Deformed Carroll particle from 2+1 gravity
aa r X i v : . [ h e p - t h ] A ug Deformed Carroll particle from 2+1 gravity
Jerzy Kowalski-Glikman ∗ and Tomasz Trze´sniewski † Institute for Theoretical Physics, University of Wroc law,Pl. Maksa Borna 9, Pl–50-204 Wroc law, Poland (Dated: October 8, 2018)We consider a point particle coupled to 2+1 gravity, with de Sitter gauge group SO (3 , AN (2) ⋉ an (2) ∗ . The former casewas thoroughly discussed in the literature, while the latter leads to the deformed particleaction with de Sitter momentum space, like in the case of κ -Poincar´e particle. However, theconstruction forces the mass shell constraint to have the form p = m , so that the effectiveparticle action describes the deformed Carroll particle.Gravity in 2+1 dimensions seems to be, at first sight, an incredibly dull theory. It does not possess anylocal degrees of freedom and local Newtonian interactions between masses as well as gravitational wavesare absent [1], [2], [3]. Therefore it can be described by a topological field theory as firmly established inthe seminal paper of Witten [4] (and slightly earlier in [5].) Remarkably, the picture changes dramaticallywhen one adds point particles to pure gravity. Then the gauge degrees of freedom of gravity at the particle’sworldlines become dynamical. By solving for [6], [7] (or integrating out in the path integral formalism [8], [9])the remaining (gauge) degrees of freedom of gravity we are left with a nontrivial particles dynamics, whichincludes not only the “pure” particles’ degrees of freedom, but also the back reaction resulting from thepresence of the gravitational field created by the particles themselves. Since in 2+1 dimensions gravitational“action at a distance” is absent, the only thing that this back reaction can do is to deform the original freeparticles’ actions.In this letter we show that one of the possible deformed actions, which can be obtained from 2+1 gravityis an action of the deformed Carroll particle with AN (2) momentum space [10], [11] and κ -Minkowski(noncommutative) spacetime [12], [13]. The Carroll particle [14] is a relativistic particle model in the limitin which the velocity of light becomes zero. Such a particle cannot move ( . . . it takes all the running youcan do, to keep in the same place . . . [15]) and its relativistic symmetry group is a particular contractionof the Poincar´e group [16], [17]. The Carroll group has attracted some attention recently, because it seemsto become potentially relevant in several distinct fields of theoretical physics, see [18] for discussion andreferences. It also arises in loop quantum cosmology [19] in the context of so-called asymptotic silence, ofwhich the Carrollian (or ultralocal) limit is a particular realization. More specifically, the Carrollian limitappears at the transition between the low-curvature (Lorentzian) regime and the high-curvature regime in ∗ [email protected] † [email protected] which the metric is Euclidean. It is supposed that at this transition the symmetry group should changefrom Poincar´e to Carroll and eventually to Euclidean.In all these cases a simple free dynamical relativistic particle model possessing Carroll symmetry isof great interest, because it exhibits physics of the Carrollian world, providing an insight into it. It isreasonable to expect that in this world some remnants of particles’ interactions should still be present, andthat the situation is similar to that of gravity coupling in 2+1 dimensions. Indeed, in the limit c → D + 1, D > J a and three translational generators P a ,satisfying [ J a , J b ] = ǫ cab J c , [ J a , P b ] = ǫ cab P c , [ P a , P b ] = − ǫ cab J c , (1)where, since the cosmological constant is absorbed into the translational generators, all generators aredimensionless. It is convenient to make use of the time plus space decomposition of spacetime and toaccordingly decompose the Chern–Simons connection one-form into A = A dt + A S . (2)The Lagrangian of the Chern–Simons theory of connection A coupled to a point particle takes the form L = k π Z D ˙ A S ∧ A S E − D C h − ˙ h E + Z (cid:28) A , k π F S − h C h − δ ( ~x ) dx ∧ dx (cid:29) , (3)where the spatial curvature F S = d A S + [ A S , A S ]. Let us pause for a moment to explain the meaning of theterms in this Lagrangian. The bracket h∗i denotes an Ad-invariant inner product on the Lie algebra of thegauge group, which in our case is defined to be h J a P b i = η ab , h J a J b i = h P a P b i = 0 . (4)The coupling constant k can be related to the physical parameters, the Planck mass κ (which in 2+1dimensions is purely classical) and the cosmological constant as follows k π = κ √ Λ . (5)Then, the first term in (3) describes the pure gravity while the second one is the particle term. Moreprecisely, h includes the translation and Lorentz transformation acting on the particle and providing it withan arbitrary position, momentum, and angular momentum, while C is a gauge algebra element characterizingthe particle at rest at the origin, C = m √ Λ J + s P , (6) where m is the mass and s the spin of the particle in the rest frame.Last but not least, the integrand in the third term in (3) can be seen as a constraint relating the curvaturewith the mass/spin of the particle k π F S = h C h − δ ( ~x ) dx ∧ dx , (7)enforced by the Lagrange multiplier A . It bounds together the gravitational and particle degrees of freedom,therefore we may use it to solve for the latter in terms of the former.In order to proceed further we divide the space into two regions: the disc D with the particle at itscenter, on which we introduce the coordinates r ∈ [0 , φ ∈ [0 , π ], and the asymptotic empty region E (with r ≥ r = 1. Then by virtue of (7) in the asymptotic region theconnection is flat and has the form A ( D ) S = γdγ − , (8)where γ is an element of the gauge group. In the particle region D the general solution of (7) can also befound and it is given by A ( E ) S = ¯ γ k C dφ ¯ γ − + ¯ γd ¯ γ − , ¯ γ (0) = h , (9)(note that ddφ = 2 π δ ( ~x ) dx ∧ dx ). Substitution of (8) into the Lagrangian yields the boundary term andthe so-called WZW term. The same is the case for the second term in (9). Contribution from the first termin (9) can be rewritten as2 (cid:10) ∂ (¯ γ C ¯ γ − dφ ) ∧ ¯ γd ¯ γ − (cid:11) = − ∂ (cid:10) C dφ ∧ ¯ γ − d ¯ γ (cid:11) − d (cid:10) C dφ ¯ γ − ˙¯ γ (cid:11) + 4 π δ ( ~x ) dx ∧ dx (cid:10) C ¯ γ − ˙¯ γ (cid:11) , (10)where the first term can be neglected being a total time derivative and the last one cancels the particleterm in (3). Summing all the contributions and adopting the opposite orientation of the boundary Γ for theterms coming from the disc D we obtain the total Lagrangian L = k π Z Γ (cid:28) ˙ γγ − dγγ − − ˙¯ γ ¯ γ − d ¯ γ ¯ γ − + 2 k C dφ ˙¯ γ ¯ γ − (cid:29) + k π Z E (cid:10) ˙ γγ − dγγ − ∧ dγγ − (cid:11) + k π Z D (cid:10) ˙¯ γ ¯ γ − d ¯ γ ¯ γ − ∧ d ¯ γ ¯ γ − (cid:11) . (11)In the next step we impose the continuity condition on the boundary Γ, A ( D ) S | Γ = A ( E ) S | Γ . Solving thisequation we find the expression γ − | Γ = N e k C φ ¯ γ − | Γ , dN = 0 , (12)where N = N ( t ) is an arbitrary gauge group element.The idea now is to use the continuity condition (12) to simplify the Lagrangian (11). Unfortunately, thiscondition is very difficult to disentangle in the case of the gauge group SO (3 , G ⋉ g ∗ , see [20].Before deriving the main result of this paper, let us pause for a moment to recall the well knownconstruction in the case of the standard contraction of de Sitter group SO (3 ,
1) to the Poincar`e group, which can be presented as SO (2 , ⋉ so (2 , ∗ ≃ SO (2 , ⋉ R . To this end we introduce the rescaled translationgenerators, ˜ P a ≡ √ Λ P a . Then (1) is replaced by[ J a , J b ] = ǫ cab J c , [ J a , ˜ P b ] = ǫ cab ˜ P c , [ ˜ P a , ˜ P b ] = − Λ ǫ cab J c (13)and h J a ˜ P b i = √ Λ η ab . If we now take the limit Λ → P a , ˜ P b ] = 0, while the remaining commutatorsare unchanged. Moreover, √ Λ in the scalar product cancels its inverse in the definition of k/ π , cf. (5)and thus no divergencies appear in this limit. Furthermore, with the help of Cartan decomposition a gaugegroup element can be written as a product g = j p = ( ι + ι a J a )(1 + ξ a ˜ P a ) , j ∈ SO (2 , , p ∈ so (2 , ∗ , (14)where the coordinates ι on SO (2 ,
1) group satisfy ι + ι a ι a = 1.Applying (14) to group elements in the Lagrangian (11) we find that the WZW terms cancel out andonly the boundary ones remain, giving L = k π Z Γ (cid:28) j − ˙ j dξ − ¯ j − ˙¯ j d ¯ ξ + 1 k C P dφ ¯ j − ˙¯ j + 1 k C J dφ h ¯ j − ˙¯ j , ¯ ξ i(cid:29) , (15)where we also neglected the total time derivative k C dφ ˙¯ ξ . Meanwhile, the sewing condition (12) factorizesinto j − = n e k C J φ ¯ j − and − ξ = Ad( n ) h − Ad( n e k C J φ ) ¯ ξ , where we write the decomposition (14) of N as N = n (1 + h ), n ∈ SO (2 , h ∈ so (2 , ∗ . C can be separated into C = C J + C P , C J = m/ √ Λ J , C P = s/ √ Λ ˜ P . Substituting the above expressions into (15) we get L = k π Z Γ d (cid:28) e − k C J φ ˙ n − n e k C J φ ¯ ξ − n − ˙ n k C P φ (cid:29) , (16)where ¯ ξ ≡ ¯ ξ a ˜ P a . Integrating (16) over φ from 0 to 2 π and noticing that ¯ ξ is a single-valued function on Γ,hence ¯ ξ (0) = ¯ ξ (2 π ), we finally obtain [7] L = κ x a (cid:16) ˙Π Π − (cid:17) a − s (cid:0) n − ˙ n (cid:1) , (17)with the new variables of particle’s position x = x a ˜ P a ≡ n ¯ ξ (0) n − and “group valued momentum” Π,defined as Π ≡ n e − πk C J n − = e − mκ n J n − . (18)Thus the Lagrangian (17) describes a deformed particle, whose momentum is now given by the group elementΠ defined above instead of the algebra element mJ .The group valued momentum Π is not arbitrary, but is given by the conjugation of e − πk C J by the Lorentzgroup element n . If we parametrize m n J n − = q a J a then from (18) we findΠ = p − κ p a J a , p + 14 κ p a p a = 1 , p = cos (cid:18) | q | κ (cid:19) , p a = 2 κ q a | q | sin (cid:18) | q | κ (cid:19) . (19)It follows that the momenta p a are coordinates on the three dimensional Anti-de Sitter space constrained bythe deformed mass shell condition. Introducing the Lagrange multiplier λ , the final action for the spinlesscase C P = 0 can be written in the components as S = Z dt (cid:18) p ˙ p a x a + 12 κ ǫ abc ˙ p a x b p c − ˙ p p a x a (cid:19) + λ (cid:16) p a p a − κ sin m κ (cid:17) , (20) where p ≡ q − κ p a p a . The detailed discussion of the properties of this action can be found in [6]. Inthe spinning case there is an additional term of the form − s ( n ˙ n − n ˙ n + ( n ˙ n − n ˙ n )).Let us now turn to the main result of this paper. The contraction of the de Sitter group SO (3 ,
1) toPoincar´e group is pretty well known and the resulting Lagrangian (17) has been derived and thoroughlyanalyzed e.g., in [6], [7]. It turns out, however, that there exist another contraction of the de Sitter groupthat to our knowledge has not been discussed in the literature. Contrary to the case considered above, wherethe translation sector of SO (3 ,
1) was “flattened”, we consider “flattening” of the Lorentz sector of SO (3 , SO (2 ,
1) and AN (2). The generators of the latter are defined as a linear combination ofthe original Lorentz and translation generators S a = P a + ǫ a b J b , (21)so that we have[ J a , J b ] = ǫ cab J c , [ J a , S b ] = ǫ cab S c − η ab J + η b J a , [ S a , S b ] = η a S b − η b S a . (22)The virtue of this decomposition is that the generators J a and S a form subalgebras of the algebra so (3 , J a ≡ √ Λ J a to obtain[ ˜ J a , ˜ J b ] = √ Λ ǫ abc ˜ J c , [ ˜ J a , S b ] = √ Λ ǫ abc S c + ( η b ˜ J a − η ab ˜ J ) , [ S a , S b ] = η a S b − η b S a , (23)which after contraction Λ → J a , ˜ J b ] = 0 , [ ˜ J a , S b ] = ( η b ˜ J a − η ab ˜ J ) , [ S a , S b ] = η a S b − η b S a . (24)It is worth mentioning that, as it was in the case of the Poicar´e algebra above, the algebra (24) is a Liealgebra of the group G ⋉ g ∗ , where G is now the group AN (2) generated by the last commutator in (24).In terms of the new generators the scalar products read D ˜ J a S b E = √ Λ η ab , D ˜ J a ˜ J b E = h S a S b i = 0 . (25)In spite of the fact that this scalar product becomes degenerate in the limit Λ →
0, in the effective particleaction Λ is cancelled out and the contraction limit is not singular.A gauge group element can now be decomposed into γ = j s = (1 + ι a ˜ J a ) e σ i S i e σ S , i = 1 , , (26)where for s we use the parametrization that proved convenient in the context κ -Poincar´e theories [21] andis related to the other parametrization s = ξ + ξ a S a via σ = 2 log( ξ + ξ ), σ i = ( ξ + ξ ) ξ i .Since it is our goal to obtain a curved momentum space after the Λ → C = C J + C S , describing the particle at rest, so as to have the mass in the S sector. Adjustingdimensions properly we get C J = s/ √ Λ ˜ J , C S = m/ √ Λ S . After these preparations we can return to the formulae (11), (12). Writing N = (1+ n ) h , with h ∈ AN (2), n ∈ so (2 ,
1) and using the factorization (26) one first finds that s − = h exp (cid:18) k C S φ (cid:19) ¯ s − . (27)Next, from the commutation relations (24), for an arbitrary s we have s exp (cid:18) k C J φ (cid:19) s − = exp (cid:18) k C J φ (cid:19) . (28)As a result one obtains the second condition u = exp (cid:18) k C J φ (cid:19) s (1 − n ) s − , (29)where we denote u ≡ ¯ j − j .We now plug the factorization (26) into our starting Lagrangian (11), finding L = k π Z Γ (cid:28) ˙ u u − (cid:18) d ¯ s ¯ s − − ¯ s k C dφ ¯ s − (cid:19) + 1 k C dφ ¯ s − ˙¯ s (cid:29) . (30)Substituting the continuity conditions for u and s into (30) and keeping in mind the limit Λ → L = k π Z (cid:28) ∂ (cid:16) ¯ s e − k C S φ h − n h e k C S φ ¯ s − (cid:17) (cid:18) − d ¯ s ¯ s − + ¯ s k C S dφ ¯ s − (cid:19) + 1 k C J dφ ¯ s − ˙¯ s (cid:29) . (31)In the last step, neglecting the total time derivative and integrating over the angular variable we eventuallyobtain the expression very similar to (17) L = k π D Π ˙Π − x E + (cid:10) C J ¯ s − ˙¯ s (cid:11) , (32)but with the deformed momenta Π being now the elements of the AN (2) group, instead of the Lorentz group SO (2 , ≡ ¯ s e πk C S ¯ s − , x ≡ ¯ s h − (0) n (0) h (0)¯ s − . (33)The Lagrangian (32) is the main result of our paper.Since the Lorentz group SO (2 ,
1) is, as a manifold, the three dimensional Anti-de Sitter space, while AN (2) is a submanifold of the three dimensional de Sitter space, we managed to obtain the momentumspace of positive, instead of the negative, constant curvature. Moreover, contrary to (17), which is definedonly in 2+1 dimension, the expression (32) can be readily generalized to any spacetime dimension.Let us now turn to the detailed discussion of the properties of Lagrangian (32). The first thing to noticeis that the first equation in (33) puts severe restrictions on the form of momentum Π. Indeed if we writeΠ = e p i /κ S i e p /κ S , (34)and take ¯ s to have the form ¯ s = e ¯ σ i S i e ¯ σ S , (35)we immediately find that p = m , p i = κ (1 − e mκ ) ¯ σ i . (36) As it was in the case considered above these equations play a role of the mass shell relation, and forcethe energy to be constant, independently of the particle dynamics. In the undeformed case (which can beobtained in the limit κ → ∞ ) such mass shell condition makes the particle effectively frozen, it can notmove, and for that reason, following [14] we call it the “Carroll particle.”In the following discussion we will consider only the spinless case. It turns out that this is in fact thegeneral case, because the spin term does not contribute nontrivially to the equations of motion. Indeed anarbitrary variation δ ¯ s = ̟ ¯ s , δ ¯ s − = − ¯ s − ̟ of this term results in a total time derivative δ (cid:10) C J ¯ s − ˙¯ s (cid:11) = (cid:10) ¯ s C J ¯ s − ˙ ̟ (cid:11) = ddt hC J ̟ i , (37)because ˜ J commutes with all S generators (cf. (24).)Let us discuss the deformed Lagrangian in more details. Expressed in components of momentum p a itreads L = x ˙ p + x i ˙ p i − κ − x i p i ˙ p + λ ( p − m ) , (38)where, as before, we introduce the Lagrange multiplier λ to enforce the mass shell constraint. The equationsof motion following from variations over x are momentum conservations ˙ p a = 0, while the ones resultingfrom the variation over momenta give ˙ x = 2 λ p = 2 λ m , ˙ x i = 0 , (39)so that indeed the Carroll particle is always at rest. Furthermore, from (33) we may also find the explicitexpressions for the components x = n − ( e ζ − ¯ σ ¯ σ i − ζ i ) n i , x i = e ζ − ¯ σ n i . Then (39) will give someconditions for coordinates of n , h = e ζ i S i e ζ S and ¯ s .The presence of the nonlinear, deformed term in the Lagrangian (38) results in the nontrivial Poissonbracket algebra of the κ -deformed phase space [22] (cid:8) x i , p j (cid:9) = δ ij , (cid:8) x , p (cid:9) = 1 , (cid:8) x , p i (cid:9) = − κ p i , (cid:8) x , x i (cid:9) = 1 κ x i , (40)Meanwhile, symmetries of the action obtained from the Lagrangian (38) form the algebra of infinitesimaldeformed Carroll transformations, containing • rotations δx i = ρ ǫ ij x j , δp i = ρ ǫ ij p j , δx = δp = 0 ; (41) • deformed boosts δx = (1 + κ − p ) λ i x i , δp i = − λ i p , δx i = δp = 0 ; (42) • deformed translations δx = a , δx i = e p /κ a i , δp a = 0 ; (43) • the spatial conformal transformation δx i = η x i , δp i = − η p i , δx = δp = 0 , (44) where ρ , λ i , a a , η are parameters of the respective transformations.Actually, as noted in [14] the undeformed Carroll particle has an infinite dimensional symmetry. Thisproperty holds in the deformed case as well, and the generator of the infinitesimal transformations δφ a = { φ a , G } , where φ is an arbitrary function on phase space, is given by G = f ( p /κ ) p ξ ( x i ) + p i ξ i ( x i ) , f (0) = 1 , (45)where f ( p /κ ) is an arbitrary function of energy, while ξ i ( x i ), ξ ( x i ) are arbitrary functions of position.Let us complete this paper with some comments.First it should be stressed that the particle model we derived (32) can be extended to any number ofspacetime dimensions D + 1, simply by replacing the group AN (2) with AN ( D ). This makes it potentiallymuch more relevant for real physical systems than the model based on Poincar´e group (17), whose applicationis strictly restricted to 2+1 dimensions.In our view, the most significant result of this paper is the derivation of a particle model with κ -deformedphase space (40) from the first principles as a deformation of the free particle model resulting from theinteraction of the particle with its own gravitational field. However, it turns out that the model we obtainedis not the κ -Poincar´e particle discussed in the context of Doubly Special Relativity [10] or Relative Locality[23], [21], but the deformed Carroll particle, with completely frozen dynamics. It is therefore still an openproblem if the κ -Poincar´e particle can be derived from the particle-gravity system as an effective deformedparticle theory. We will revisit this issue in the forthcoming paper.There is a curious similarity between Carrollian relativity, being the relativistic theory obtained in thelimit c →
0, in which no local interactions are possible and the very similar feature of the 2+1 gravity andthe topological limit of gravity in 3+1 dimensions [24], [25], [26]. Especially in the 3+1 case it would beof interest to find out if gravity is described by a topological field theory in the Carrollian limit (for somediscussion of this issue see [27].)As already mentioned, the Carrollian limit appears in many distinct areas of theoretical physics, thecommon feature of whose is the presence of gravity in one form or another. The free and interacting particlemodels are extremely useful in that they help to grasp the underlying physics. It seems that the deformedmodel presented here, which already takes into account self-gravitational interactions might be of greatinterest, especially in the context of cosmological investigations [19].
ACKNOWLEDGMENT
This work was supported by funds provided by the National Science Center under the agreement DEC-2011/02/A/ST2/00294. TT acknowledges the support by the Foundation for Polish Science InternationalPhD Projects Programme co-financed by the EU European Regional Development Fund and the additional funds by the European Human Capital Program.[1] A. Staruszkiewicz, “Gravitation Theory in Three-Dimensional Space,” Acta Phys. Polon. , 735(1963).[2] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional Einstein Gravity: Dynamics of Flat Space,”Annals Phys. , 220 (1984).[3] S. Deser and R. Jackiw, “Three-Dimensional Cosmological Gravity: Dynamics of Constant Curvature,”Annals Phys. , 405 (1984).[4] E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B , 46 (1988).[5] A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De SitterSupergravity Theories,” Phys. Lett. B , 89 (1986).[6] H.-J. Matschull and M. Welling, “Quantum mechanics of a point particle in (2+1)-dimensional gravity,”Class. Quant. Grav. , 2981 (1998) [gr-qc/9708054].[7] C. Meusburger and B. J. Schroers, “Poisson structure and symmetry in the Chern-Simons formulationof (2+1)-dimensional gravity,” Class. Quant. Grav. , 2193 (2003) [arXiv:gr-qc/0301108].[8] L. Freidel and E. R. Livine, “Ponzano-Regge model revisited III: Feynman diagrams and effective fieldtheory,” Class. Quant. Grav. , 2021 (2006) [hep-th/0502106].[9] L. Freidel and E. R. Livine, “Effective 3-D quantum gravity and non-commutative quantum fieldtheory,” Phys. Rev. Lett. , 221301 (2006) [hep-th/0512113].[10] J. Kowalski-Glikman and S. Nowak, “Doubly special relativity and de Sitter space,” Class. Quant.Grav. , 4799 (2003) [hep-th/0304101].[11] M. Arzano and J. Kowalski-Glikman, “Kinematics of a relativistic particle with de Sitter momentumspace,” Class. Quant. Grav. , 105009 (2011) [arXiv:1008.2962 [hep-th]].[12] J. Lukierski, H. Ruegg and W. J. Zakrzewski, “Classical quantum mechanics of free kappa relativisticsystems,” Annals Phys. , 90 (1995) [hep-th/9312153].[13] S. Majid and H. Ruegg, “Bicrossproduct structure of kappa Poincare group and noncommutativegeometry,” Phys. Lett. B , 348 (1994) [hep-th/9405107].[14] E. Bergshoeff, J. Gomis and G. Longhi, “Dynamics of Carroll Particles,” arXiv:1405.2264 [hep-th].[15] Lewis Carroll, Through the Looking Glass and what Alice Found There,
London: MacMillan (1871).[16] H. Bacry and J. Levy-Leblond, “Possible kinematics,” J. Math. Phys. , 1605 (1968).[17] C. -G. Huang, Y. Tian, X. -N. Wu, Z. Xu and B. Zhou, “Geometries for Possible Kinematics,” Sci.China G (2012) 1978 [arXiv:1007.3618 [math-ph]].[18] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei:two dual non-Einsteinian concepts of time,” Class. Quant. Grav. , 085016 (2014) [arXiv:1402.0657[gr-qc]]. [19] J. Mielczarek, “Asymptotic silence in loop quantum cosmology,” AIP Conf. Proc. , 81 (2012)[arXiv:1212.3527].[20] C. Meusburger and B. J. Schroers, “The Quantization of Poisson structures arising in Chern-Simonstheory with gauge group G x g*,” Adv. Theor. Math. Phys. , 1003 (2004) [hep-th/0310218].[21] J. Kowalski-Glikman, “Living in Curved Momentum Space,” Int. J. Mod. Phys. A , 1330014 (2013)[arXiv:1303.0195 [hep-th]].[22] G. Amelino-Camelia, J. Lukierski and A. Nowicki, “kappa-deformed Covariant Phase Space andQuantum-Gravity Uncertainty Relations,” Phys. Atom. Nucl. , 1811 (1998) [Yad. Fiz. (1998)1925] [hep-th/9706031].[23] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, “The principle of relative locality,”Phys. Rev. D , 084010 (2011) [arXiv:1101.0931 [hep-th]].[24] L. Freidel and A. Starodubtsev, “Quantum gravity in terms of topological observables,” hep-th/0501191.[25] L. Freidel, J. Kowalski-Glikman and A. Starodubtsev, “Particles as Wilson lines of gravitational field,”Phys. Rev. D , 084002 (2006) [gr-qc/0607014].[26] J. Kowalski-Glikman and A. Starodubtsev, “Effective particle kinematics from Quantum Gravity,”Phys. Rev. D , 084039 (2008) [arXiv:0808.2613 [gr-qc]].[27] G. Dautcourt, “On the ultrarelativistic limit of general relativity,” Acta Phys. Polon. B29