Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin Doubly Special Relativity
aa r X i v : . [ h e p - t h ] A ug Five-Dimensional Mechanics as the Starting Point for the Magueijo-Smolin DoublySpecial Relativity
B. F. Rizzuti ∗ and A. A. Deriglazov Depto. de F´ısica, ICE, Universidade Federal de Juiz de Fora, MG, Brazil andDepto. de Matem´atica, ICE, Universidade Federal de Juiz de Fora, MG, Brazil
We discuss a way to obtain the doubly special relativity kinematical rules (the deformed energy-momentum relation and the nonlinear Lorentz transformations of momenta) starting from a singularLagrangian action of a particle with linearly realized SO (1 ,
4) symmetry group. The deformedenergy-momentum relation appears in a special gauge of the model. The nonlinear transformationsof momenta arise from the requirement of covariance of the chosen gauge.
PACS numbers: 03.30.+p; 11.30.Cp
Keywords:
Nonlinear Lorentz transformations, Dou-bly special relativity models
I. INTRODUCTION
Various doubly special relativity (DSR) proposals havereceived a great amount of attention in the last years[1–18]. They have been formulated on the base of nonli-near realizations of the Lorentz group in four-dimensionalspace of the particle momentum [2]. It can be achievedintroducing, in addition to the speed of light, one moreobserver independent scale , ζ , the latter is associatedto the Planck scale (for a recent review, see [17]). Inturn, the nonlinear realization implies deformed energy-momentum dispersion relation of the form η µν p µ p ν = − m c + f ( ζ, p ) . (1)It is supposed that in the limit ζ → p µ p µ = − m c .The attractive motivations for such kind of modifi-cation have been discussed in the literature. Thereis evidence on discreteness of space-time from non-perturbative quantum gravity calculations [20]. Themodified energy-momentum relation implies correctionsto the GZK cut-off [21], so DSR may be relevant forstudying the threshold anomalies in ultra-high-energycosmic rays [3, 22]. Astrophysical data of gamma-raybursts can be used for bounding possible corrections to p µ p µ = − m c , see [14]. In the recent work [15] it wassuggested that experiments with cold-atom-recoil maydetect corrections to the energy-momentum relations,and f = 0 in (1) should be interpreted as a quantumgravity effect. ∗ Electronic address: [email protected]; On leave of ab-sence from Instituto de Sa´ude e Biotecnologia, ISB, UniversidadeFederal do Amazonas, AM, Brazil. The idea of another invariant scale in space-time, besides c , isvery old. Snyder has constructed a Lorentz invariant space-timethat admits an invariant length, in one of the first attempts toavoid divergence problems [19]. In this work we discuss the initial Magueijo-Smolin(MS) proposal [2], which states that all inertial observersshould agree to take the deformed dispersion relation forthe conserved four-momentum of a particle p = − m c (1 + ζp ) . (2)This is invariant under the following nonlinear transfor-mations: p ′ µ = Λ µ ν p ν ζ ( p − Λ ν p ν ) . (3)However, the list of kinematical rules of the model is notcomplete, which raised a lively and controversial debateon the status of DSR [17]. One of the problems consistsof the proper definition of total momentum for many par-ticle system . Due to non-linear form of the transfor-mations, ordinary sum of momenta does not transformas the constituents. Different covariant composition ru-les proposed in the literature lead to some astonishingeffects, like the ”soccer ball problem”and the ”rainbowgeometry”[8, 16].To understand these controversial properties, it wouldbe desirable to have in our disposal the relativistic par-ticle model formulated in the position space, which leadsto DSR relations (2), (3) in the momentum space. Des-pite a lot of efforts [6, 10–13], there appears to be nowholly satisfactory solution of the problem to date. It isthe aim of the present work to construct the model thatcould be used as a laboratory for simulations of the DSRkinematics.Nonlinear realizations of the Lorentz group on thespace of physical dynamical variables often arise afterfixation of a gauge in a theory with the linearly rea-lized Lorentz group on the initial configuration space.Adopting this point of view, we study in Section 2 a sin-gular Lagrangian on five-dimensional position space x A , In [9] we observed that MS-type kinematics can be related withlinear realization of Lorentz group in five-dimensional positionspace. On this base, an example of DSR model free of the pro-blem of total momentum has been constructed. A = ( µ, µ = 0 , , ,
3, with linearly realized SO (1 , SO (1 , p , xp , x . We reject x as itwould lead to a curved space-time . So, we look for themodel with the constraints p = 0, xp = 0. They cor-respond to a particle with unfixed four-momentum, andwithout five-dimensional translation invariance. In Sec-tion 3 we show that the MS deformed energy-momentumrelation arises by fixing an appropriate gauge (for one ofthe constraints), and the nonlinear transformation law ofmomenta is dictated by covariance of the gauge. Section4 is left for conclusions. II. SO (1 , -INVARIANT MECHANICS The motion of a particle in the special relativity theorycan be described starting from the three-dimensional ac-tion − mc R dt q − ( dx i dx ) . It implies the Hamiltonianequations dx i dx = p i p ~p + m c , dp i dx = 0 . (4)The problem here is that the Lorentz transformations, x ′ µ = Λ µν x ν , act on the physical dynamical variables x i ( x ) in a higher nonlinear way. To improve this,we pass from the three-dimensional to four-dimensionalformulation introducing the parametric representation x i ( τ ), x ( τ ) of the particle trajectory x i ( x ). Usingthe relation dx i dx = ˙ x i ( τ )˙ x ( τ ) , the action acquires the form − mc R dτ p − η µν ˙ x µ ˙ x ν . It is invariant under the localtransformations which are arbitrary reparametrizationsof the trajectory, τ → τ ′ ( τ ). In turn, in the Hamil-tonian formulation the reparametrization invariance im-plies the Dirac constraint which is precisely the energy-momentum relation ( p µ ) = − m c . Presence of theconstraint becomes evident if we introduce an auxiliaryvariable e ( τ ) and rewrite the action in the equivalentform, S = R dτ ( e ( ˙ x µ ) − e m c ). Then equation ofmotion for e implies the Lagrangian counterpart of theenergy-momentum relation, δSδe ∼ ˙ x + e m c = 0. Be-sides the constraint, the action implies the equations ofmotion ˙ x µ = ep µ , ˙ p µ = 0. The auxiliary variable e isnot determined by these equations and enter into solu-tion for x µ ( τ ) as an arbitrary function. The ambiguityreflects the freedom which we have in the choice of para-metrization of the particle trajectory. By construction,the ambiguity is removed excluding the parameter τ fromthe final answers. Equivalently, we can impose a gauge There are proposals considering the de Sitter as the underlyingspace for DSR theories [4–6, 18]. to rule out the ambiguity as well as the extra variables.The most convenient gauge is e = 1, x = p τ , as it leadsdirectly to the equations (4) for the physical variables.In resume, to avoid a nonlinear realization of the Lo-rentz group in special relativity, we elevate the dimen-sion of space from 3 to 4. In the DSR case, the Lorentztransformations are non linear in the four-dimensionalspace. So, by analogy with the previous case, we startfrom a theory with the linearly realized group in five-dimensional space. Consider the action S = Z dτ m η AB Dx A Dx B , (5)where η AB = ( − , +1 , +1 , +1 , +1), Dx A stands for the”covariant derivative”, Dx A ≡ ˙ x A − gx A , and g ( τ ) is anauxiliary variable. The action is invariant under SO (1 , x A → x ′ A = Λ AB x B . (6)There is also the local symmetry with the parameter γ ( τ ), τ → τ ′ ( τ ); dτ ′ dτ = γ ( τ ) ,x A ( τ ) → x ′ A ( τ ′ ) = γ ( τ ) x A ( τ ) ,g ( τ ) → g ′ ( τ ′ ) = ˙ γ ( τ ) γ ( τ ) + g ( τ ) γ ( τ ) . (7)The transformation law for g implies a simple transfor-mation law of the covariant derivative, Dx A → γ Dx A .Hence g play the role of the gauge field for the symmetry.The presence of local symmetry indicates that the mo-del presents constraints in the Hamiltonian formulation.So we apply the Dirac method [23] to analyze the ac-tion (5). Introducing the conjugate momenta, we findthe expressions p A = ∂L∂ ˙ x A m ( ˙ x A − gx A ) , p g = ∂L∂ ˙ g = 0 . (8)Hence there is the primary constraint, p g = 0. The cano-nical Hamiltonian H and the complete Hamiltonian H are given by the expressions H = 12 m p A + gp A x A , H = H + λp g , (9)where λ is the Lagrange multiplier for the primary cons-traint. The Poisson brackets are defined in the standardway, and equations of motion follow directly˙ x A = p A m + gx A , ˙ p A = − gp A , ˙ g = λ. (10)From preservation in time of the primary constraint, ˙ p g =0, we find the secondary constraint p A x A = 0 . (11)In turn, it implies the tertiary constraint p A = 0 . (12)The Dirac procedure stops on this stage, all the cons-traints obtained belong to the first class.Since we deal with a constrained theory, our first taskis to specify the physical-sector variables [26]. The initialphase space is parameterized by 12 variables x A , p B , g , p g . Taking into account that each first-class constraintrules out two variables, the number of phase-space phy-sical variables is 12 − × λ , which enters as an arbitrary func-tion into solutions to the equations of motion. Accordingto the general theory [23–25], variables with ambiguousdynamics do not represent the observable quantities. Forour case, all the initial variables turn out to be ambi-guous.To construct the unambiguous variables, we note thatthe quantities π µ = p µ p , y µ = x µ x , obey ˙ π µ = 0,˙ y µ = em ( π µ − y µ ), where e ≡ p x . Since these equationsresemble those for a spinless relativistic particle, the re-maining ambiguity due to e has the well-known interpre-tation, being related with reparametrization invarianceof the theory. In accordance with this, we can assumethat y µ ( τ ) represent the parametric equations of the tra-jectory y i ( t ). The reparametrization-invariant variable y i ( t ) has deterministic evolution dy i dt = c π i − y i π − y .We can also look for the gauge-invariant combinationson the phase-space. The well known remarkable pro-perty of the Hamiltonian formalism is that there are thephase-space coordinates for which the Hamiltonian va-nishes [25]. In these coordinates trajectories look like thestraight lines. For the case, the unambiguous variableswith this property are π µ , ˜ x µ ≡ y µ − π µ . III. DSR GAUGE
In this Section we reproduce the MS DSR kinematicsstarting from the SO (1 ,
4) model. First, we obtain theMS dispersion relation (2) imposing a particular gaugein our model. Generally, neither the global nor the localsymmetries survive separately in the gauge fixed version.But we can look for their combination that does not spoilthe gauge condition. Following this line, we arrive at theMS transformation law of the momenta (3).According the Dirac algorithm, each first class cons-traint must be accompanied by some gauge condition ofthe form h ( x, p ) = 0, where the function h must be cho-sen such that the system formed by constraints and gau-ges is second class. The constraints and the gauges thencan be used to represent part of the phase space variablesthrough other. Equations of motion for the remaining va-riables are obtained by substituting the constraints andgauges into the equations already found. Let us choose the gauge g = 0 for the constraint p g = 0.This gauge fixes the local symmetry, as it should be, g ′ = ˙ γγ + gγ (cid:12)(cid:12)(cid:12) g =0 ⇒ ˙ γ = 0 . (13)We are, then, left with two constraints. To obtaina deformed dispersion relation, we impose the gauge p = mch ( ζ, p ) for the constraint p A x A = 0. Usingthis expression in the constraint (12), we obtain p µ p µ = − m c h ( ζ, p ) . (14)We wrote the function h depending on the arguments ζ and p but one is free to chose the particular dispersionrelation he wants. We point out that the scale ζ is gauge-noninvariant notion in this model .We now turn to the induced nonlinear Lorentz trans-formation of momenta. Under the symmetries (6), (7),the conjugated momentum p A = mDx A transforms as p A → p ′ A = 1 γ Λ AB p B . (15)For SO (1 ,
3) -subgroup Λ AB = (cid:18) Λ µν
00 1 (cid:19) , (16)we have p µ → p ′ µ = 1 γ Λ µν p ν , p → p ′ = 1 γ Λ A p A = 1 γ p . (17)Now, as it often happens in gauge theories, global symme-try of the gauge-fixed formulation is a combination of theinitial global symmetry and local symmetry with speci-ally chosen parameter γ . Since the gauge p = mch ( ζ, p )is not preserved by the transformations (6) and (7) sepa-rately, one is forced to search for their combination, (17),which preserves the gauge. Imposing the covariance ofthe gauge p = mch ( ζ, p ) ⇔ p ′ = mch ( ζ, p ′ ) , (18)we obtain the equation for determining γh ( ζ, p ) = γh ( ζ, γ Λ µ p µ ) . (19)In the gauge g = 0, we have p A = const. on-shell, soEq. (19) is consistent with (13). Eqs. (17) with this It is worth noting that a gauge-fixed formulation, considered ir-respectively to the initial one, generally has the physical sectordifferent from those of the initial theory [24]. We discuss only the induced Lorentz transformations. The re-maining transformations are boosts in the fifth dimension. Inthe gauge-fixed formulation they produce the nonlinear transfor-mations which play the role of four-dimensional translations. γ provides a non linear realization of the Lorentz groupwhich leaves invariant the deformed energy-momentumrelation (14).Let us specify all this for MS model. If we fix the gauge p = mc (1+ ζp ), the constraint p A = 0 acquires the formof MS dispersion relation (2). Enforcing covariance of thegauge, the equation (19) for determining γ reads1 + ζp = γ (1 + ζ γ Λ µ p µ ) . (20)So, γ is given by γ = 1 + ζ ( p − Λ µ p µ ) . (21)Using this γ in Eq. (17), we see that the momenta p µ transform according to Eq. (3). IV. CONCLUDING REMARKS
We have constructed an example of the relativistic par-ticle model (5) on five-dimensional flat space-time withlinearly realized SO (1 ,
4) group of global symmetries andwithout the five-dimensional translation invariance. Dueto the local symmetry presented in the action, the num-ber of physical degrees of freedom of the model is thesame as for the particle of special relativity theory. Wehave applied the model to simulate kinematics of theMagueijo-Smolin doubly-special-relativity proposal. Itwas done by an appropriate fixation of a gauge for theconstraint (11), that leads to the MS deformed disper-sion relation (2). The nonlinear transformation law ofmomenta (3) was found from the requirement of covari-ance of the gauge-fixed version. We finish with the comment on a transformation lawfor the spatial coordinates. Using the parameter γ obtai-ned in Eq. (21), the transformation of the configuration-space coordinates can be found from (6) and (7) x µ → x ′ µ = [1 + ζ ( p − Λ µ p µ )]Λ µν x ν , (22) x → x ′ = [1 + ζ ( p − Λ µ p µ )] x . (23)The component x is affected only by a scale factor. Thecoordinates x µ transform as usually happens in DSRtheories: we have a transformation law that is energy-momentum dependent. These transformations were ob-tained in the work [7] from the requirement that the freefield defined on DSR space (2) should have plane-wavesolutions of the form φ ∼ Ae − ip µ x µ , then the contraction p µ x µ must remain linear in any frame. We point out thatit turns out to be true in our model η µν p ′ µ x ′ ν = η µν ( 1 γ Λ µα p α )( γ Λ νβ x β ) = η αβ p α x β . (24)Eq. (22) leads also to the energy-dependent metric of theposition space [8]. There are some attempts to interpret p in this case, see [7, 8]. Acknowledgments
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