5-Dimensional SO(1, 4)-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal
55-Dimensional SO ( , ) -Invariant Action as anOrigin to the Magueijo-Smolin Doubly SpecialRelativy Proposal B. F. Rizzuti ∗ and G. F. Vasconcelos Jr. † Departamento de F´ısica, Universidade Federal de Juiz de Fora, MG, 36.036-900,Brazil
Abstract
In this paper we discuss how the Magueijo-Smolin Doubly Special Rel-ativity proposal may obtained from a singular Lagrangian action. The de-formed energy-momentum dispersion relation rises as a particular gauge,whose covariance imposes the non-linear Lorentz group action. Moreover,the additional invariant scale is present from the beginning as a couplingconstant to a gauge auxiliary variable. The geometrical meaning of thegauge fixing procedure and its connection to the free relativistic particleare also described.
The idea to introduce another invariant scale into relativistic regimes is not new.In [1], the author presents a Lorentz invariant quantized spacetime. It was anattempt to avoid divergences in field theories. More recently, quantum grav-ity (QG) calculations seem to reinforce the existence of a fundamental scale,implying discreteness of areas and volumes [2]. In particular, in [3] differentapproaches to quantum gravity indicate a lower bound to distance measure-ments. This convergence would confirm that at small scale distances, the veryspacetime structure would be discrete. This is not the only issue to be tackledthroughout the road to a consistent QG theory though. When gravity is takeninto account spacetime becomes alive as a dynamical quantity. Can we try toinsert this allegedly new scale in a partial regime where gravity/curvature ef-fects can be neglected? The answer gives rise to the so-called Doubly SpecialRelativity (DSR) models [4, 5]. They have first appeared a couple of years afterthe enlightening afore mentioned paper [3]. With different motivations, such as ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] F e b osmological [6], physical [7] or even mathematical [8], the central concept ofthese trials was the same: to insert another invariant scale, besides the speed oflight, while keeping an intermediate regime with fixed spacetime as background.A list of facts and myths about the DSR proposals may be found in [9].One of the most celebrated DSR models was proposed by J. Magueijo andL. Smolin [5], which we call MS for short. One of its issues was the lack ofa fully covariant spacetime description, since it was set from the beginning onthe energy-momentum space. Actually, the construction of DSR Lagrangianmodels starting from a four-dimensional spacetime is a rather delicate issue,see, for example [10]. MS have specified a modified dispersion relation, p µ p µ = m c ( + ξp ) , (1)accompanied by the non-linear representation of the Lorentz group p ′ µ = Λ µν p ν + ξ ( p − Λ ν p ν ) , (2)which keeps (1) unaltered. The variable ξ stands for the invariant scale, relatedto the Planck length. In fact, p µ = (− ξ , , , ) remains intact under (2). Apossible experimental confirmation to such Lorentz violation scenario wouldgive a possible fingerprint of QG effects, as explained previously.This work is intended to show how the MS kinematic rules derived from(1) and (2) can be deduced from a singular particle model. This path hasalready been taken in a previous work [11]. For the latter, the invariant scale ξ enters into the game by an ad hoc gauge fixation. This rather non-trivialchoice is made only to produce the expected dispersion relation (1). In thiswork a slightly different Lagrangian from the one in [11] will be presented. Themodel lives in a flat 5-dimensional spacetime whereas the corresponding phasespace is constrained to a constant curvature hypersurface. In the current case,the parameter ξ is present since the beginning. Moreover, we present a clearmeaning for it. In effect, it is nothing but the coupling constant that connectsa gauge auxiliary variable to the dynamical sector of the action.The paper is divided as follows. In Section 2 we present a singular Lagrangianaction with global SO ( , ) invariance as well as a local symmetry. This model isbuilt on a five-dimensional position space and, as a consequence of its singularity,is endowed with a set of first class constraints, intrinsically connected to thegauge symmetries [12]. In Section 3 we apply Dirac’s hamiltonization procedurefor singular theories [13, 14] and obtain the Hamiltonian equations of motion,together with the explicit first class constraints of the system. Section 4 isdesigned to shed some light on the physical sector of the model. In Section 5we discuss how the free relativistic particle and the Magueijo-Smolin proposalcan be related by specific gauge choices. Moreover, in Section 6 we proposegeometrical interpretations of the previous results and Section 7 is left for theconclusions. 2 The 5-dimensional Lagrangian Model
It is a standard procedure to construct mechanical or field models with morevariables than the physical ones present on them. This way, a fully global linearLorentz group covariance can be guaranteed. Consequently, the price to bepaid is that the models carry constraints between degrees of freedom to assurethat not all among the variables are observables. We point out that the actionof the group of covariance acts in a slightly non-linear way upon the physicalsector of the models. Standard examples of such models are the free relativisticparticle [15] and the electrodynamics Lagrangian action [12].For the MS DSR, as exposed through (2), one starts with a non-linear real-ization of the Lorentz group on the four dimensional space of energy-momentum,parametrized by { p µ } . Thus, our suggestion here is to start with a five dimen-sional singular Lagrangian model with a set of first class constraints. As it willbecome clear in a while, this Lagrangian has global SO ( , )− invariance. Thenon-linear SO ( , )− action will be achieved by slicing the phase space throughthe gauge fixing, for the (first class) constraints. Throughout the paper, unlessstated otherwise, dots over quantities mean derivatives with respect to τ , thatis, ˙ γ ∶= dγdτ . The construction proceeds as follows.Consider the five dimensional configuration space para- metrized by { x A , g } ,where the indices A , B , C , ... take the values 0, 1, 2, 3, 5, and g is an auxiliaryvariable. Our particle model is described by the action S = ∫ dτ [ m η AB D x A D x B − ξg ] ∶= ∫ dτ L. (3)Here τ is an (arbitrary) evolution parameter and m shall be interpreted as therest mass of the particle. We also point out that D ∶= ddτ − g can be seen as ananalogue of a covariant derivative to gauge theories, with gauge field g . Thisfact will be confirmed soon. However, we firstly note that the factor ξ in secondterm in (3) is interpreted as a coupling constant that connects the gauge field g with the dynamical sector of the model. The configuration space is endowedwith the (pseudo)-metric η AB = (+ , − , − , − , − ) . Additionally, it is clear thatthe action (3) has global SO ( , ) -invariance, x A → x ′ A = Λ AB x B ; ∀ Λ ∈ SO ( , ) . (4)Besides the global symmetry, the action is also invariant under the local exacttransformations τ → τ ′ ( τ ) = Γ ( τ ) ; dτ ′ dτ = γ ( τ ) , (5) x A ( τ ) → x ′ A ( τ ′ ) = γ ( τ ) x A ( τ ) , (6) g ( τ ) → g ′ ( τ ′ ) = ˙ γ ( τ ) γ ( τ ) + g ( τ ) γ ( τ ) , (7)parametrized by the arbitrary function γ . In effect, we first note that the3ransformation of the derivative D x A under these is D x A → D ′ x ′ A = γ D x A . (8)In turn, this calculation and the simple result (8) justify our claim that D maybe identified as a covariant derivative, with g being the corresponding gaugefield. This is the very structure one may find in the case of electrodynamics andits interaction with the electron field, when the global U ( ) transformation isgraduated to a local one [16]. Now, if one plugs (5), (6), (7) and (8) back onthe action (3) then the result is S → S ′ = ∫ L ′ dτ ′ = ∫ dτ [ L + ddτ (− ξlnγ )] . (9)Since L and L ′ differ by a total derivative term, the transformations (5), (6)and (7) are local symmetries of the action S , as stated previously.The presence of gauge symmetries is linked to the appearance of first classconstraints through the Dirac recipe for singular systems [12]. This is the stan-dard procedure to investigate singular mechanical models and shall be done inthe next section. In this section we will construct the Hamiltonian version of the action (3). Thisis a singular model since the Hessian matrix, whose components are, ∂ L∂ ˙ Y a ∂ ˙ Y b ; Y a = { x A , g } , (10)has a null determinant and Dirac’s hamiltonization prescription can be used.Among the advantages of using the Hamiltonian over the Lagrangian formula-tion, the main one is to uncover constraints between degrees of freedom as partof the Hamiltonian equations.The first step in the prescription consists of defining the conjugate momenta: p a ∶= ∂L∂ ˙ Y a . With more details, p A = ∂L∂ ˙ x A = mη AB ( ˙ x B − gx B ) , (11) p g = ∂L∂ ˙ g = . (12)They are used as algebraic equations to obtain the velocities in terms of config-uration and momenta variables. From Eq. (11) we have˙ x A = η AB ( p B m + gx B ) , (13)4here η AB ( η CD ) may be used to raise (lower) indices. Eq. (12), in turn, is aprimary constraint. With these quantities in hand we are now able to write theHamiltonian of the system, H ( Y a , p a ) = ( p a ˙ Y a − L ) ∣ (11) , (12) + v g p g = m η AB p A p B + gη AB p A x B + ξg + v g p g . (14)It is defined on the extended phase space parametrized by { Y a , p a , v g } , where v g is the Lagrange multiplier for the constraint T ∶= p g =
0. The Poisson bracketsare defined canonically, { ., . } = ∂∂Y a ∂∂p a − ∂∂p a ∂∂Y a , (15)allowing us to write the equations of motion˙ x A = { x A , H } = m η AB p B + gx A ; (16)˙ p A = { p A , H } = − gp A ; (17)˙ g = { g, H } = v g ; (18) T = p g = . (19)Due to a consistency condition, it is expected that T remains equal tozero throughout the passage of time. Henceforth, one finds the equations of thefollowing stages of the Dirac procedure as algebraic consequences of this system.In effect, 0 = { T , H } ⇒ T ∶= p A x A + ξ = . (20)By applying the same line of reasoning one finds a third-stage constraint,0 = { T , H } ⇒ T ∶= η AB p A p B = . (21)The procedure stops at this step, since the time evolution of T does not bringany new information. Therefore, the complete set of constraints of this modelis { T , T , T } . Moreover, they are all first class constraints. Indeed, we have { T , T } = { T , T } =
0; (22) { T , T } = T . (23) We begin by pointing out that throughout this section and the rest of the paperthe Greek letters µ, ν, ... shall take values in { , , , } , whilst the letters i, j, k, ... shall take values in { , , } , unless stated otherwise. Our model is describedby a singular Lagrangian, subject to the local symmetries (5), (6) and (7). Itimplies that all our initial variables have ambiguous evolution and, thus, are5ot suitable candidates to be observables. The lack of a single solution forthe equations of motion is exposed through the Dirac procedure. In fact, theLagrange multiplier v g in (14) cannot be found as an algebraic consequence ofthe equations of motion. Thus, g enters into the game as an arbitrary functionof the evolution parameter τ . The same train of though applies to both x A and p B , since the corresponding equations of motion have an explicit dependenceon g . In order to skirt this issue we separate the fifth dimension, defining thevariables z µ = x µ x , (24) π µ = p µ p , (25)These quantities are not chosen randomly. We simply follow the same stepstaken on the standard analysis of the free relativistic particle model [15], or,by the same token, the semiclassical spinning particle model [17], in order toremove the so discussed arbitrariness our model possesses. First of all, we pointout that these quantities remain unaltered under the local symmetries, z µ → z ′ µ = x ′ µ x ′ = z µ , (26) π µ → π ′ µ = p ′ µ p ′ = π µ . (27)Therefore, they are suitable candidates for observables of our model. In partic-ular, we may promptly write the associated equations of motion˙ z µ = m λ ( π µ − z µ ) , (28)˙ π µ = , (29)where λ ∶= p x . Evidently, the equations of motion for the ( z, π ) -sector re-semble those of the free relativistic particle. The ambiguity due to the arbi-trary parameter λ is related to the reparametrization invariance of the theory.Henceforth, z µ ( τ ) can be interpreted as the parametric equations of the phys-ical variables z i ( z ) . The latter may be obtained by inverting the expression z = z ( τ ) ↔ τ = τ ( z ) and substituting it back in z i ( τ ( z )) ≡ z i ( z ) . If we set,as usual, z = ct and consider c as the speed of light, we get dz i dz = ˙ z i ˙ z = π i − z i π − z ⇒ dz i dt = c π i − z i π − z . (30)In terms of the π µ variables, the constraint η AB p A p B = η µν p µ p ν − ( p ) = ⇒ η µν π µ π ν = . (31)Finally, we may resolve π from (31), π = √ + δ ij π i π j (32)6here δ ij is the Kronecker delta. Hence, in the limit ∣ π i ∣ → +∞ , we obtain dz i dt = c π i − z i √ + δ ij π i π j − z → c ˆ π i , (33)where ˆ π i ∶= π i ∣ π i ∣ . This result reveals that the physical variables in our systemdescribe a free particle which, particularly, is moving in a straight line with aspeed bounded above by c .To conclude this section, we point out that our model bears the same numberof degrees of freedom of DSR particles. In fact, our extended phase spaceis parametrized by { x A , p B , g, p g } , counting a total of 12 degrees of freedom.Taking into account that each first class constraint rules out 2 spurious degreesof freedom (after gauge fixation), we are left with 12 − × = x i ( x ) , and the 3 remaining ones come from the deformeddispersion relation (1). Needles to say, the same arguments hold true for thefree relativistic particle. Our next task consists of slicing out the momentum sector of the phase space.This means that particular gauges will be fixed for our first class constraintsin the form of hyperplanes. In turn, they will reproduce both the ( i ) freerelativistic particle (FRP) and ( ii ) the Magueijo-Smolin DSR proposal. Therelation between these two models shall be discussed in the next section.Furthermore, as it is usual in gauge theories, the local symmetries are notpreserved after the gauges are fixed. Nonetheless, one can search for theircombinations that retain the gauge condition. Following this prescription, bothcases ( i ) and ( ii ) may be derived. To reproduce the FRP dynamics we fix the following gauges for the constraints T = p g and T = p A x A + ξ , forming second class pairs of constraints: g = { g, p g } = , (34) p = mc ; { p − mc, p A x A + ξ } = mc. (35)Moreover, we impose invariance (both global and local) for the gauge conditions.First, for the g − sector, we have g = ⇔ g ′ = ˙ γ ( τ ) γ ( τ ) + g ( τ ) γ ( τ ) = ⇒ γ = const. (36)Under local and global transformations, the momenta p A = m D x A transform as p A → p ′ A = γ Λ AB p B , Λ AB ∈ SO ( , ) . (37)7e restrict ourselves to the subgroup of SO ( , ) whose elements are Λ µν ∈ SO ( , ) , such that Λ AB = ⎛⎜⎜⎜⎜⎜⎝ µν ⎞⎟⎟⎟⎟⎟⎠ (38)since for p = mc the remaining sector of Λ AB induces boosts in the fifth dimen-sion, which can be seen as translations in four dimensions, that is, p ′ µ = Λ µA p A = Λ µν p ν + P µ . (39)Here P µ = Λ µ mc = const. . We follow the same steps for the p A -sector, p = mc ⇔ p ′ = mc = γ Λ A p A = γ mc ⇒ γ = , (40)which fixes the function γ .Finally, we may assemble the results that have been obtained previously.The dynamics reads ˙ x = c, ˙ x µ = m p µ and ˙ p µ = p µ coordinates are restricted to the mass-shell relation η µν p µ p ν = m c , (42)which was obtained by substituting the gauge (35) back into the constraint η AB p A p B =
0. With γ =
1, the momenta p µ transform as p ′ µ = Λ µν p ν . (43)The equations (41), (42) and (43) allow us to interpret the gauge fixed versionas a free relativistic particle, with mass m and momenta p µ . This gauge isequivalent to the physical sector, described in Section 4. In this case, the fifthdimension in the configuration space is just the arbitrary evolution parameter, x = cτ + const. . We follow the same steps we have made so far. Once again, we take g = T = p g . However, instead of the hyperplane given by (35), wework with a slightly rotated version of it and fix the MS DSR gauge, p = mc ( + ξp ) , (44)for the constraint T = p A x A + ξ , forming a second class pair { p − mcξp − mc, p A x A + ξ } = mc ≠ . (45)8nvariance of g implies, as before, the fixation of the local parameter γ = const. . It can be fully determined by imposing the invariance of the MS DSRgauge, p = mc ( + ξp ) ⇔ p ′ = mc ( + ξp ′ ) ⇒ γ = + ξ ( p − Λ µ p µ ) . (46)Thus, we are left with a free particle bounded to the MS DSR kinematicalpredictions, ˙ x = c ( + ξp ) , ˙ x µ = m p µ and ˙ p µ = , (47)where the momenta is restricted to the deformed dispersion relation η µν p µ p ν = m c ( + ξp ) (48)and transforms accordingly, p ′ µ = Λ µν p ν + ξ ( p − Λ µ p µ ) . (49)As claimed, equations (47), (48) and (49) follow from the non-standard gauge(44) and represents the MS DSR proposal. Once more, the x coordinate isproportional to the arbitrary parameter τ . We highlight that eq. (47) doesnot imply that x evolves with a speed faster than the speed of light. τ canbe reparametrized and has no physical interpretation. A change of scale couldsuppress ˜ c ∶= c ( + ξp ) : τ → τ ′ = c ˜ c τ ⇒ x = cτ + const. . It is usual to set DSR models initially on the space of conserved energy - momen-tum. With this perspective in mind let us analyze the p A -sector of our proposal.To begin with, it is governed by the constraint T = η AB p A p B =
0. If we mergethe p i coordinates among the indices A, B , then we can sketch its geometricalrepresentation, which is a 5-dimensional cone. The FRP gauge is nothing butthe plane p = mc , which, when intersecting the previous structure, results inthe standard hyperboloid on the momenta coordinates { p µ } , described by theusual mass-shell condition. This intersection ({ T = } ∩ { p = mc }) is shown inFig. 1. On the other hand, the MS DSR gauge corresponds to the hyperplanecharacterized by p = mc ( + ξp ) , which is also exhibited in the same figure.The angle between the planes that define both gauges is given by θ ∶= arctan ( mcξ ) . (50)It is clear from the Fig. 1 that the rotation of one of the gauges around the p i -sector by this angle generates the other.Let us now simulate a scenario where DSR effects could be feasible. Weconsider, for instance, a proton close to the GZK threshold [18, 19], p GZK = × eVc . For this estimate, we assume ξ = × p GZK , as ξp should be muchlesser than 1. Therefore, in this case one finds θ ≈ × − . (51)Since θ <<
1, our calculation indicates that it would be difficult to detect a DSReffect when comparing it to measurements obtained according to the standardrelativistic kinematics predictions.
In this work we proposed a singular Lagrangian model (3) on a five-dimensionalspacetime which, after fixing specific gauges, reproduces both the free relativisticparticle and the Magueijo-Smolin doubly special relativistic kinematics. TheLagrangian was chosen so that it would be globally invariant under the SO ( , ) group of symmetries. In turn, the local symmetries of the configuration spacevariables were presented. Due to the singularity of the system, its Hamiltonianformulation was constructed according to Dirac’s hamiltonization procedure.This allowed us to explicitly uncover the full set of first class constraints betweendegrees of freedom from the Hamiltonian equations of motion.10ince the model presents first class constraints, it was discussed that not allof its variables could be observables of the system. In effect, we discussed thatthe initial set of variables had ambiguous evolution and were not in the physicalsector of the model. However, separation of the fifth spacial dimension fromthe rest of the variables allowed us to write suitable candidates for observablesof the model. Besides that, it was discussed that the equations of motion ofthese quantities resembled those of the free relativistic particle on flat four-dimensional spacetime. Moreover, it was shown that such particle obeyed thestandard relativistic dispersion relation and its speed was bounded above bythe speed of light, while bearing the same number of degrees of freedom of DSRparticles.Finally, after fixing particular gauges for the first class constraints both theFRP and the MS DSR dynamics could be reproduced from this model. In turn,the MS gauge implied the model reduced to a free relativistic on the deformedfour-dimensional space. Moreover, this approach of slicing out the momentumsector of the phase space with planes led us to a geometrical interpretation ofthe relation of these two systems: the scale ξ defines the angle between the twoplanes, according to (50).Although new perspectives indicate that the MS is just a particular typeof a broader class of DSR proposals [20], our results show that the MS DSRturns out to be one particular gauge of a FRP, living on a hypersurface ofconstant curvature. These last two observations may suggest that the rise of newobservable physical effects within this context would be demanding to detect. Acknowledgement
This work is supported by XXIX PIBIC/CNPq/UFJF - 2020/2021, projectnumber ID-47862.
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