Taming nonlocality in theories with Planck-scale-deformed Lorentz symmetry
Giovanni Amelino-Camelia, Marco Matassa, Flavio Mercati, Giacomo Rosati
aa r X i v : . [ g r- q c ] F e b http://dx.doi.org/10.1103/PhysRevLett.106.071301 Taming nonlocality in theories with Planck-scale-deformed Lorentz symmetry
Giovanni AMELINO-CAMELIA,
Marco MATASSA, Flavio MERCATI,
1, 2 and Giacomo ROSATI
1, 2 Dipartimento di Fisica, Università di Roma “La Sapienza", P.le A. Moro 2, 00185 Roma, EU INFN, Sez. Roma1, P.le A. Moro 2, 00185 Roma, EU
We report a general analysis of worldlines for theories with deformed relativistic symmetries and momentumdependence of the speed of photons. Our formalization is faithful to Einstein’s program, with spacetime pointsviewed as an abstraction of physical events. The emerging picture imposes the renunciation of the idealizationof absolutely coincident events, but is free from some pathologies which had been previously conjectured.
Over the last decade there has been considerable interest inthe quantum-gravity literature about the possibility [1] of de-formations of Lorentz symmetry that would allow the intro-duction of a momentum dependence of the speed of photons v = 1 − ℓ p (1)as a relativistic law, with an observer-independent length pa-rameter ℓ usually assumed to be roughly of the order ofthe Planck length. This is the most studied possibility fora “Doubly-Special Relativity" (DSR) [1–5]. The interest itattracts is mostly due to associated features that emerge inenergy-momentum space, which find some support in pre-liminary results obtained within the Loop Quantum Gravityapproach [6] and in some models based on spacetime non-commutativity [7]. But the development of this research pro-gram must face the challenge of several indirect arguments(see, e.g. , Ref. [8, 9] and references therein) suggesting that alogically consistent formulation of (1) is not possible within afully conventional description of spacetime.The possibility of novel properties for spacetime was ex-pected at the onset [1] of DSR research, since the motiva-tion for the proposal came from some aspects of the quantum-gravity problem, which also suggest that there might be someabsolute limitations to localizability of an event. But the factit was expected does not make it any less of a challenge: whatcould replace the classical points of spacetime?Those who looked at DSR research from the outside havebeen understandably rather puzzled (see, e.g. , Ref. [10]) aboutsome of the implications of renouncing to an ordinary space-time picture. In particular, the recent Ref. [11] ventured tomake a bold claim: even without adopting any specific formal-ization, using only the bare idea of momentum-dependence ofthe speed of photons, one could robustly estimate the natureand size of the nonlocal effects that should be produced. And,still according to Ref. [11], this could be used to constrain ℓ to | ℓ | < − m , i.e. at a level which is 23 orders of magni-tude beyond the one of direct experimental bounds based onthe momentum dependence of the speed of photons [12, 13].The claim reported in Ref. [11] clearly renders even moreurgent for DSR research to establish what are the actual im-plications for nonlocality. We start by observing that the ar-gument presented in Ref. [11] did not make use of the well-established results on DSR-deformed boosts, but rather reliedon assumptions that fail to be consistently relativistic. As shown in Fig. 1 the assumptions of Ref. [11] amount to adopt-ing undeformed rules of boost transformation for the coordi-nates of the emission points of particles but deformed boosttransformations for the velocities of the particles. Evidentlysuch criteria of “selective applicability" of deformed boostscannot produce a consistently relativistic picture. FIG. 1. In the argument of Ref. [11] a key role is played by theassumption that a photon which Alice sees emitted in P A (from asource at a distance L from Alice) with speed v (and momentum p ) should be seen by boosted Bob as a photon emitted from P B ,obtained by classical/undeformed boost of P A , with speed obtainedfrom the speed v with a deformed boost. In this figure we expose thelogical inconsistency of such criteria of “selective applicability" ofdeformed boosts by allowing for a second photon which accordingto Alice also has speed v and is emitted from a point P A ′ such thatthe two photons share the same worldline: a single worldline wouldbe mapped by a relativistic boost into two wildly different worldlines. The picture proposed in Ref. [11] clearly needed to be re-vised. We here report a deductive result of characterizationof the nonlocality produced by DSR boosts. We derive it rig-orously from the formalizations of DSR-deformed boosts thathave been proposed in the DSR literature. We succeed, whereothers had failed, primarily as a result of using as guidanceEinstein’s insight on the proper characterization of a space-time point, to be viewed as the abstraction of an event of cross-ing of worldlines. This leads us to a fully relativistic charac-terization of the concept of locality, as a concept that pertainsthe coincidence of events: from a relativistic perspective themain locality issue concerns whether events that are coinci-dent for one observer are also coincident for other observers.We set to both Planck constant ~ and the speed-of-lightscale c (speed of photons in the low-momentum limit). Themodulus of a spatial 3-vector, with components W j , is de-noted by W ( W = W j W j ). And we work in leading order in ℓ , since (1) is assumed [1, 6] to be valid only for p ≪ / | ℓ | .We also take on the challenge of a full 3+1-dimensionalanalysis. Most of the previous DSR literature, includingRef. [11], is confined to 1+1-dimensional frameworks, asa way to temper the complexity of dealing with deformedboosts. It is natural to expect that the core implications forthe nonlocality produced by DSR boosts would be alreadyuncovered in a 1+1-dimensional analysis, but our ability tocharacterize transverse boosts is a valuable addition, and pro-vides further evidence of the robustness of the approach wedeveloped. Another significant strength of our setup is that itapplies to all previously considered deformations of Lorentzsymmetry compatible with (1). Previous DSR studies of (1)not only failed to offer an explicit analysis of worldlines, butwere also often assuming a specific ansatz for the formaliza-tion of the symmetry deformation.The derivation of the worldlines is here achieved within aHamiltonian setup which was already fruitfully applied [14,15] to other DSR scenarios for the introduction of the secondrelativistic scale ℓ , but was not previously implemented for aDSR description of the speed law (1) for massless particles.We start by introducing canonical momenta conjugate to thecoordinates x j and t : { Π j , x k } = − δ jk , { Ω , t } = 1 . Wemust then specify a form of the DSR-deformed mass Casimir C , which will play the role [14, 15] of Hamiltonian. We havea two-parameter family of O ( ℓ ) possibilities C = Ω − Π + ℓ (cid:0) γ Ω + γ ΩΠ (cid:1) , (2)upon enforcing analyticity of the deformation and invarianceunder classical space-rotation transformations. The types ofdeformed boosts that were previously considered in the DSRliterature have the property of being compatible with such adeformed Casimir, for some corresponding choices of γ , γ .Hamilton’s equations give the conservation of Π j and Ω along the worldlines ˙Π j = ∂ C ∂x j = 0 , ˙Ω = − ∂ C ∂t = 0 , (3)where ˙ f ≡ ∂f /∂τ and τ is an auxiliary worldline parameter.The worldlines can then be obtained observing that ˙ x j = − ∂ C ∂ Π j ⇒ x j ( τ ) = x (0) j + (2Π j − ℓγ ΩΠ j ) τ ˙ t = ∂ C ∂ Ω ⇒ t ( τ ) = t (0) + (cid:2)
2Ω + ℓ ( γ Π + 3 γ Ω ) (cid:3) τ . (4)Eliminating the parameter τ and imposing the Hamiltonianconstraint C = 0 (massless case) one finds that x j = x (0) j + Π j Π ( t − t (0) ) − ℓ ( γ + γ )Π j ( t − t (0) ) , (5)which reproduces (1) for γ + γ = 1 . Note that this derivationof worldlines compatible with (1) is insensitive to the possi-bility of a different DSR description for the canonical mo-mentum Π j and for the “momentum" p j , intended as the DSR generalization of the concept of space-translation charge. In-deed, Π j enters only at order ℓ and of course, since we areworking in leading order, we must take ℓ Π j = ℓp j (while themodulus of p j and Π j may differ [5, 16] at order ℓ ).We must now enforce covariance of the worldlines underDSR-deformed boosts. The form of the correction terms in-troduced in (2) suggests that the type of deformed boosts con-sidered in the DSR literature should be well suited: N j = − t Π+ x j Ω+ ℓ [ α t ΩΠ j + α Π x j + α Ω x j + α x k Π k Π j ] Note that this four-parameter family of O ( ℓ ) deformed boosts,which enforces compatibility with undeformed space rota-tions, includes, as different particular cases, all the proposalsfor deformed boosts that were put forward in this first decadeof DSR research [1–5, 17]. The compatibility between boosttransformations and form of the Casimir is encoded in the re-quirement that the boost charge is conserved ˙ N j = {C , N j } = ∂ C ∂ Ω ∂ N j ∂t − ∂ C ∂ Π k ∂ N j ∂x k = 0 , (6)which straightforwardly leads to the following constraints onthe parameters γ , γ , α , α , α , α : α + 2 α = γ , α + 2 α − γ − γ = 0 . (7)Combining these with the requirement γ + γ = 1 derivedabove, we finally arrive at a three-parameter family of Hamil-tonian/boost pairs C = Ω − Π + ℓ (cid:0) γ Ω + (1 − γ ) Ω Π (cid:1) N j = − t Π j + x j Ω + ℓ αt ΩΠ j − ℓ ( γ + β − / x k Π k Π j + ℓx j (cid:0) β Π + (1 + γ − α ) Ω (cid:1) , where γ = γ / , α = α , β = α . For any given choice of γ, α, β relativistic covariance is ensured and we have a rigor-ous Hamiltonian derivation of worldlines for which the speedlaw (1) is verified. We have so far focused on massless parti-cles, but one also easily obtains the worldlines of particles ofany mass by enforcing the Hamiltonian constraint C = m : x j = x (0) j + Π j √ Π + m ( t − t (0) ) − ℓ Π j ( t − t (0) ) . (8)The covariance of these worldlines under undeformed spacerotations is manifest. The covariance under γ, α, β -deformedboosts, ensured by construction, can also be verified by com-puting explicitly the action of an infinitesimal deformed boostwith rapidity vector ξ j ( A ′ = A + ξ j { A, N j } ) Π ′ j = Π j − ξ j Ω − ℓ ξ j (cid:0) β Π + (1 + γ − α ) Ω (cid:1) − ℓ (1 / − γ − β ) ξ k Π k Π j (9) t ′ = t − ξ j x j − ℓ (cid:0) αtξ j Π j + 2 (1+ γ − α ) Ω ξ j x j (cid:1) (10) x ′ j = x j − tξ j + ℓ (cid:0) αt Ω ξ j + 2 βξ k x k Π j (cid:1) − ℓ ( γ + β − / (cid:0) ξ k Π k x j + x k Π k ξ j (cid:1) (11)Using these one easily verifies that when Alice has the particleon the worldline (8) Bob sees the particle on the worldline x ′ j = x ′ (0) j + Π ′ j √ m + Π ′ ( t ′ − t ′ (0) ) − ℓ Π ′ j ( t ′ − t ′ (0) ) , consistently with the relativistic nature of our framework.We are now ready to exploit our technical results for a“physical" characterization of the nonlocality produced byDSR boosts. The observations we shall make on nonlocal-ity apply equally well to all choices of γ , α , β . We noticehowever that by enforcing the condition α − β − γ = 1 / onehas the welcome [7, 15] simplification of undeformed Pois-son brackets among boosts and rotations (“the Lorentz sectoris classical" [7, 15]). And in particular for the case γ = 1 / , α = 1 , β = 0 , on which we focus for our graphical illustrations,the laws of transformation take a noticeably simple form: Π ′ j = Π j − ξ j (cid:0) Ω + ℓ Ω / (cid:1) (12) t ′ = t − ξ j x j − ℓ (cid:0) tξ j Π j + Ω ξ j x j (cid:1) (13) x ′ j = x j − (1 − ℓ Ω) tξ j (14)This case preserves much of the simplicity of classical boostsfor what concerns boosts acting transversely to the direction ofmotion. We do not expect anything objectively pathological inthe richer structure that other choices of γ , α , β produce (see(10)-(11)) for such transverse boosts. But it is nonethelessnoteworthy that there are candidates for the DSR deformedboosts that have properties as simple as codified in (13)-(14).In what follows we shall not offer any additional commentson transverse boosts (and our figures focus on boosts alongthe direction of motion). But it is easy to verify using (10)-(11) (and even easier using (13)-(14)) that boosts acting trans-versely to the direction of motion lead to features of nonlocal-ity that are of the same magnitude and qualitative type as theones we visualize for boosts along the direction of motion.Let us now move on to reconsidering the issues raised inour Fig. 1, and the shortcomings of the analysis reported inRef. [11]. Having managed to derive constructively quantita-tive formulas for the action of the deformed boosts advocatedin the DSR literature, we can now more definitely observe thatthe assumptions made for the analysis reported in Ref. [11] areinconsistent with the fact, here shown in Eqs. (10)-(11), thatthe deformed boosts still act, like ordinary Lorentz boosts, inway that is homogeneous in the coordinates. A boost con-nects two observers with the same origin of their referenceframes and, as shown in Fig. 2, the differences between DSR-deformed boosts and classical boosts are minute for points thatare close to the common origin of the two relevant referenceframes, but gradually grow with distance from that origin.As shown by two of the worldlines in Fig. 3, when an ob-server Alice is local to a coincidence of events (the violet anda red photon simultaneously crossing Alice’s worldline) allobservers that are purely boosted with respect to Alice, andtherefore share her origin, also describe those two events ascoincident. This in particular addresses the “box problem"raised in Ref. [11], which concerned the possibility of a lossof objectivity of coincidences of events as witnessed by lo-cal observers: we have found that, at least in leading order in ℓ and ξ , in the DSR framework “locality", a coincidence ofevents, preserves its objectivity if assessed by local observers.The element of nonlocality that is actually produced byDSR-deformed boosts is seen by focusing on the “burst" ofthree photon worldlines also shown in Fig. 3, whose crossings FIG. 2. We here show a hard-photon worldline as seen by Alice (solidbue), by DSR-boosted Bob (dashed blue) and by classically-boostedBob (dashed-black). In spite of assuming (for visibility) the unreal-istically huge
Π = 0 . /ℓ , ξ = 0 . , the difference between DSRboosts and undeformed boosts is minute near the origin. But accord-ing to Bob’s coordinates the emission of the hard particle appears tooccur slightly off the (thick) worldline of the source.FIG. 3. A case with two hard (violet) worldlines, with momen-tum Π v = 0 . /ℓ , a “semi-hard" (blue) worldlline with momen-tum Π b = Π v / , and a ultrasoft worldline (red, with Π r ≪ /ℓ ).According to Alice (whose lines are solid, while boosted Bob hasdashed lines) three of the worldlines give a distant coincidence ofevents, while two of the worldlines cross in the origin. establish a coincidence of events for Alice far from her ori-gin, an aspect of locality encoded in a “distant coincidence ofevents". The objectivity of such distant coincidences of eventsis partly spoiled by the DSR deformation: the coincidence isonly approximately present in the coordinates of an observerboosted with respect to Alice. But we stress that in Figs. 2and 3 we used, for visibility, gigantically unrealistic valuesof photon momentum (up to ∼ . /ℓ ): it should nonethelessbe noticed that even distant coincidence is objective up to avery good approximation, if indeed, as assumed in the DSRliterature[1–5], the observer-independent length scale ℓ is assmall as the Planck length ( ∼ − m ). On terrestrial scalesone might imagine hypothetically to observe a certain parti-cle decay with two laboratories, with a large relative boost of,say, ξ ∼ − , with idealized absolute accuracy in trackingback to the decay region the worldlines of two particles thatare the decay products. As one easily checks from (10)-(11),the peculiar sort of nonlocality we uncovered is of size ξℓL Π .Therefore even if the distance L between the decay regionand the observers is of, say, m , and the decay productshave momenta of, say, GeV , one ends up with an apparentnonlocality of the decay region which is only of ∼ − m .Another interesting case is the one of a typical observationof a gamma-ray burst, with GeV particles that travel for, say, s before reaching our telescopes. For two telescopes witha relative boost of ξ ∼ − the loss of coincidence of eventsat the source is ∼ m , well below the sharpness we are ableto attribute [12] to the location of a gamma-ray burst.We should stress that actually, in light of the results we ob-tained, in such a DSR framework two relatively boosted ob-servers should not dwell about distant coincidences, but ratherexpress all observables in terms of local measurements (whichis anyway what should be done in a relativistic theory). Forexample, for the burst of three photons shown in Fig. 3 the momentum dependence of the speed of photons is objectivelymanifest (manifest both for Alice and Bob) in the linear cor-relation between arrival times and momentum of the photons.Also insightful is the comparison of the loss of objectivityof coincidences of distant events, which we uncovered herefor DSR boosts, with the loss of objectivity of simultaneitythat was required by the replacement of Galileian boosts withLorentz boosts. With absolute time of course any statement ofsimultaneity was objective. With the introduction of Lorentzboosts, which are obtained deforming Galileian boosts, simul-taneity is no longer objective in general, but it remains objec-tive for events occurring at the same spatial position, whetheror not that spatial position is where the observer is located(the origin). With one more step of deformation of boosts,the DSR proposal, the realm of objectivity of simultaneity isfarther reduced: simultaneity of events is only objective if theevents are coincident according to a local observer, and thisis only manifest in the coordinate systems of other observersthat are also local to the coincidence of events.Amusingly it appears that the possibility of coincidentevents was cumbersome already for Einstein, as shown by afootnote in the famous 1905 paper [19]: “ We shall not discuss here the imprecision inherent inthe concept of simultaneity of two events taking placeat (approximately) the same location, which can beremoved only by abstraction. " We conjecture that the proper description of the quantum-gravity realm, whether or not there will be a role for DSRconcepts, will impose the renunciation of the idealization ofthe possibility of exact and absolute coincidence of events. [1] G. Amelino-Camelia, Int. J. Mod. Phys.
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