ,

    (20)4 = cosh P0

    P 22

    eP0 .

    Let us now observe that position and boost operatorsX and Ni form the basis of the SO(4,1) algebra, andso their commutators with A, A= 0, . . . ,4, are givenby

    [X0, 4] = i0, [X0, 0] = i

    4,

    [X0, i ] = 0, [Xi,4] = [Xi,0] = ii,

    [Xi,j ] = iij (0 4), [Ni,0] = ii,

    [Ni,j ] = iij 0, [Ni,4] = 0.

  • J. Kowalski-Glikman / Physics Letters B 547 (2002) 291296 295

    Using now Leibnitz identity one can read off fromthese equations the form of non-vanishing cross com-mutators

    [P0,X0] = i, [Pi ,X0] = iPi ,

    [Pi ,Xj ] = iij e2P0/ + i2( P 2ij 2PiPj ),

    (21)[P0,Xi] = 2iPi .

    Remarkably, the action of boosts on momenta has theform

    [Ni,Pj ] = iij(

    2(e2P0/ 1) 1

    2P 2)

    i 1PiPj ,

    (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 with re-placed by ). Let us observe now that since 4 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

    4 = 12((

    2 sinhP02

    )2 P 2eP0/

    )

    (23)= 12(C + 22),

    where C is the quadratic Casimir of DSR in bi-crossproduct basis.

    Of course, one could consider another bases, de-fined by the prescription

    G(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 of )

    0 = sinh P0+

    P 22

    eP0/ , i =Pi,

    (24)4 = cosh P0

    P 22

    eP0/

    which leads to

    [P0,X0] = i, [Pi ,Xj ] = iij(

    1P 22

    )eP0/ ,

    (25)[P0,Xi ] = 2iPieP0/ ,

    [Ni,Pj ] = iij( sinh

    P0 P

    2

    2eP0/

    ),

    (26)[Ni,P0] = iPieP0/ .The quadratic Casimir of the algebra (24) is againrelated to 4 and has the form

    (27)C =(

    2 sinhP02

    )2+ P 2eP0/ .

    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 tran...