Ann. Phys. (Berlin) 522, No. 12, 924 – 940 (2010) / DOI 10.1002/andp.201000105 Doubly special relativity in de Sitter spacetime S. Mignemi1,2,∗ 1 2 Dipartimento di Matematica, Università di Cagliari, viale Merello 92, 09123 Cagliari, Italy INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato (Cagliari), Italy Received 10 August 2010, revised 13 September 2010, accepted 28 September 2010 by F. W. Hehl Published online 26 October 2010 Key words Doubly special relativity, de Sitter space, noncommutative spacetime, de Sitter group. We discuss the generalization of Doubly Special Relativity to a curved de Sitter background. The model has three fundamental observer-independent scales, the velocity of light c, the de Sitter radius α, and the Planck energy κ, and can be realized through a nonlinear action of the de Sitter group on a noncommutative position space. We consider different choices of coordinates on the de Sitter hyperboloid that, although equivalent, may be more suitable for treating different problems. Also the momentum space can be described as a hyperboloid embedded in a five-dimensional space, but in this case different choices of coordinates lead to inequivalent models. We investigate the kinematics and the Hamiltonian dynamics of some specific models and describe some of their phenomenological consequences. Finally, we show that it is possible to construct a model exhibiting a duality for the interchange of positions and momenta together with the interchange of α and κ. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Since the early years of general relativity, de Sitter and anti-de Sitter spacetimes have acquired a fundamental importance, both theoretical and phenomenological, especially in the context of cosmology. Indeed, recent astrophysical observations seem to indicate that our universe has positive cosmological constant [1]. In spite of their relevance, there is not much literature about the extension of the kinematics of special relativity to de Sitter or anti-de Sitter backgrounds1. Geometrically, de Sitter space is defined as a space of constant positive curvature. Its isometries are generated by the de√Sitter algebra, that can be considered as a deformation of the Poincaré algebra with a parameter α = 1/ Λ with the dimension of length2. Of course, several geometric and algebraic properties of de Sitter space differ from those of Minkowski space. For example, contrary to flat space, in de Sitter space the generators of the translations cannot be identified with the canonical momenta, as should be obvious because of the position dependence of the de Sitter Hamiltonian. Moreover, there is no natural parametrization of the space and, depending on the specific problem one is studying, different systems of coordinates can be more suited for the calculations. A different kind of deformation of special relativity is given by the more recent proposal of deformed (or doubly) special relativity (DSR) [6]. This theory is based on the generalization of the standard energymomentum dispersion law of particles P 2 = m2 . The deformation is achieved by modifying the action of the Lorentz group on momentum space through the introduction of a new observer-independent constant κ, with the dimensions of energy (usually identified with the Planck energy). In this framework, the transformation law of momenta becomes nonlinear, and that of positions momentum dependent. Special relativity ∗ E-mail: smignemi@unica.it 1 To our knowledge, this is discussed only in [2, 3] for a specific choice of coordinates, and more recently in [4, 5] from a different point of view. 2 In the following, we denote by Λ one third of the cosmological constant. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Ann. Phys. (Berlin) 522, No. 12 (2010) 925 is recovered in the limit κ → ∞. Different choices of the deformed dispersion law correspond to different DSR models and, even imposing suitable physical constraints, there exist in principle infinite inequivalent realizations of the DSR axioms. The physical motivations for the introduction of DSR are given by the possibility of explaining some anomalies observed in high-energy cosmic ray distribution [7] by means of deformed dispersion relations, and by the theoretical requirement that the Planck energy, which sets the scale for quantum gravity, be invariant under the symmetries of spacetime3. Algebraically, DSR theories can be realized in two equivalent ways, either identifying the generators of the translations with the phase space momenta and then deforming the Poincaré algebra, as in the κPoincaré algebra approach [10], or by maintaining the canonical form of the Poincaré algebra, but making it act nonlinearly on momentum space, as in the Magueijo-Smolin (MS) model [11]. The second approach is especially convenient in the case of the de Sitter algebra, where, as mentioned above, even classically the generators of the translations do not coincide with the canonical momenta. From a physical perspective, the geometry of the spacetime on which DSR models act has a fundamental importance. Unfortunately, however, these models are usually defined only in momentum space and the spacetime geometry is not fixed uniquely from their postulates. Although it is possible to define DSR in ordinary spacetime, its most natural realization appears nevertheless to be in terms of noncommutative geometry, with momentum-dependent metric [12–14], and this is the approach that we adopt in this paper. The simplest way to construct a DSR model starting from canonical special relativity was suggested in [15]: one can define the physical momenta as functions of auxiliary variables which transform in the standard way under Lorentz transformations. The deformed transformation laws and dispersion relations of the physical momenta then follow from this definition. More recently, it has been shown that also the definition of a suitable noncommutative position space can be obtained by an analogous procedure [16]. Algebraically, de Sitter space and the DSR momentum space have a very similar structure, both being realized by imposing a quadratic constraint on the coordinates of a five-dimensional space [17]. However, their physical interpretation is different: de Sitter space has a natural Riemannian structure, and one can choose arbitrary coordinates on it; momentum space has no such structure, and different parametrizations cannot be interpreted as physically equivalent, unless further structure is added. In fact, they lead to inequivalent DSR models with different dispersion relations. Of course, a rigorous discussion of this topic would require a precise definition of the physical momentum. Unfortunately, to our knowledge, an operational definition of momentum measurements in DSR is still lacking. The aim of this paper is to extend DSR models to the case of a background de Sitter spacetime. The first example of a DSR deformation of the de Sitter algebra, limited to the momentum sector of phase space, was given in [18]. A different realization, based on the formalism of quantum groups and extended to the full phase space was proposed in [19]. Later, the authors of [20] gave still another realization and discussed some of its phenomenological consequences. However, all these approaches were purely algebraic, since a metric structure on de Sitter space was not introduced. This lack may lead to ambiguities in the interpretation of the spacetime structure and in the physical predictions of the models. After the completion of the present work, a more thorough investigation of DSR in a de Sitter background in the context of the κ-Poincaré approach has appeared [21, 22], mainly focused on the interplay between Planck-scale and curvature effects. In particular, in [22] the possibility to test the effect of the deformed Poincaré symmetry on the expansion of the universe by observations of gamma-ray bursts has been discussed. In this paper, we shall limit ourselves to consider a classical setting based on a phase space realization of the theory and on the Hamiltonian dynamics of a free particle. We shall discuss the formal aspects of the theory, without exploring phenomenological implications, except for the extension of some standard 3 More precisely, in many DSR models, it is not the energy κ that is left invariant by the deformed transformations, but some components of the 4-momentum. An extreme example of this fact is given by the Snyder model [8, 9], where Lorentz transformations act in the canonical (linear) way, and only the action of the translation generators is nonlinear. www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 926 S. Mignemi: Doubly special relativity in de Sitter spacetime predictions of flat-space DSR models enforced by the deformed addition law of momenta. More specific effects related to the cosmological nature of de Sitter spacetime and to the effective energy dependence of the cosmological constant are left for future work. Finally, it is interesting to remark that the deformed de Sitter algebra has two invariant scales, beyond the speed of light. These are the cosmological constant Λ and the Planck energy κ (or equivalently, the de Sitter radius α ∼ 1025 m and the Planck length 1/κ ∼ 10−35 m). The two scales differ by 60 orders of magnitude and are related to the opposite extrema of the range of observable physical phenomena. The origin of such difference is not explained by the present physical theories. One of the models discussed in this paper possesses a duality for the interchange of α and κ together with the interchange of positions and momenta. The paper is organized as follows: in Sect. 2 we discuss the de Sitter algebra and different parametrizations of de Sitter space, realized as a hyperboloid embedded in five-dimensional spacetime. In Sect. 3 we discuss the dynamics of a free particle in de Sitter space. Sections 4 and 5 are devoted to the study of the generalization of the MS model to de Sitter space, using the results of the previous sections. In Sect. 6, a different generalization of DSR in de Sitter space is considered, related to the Snyder model. An alternative realization is given in Sect. 7. In Sect. 8, some physical implications of our results are discussed. Although we shall not consider this issue in detail, all our result can be straightforwardly extended to the anti-de Sitter case, by changing the sign of the cosmological constant. We use the following notations: A, B = 0, . . . , 4; μ, ν = 0, . . . , 3; i, j = 1, . . . , 3. Except when dealing explicitly with the spacetime metric gμν , we always use lower indices, for example Xμ ≡ ημν X μ , where X μ are the natural (contravariant) coordinates. The manipulations of indices are always performed with the flat metric ημν = diag (1, −1, −1, −1), and not with the metric gμν . The product between two 4-vectors ημν V μ W ν = η μν Vμ Wν is denoted by V ·W , and if V = W by V 2 . For 5-vectors, we write the indices explicitly. 2 de Sitter space We review some properties of de Sitter space and its symmetry group, which are not easily found in the literature. In particular, we discuss some coordinate systems that will be useful in the following. Unfortunately, contrary to Minkowski space, de Sitter space does not admit a natural choice of coordinates. In particular, as we shall see, different quantities have simpler expressions in different coordinate systems. We shall therefore alternate between them, depending on the subject under consideration. 2.1 Generalities 2 = −α2 embedded in It is well known that de Sitter space can be realized as a hyperboloid of equation ξA 5-dimensional flat space, with coordinates ξA and metric tensor ηAB = diag (1, −1, −1, −1, −1). The isometries of de Sitter space are generated by the de Sitter algebra. This can be identified with the Lorentz algebra so(1, 4) of the 5-dimensional space, which leaves invariant the hyperboloid. In terms of the 5-dimensional canonical positions ξA and momenta πA , the generators of the Lorentz algebra read JAB = ξA πB − ξB πA , and obey the Poisson brackets {JAB , JCD } = ηBC JAD − ηBD JAC + ηAD JBC − ηAC JBD . (2.1) In mathematical language, de Sitter space can be described as a coset space SO(1, 4)/ SO(1, 3). In order to parametrize it, one must single out one of the (equivalent) spatial coordinates, say ξ4 . The position coordinates Xμ of de Sitter space are then obtained through a projection from the hyperboloid to a plane orthogonal to the ξ4 axis. Moreover, the canonically conjugate momenta Pμ can be defined in such a way that they obey canonical Poisson brackets with the Xμ , according to the 5-dimensional symplectic structure. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 927 Under projection, the original 5-dimensional Lorentz algebra so(1, 4) reduces to the √ 4-dimensional Lorentz algebra so(1, 3) generated by the Jμν , while the remaining generators Tμ = Λ J4μ are interpreted as generators of de Sitter translations. The de Sitter algebra can therefore be written as {Jμν , Jρσ } = ηνσ Jμρ − ηνρ Jμσ + ημρ Jνσ − ημσ Jνρ , {Jμν , Tλ } = ημλ Tν − ηνλ Tμ , {Tμ , Tν } = −ΛJμν . (2.2) Hence, the Lorentz subalgebra of the 4-dimensional de Sitter algebra is identical to the flat space Lorentz algebra. In particular, if one writes its generators Jμν in terms of the 4-dimensional coordinates Xμ and their canonically conjugate momenta Pμ , one gets the canonical expression Jμν = Xμ Pν − Xν Pμ . Thus the positions and the momenta obey the usual transformation laws under Lorentz transformations, {Jμν , Xλ } = ημλ Xν − ηνλ Xμ , {Jμν , Pλ } = ημλ Pν − ηνλ Pμ . (2.3) As we shall see, the realization of the translation generators Tμ depends instead on the specific choice of coordinates on the hyperboloid. 2.2 Natural coordinates The de Sitter hyperboloid can be parametrized by arbitrary coordinates and, contrary to the case of flat space, there is no privileged system of coordinates for de Sitter space. The systems commonly used in the applications to general relativity single out the time coordinate, while the most interesting for our purposes are isotropic in space and time. Since these systems of coordinates are not very well known, we shortly review their properties. √ We start by considering the natural parametrization, given by X̂μ = ξμ . It follows that Λ ξ4 = 1 + ΛX̂ 2 , and the metric induced on the hyperboloid by the 5-dimensional flat metric reads ĝμν = ημν − Λ X̂μ X̂ν 1 + ΛX̂ 2 , ĝ μν = η μν + Λ X̂ μ X̂ ν . (2.4) A spacelike coordinate singularity, that can be interpreted as a cosmological horizon, occurs at X̂ 2 = −1/Λ. √ From the definition of Tμ = Λ J4μ , it is easy to see that under translations {Tμ , X̂ν } = − 1 + ΛX̂ 2 ημν . (2.5) The nontrivial effect of translations is of course due to the curvature of the space. From (2.5) it is evident that the translation generators Tμ do not coincide with the momenta P̂μ = πμ canonically conjugate to X̂μ . In fact, Tμ = 1 + ΛX̂ 2 P̂μ , (2.6) and the momenta transform as ΛX̂ν P̂μ {Tμ , P̂ν } = . 1 + ΛX̂ 2 www.ann-phys.org (2.7) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 928 S. Mignemi: Doubly special relativity in de Sitter spacetime 2.3 Conformal coordinates The parametrization that yields √ the simplest form of the metric is given by√conformal (stereographic) coordinates X̃μ = 2ξμ /(1+ Λ ξ4 ), with inverse ξμ = X̃μ /(1−ΛX̃ 2/4), and Λ ξ4 = (1+ΛX̃ 2/4)/(1− ΛX̃ 2 /4). In these coordinates the metric takes the diagonal form 2 ημν ΛX̃ 2 μν , g̃ = 1 − η μν . (2.8) g̃μν = 4 (1 − Λ X̃ 2 /4)2 The coordinate singularity at X̃ 2 = 4/Λ corresponds to the infinity of the hyperboloid. Under translations the position coordinates transform as ΛX̃ 2 Λ ημν + X̃μ X̃ν , {Tμ , X̃ν } = − 1 + 4 2 √ and hence, in terms of the canonical momenta P̃μ = (1 + Λ ξ4 ) πμ /2, ΛX̃ 2 Λ P̃μ − X̃·P̃ X̃μ . Tμ = 1 + 4 2 (2.9) (2.10) The momenta transform as Λ {Tμ , P̃ν } = − (X̃·P̃ ημν + X̃μ P̃ν − X̃ν P̃μ ). 2 (2.11) 2.4 Beltrami coordinates Another useful√parametrization of the de Sitter given by Beltrami (projective) coordinates [2, √hyperboloid is√ √ 3], X̄μ = ξμ / Λ ξ4 , with inverse ξμ = X̄μ / 1 − ΛX̄ 2 and Λ ξ4 = 1/ 1 − ΛX̄ 2 . In these coordinates, the metric has the form μν (1 − ΛX̄ 2 )ημν + ΛX̄μ X̄ν μν 2 μ ν X̄ . (2.12) , ḡ = (1 − Λ X̄ ) η − Λ X̄ ḡμν = (1 − ΛX̄ 2 )2 Again, the coordinate singularity at X̄ 2 = 1/Λ corresponds to the infinity of the hyperboloid. Under translations, the coordinates X̄μ transform as {Tμ , X̄ν } = −ημν + ΛX̄μ X̄ν . (2.13) Tμ = P̄μ − Λ X̄·P̄ X̄μ , (2.14) {Tμ , P̄ν } = −Λ(X̄·P̄ ημν + X̄μ P̄ν ). (2.15) √ In terms of the canonical momenta P̄μ = Λ ξ4 πμ , the translation generators read and 3 Motion in de Sitter space We consider now the motion of a free particle in de Sitter space in the coordinate systems introduced in the previous section, using the Hamiltonian formalism. For simplicity, we do not explicitly impose the Hamiltonian constraint, since it is not essential for our considerations, but this can be done straightforwardly. We observe also that one may write the dynamics in terms of the conserved quantities Tμ instead of Pμ . However since the Tμ do not satisfy canonical Poisson brackets with the position coordinates Xμ , the formalism would become more involved and the generalization to DSR uneasy. Moreover, it is not evident to us which of the variables should be identified with the physically measured momentum. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 929 3.1 Generalities4 The lagrangian of a free particle of mass m, invariant under de Sitter transformations, is given by L= m gμν Ẋ μ Ẋ ν , 2 (3.1) where a dot denotes derivative with respect to an evolution parameter τ . Varying with respect to Xμ one obtains the geodesics equations. Alternatively, defining the canonically conjugate momenta ∂L = m gμν Ẋ ν , ∂ Ẋ μ Pμ ≡ (3.2) 1 μν (with inverse Ẋ μ = m g Pν ), one can define the Hamiltonian as H = Pμ Ẋ μ − L = 1 μν g Pμ Pν . 2m (3.3) The equations of motion in Hamiltonian form are Ẋ μ = {X μ , H} = g μν Pν , Ṗμ = {Pμ , H} = − ∂g λν Pλ Pν , ∂X μ and can be derived by varying the action dτ (Ẋ μ Pμ − H). (3.4) (3.5) An equivalent way to obtain the Hamiltonian is to identify it with the quadratic Casimir invariant of the de Sitter group, JAB JAB = Jμν Jμν − Λ2 Tμ Tμ , written in terms of the four-dimensional phase space variables. It is easy to see that in this way one recovers the previous results. 3.2 Conformal coordinates These coordinates give the simplest relation between velocity and momentum. The Hamiltonian takes the form (from now on we put m = 1), 2 ΛX̃ 2 1 P̃ 2 , 1− H̃ = 2 4 (3.6) with field equations X̃˙ μ = ΛX̃ 2 1− 4 2 P̃μ , 2 Λ X̃ Λ P̃ 2 X̃μ . 1− P̃˙μ = 2 4 (3.7) In these coordinates, the 3-velocity vi is given by vi ≡ X̃˙ i P̃i = , X̃˙ 0 P̃0 (3.8) and hence the relation between 3-velocity and momenta is the same as in flat space. 4 In this subsection we restore the difference between upper and lower indices. www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 930 S. Mignemi: Doubly special relativity in de Sitter spacetime The field equations may be integrated by substituting the first Eq. (3.7) into the second. However, it is more convenient to use the conservation law associated with the translations, which gives a first integral Tμ = Aμ , with Aμ a constant vector. From (2.10) and (3.7), Λ (1 + ΛX̃ 2 /4)X̃˙ μ − Λ X̃·X̃˙ X̃μ /2 ΛX̃ 2 P̃μ − X̃·P̃ X̃μ = = Aμ . 1+ 4 2 (1 − ΛX̃ 2 /4)2 (3.9) Inverting, one obtains 1 − ΛX̃ 2 /4 X̃˙ μ = 1 + ΛX̃ 2 /4 Λ ΛX̃ 2 Aμ − A·X̃ X̃μ , 1− 4 2 (3.10) and therefore vi = (1 − ΛX̃ 2 /4)Ai − Λ A·X̃ X̃i /2 . (1 − ΛX̃ 2 /4)A0 − Λ A·X̃ X̃0 /2 (3.11) 3.3 Beltrami coordinates These coordinates have the nice property that 3-dimensional geodesics are straight lines. The Hamiltonian takes the form H̄ = 1 (1 − ΛX̄ 2 )[P̄ 2 − Λ(X̄·P̄ )2 ], 2 (3.12) with equations of motion X̄˙ μ = (1 − ΛX̄ 2 )(P̄μ − Λ X̄·P̄ X̄μ ), P̄˙μ = Λ P̄ 2 − Λ(X̄·P̄ )2 X̄μ + (1 − ΛX̄ 2 )X̄·P̄ P̄μ ) . (3.13) Hence the 3-velocity can be written as vi ≡ X̄˙ i P̄i − ΛX̄·P̄ X̄i = . ˙ X̄0 P̄0 − ΛX̄·P̄ X̄0 (3.14) Its form no longer coincides with its flat space analogous. The Eqs. (3.13) are rather involved, but one can exploit the conservation law Ṫμ = 0 to obtain a first integral, P̄μ − ΛX̄·P̄ X̄μ = X̄˙ μ = Aμ , 1 − ΛX̄ 2 (3.15) for constant Aμ . Inverting, one obtains X̄˙ μ = (1 − ΛX̄ 2 )Aμ , (3.16) and hence vi = Ai . A0 (3.17) Therefore, free particles have constant 3-velocity and their trajectories in 3-space are straight lines. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 931 3.4 Natural coordinates These coordinates do not give rise to particularly simple expressions. Therefore, we just summarize the main results. The Hamiltonian has the form Ĥ = 1 2 [P̂ + Λ(X̂·P̂ )2 ], 2 (3.18) and yields the equations of motion ˙ X̂μ = P̂μ + Λ X̂·P̂ X̂μ , ˙ P̂μ = −Λ X̂·P̂ P̂μ , (3.19) with 3-velocity vi = P̂i + ΛX̂·P̂ X̂i P̂0 + ΛX̂·P̂ X̂0 (3.20) . One can again exploit the conservation law Ṫμ = 0 to obtain a first integral, P̂μ + ΛX̂·P̂ X̂μ = ˙ ˙ (1 + ΛX̂ 2 )X̂μ − ΛX̂·X̂ X̂μ = Aμ . 1 + ΛX̂ 2 (3.21) ˙ Inverting, one obtains X̂μ and then vi = Ai − Λ A·X̂ X̂i A0 − Λ A·X̂ X̂0 (3.22) . 4 The MS model in de Sitter space DSR theories in flat space can be implemented in two different ways. One can either deform the Poincaré algebra [10, 12], imposing nonlinear Poisson brackets between the generators, or maintain the canonical form of the algebra, but modify its action on the momentum variables [11, 15]. The first approach has been considered in [18–20] in order to derive a deformed de Sitter algebra. However, for the discussion of the extension of DSR models to the full phase space, especially in the case of a de Sitter background, the second approach appears to be more useful. The MS model was introduced in [11] and is characterized by a deformed dispersion relation p2 /(1 − p0 /κ)2 = m2 . A remarkable property of this model is that the Planck energy κ is left invariant under the deformed Lorentz transformations. The covariant realization of the model in a noncommutative position space was discussed in [13, 14]. In [16] it was observed that the representation of the MS algebra in phase space can be obtained in a straightforward way from the Poincaré algebra acting canonically on a space of coordinates Xμ , Pμ , by performing the substitution Xμ = (1 − p0 /κ) xμ , Pμ = pμ , 1 − p0 /κ (4.1) pμ = Pμ . 1 + P0 /κ (4.2) with inverse xμ = (1 + P0 /κ) Xμ , Here xμ , pμ are interpreted as physical observables, in contrast with the auxiliary variables Xμ , Pμ .5 5 Note that this cannot be considered simply as a change of variables. The transformation is noncanonical and the structure of spacetime is totally different from the classical one. www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 932 S. Mignemi: Doubly special relativity in de Sitter spacetime The symplectic structure of phase space is then deformed and takes the form [13, 14], xi {xi , xj } = 0, {p0 , pi } = {pi , pj } = 0, {x0 , xi } = , κ p0 {xi , pj } = −δij , {x0 , p0 } = 1 − , κ pi {xi , p0 } = 0. {x0 , pi } = − , (4.3) κ In particular, the position coordinates xμ do not commute. One can apply the same procedure to the case of a de Sitter background. In this context it is useful to rewrite the de Sitter algebra in the form {Ni , Nj } = ijk Mk , {Mi , Nj } = ijk Nk , {Ti , Tj } = −Λ ijk Mk , {Mi , Tj } = ijk Tk , {Ni , Tj } = δij T0 , {Mi , Mj } = ijk Mk , {T0 , Tj } = −Λ Nj , {Mi , T0 } = 0, {Ni , T0 } = Ti , (4.4) where Mk = 12ijk Jij are the generators of rotations and Ni = J0i the generators of boosts. The Poisson brackets between phase space variables maintain the form (4.3). Also the deformed action of the Lorentz subalgebra on coordinates and momenta is the same as in the flat space MS model [14], {Mi , xj } = ijk xk , {Mi , x0 } = 0, {Ni , xj } = δij x0 + pi xj /κ, {Ni , x0 } = xi + pi x0 /κ. {Mi , pj } = ijk pk , {Mi , p0 } = 0, {Ni , pj } = δij p0 − pi pj /κ, {Ni , p0 } = pi − pi p0 /κ. (4.5) The action of translations on coordinates and momenta depends instead on the specific coordinates chosen for de Sitter space. For example, in the natural parametrization, Tμ = (1 − p̂0 /κ)−2 + Λx̂2 p̂μ , (4.6) and x̂0 x̂ν p̂μ Λ −2 2 {Tμ , x̂ν } = − (1 − p̂0 /κ) + Λx̂ ημν − , κ (1 − p̂0 /κ)−2 + Λx̂2 Λ (x̂ν − p̂ν x̂0 /κ) p̂μ {Tμ , p̂ν } = . (1 − p̂0 /κ)−2 + Λx̂2 (4.7) An interesting physical implication of this model is that the cosmological constant becomes effectively energy dependent. Consider for example natural coordinates and define, in analogy with (4.2), x̂4 = (1 + P̂0 /κ) X̂4 , with X̂4 = ξ4 . Then x̂2A = −α2 /(1 − p̂0/κ)2 ≡ −1/Λ(p̂0). In particular, for p̂0 → κ, Λ(p̂0 ) → 0, i.e. particles with energy close to the Planck energy do not experience the curvature of spacetime. 5 Dynamics of the MS model in de Sitter space Also the Hamiltonian of a free particle can be obtained by substituting (4.1) into the undeformed Hamiltonian [16]. The equations of motion can then be obtained by taking into account the deformed symplectic structure (4.4), namely, p0 ∂H pi ∂H xi ∂H − + , ẋ0 = {x0 , H} = 1 − κ ∂p0 κ ∂pi κ ∂xi c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) ẋi = {xi , H} = − and ∂H xi ∂H − , ∂pi κ ∂x0 933 (5.1) p0 ∂H , ṗ0 = {p0 , H} = − 1 − κ ∂x0 ṗi = {pi , H} = ∂H pi ∂H + . ∂xi κ ∂x0 (5.2) Equivalently, the Hamilton equations can be obtained by varying the action in which the substitution (4.1) has been done. For example, in conformal coordinates the Hamiltonian is given by 1 H̃ = Δ̃2 p̃2 , 2 (5.3) where Δ̃ = 1 Λ − (1 − p̃0 /κ) x̃2 . 1 − p̃0 /κ 4 (5.4) The Hamilton equations then read Λ x̃˙ μ = Δ̃2 p̃μ + (1 − p̃0 /κ) Δ̃ p̃2 x̃0 x̃μ , 2κ (5.5) Λ p̃˙ μ = (1 − p̃0 /κ) Δ̃ p̃2 (x̃μ − x̃0 p̃μ /κ). 2 (5.6) and They can also be recovered from the action 1 ˙ 2 2 2 I = dτ X̃μ P̃μ − (1 − ΛX̃ /4) P̃ 2 1 2 2 p̃μ d (1 − p̃0 /κ) x̃μ − Δ̃ p̃ . = dτ 1 − p̃0 /κ dτ 2 (5.7) The Hamilton equations (5.5) have acquired complicated terms proportional to Λ/κ, and are no longer linear in the momentum, so that it is not easy to invert them in order to obtain p̃μ in terms of x̃˙ μ . Because of this, it is difficult to obtain the equations of motion in second order form, even using the conservation law for Tμ . Moreover, the property that the velocity has the same expression as in the undeformed case, valid for the flat space MS model [14], does not extend to the de Sitter case. In fact, this property was proven in [23] to hold for position-independent Hamiltonians. If one wishes to maintain its validity, one should look for a different deformation of the symplectic structure. For the same reason, contrary to the flat space MS model, the evolution parameter dτ cannot be identified with the line element invariant under the deformed transformations, which reads ds2 = dx̃2 (1 − p̃0 /κ)2 dx̃2 = 2 . Δ̃2 1 − Λ4 (1 − p̃0 /κ)2 x̃2 (5.8) An analogous calculation can be performed in natural coordinates. The deformed Hamiltonian is 1 p̂2 2 Ĥ = + Λ(x̂·p̂) , (5.9) 2 (1 − p̂0 /κ)2 www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 934 S. Mignemi: Doubly special relativity in de Sitter spacetime with Hamilton equations x̂˙ μ = (1 − p̂0 /κ)−2 p̂μ + Λ(1 − p̂0 /κ) x̂·p̂ x̂μ , p̂˙ μ = −Λ(1 − p̂0 /κ) x̂·p̂ p̂μ . (5.10) The invariant metric reads ds2 = (1 − p̂0 /κ)2 dx̂2 − Λ(x̂·dx̂)2 (1 − p̂0 /κ)−2 + Λx̂2 (5.11) Also in this case one finds the same problems as with conformal coordinates. It is also interesting to notice that the metric (5.11) exhibits a momentum-dependent cosmological horizon at Λx̂2 = −(1 − p̂0 /κ)−2 . This property is of course related to the momentum dependence of the cosmological constant discussed at the end of previous section. The metrics (5.8) and (5.11) can be considered as examples of rainbow metrics [24]. Finally, we notice that in the limits κ → ∞ and Λ → 0 one recovers the ordinary de Sitter space and the flat space MS model, respectively, while the limit p0 → κ is analogous to that of the MS model [11]. 6 DSR in de Sitter space in a Snyder-like basis It is known that DSR theories can be realized in several different ways. An interesting realization is given by the so-called Snyder basis [8], which is characterized by the dispersion relation P 2 /(1 − P 2/κ2 ) = m2 . In a DSR interpretation [9] this implies that the rest mass of particles must always be less than κ. Another important property of this basis is that only the action of the translations is deformed, while that of the Lorentz group is not affected. This example illustrates the fact that the most relevant characteristic for the implementation of DSR is the deformation of the action of translations (and hence a modified composition law of momenta) rather than that of Lorentz transformations, as usually postulated. 6.1 Minkowski space Let us briefly review the case of flat spacetime [9]. It is easy to see that, in analogy with our previous treatment of the MS model, the easiest way to recover the Snyder realization of DSR is to define new coordinates Xμ , Pμ starting from the canonical Xμ , Pμ , which are thus interpreted as auxiliary variables, Xμ = 1 + ΩP 2 Xμ , Pμ = √ Pμ , 1 + ΩP 2 where Ω = 1/κ2 is the Planck area6. The inverse transformations are Pμ Xμ = 1 − Ω P 2 Xμ , Pμ = √ , 1 − Ω P2 (6.1) (6.2) One has then, {Xμ , Xν } = −Ω(Xμ Pν − Xν Pμ ), {Pμ , Pν } = 0, {Xμ , Pν } = ημν − Ω Pμ Pν . (6.3) the canonical form (2.3). The translation generaThe Lorentz transformations acting on Xμ , Pμ maintain √ tors Tμ must instead be identified with Pμ = Pμ / 1 − Ω P 2 . Their action changes accordingly, ημν , {Tμ , Xν } = √ 1 − ΩP 2 6 {Tμ , Pν } = 0. (6.4) In principle one may choose a negative sign for Ω, obtaining an inequivalent model with rather different properties [9]: for example, in that case the mass does not admit a maximal value, but in the quantum theory a nonzero minimal uncertainty for the position appears due to the modified commutation relations, as in the model discussed in [25]. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 935 The invariant Hamiltonian for a free particle can be written as H= P2 P2 1 = , 2 2 1 − Ω P2 (6.5) with equations of motion X˙μ = Pμ , 1 − Ω P2 Ṗμ = 0. (6.6) It follows that Ẋμ = Aμ is constant. The 3-velocity is then given by vi = Pi Ai = , P0 A0 (6.7) and the 3-dimensional geodesics are straight lines. Moreover, it is easy to verify that the invariant line element ds2 = (1 − Ω P 2 )dX 2 can be identified with dτ 2 , with τ the evolution parameter. 6.2 de Sitter space Let us now extend the above construction to the case of de Sitter space in the Beltrami coordinates of Sect. 2.4. The substitution (6.1) yields X̄μ = 1 + ΩP̄ 2 X̄μ = Ω π 2 + (Λξ42 )−1 ξμ , πμ P̄μ P̄μ = √ = , 2 2 Ω π + (Λξ42 )−1 1 + ΩP̄ (6.8) where ξA are as usual the coordinates of the five-dimensional embedding space. The new phase space coordinates X̄μ , P̄μ satisfy the Poisson brackets (6.3). Using the definition of ξ4 and inverting (6.8), one gets ξμ = where Φ= X¯μ , Φ πμ = ΦP̄μ , (6.9) (1 − Ω P̄ 2 )−1 − ΛX̄ 2 . (6.10) In terms of the variables X̄μ , P̄μ , the Lorentz generators of the de Sitter algebra (2.2) have canonical form, while the translation generators read 1 Tμ = √ P̄μ − Λ(1 − Ω P̄ 2 )X¯ ·P̄ X¯μ , 1 − Ω P̄ 2 (6.11) and 1 {Tμ , X¯ν } = −√ ημν − Λ(1 − Ω P̄ 2 )2 X̄μ X¯ν + ΛΩ(1 − Ω P̄ 2 )X¯ ·P̄ P̄μ X¯ν ) , 2 1 − Ω P̄ (6.12) {Tμ , P̄ν } = −Λ 1 − Ω P̄ 2 X̄ ·P̄(ημν − Ω P̄μ P̄ν ) − (1 − Ω P̄ 2 )X¯μ P̄ν . One can also define a Hamiltonian, invariant under the full deformed de Sitter group, H= 1 2 2 Φ P̄ − Λ(1 − Ω P̄ 2 )(X¯ ·P̄)2 . 2 www.ann-phys.org (6.13) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 936 S. Mignemi: Doubly special relativity in de Sitter spacetime The Hamilton equations take an extremely involved form, and we shall not report them here. The invariant metric for this model is gμν = Φ2 ημν + ΛX¯μ X̄ν . Φ4 (6.14) No coordinate singularities occur at finite distance. In the limits Ω → 0 and Λ → 0 one recovers the ordinary de Sitter space and the flat space Snyder model of previous section, respectively. 7 A different Snyder-like realization The Snyder realization of DSR in de Sitter space given in the previous section is rather awkward. In this section, we consider a slightly different realization, which takes a more symmetric form and gives rise to more elegant formulas. The algebra of this model displays some similarities with that proposed in [20]. As in that case, in the present realization the momenta coincide in the limit Ω → 0 with the de Sitter translation generators rather than with the canonical momenta, and hence have nontrivial Poisson brackets with the position coordinates in this limit. We define 1 + Ωπ 2 1 + Λξ 2 P̆ ξ = πμ , (7.1) X̆μ = μ μ 1 + Λξ 2 1 + Ωπ 2 with inverse ξμ = 1 − Ω P̆ 2 ˘ Xμ , 1 − Λ X̆ 2 πμ = 1 − Λ X̆ 2 1 − Ω P̆ 2 P̆μ . (7.2) The coordinates (7.1) satisfy the Poisson brackets {X̆μ , X̆ν } = − {P̆μ , P̆ν } = − Ω (1 − Λ X̆ 2 ) 1 − ΛΩ X̆ 2 P̆ 2 Λ (1 − Ω P̆ 2 ) 1 − ΛΩ X̆ 2 P̆ 2 {X̆μ , P̆ν } = ημν − (X̆μ P̆ν − X˘ν P̆μ ), (X̆μ P̆ν − X˘ν P̆μ ), Λ (1 − Ω P̆ 2 )X˘μ X˘ν + Ω (1 − Λ X̆ 2 )P̆μ P̆ν 1 − ΛΩ X̆ 2 P̆ 2 . (7.3) The Lorentz generators of the de Sitter algebra have canonical form, while the dilatation generators are 1 − ΛΩ X̆ 2 P̆ 2 P̆μ , (7.4) Tμ = 1 − Ω P̆ 2 and their action is given by 2 2 ˘ 2 P̆ 2 ˘ν X 1 − ΛΩ X̆ ) ) X · P̆ P̆ Λ(1 − Ω P̆ (1 − Λ X̆ μ X˘μ X̆ν + Ω {Tμ , X˘ν } = − ημν − , 1 − Ω P̆ 2 1 − ΛΩ X̆ 2 P̆ 2 1 − ΛΩ X̆ 2 P̆ 2 2 ˘ P̆ 1 − Ω P̆ 2 ) X · P̆ P̆ (1 − Λ X̆ μ ν X̆μ P̆ν − X˘ν P̆μ + Ω {Tμ , P̆ν } = −Λ . (7.5) 1 − ΛΩ X̆ 2 P̆ 2 1 − ΛΩ X̆ 2 P̆ 2 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 937 The Hamiltonian of a free particle can be obtained from the Casimir invariant of the de Sitter algebra, and takes the form 1 1 − ΛX̆ 2 2 H= P̆ + Λ(X̆ ·P̆)2 . (7.6) 2 1 − Ω P̆ 2 Taking into account the symplectic structure (7.3), the equations of motion ensuing from the Hamiltonian are P̆μ ΛΩ X̆ ·P̆ P̆ 2 X˘μ ˙ 2 X̆μ = (1 − Λ X̆ ) − , 1 − Ω P̆ 2 1 − ΛΩ X̆ 2 P̆ 2 2 ΛΩ(1 − Λ X̆ ) ˘ ˙ X ·P̆ P̆ 2 P̆μ . P̆μ = 1 − ΛΩ X̆ 2 P̆ 2 (7.7) Also in this case, there does not seem to exist a simple relation between velocity and momentum. The 4-dimensional metric can be derived in the usual way from the 5-dimensional flat metric subject to the constraint 1 − ΛΩ X̆ 2 P̆ 2 ξ4 = 1 + Λξ 2 = , (7.8) 1 − Λ X̆ 2 and reads gμν = 1 − Ω P̆ 2 1 − Λ X̆ 2 ημν + Λ 1 + Ω(1 − Λ X̆ 2 )P̆ 2 (1 − Λ X̆ 2 )(1 − ΛΩ X̆ 2 P̆ 2 ) ˘ Xμ X̆ν . (7.9) Also in this case there is no evident relation between the metric and the differential dτ of the evolution parameter. From (7.2) it is evident that the coordinate singularity at X̆ 2 = 1/Λ corresponds to points at infinity in the hyperboloid. Moreover, the metric (7.9) presents a second momentum-dependent coordinate singularity at ΛX̆ 2 = 1/ΩP̆ 2, or better X̆ 2 P̆ 2 = 1/ΛΩ. However, for such values of X˘ and P̆ the model is ill-defined (see (7.3)): this region is anyway far beyond the range of physically observable phenomena, since 1/ΛΩ ∼ 10120 . In the limit Λ → 0 one of course recovers the flat-space Snyder model of previous section, while in the limit Ω → 0 one gets the standard de Sitter space, although with noncanonical Poisson brackets between positions and momenta (since the momenta are identified generators in this limit). √ with the translation √ √ and More interesting are the limits X̆ → 1/ Λ and P̆ → 1/ Ω. For X˘ → 1/ Λ, one is close to infinity, √ the symplectic structure reduces to the undeformed one obtained in the limit Ω = 0. The limit P̆ → 1/ Ω corresponds instead to the extremal value of the momentum. In this limit, the symplectic structure is that of the flat Snyder model, Λ = 0, and the metric and the Hamiltonian become singular. It is also interesting to notice that the Poisson brackets (7.3) lead after quantization to generalized commutation relations of the most general kind proposed in [25] that, in case of negative Λ and Ω, imply the existence of both a minimal length and momentum. Another interesting property of this model is the existence of a duality for the exchange of X̆ ↔ P̆, together with Λ ↔ Ω. This duality connects the high-energy/short-distance regime, governed by the Planck area Ω, with the low-energy/long-distance regime, governed by the cosmological constant Λ. 8 Phenomenological implications We briefly review some immediate phenomenological consequences of our model, analogous to those of flat space DSR. We also mention some effects that could be associated to the cosmological character of www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 938 S. Mignemi: Doubly special relativity in de Sitter spacetime de Sitter spacetime, but a more thorough investigation of this topic is left for future work. The discussion relies on our specific interpretation of the DSR formalism, that, in particular, implies a constant speed of light. Other interpretations may lead to different predictions (see for example [26]). 8.1 Addition law of momenta One of the main phenomenological predictions of DSR is a deformation of the addition law of momenta. For consistency with the DSR postulates, in fact, the total energy or momentum of a composite system should not exceed κ. This fact poses the problem of the transition to macroscopic systems, where such limit cannot be valid, but this is beyond the scope of the theory, at least if it is considered just as a phenomenological model describing quantum gravity effects. The main application of the deformed addition law is to the scattering of ultra-high energy particles by the background cosmological radiation, which may give rise to observable effects [7]. To obtain the addition law, one writes the conserved quantity Tμi for the scattered particles as a function i Tμ . The total momentum ptμ is of their physical momenta piμ as Tμi = Tμi (piμ ), and then adds, Tμt = t t t finally obtained by inverting the previous relation and writing pμ = pμ (Tμ ) [15, 25]. In de Sitter relativity, Tμ (pμ ) is a (position-dependent) linear function of pμ and therefore the physical momenta add in the usual linear way, at least if x is fixed, as in the case of point interactions (in any case corrections could be relevant only at cosmological scales). In de Sitter DSR the function Tμ (pμ ) is nonlinear and position dependent. However, for point interactions one can always choose the origin as the collision point. In this case, the addition law reduces to that of the corresponding flat space DSR model. For example, in the MS model of Sects. 4–5, one obtains the addition law [15] i p /(1 − pi0 ) μ i . (8.1) ptμ = 1 + p0 /(1 − pi0 ) Its physical consequences have been thoroughly investigated in [27]. If the scattering point is not at the origin, Tμ (pμ ) is given by (4.6), and its inversion is nontrivial, so that an explicit expression for the addition law is awkward. Due to the homogeneity of de Sitter space, one may however always perform a translation in order to carry x into the origin. 8.2 Proper time The possibility of defining an invariant metric allows one to easily calculate the correction to the lifetime of a massive particle. In particular, in conformal coordinates the metric has the simple expression ds2 = A2 (x, p) dx2 , (8.2) and, calling xμ , pμ the laboratory coordinates and xμ , pμ the rest coordinates of the particle, one has A2 (x , p )dt2 = A2 (x, p)(1 − v2 )dt2 , (8.3) where we have put x0 = t. Hence, dt = A(x , p ) γ dt A(x, p) (8.4) with γ = (1 − v2 )−1/2 . This gives rise to corrections to the relativistic formula. For example, for the model of Sects. 4–5, A(x, p) = 1/Δ̃, (cf. (5.8)) and (1 − M/κ) dt (1 − p0 /κ) dt = , γ 1 − 4γΛ2 (1 − p0 /κ)2 t2 1 − Λ4(1 − M κ)2 t2 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (8.5) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 12 (2010) 939 where M ≡ p0 is the rest energy of the particle (not to be confused with its Casimir mass m). Integrating, one gets the same relation that holds in the flat case [28], (1 − M/κ) p0 − M Δt = γ Δt ∼ 1 + (8.6) γ Δt . (1 − p0 /κ) κ One has therefore energy-dependent corrections to the formula relating proper time to laboratory time, and hence to the observed lifetime of high-energy particles. The corrections are however of order p0 /κ and are only relevant for Planck-scale energies. 8.3 Cosmology As mentioned before, in de Sitter DSR the cosmological constant becomes effectively energy dependent. This can have important consequences on the cosmological models, especially in the first stages of the cosmological evolution, when Planck-scale energies are involved. Moreover, also astrophysical observations may be affected, since ultra-high energy particles experience a smaller value of the cosmological constant. Also the cosmological horizon is shifted, depending on the energy of the particles. This may have implications on the horizon problem, as observed in [11]. Recently, the interplay between curvature effects and DSR has been investigated in great detail in [21, 22] in the context of the κ-Poincaré formalism. It must be noticed however that, contrary to our approach, in such framework the speed of light is momentum dependent, so that not all phenomenological predictions of those papers can be extended to our models. Finally, it has been argued that, due to the modified dynamics, correction to the galaxy dynamics may arise [20]. Also these, however, strongly depend on the specific realization of DSR in de Sitter space adopted. 9 Conclusions It is known that DSR models can be derived from a 5-dimensional momentum space of coordinates πA , 2 subject to the constraint πA = −κ2 [17]. This is similar to the de Sitter constraint for the spacetime coordinates. However, the physical interpretation is quite different. First of all, de Sitter spacetime inherits a metric structure from the 5-dimensional space and this allows one to define a curvature. Different systems of coordinates are physically equivalent. The momentum space, instead, does not possess a metric structure and different coordinates cannot be considered physically equivalent, unless one adds further structure. In fact, different realizations of DSR lead to different physical theories. Moreover, the mere existence of a de Sitter group of transformations on a four-dimensional manifold does not automatically imply that this can be identified with de Sitter space. With these remarks in mind, one may try to construct a realization of a deformed de Sitter relativity starting from five-dimensional space, similarly to what has been done for flat space [29]. Unfortunately, however, it is not possible to impose contemporary constraints on the five-dimensional positions and momenta, and one is forced to start from a six-dimensional space. The construction of a Hamiltonian formalism in six-dimensional phase space with coordinates ΞM and momenta ΠM , subject to the constraints Ξ2M = −α2 , Π2M = −κ2 has recently been investigated in [30]. From the study of DSR in de Sitter space one can also learn some lessons concerning the flat space limit. First of all, it is useful to distinguish the translation generators, that dictate the conservation laws for the momentum, from the physical momentum, identified with the phase space momentum variables. This observation also gives a physical meaning to the auxiliary variables obeying canonical transformation laws introduced in [15], whose interpretation was unclear: they are simply the generators of translations. Moreover, it appears that the distinguishing feature of DSR is not the deformation of the Lorentz symmetry, as usually postulated, but rather that of the translation symmetry, as exemplified by the Snyder model discussed in Sects. 6 and 7 [9]. Of course, a complete discussion of this topic requires an operational www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 940 S. Mignemi: Doubly special relativity in de Sitter spacetime definition of the momentum of a particle. In particular, for our interpretation to be valid, this should allow to identify the measured value of the momentum with the canonical momentum rather than with the generator of translations. It would also be interesting to investigate the implication for cosmology of the energy dependence of the cosmological constant, that could give rise to interesting deviations from standard picture at the early stages of the evolution of the universe. This study may help to establish which one, among the infinite possible realizations of DSR in de Sitter space some of which have been discussed in this paper, is closer to physical reality. Although we have not considered this subject in detail, it is also worth noticing that all our considerations can be easily extended to the case of anti-de Sitter space, by simply changing the sign of the cosmological constant Λ. Acknowledgements I wish to thank J.P. Pereira for drawing my attention to [4]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). R. L. Mallett and G. N. Fleming, J. Math. Phys. 14, 45 (1973). H. Y. Guo, C. G. Huang, Z. Xu, and B. Zhou, Phys. Lett. A 331, 1 (2004). R. Aldrovandi, J. P. Beltrán Almeida, and J. P. Pereira, Class. Quantum Gravity 24, 1385 (2007). S. Cacciatori, V. Gorini, and A. Kamenshchik, Ann. Phys. 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