Observers and their notion of spacetime beyond special relativity
AArticle
Observers and their notion of spacetime beyondspecial relativity
José Manuel Carmona * ID , José Luis Cortés ID and José Javier Relancio ID Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain; [email protected] (J.M.C);[email protected] (J.L.C.); [email protected] (J.J.R.) * Correspondence: [email protected]
Abstract:
It is plausible that quantum gravity effects may lead us to a description of Nature beyondthe framework of special relativity. In this case, either the relativity principle is broken or it ismaintained. These two scenarios (a violation or a deformation of special relativity) are very different,both conceptually and phenomenologically. We discuss some of their implications on the descriptionof events for different observers and the notion of spacetime.
Keywords:
Lorentz invariance violation; Doubly Special Relativity; Noncommutative spacetime;Photon time delay experiments; Relative locality
1. Introduction
Poincaré invariance is at the root of our modern theories of particle physics. It is indeed thesymmetry of the classical spacetime of special relativity (SR), which is one of their basic ingredients.These theories, however, are nowadays understood as low-energy limits of more fundamental, not yetknown, constructions, where these essential ingredients might no longer be valid (see, for example,the different contributions in Ref. [1]). In particular, effects coming from a quantum theory of gravity(such as the creation and evaporation of virtual black holes [2]) are expected to modify the classicalspacetime picture, and therefore, also its symmetries. In this case, Poincaré invariance would thenbe a symmetry of spacetime in the low-energy limit. This reasoning makes plausible an intermediateregime where gravity and quantum effects can be neglected (¯ h → G →
0) but there is still a footprintof these effects ( M P = √ ¯ hc / G (cid:54) =
0, where M P is the Planck mass) producing a departure of SR.Once one accepts that SR may be an approximate symmetry, one can ask about the possibility thatthe scale Λ controlling departures from SR be much lower than M P , Λ (cid:28) M P . The phenomenologicalconsistency of this possibility depends in fact on how one goes beyond the framework of specialrelativity, with two main options: either there is no a relativity principle, or there is one. In the first case,we speak of Lorentz invariance violation (LIV) [3], while in the second case, we speak of a deformedspecial relativity (or doubly special relativity, DSR [4]).Conceptually, LIV and DSR are very different. In LIV there is a special observer, that is, aprivileged system of reference; in DSR, however, there is a class of equivalent observers (inertialreference systems), which are related through Λ -deformed Lorentz transformations. In the case ofLIV, its dynamics is usually studied in the framework of local effective field theory (EFT), in whichmodified dispersion relations (MDR) for particles appear, thereby affecting the kinematics of SR;energy-momentum conservation laws are however unaltered in this context. This is however differentin the case of DSR, since non-linear Λ -deformed Lorentz transformations are incompatible with linearenergy-momentum conservation laws: the kinematics of DSR therefore contains MDR and MCL(modified composition laws); on the other hand, there is not a clear dynamical framework of DSR. a r X i v : . [ h e p - t h ] J un of 10 The conservation of the total momentum P of a system of particles is a reflection of an invarianceunder the transformations generated by P (translations). In the LIV case, and for the EFT framework, the total momentum is just the sum of the momenta of each of the particles, P = ∑ p I , and translationsare constant displacements, x I → x I + a ∀ I . As we argued in the previous subsection, the totalmomentum of a system of particles in DSR is a non-linear composition of momenta, P = ⊕ p I , andthen translations (the transformations generated by P ) are momentum-dependent displacements, x I → x I + f ( { p } ) .As a consequence of the above, LIV (in the EFT framework) has the property of absolute locality,as in SR: if two worldlines meet at a point, this will also happen for translated observer. In DSR,however, translations are not constant displacements, but they are momentum-dependent. Therefore,a local interaction for one observer is seen as non-local for a DSR-translated observer.Nontrivial translations in DSR are in fact needed to avoid inconsistencies with tests of locality [5,6].These inconsistencies derived from the hypothesis that crossing worldlines must also cross for aLorentz-transformed observer. However, this hypothesis can be proved only in the case of absolutelocality: in such a case, the crossing of two wordlines is an absolute fact for any observer, andin particular, for an observer whose origin coincides with the point of crossing. Since a Lorentztransformation does not affect the origin of coordinates and there exists absolute locality, the twolines will cross for any Lorentz-transformed observer. The failure of this hypothesis leading toinconsistencies with tests of locality, therefore, implies a loss of absolute locality.On the other hand, if we define an interaction as the crossing of worldlines when the interactionpoint coincides with the origin of the observer, this is a relativistic invariant definition. Therefore,in DSR, which is a relativistic theory, all observers can define a spacetime region where interactionscorrespond to crossing of worldlines with a very good approximation. Far away from this region,however, interactions are seen as highly non-local. This property of DSR theories has been named as relative locality [7]. Note that relative locality was first described ([7]) as a property arising in a modelof classical worldlines from a variational principle (we will review this construction in Sec. 2), but, aswe have shown here, the loss of absolute locality is a simple and generic consequence of a MCL.The property of SR that an interaction corresponds to a crossing of worldlines is therefore nolonger valid in DSR. This means that while, as we will see in Sec. 2, interactions may still be used todefine points in a spacetime, the structure of this spacetime in DSR will be more complicated than inSR or in LIV. The new high-energy effects (effects that are suppressed when Λ → ∞ ) require astrophysicalobservations sensitive enough to the modifications to kinematics and/or cumulative effects, such aslong distance propagations; the phenomenology, however, is radically different in the LIV and DSRscenarios.LIV affects strongly thresholds of reactions (with corrections of order E / m Λ ) and is ableto forbid/allow decays at high energy that are allowed/forbidden by SR, what can then produceobservable spectrum changes after a long propagation. In contrast, DSR cannot change the low-energy(allowed or forbidden) character of a decaying particle at high energy because of the relativity principle,and reaction thresholds are also much less changed because of cancellations between the modificationsin the dispersion relation and the composition laws (essentially, changes are described in terms of therelativistic invariant of the reaction, not in terms of one of the energies) [8]. Note that it is possible to consider LIV with a MCL, which goes beyond the EFT framework. In such a case, the MDR andthe MCL would not be related by the compatibility relationship that the relativity principle establishes in DSR theories. of 10
DSR phenomenology is therefore much more limited that LIV phenomenology, and only involvesamplification mechanisms in the propagation of stable particles: photons and neutrinos. Since vacuumbirrefringence is not a generic feature in DSR, this leaves time of flight as the only window to leadingorder modifications of SR which can be compatible with a relativity principle. The analysis of time offlight in the LIV and the DSR cases is however very different: in the LIV case, it only depends on theparticle dispersion relation; in the DSR case, it also depends on the spacetime structure.In this work, we will review a recently proposed [9] model for the nontrivial spacetime structurethat arises in the description of physical measurements made by translated observers in DSR theories,and in view of this result, we will update the time delay calculation presented in Ref. [10], giving anew perspective on the problem.
2. Spacetime through local interactions
Local interactions in field theory are associated to linear momentum composition laws, sincethe product of fields at a single point gives a δ ( ∑ p i ) in momentum space. Nonlinear momentumcomposition laws will be, therefore, associated with a change in the concept of locality, and lie beyondthe EFT framework. Indeed, we have argued that a MCL of momenta is associated with a loss of thenotion of absolute (observer-independent) locality of interactions. Although there is no at present aconsistent field theory treatment of DSR theories, the loss of absolute spacetime locality can be easilyincorporated in a classical model of worldlines through a variational principle: this is the relativelocality framework [7].Take N interacting particles, numbering from 1 to N the incoming worldlines, and from N + N the outgoing worldlines: S total = S infree + S outfree + S int S infree = N ∑ J = (cid:90) − ∞ d τ (cid:16) x µ J ˙ p J µ + N J (cid:16) C ( p J ) − m J (cid:17)(cid:17) S outfree = N ∑ J = N + (cid:90) ∞ d τ (cid:16) x µ J ˙ p J µ + N J (cid:16) C ( p J ) − m J (cid:17)(cid:17) S int = (cid:77) N + ≤ J ≤ N p J ν ( ) − (cid:77) ≤ J ≤ N p J ν ( ) ξ ν , (1)where the N J are Lagrange multipliers that ensure the satisfaction of the modified dispersion relation C ( p J ) = m J . The parameter τ along each worldline has been chosen such that τ = N worldlines are determined by the condition δ S total = δξ µ , δ x µ J , δ p J µ . From the variation δ p J µ ( ) one finds x µ J ( ) = ξ ν ∂ P ν ∂ p J µ ( ) ∀ J , (2)and the variation δξ µ leads to identify the total momentum P = (cid:76) ≤ J ≤ N p J ( ) = (cid:76) N + ≤ J ≤ N p J ( ) ,conserved in the interaction.From Eq. (2) we get absolute locality, x µ J ( ) = ξ µ , if and only if P is the linear addition ofmomenta. We can ask then the following question: can one define a new spacetime ˜ x from thecanonical spacetime ( x , p ) , { p ν , ˜ x µ } = δ µν , such that we recover absolute locality in this new spacetime? of 10 As a first attempt, let us introduce a nontrivial spacetime ( ˜ x ) through the definition˜ x µ = x ν ϕ µν ( p / Λ ) , (3)where ϕ µν is such that ϕ µν ( ) = δ µν . This new spacetime is ‘noncommutative’, in the sense { ˜ x µ , ˜ x ν } = − x σ ϕ µρ ∂ϕ νσ ∂ p ρ + x ρ ϕ νσ ∂ϕ µρ ∂ p σ , (4)and the Poisson brackets in the new phase space are now: { p ν , ˜ x µ } = ϕ µν .Now let us consider a process in which we have a particle in the initial state with momentum k ,and two particles in the final state with momenta p and q , such that k = p ⊕ q . Since the coordinates ofthe two outgoing worldlines at the interaction are y µ ( ) = ξ ν ∂ ( p ⊕ q ) ν ∂ p µ , z µ ( ) = ξ ν ∂ ( p ⊕ q ) ν ∂ q µ , (5)and the coordinate of the first particle is x µ ( ) = ξ µ , the condition to have a local interaction in thenoncommutative spacetime is ϕ µν ( p ⊕ q ) = ∂ ( p ⊕ q ) ν ∂ p ρ ϕ µρ ( p ) = ∂ ( p ⊕ q ) ν ∂ q ρ ϕ µρ ( q ) , (6)where we have imposed a common interaction point˜ x µ ( ) = ξ ν ϕ µν ( p ⊕ q ) = ˜ y µ ( ) = ˜ z µ ( ) . (7)However, taking the limits p → q → ϕ µν ( p ) = lim q → ∂ ( q ⊕ p ) ν ∂ q µ = lim q → ∂ ( p ⊕ q ) ν ∂ q µ (8)which is valid in the case of a commutative MCL, but not for more general cases. Moreover, in thiscase the ˜ x are commutative coordinates, and can then be seen as just a choice of spacetime coordinatesin a canonical phase space [9]. Since in relative locality the amount of non-locality depends on the total momentum of the process,it is natural to consider that the noncommutative spacetime in which the interaction is local shoulddepend on the momenta of the rest of worldlines. For the two out-going particles,˜ y µ = y ν ϕ µ L ν ( p , q ) , ˜ z µ = z ν ϕ µ R ν ( p , q ) . (9)The condition to have an event defined by the interaction is then ϕ µν ( p ⊕ q ) = ∂ ( p ⊕ q ) ν ∂ p ρ ϕ µ L ρ ( p , q ) = ∂ ( p ⊕ q ) ν ∂ q ρ ϕ µ R ρ ( p , q ) . (10)Taking again the limits p → q → ϕ µ L σ ( p , 0 ) = ϕ µσ ( p ) , ϕ µ R σ ( q ) = ϕ µσ ( q ) . (11)However, the MCL does not unequivocally determine in this case the functions ϕ , ϕ L , ϕ R . of 10 A specially simple case is if one takes ϕ µ L ρ to depend only on one momentum; then one has ϕ µ L ρ ( p , q ) = ϕ µ L ρ ( p , 0 ) = ϕ µρ ( p ) , and ϕ µν ( p ⊕ q ) = ∂ ( p ⊕ q ) ν ∂ p ρ ϕ µρ ( p ) , (12)which can be used to determine the MCL for a given one-particle noncommutative spacetime ϕ . Thiscase gives, in particular, ϕ µν ( p ) = lim q → ∂ ( q ⊕ p ) ν ∂ q µ , (13)which has a simple interpretation: the transformation generated by the noncommutative spacetimecoordinates is a displacement in momentum space defined by the MCL: δ p µ = − (cid:101) ν { ˜ x ν , p µ } = (cid:101) ν ϕ νµ ( p ) = (cid:101) ν lim q → ∂ ( q ⊕ p ) µ ∂ q ν = (cid:2) ( (cid:101) ⊕ p ) µ − p µ (cid:3) . (14) κ -Poincaré The Hopf algebra κ -Poincaré offers also a link between a MCL (that can be read from the coproductof the algebra) and a noncommutative spacetime (in this case, κ -Minkowski). We can show that thislink is exactly the one described above.In the bicrossproduct basis of κ -Poincaré [11], the coproduct of the generators of translations P µ reads ∆ ( P ) = P ⊗ I + I ⊗ P , ∆ ( P i ) = P i ⊗ I + e − P / Λ ⊗ P i , (15)which defines the momentum composition law ( p ⊕ q ) = p + q , ( p ⊕ q ) i = p i + e − p / Λ q i . (16)Phase-space Poisson brackets are obtained through the method of ‘pairing’ [12], which gives { ˜ x , ˜ x i } = − ˜ x i Λ , { ˜ x , p } = − { ˜ x , p i } = p i Λ , { ˜ x i , p j } = − δ ij , { ˜ x i , p } = ϕ ( p ) = ϕ i ( p ) = − p i Λ , ϕ ij ( p ) = δ ij , ϕ i ( p ) =
0. (18)The functions ϕ µν ( p ) and the composition law ( p ⊕ q ) above are indeed related by Eq. (12). Thelocality condition gives then a physical interpretation of the ‘pairing’ procedure that introducesspacetime in κ -Poincaré in a specific way.From this example, κ -Minkowski spacetime can be seen as the (one-particle) noncommutativespacetime that emerges from a locality condition in a classical model which generalizes SR inmomentum space through the algebra of κ -Poincaré. This is a rather different approach from theintroduction of noncommutativity as the implementation of a possible minimal length in a quantumspacetime.Note that in the general case, the implementation of locality is compatible with an independentchoice for a one-particle noncommutative spacetime (the ϕ function) and a MCL, while these twoingredients are related in the previous example. From this perspective, then, the example of κ -PoincaréHopf algebra is a very particular case of the implementation of locality leading to a relativistic of 10 generalization of SR. In this case, the two particles coming out of the interaction propagate in (different)noncommutative spacetimes that are defined by the functions ϕ µ L ν ( p , q ) = ϕ µν ( p ) and ϕ µ R ν ( p , q ) , ϕ R ( p , q ) = ϕ iR ( p , q ) = ϕ R i ( p , q ) = − e p / Λ p i + q i Λ , ϕ iR j ( p , q ) = e p / Λ δ ij . (19)From this, one obtains the two-particle spacetime Poisson brackets that are different from zero: { ˜ y , ˜ y i } = − ˜ y i Λ , { ˜ z , ˜ z i } = − ˜ z i Λ , { ˜ y , ˜ z i } = { ˜ z , ˜ y i } = − ˜ z i Λ . (20) We have obtained the locality condition Eq. (10) for an interaction with two particles in the finalstate. It is possible, however, to generalize this condition to an arbitrary number of particles. As anexample, let us consider an initial state where all except one of the momenta are arbitrarily small and afinal state of three particles with momenta ( k , p , q ) and a total momentum P = ( k ⊕ p ) ⊕ q . Then onehas X ρ ( ) = ξ ρ , x ν ( ) = ξ ρ ∂ P ρ ∂ k ν , y ν ( ) = ξ ρ ∂ P ρ ∂ p ν , z µ ( ) = ξ ρ ∂ P ρ ∂ q ν . (21)We can introduce new spacetime coordinates˜ X µ = X ν ϕ µν ( P ) , ˜ x µ = x ν ϕ µ ν ( k , p , q ) , ˜ y µ = y ν ϕ µ ν ( k , p , q ) , ˜ z µ = z ν ϕ µ ν ( k , p , q ) . (22)If we choose the functions ϕ i such that ϕ µρ ( P ) = ∂ P ρ ∂ k ν ϕ µ ν ( k , p , q ) = ∂ P ρ ∂ p ν ϕ µ ν ( k , p , q ) = ∂ P ρ ∂ q ν ϕ µ ν ( k , p , q ) , (23)then one has all the particles at the same point in the interaction˜ X µ ( ) = ˜ x µ ( ) = ˜ y µ ( ) = ˜ z µ ( ) , (24)and locality is implemented in the new spacetime. Using now the chain rule for the partial derivatives: ∂ P ρ ∂ k ν = ∂ P ρ ∂ ( k ⊕ p ) σ ∂ ( k ⊕ p ) σ ∂ k ν ∂ P ρ ∂ p ν = ∂ P ρ ∂ ( k ⊕ p ) σ ∂ ( k ⊕ p ) σ ∂ p ν , (25)one has ϕ µρ ( P ) = ∂ P ρ ∂ ( k ⊕ p ) σ ∂ ( k ⊕ p ) σ ∂ k ν ϕ µ ν ( k , p , q ) = ∂ P ρ ∂ ( k ⊕ p ) σ ∂ ( k ⊕ p ) σ ∂ p ν ϕ µ ν ( k , p , q ) = ∂ P ρ ∂ q ν ϕ µ ν ( k , p , q ) .(26)This requires that ∂ ( k ⊕ p ) σ ∂ k ν ϕ µ ν ( k , p , q ) = ∂ ( k ⊕ p ) σ ∂ p ν ϕ µ ν ( k , p , q ) = ϕ µ L σ (( k ⊕ p ) , q ) , ϕ µ ν ( k , p , q ) = ϕ µ R σ (( k ⊕ p ) , q ) ,(27)which can be used to determine ϕ , ϕ and ϕ .Therefore, a given composition law of momenta ( p ⊕ q ) and a noncommutative one-particlespacetime ( ϕ ) allow one to determine the spacetime of a two particle system (the fuctions ϕ L , ϕ R ), andEqs. (27) give the spacetime of a three particle system implementing locality. of 10
3. Phenomenology in the relative locality framework
In the previous section we have shown that, even in the framework of relative locality, whereworldlines of particles do not generically cross in the x -spacetime (where x is canonically conjugatedto the momentum variable p ), interactions may serve to define a notion of spacetime. Indeed, eachinteraction has a set of four coordinates ξ µ associated to it, as defined in the action Eq. (1). Thesecoordinates might serve as a definition for spacetime. Thus, if observer A sees an interaction ocurring“at” coordinates ξ µ A , a translated observer (with parameters of translation a µ ) will associate thatinteraction with coordinates ξ µ B = ξ µ A + a µ . (28)The description of worldlines is however not simple in such a spacetime, because outgoing(incoming) worldlines begin (end) at different points of that spacetime, which depend on the whole setof momenta that participate in the interaction. The use of the noncommutative spacetime introducedin the previous subsections makes possible to have a common interaction point of coordinates ζ µ ,whose relation with the ξ µ coordinates is [see Eq. (7)]: ζ µ = ξ ν ϕ µν ( P / Λ ) , (29)where P is the total momentum of the interaction.Coordinates ζ µ might serve as well as coordinates ξ µ to define a notion of spacetime, and in thiscase they have the advantage to maintain absolute locality for interactions, that is: every observeragrees that wordlines cross at interactions. In such a spacetime, the coordinates of an interactionpoint for observer A , ζ µ A , are related to the coordinates of a translated observer B (with parameters oftranslation b µ ) by ζ µ B = ζ µ A + b µ . (30)Compatibility of Eqs. (28) and (30) requires, according to Eq. (29), the relation a ν ϕ µν ( P / Λ ) = b µ (31)between the parameters of the translation in the two spacetimes. Then, if one identifies the parametersof a translation with a fixed distance between the two observers, it is clear that the two notionsof spacetime defined above are different alternatives that have in fact different phenomenologicalconsequences. Any physical measurement involves at least two interactions: in the simplest case, the emission ofa photon by a (static) source and the detection of this photon by a (static) detector. Two observers forwhich each of the interactions is respectively local in the canonical spacetime, can then be brought intoplay to describe this measurement: observer A , at the source, and observer B , at the detector.Let us first consider a model of spacetime that derives directly from the noncommutativity, that is,the model that is defined by Eq. (30). When we say that source and detector are in relative rest andseparated by a distance L , we have that (cid:126) ζ B = (cid:126) ζ A − (
0, 0, L ) , (32)where we have taken that the propagation of the photon takes place along the third spatial direction.Let us now consider the detection of a photon which is produced by a certain source in somemore or less complicated process (for example, a gamma-ray burst). Detection of this photon takes of 10 place for observer B at (cid:126) ˜ x d = (
0, 0, 0 ) and is emitted at (cid:126) ˜ x e = (
0, 0, − L ) , which corresponds to the originof spatial coordinates for observer A , according to Eq. (32).The time coordinates at the detection and emission of the photon will be related by˜ x d = ˜ x e + L ˜ v , (33)where ˜ v is the velocity of propagation of the photon. Using the expression of the space-time coordinatesin terms of canonical coordinates ˜ x µ = x ν ϕ µν ( { p } ) , (34)where ( { p } ) represents the set of momenta involved in the interaction (as in Eq. (9)), one gets˜ v ( { p } ) = ( x d − x e ) ϕ + ( x d − x e ) ϕ ( x d − x e ) ϕ + ( x d − x e ) ϕ = ϕ + v ( p ) ϕ ϕ + v ( p ) ϕ , (35)where v ( p ) is the velocity in the canonical spacetime, which can be derived from the action Eq. (1): v ( p ) = ∂ C ( p ) / ∂ p ∂ C ( p ) / ∂ p . (36)A difficulty remains: the fact that ϕ µν ( { p } ) could depend on all the momenta involved in theinteraction would make impossible to calculate in practice the time interval between the emission anddetection of the photon. However, we have seen that it is possible to implement the locality conditionEq. (10) in such a way that the function ϕ µν for one of the particles depends only on its own momentum p (this choice gave the relationship between the MCL and the one-particle noncommutative spacetimeEq. (12)). If we adopt this choice for the emitted photon which will arrive to the detector, ˜ v is a functionof p only, and can then be calculated for a given noncommutative spacetime ϕ µν ( p ) .Within this scheme, Eq. (33) gives a time delay ˜ T ˜ T . = ˜ x d − ˜ x e − L = L (cid:18) v − (cid:19) . (37)As a particular example, one can compute ˜ v in the bicrossproduct basis of κ -Poincaré using Eq. (18)and the velocity in the canonical spacetime Eq. (36) determined by the Casimir in this basis [11]. Theresult is ˜ v =
1, and, therefore, there is no time delay in this basis. In fact, the different bases of κ -Poincarécorrespond to different choices of canonical phase-space coordinates with the same noncommutativespacetime [9]; then, the absence of time delay is a property of κ -Poincaré, independent of the choice ofbasis.In Ref. [10] a different expression for a ‘time delay’ was obtained. It used in fact a differentdefinition of the distance L between the detector and the source (difference of canonical spacecoordinates instead of noncommutative space coordinates). The absence of time delay was in thatcase a consequence of a velocity of propagation in canonical spacetime independent of energy ( v = Let us now consider the second model of spacetime, in which space-time coordinates are definedwith the vertex coordinates ξ µ of an interaction, Eq. (2). One can follow step by step the definition of thedistance L between source and detector in relative rest in the previous section, replacing everywherethe coordinates ˜ x µ d by ξ µ d and ˜ x µ e by ξ µ e . of 10 The computation of the time delay T . = ξ d − ξ e − L requires the relations x µ d = ξ ν d ∂ P ν ∂ p µ x µ e = ξ ν e ∂ P ν ∂ p µ , (38)and the equation of the worldline in canonical spacetime corresponding to the velocity Eq. (36). Theresult is T = L (cid:20) ( ∂ C / ∂ p )( ∂ P / ∂ p ) − ( ∂ C / ∂ p )( ∂ P / ∂ p )( ∂ C / ∂ p )( ∂ P / ∂ p ) − ( ∂ C / ∂ p )( ∂ P / ∂ p ) − (cid:21) . (39)The derivatives ∂ P ν / ∂ p µ appearing in the previous expression inevitably depend on all themomenta involved in the process, making impossible to determine the time delay in terms of theenergy of the photon. In particular, it is not possible to have zero time delay, in contrast with theprevious choice of spacetime.
4. Conclusions
The phenomenology beyond SR is very different in the cases of absence of a relativity principle(LIV) and presence of a relativity principle (DSR). In contrast to a number of possible footprints in theLIV case, time of flight of stable particles may be the only phenomenological window in the DSR case.Moreover, the phenomenological analysis of time of flight is also different for the LIV and DSR cases.The concept of absolute locality in canonical spacetime, which is valid in SR, is only possiblein a LIV framework. DSR necessarily requires a modification in the notion of locality, that can beincorporated through a new (noncommutative) spacetime defined from the canonical phase space.A definition of spacetime from interactions in the case of DSR requires to go beyond the canonicalspacetime. The modification of the composition of momenta leads to a distribution of the canonicalspacetime coordinates of the different particles in the interaction, which are fixed in terms of a vertex.We have presented two options for the spacetime in DSR. One possibility is to consider new(noncommutative) space-time coordinates for each particle, such that the interaction is local in thisnew spacetime. A second possibility is to consider directly the vertex coordinates as the coordinates ofspacetime.In both cases one has in general a time delay depending on all the momenta in the process, andthen one does not have a definite prediction. But in the case of a noncommutative spacetime we haveshown that it is possible to find a model where one does not have time delays. This opens up thepossibility to consider a scale in DSR much smaller than the Planck scale, leading to a new perspectivein quantum gravity phenomenology.Finally, we note that it is plausible that the recovery of locality in the new spacetime might serveas a starting point for a generalization of field theory based on modified composition laws.
Acknowledgments:
This work is supported by the Spanish MINECO FPA2015-65745-P (MINECO/FEDER) andSpanish DGIID-DGA Grant No. 2015-E24/2. We thank Flavio Mercati for the organization of the conference
Observers in Quantum Gravity , that took place in Rome from on 22-23 January 2018. This paper is based on a talkgiven at the mentioned conference.
Abbreviations
The following abbreviations are used in this manuscript:SR Special RelativityLIV Lorentz Invariant ViolationDSR Deformed Special RelativityEFT Effective Field TheoryMDR Modified Dispersion RelationMCL Modified Composition Law
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