EFTA01114598.pdf

The principle of relative locality Giovanni Amelino-Camelit, Laurent Freider, Jerzy Kowalski-Glikmanb, Lee Smolin` ° Dipartimento di Fisica, University "La Sapienza" and Sez. Romal INFN, P.le A. Moro 2, 00185 Roma, Italy bInstitute for Theoretical Physics, University of Wroclaw, Pl. Maxa Boma 9, 50-204 Wroclaw, Poland `Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada (Dated: January 5. 2011) We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in space- times are constructed by observers, but different observers, separated from each other by translations. construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry. such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle C•1 all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature. Ct ti I. INTRODUCTION powerful generalization of special relativistic physics which is motivated by general considerations of the unification of arXiv:submit/0174271 [hep-th] How do we know we live in a spacetime? And if so how gravity with quantum physics. In this work we link the no- do we know we all share the same spacetime? These are the tion of locality with assumptions made about the geometry of fundamental questions we are investigating in this note. momentum space. We propose a new framework in which we can relax in a controlled manner the concept of absolute lo- We local observers do not directly observe any events cality by linking this to a new understanding of the geometry macroscopically displaced from our measuring instruments. of momentum space. In this framework there is no notion of As naive observers looking out at the world, and no less as par- absolute locality, different observers see different spacetimes, ticle physicists and astronomers, we are basically "calorime- and the spacetimes they observe are energy and momentum ters" with clocks. Our most fundamental measurements are dependent. Locality, a coincidence of events, becomes rela- the energies and angles of the quanta we emit or absorb, and tive: coincidences of events are still objective for all local ob- the times of those events. Judging by what we observe, we servers, but they are not in general manifest in the spacetime live in momentum space, not in spacetime. coordinates constructed by distant observers. The idea that we live in a spacetime is constructed by in- One way to motivate this new physical framework is by ference from our measurements of momenta and energy. This thinking about the symmetry of the vacuum. The most ba- was vividly illustrated by Einstein's procedure to give space- sic question that can be asked of any physical system is what time coordinates to distant events by exchanges of light sig- is the symmetry of the ground state that governs its low lying nals [1]. When we use Einstein's procedure we take into ac- excitations. This is no less true of spacetime itself, moreover count the time it takes the photons to travel back and forth in general relativity, and presumably in any description of the but we throw away information about their energy, resulting quantum dynamics of spacetime, the symmetry of the ground in a projection into spacetime. When we do this we presume state is dynamically determined. We also expect that the clas- that the same spacetime is reconstructed by exchanges of light sical spacetime geometry of general relativity is a semiclassi- signals of different frequencies. We are also used to assuming cal approximation to a more fundamental quantum geometry. that different local observers, distant from each other, recon- In this paper we show how simple physical assumptions about struct the same spacetime by measurements of photons they the geometry of momentum space may control the departure send and receive. of the spacetime description from the classical one. But why should the information about the energy of the We will first restrict attention to an approximation in which photons we use to probe spacetime be inessential? Might that ft GNewu,„ both may be neglected while their ratio just be a low energy approximation? And why should we pre- sume that we construct the same spacetime from our observa- fi tions as observers a cosmological distance from us? (I) GNewron P In this paper we show that absolute locality, which postu- lates that all observers live in the same space time, is equiva- is held fixed'. In this approximation quantum and gr,tvita- lent to the assumption that momentum space is a linear man- ifold. This corresponds to an idealization in which we throw away the information about the energy of the quanta we use to probe spacetime and it can be transcended in a simple and I We work in units in which < = I. EFTA01114598  2 tional effects may both be neglected, but there may be new ometry of momentum space2 T from measurements made of phenomena on scales of momentum or energy given by nip. the dynamics of interacting particles. We assume that to each At the same time, because = N/Cr e„,„,, —r 0 we expect no choice of calorimeter and other instruments carried by our ob- features of quantum spacetime geometry to be relevant. server there is a preferred coordinate on momentum space, lea. But we also assume that the dynamics can be expressed co- Since our approximation gives us an energy scale, but not variantly in terms of geometry of fP and do not depend on the a length scale, we will begin by presuming that momentum choice of calorimeter's coordinates. We note that the k„ mea- space is more fundamental than spacetime. This is in accord sure the energy and momenta of excitations above the ground with the operational point of view we mentioned in the open- state, hence the origin of momentum space, lea = 0, is physi- ing paragraph. So we begin in momentum space by asking cally well defined. how it may be deformed in a way that is measured by a scale Our local observer can make two kinds of measurements. mp. Once that is established we will derive the properties One type of measurement can be done only with a single par- of spacetime from dynamics formulated in momentum space. ticle and it defines, as we will see, a metric on momentum For convenience we work in first in the limit just described, space. The other type of measurement involve multi particles after which we will briefly turn on h. and defines a connection. A key mathematical idea underly- By following this logic below, we will find that physics ing our construction is that a connection on a manifold can may be governed by a novel principle, which we call the be determined by an algebra Rh in the present case this will Principle of Relative Locality. This states that, be an algebra that determines how momenta combine when particles interact. Physics takes place in phase space and there is no invariant global projection that gives a description of processes in A. The metric geometry of momentum space spacetime. From their measurements local observers can construct descriptions of particles moving and interacting First we describe the metric geometry. Our local observer in a spacetime, but different observers construct different can measure the rest energy or relativistic mass of a parti- spacetimes, which are observer-dependent slices of phase cle which is a function of the four momenta. She can also space. measure the kinetic energy K of a particle of mass in moving with respect to her, but local to her. We postulate that these In the next section we introduce an operational approach measurements determine the metric geometry of momentum to the geometry of spacetime, which we build on in section space. We interpret the mass as the geodesic distance from the III to give a dynamics of particles on a curved momentum origin, this gives the dispersion relation space. We see how a modified version of spacetime geometry is emergent from the dynamics which is formulated on mo- D(p) a &(p,0) = m2. (2) mentum space. In these sections we consider a general mo- The measurement of kinetic energy defines the geodesic dis- mentum space geometry, which illuminates a variety of new tance between a particle p at rest and a particle p' of identical phenomena that might be experimentally probed correspond- mass and kinetic energy K, that is D(p) = D(p') = in and ing to the curvature, torsion and non-metricity of momentum space. However, one advantage of this approach is that with 132(p, pi) = —2InK• (3) a few reasonable physical principles the geometry of momen- tum space can be reduced to three choices, depending on the The minus sign express the fact that the geometry of momen- sign of a parameter. As we show in section IV, this gives this tum space is Lorentzian. From these measurements one can framework both great elegance and experimental specificity. reconstruct a metric on In section V we make some preliminary observations as to dk2 =lith(k)dkakb. (4) how the geometry of momentum space may be probed exper- imentally, after which we conclude. B. The algebra of Interactions Now we describe the construction of the connection on mo- mentum space. This is determined by processes in which II. AN OPERATIONAL APPROACH TO THE GEOMETRY OF MOMENTUM SPACE 2 By which we mean the space of relativistic four-momenta denoted pa with We take an operational point of view in which we describe a =0.1.2.3. physics from the point of view of a local observer who is I In the standard case of physics in Minkowski spacetime. h'0 is the dual Minkowski metric and 5‘,4)= Fj lea. A scale nip may be introduced by equipped with devices to measure the energy and momenta deforming the geometry of !P so that it is curved. The correspondence of elementary particles in her vicinity. The observer also has principle (to be introduced below) assures that we recover the standard fiat a clock that measures local proper time. We construct the ge- geometry of P in the limit ntp cc. EFTA01114599  3 n particles interact, nk, incoming and new outgoing, with is equivalent to saying there is a definite microscopic causal n= n„„,. This proceeds by the construction of an al- structure. That is, causal structure of the physics maps to non- gebra, which then determines the connection. associativity of the combination rule for momentum which in Associated to each interaction there must be a combination turn maps to curvature of momentum space. The curvature rule for momentum, which will in general be non-linear. We of momentum space makes microscopic causal orders distin- denote this rule for two particles by guishable, and hence meaningful. This gives rise to proposals to measure the curvature of momentum space which we will (p,q) i pi„= (peg). (5) discuss below. Hence the momentum space P has the structure of an algebra defined by the product rule e. We assume that more compli- cated processes are built up by iterations of this product — but to begin with we assume neither linearity, nor commutativity fpeqlek pe(q$k) nor associativity. We will also need an operation that turns outgoing momenta into incoming momenta. This is denoted, p ep and it sat- isfies4 R (ep) e p = 0 (6) Then we have the conservation law of energy and momen- tum for any process, giving, for each type of interaction, four functions on PR, depending on momenta of interacting parti- cles, which vanish z(e). 0 (7) C. From the algebra of interactions to the connection on FIG. I: Curvature of the connection on momentum space produces momentum space non-associativity of composition rule. Corresponding to the algebra of combinations of momenta To determine the connection, torsion and curvature away there is a connection on P. The geometry of momentum space from the origin of momentum space we have to consider trans- is studied in detail in [21, but the basics are as follows. The lating in momentum space, ie we can denote algebra of the combination rule determines a connection on by pekq=k9((ekep)ET3(ekeq)) (II) a a aPa aqb (Peq).1q.p.= —17.b(o) (8) the identity for this product is at Ok = k. Then a a The torsion of It is a measure of the asymmetric part of the am, aqb (P ek q)clq.p.k = —reNk) (12) combination rule a a (9) Thus, the action of adding an infinitesimal momentum dqa — a— m ((p q), — (pa) fir)aq.p=O =Tab(o) pa aT from particle .1to a finite momentum pa of particle I defines a parallel transport on P. Similarly the curvature of P is a measure of the lack of asso- ciativity of the combination rule e dqa = pa+ dqbtp) (13) where T(p) is the parallel transport operation from the identity a a s top. It can be expanded around p = 0 2— — — (p ( e e k — p ED (qek))d = Rah5(0) aPia agmake (10) ti,;(17) = 8", - rtf Pc — Paledpcpd + • (14) where the bracket denote the anti-symmetrisation. We note that there is no physical reason to expect a com- with bination rule for momentum to be associative, once it is non- linear. Indeed, the lack of associativity means there is a phys- = adplir (15) ical distinction between the two processes of Figure 1, which The corresponding conservation law thus has the form to sec- ond order more generally (ep)a (pek) = k. where e is a left inverse. zoo =EX,- I E (16) And .1O10 EFTA01114600  4 where 5(1) is the set of particles that interact with the /1th Note that we do not need a spacetime or spacetime metric particle. to describe how these particles interact. If we consider the We will shortly study the consequences of curvature and process with n interacting particles we want to impose con- torsion on momentum space for the dynamics of particles. We servation of the non-linear quantities, X. We do this by in- will see that the meaning of the curvature of momentum space troducing a lagrange multiplier to guarantee conservation (7). is that it implies a limitation of the usefulness of the notion The action is that processes happen in an invariant spacetime, rather than in phase space. The hypothesis of a shared, invariant spacetime, sad= siff„,+si" (20) in which all observers agree on the locality of distant interac- tions, turns out to be a direct consequence of the linearity of where for the incoming particles the usual conservation laws of energy and momentum. When we deform the conservation laws by making them non-linear, to this gives rise to relative locality. Sjiree = j ds (xjkl, + %C/(14) (21) while for the outgoing particles HL THE EMERGENCE OF SPACETIME FROM THE DYNAMICS OF PARTICLES Si ft, = L eds(xli;+gt(1CI (10) (22) We take the point of view that spacetime is an auxiliary concept which emerges when we seek to define dynamics in The interaction contribution to the action is simply a lagrange momentum space. If we take the momenta of elementary par- multiplier times the conservation law (7). ticles to be primary, they themselves need momenta, so that a canonical dynamics can be formulated. The momenta of the = gak(O)Le (23) momenta are quantities xi that live in the cotangent space of We have set the interaction to take place at affine parameter r at a point ea. e s= 0 for each of the particles. At this point can be just con- sidered to be a lagrange multiplier to enforce the conservation of momentum (7) at the interaction where for each particle A. Variational principle s = 0. We vary the total action. After an integration by parts in Given these we can define the free particle dynamics by each of the free actions we have &squid Sj n = f ds(Ajici, -F %CV)) (17) where s is an arbitrary time parameter and % is the Lagrange multiplier imposing the mass shell condition CV) U' — m3. (18) Ef'(84g-se,[4-9465]+89,60k))+gt SI (24) Here t contains both the result of varying S" and the bound- We emphasize that the contraction does not involve a ary terms from the integration by parts. SI.2 are 0,0o, —co de- metric, and the dynamics is otherwise given by constraints pending on whether the term is incoming or outgoing. Before which are functions only of coordinates on P and depend only examining the boundary terms we confirm we have the desired the geometry of P. This leads to the Poisson brackets, free parts of the equations of motion la = 0 We then have a phase space, r of a single particle, which is the cotangent bundle of P. We note that there is neither = an invariant projection from r to a spacetime, M nor is there =0 (25) defined any invariant spacetime metric. Yet this structure is sufficient to describe the dynamics of free particles. The fact We fix elk: = 0 at s = ±-oo and examine the remainder of the that there is no invariant projection to a spacetime is related to variation the non linearity of momentum space. Indeed under a non lin- ear redefinition p -3 F(p) the conjugated coordinates is given by x (ap/aF)x, so the new canonical coordinate appears to = xvo.se —(.4 co— zbt (26) be momentum space dependent. This is this mixing between "spacetime" and momentum space that is the basis of the rel- Here 4 and kla are taken for each particle at the parameter ative locality. We can call the .4 Hamiltonian spacetime co- time s = 0. This has to vanish if the variational principle is ordinates because they are defined as being canonically con- to have solutions. From the vanishing of the coefficient of Sza jugate to coordinates on momentum space. we get the four conservation laws of the interaction (7). From EFTA01114601  5 the vanishing of the coefficient of flea we find 4n conditions B. The physical meaning of relative locality which hold at the interaction Is this a real, physical non-locality or a new kind of coor- 4(0) = ? Ski (27) dinate artifact? It is straightforward to see that it is the latter, because the Axj(0) can be made to vanish by making a trans- Using (16) this gives conditions lation to the coordinates of another observer. In a canonical -4(0)= e - £, (28) formulation, translations are generated by the laws of conser- LE.w) vation of energy and momentum, (7). Given any local ob- This tells us that to leading order, in which we ignore the servable in phase space O observed by a local observer, Alice, curvature of momentum space, all of the worldlines involved we can construct the observable as seen in coordinates con- in the interaction meet at a single spacetime event, e. The structed by another observer, Bob, distant from Alice, by a choice of za is not constrained and cannot be, for its variation translation labeled by parameters b°. gives the conservation laws (7). Thus, we have recovered the usual notion that interactions of particles take place at single 86O=bb{7Cb,O} (32) events in spacetime from the conservation of energy and mo- mentum. This is good because in quantum field theory con- Since momentum space is curved, and is non-linear, it fol- servation implies locality, and it is good to have a formulation lows that the "spacetime coordinates" 4 of a particle translate of classical interactions where this is also the case. in a way that is dependent on the energy and momenta of the However when we include terms proportional to z°, which particles it interacts with, 4 -)4,(0) .4(0)+44(0) where is to say when the observer is not at the interaction event, we see that the relationship between conservation of energy and 864(0) = bb{ „.r`n = -F bb E rtg+ (33) momentum and locality of interactions is realized a bit more subtly. The interaction takes place when the condition (28) is satisfied, that is at n separate events, separated from z" by This is a manifestation of the relativity of locality, ie local intervals spacetime coordinates for one observer mix up with energy and momenta on translation to the coordinates of a distant ob- A4(0)= zbLow E (29) server. This mixing under translations effect also entirely accounts These relations (28), (29) illustrate concisely the relativity for the separation of an interaction into apparently distinct of locality. For some fortunate observers the interaction takes events, because with bb = we see that At of (27) is equal place at the origin of their coordinates, so that e = 4(0) = 0 to 44(0) of (33). Thus, the observer whose new coordinates in which case the interaction is observed to be local. Any other we have translated to observes a single interation taking place observer, translated with respect to these, has a non-vanishing at 4 —> if (0) = 0. e and hence sees the interaction to take place at a distant set Thus, if I am a local observer and see an interaction to take of events. These are centered around z" but are not precisely place via a collision at my origin of coordinates, a distant ob- at the same values of the coordinates. That is the coordinates server will generally see it in their coordinates as spread out of particles involved in an interaction removed from the origin in space-time by (27). And vice versa. There is not a physi- of the observer by a vector e are spread over a region of order cal non-locality, as all momentum conserving interactions are seen as happening at a single spacetime event by some family Ax.;: IzIIrIk (30) of observers, who are local to the interaction. But it becomes The relationship (27) possess a very nice mathematical mean- impossible to localize distant interactions in an absolute man- ing too. Since the momentum space is in general curved the ner. Furthermore, all observers related by a translation agree proper way to define the conjugate coordinates is as elements about the momenta of the particles in the interaction, because of the cotangent bundle of T. The cotangent space based at under translations (32) sbkf, = 0. pl and the cotangent space based at 0 are different spaces in Note that if the curvature and torsion vanish there is no the general curved case. This expresses mathematically the mixing of spacetime coordinates with momenta under trans- relativity of locality. The hamiltonian particle coordinate xi lations, so there is an invariant definition of spacetime. Thus, represent an element of ciT while the interaction coordi- the flatness of momentum space is responsible for the notion nate being dual to the conservation law represent an element of an absolute spacetime, just as the additivity of velocity al- of TcrP. (27) represent a relation between these two spaces lows Newtonian physics to have an absolute time. and remarkably it can be shown 121 that indeed the coefficient Note also that the translations of spacetime coordinates de- avak evaluated when = 0 is the parallel transport opera- fine sections S on r. which extend from the origin of local tor, more precisely coordinates. These tell us how to translate events at the origin of the coordinates of an observer to coordinates measured by aigk P)aIp=sk = (1(k)-1)ab (31) a distant observer. These sections provide local and energy where t(k) is the parallel transport operator of vectors from 0 dependent definitions of space-time, relative to observers and to k introduced earlier therefore t(k)- I is the parallel transport energy scales. These sections are defined by Hamiltonian vec- operator of covectors from 0 to k. tor fields on r which are defined acting on functions f on r' EFTA01114602  6 by characterizing the limitation on the sharpness of coordinates of particles with energy po. Taking small po helps reduce the vbj= {hagc„ VIE(' af (34) - s ri:k5 +..) — relative-locality features but increases the quantum mechani- cal uncertainty. KE5(J) To understand the implications of this in more detail we will We can check that these commute and hence define submani- next specialize from the general case by imposing physical folds of T. We can define an inverse metric on the sections S conditions which restrict the geometry of momentum space. defined by gab(x,k). g(dx°,dxb). ha (k)(X.,"}{X,xb} (35) IV. SPECIALIZING THE GEOMETRY We note that this metric on the sections S is momentum dependent. Thus, we arrive at a description of the geometry As we have seen, the geometry of momentum space can of spacetime which is energy dependent. This metric is in code several kinds of deformations of the energy-momentum fact just the fiber metric at the point k where the fiber is the conservation laws, which take advantage of the flexibility to choose the metric, torsion, curvature and non-metricity of the cotangent plane at this points. But note that we can take all the kip = 0 in which case connection. This gives us an arena within which we can for- e(x,k = 0) =lird is the Minkowski metric. So observers mulate and test new physical principles. These impose con- straints on the choice of the geometry of P. To illustrate this who probe spacetime with zero momentum probes will see Minkowski spacetime. However, the coordinates of this in- we next turn from the general case to show how a set of sim- variant zero momentum section are non-commutative. ple principles restricts us to a one parameter set of momentum space geometries, and consequently, an almost unique set of (36) experimental predictions. Consider the following four increasingly strong physical Indeed, if one wants to describe the spacetime as probed by principles: the zero momentum probes this means that we desire to model the spacetime as the cotangent space of the origin. In order to t. The correspondence principle: Special relativity de- achieve this we need to parallel transport the event at p back scribes accurately all processes involving momenta to events at 0. This means that we interpret the coordinates z small compared to some mass scale me. While it is nat- as being covariant covector fields defined by ural to presume that me x mp, the scale me should be determined experimentally. (37) 2. The weak dual equivalence principle: The algebra of where 4 is the coordinate dual top with respect to the pois- combination of momenta, and hence the geometry of son bracket and living in the cotangent space at p. The Pois- P are universal; they do not depend on which kinds of son commutator can now be evaluated, it is related to the Lie particles are involved in interactions. bracket of the covariantly constant vector field on P and its expansion is given by 3. The strong dual equivalence principle or E = mc2: There is equivalence between the rest mass energy de- {z°d'} = @PM - tl,palvas fined by the metric and the inertial mass which involves = Ttel +ler dpeel + • • • (38) the connection. where we have expanded around p= 0 in the second equality. 4. Maximal symmetry. The geometry of momentum space However, when we go to zero momentum we can no longer is isotropic and homogeneous: invariant under Lorentz neglect the limitations on local measurements coming from transformations and invariant under "translations". the uncertainty principle of quantum mechanics. Quantum mechanically a particle of energy po can only be localized The geometry of momentum space is discussed in more de- with accuracy not greater than h/po, and this combines with tail in RI where it is precisely shown how these principles the relativity of locality expressed by (30) to give uncertainty lead to a unique geometry. This unique geometry is character- relation of the form ized by a constant: the dual cosmological constant which as the dimension of an inverse mass square. When this dual cos- mological constant is zero we recover usual special relativity, Ar ≥ p—o + Ixl Wino (39) when it is non zero, momentum space is genuinely curved. The first principle implies first that the metric of momen- tum space is a Lorentzian metric ( which we already have implicitly assumed). It also implies that the torsion and non- 5 We am usually familiar with such a phenomenon in the dual picture where gravity is turned on and spacetime is curved. in which case momentum metricity ofrir must be at least of order limp, while curva- space is represented by covector fields and the metric induced on each fiber ture must be of order of I /m2 is dependent on the spacetime point. It can be said that relative locality is The weak equivalence principle implies that the combina- a dual gravity. tion of momenta do not depend on the colors or charges of EFTA01114603  7 particles and is the same as the composition for identical par- well known that around the identity 0 we can set up a special ticles. For identical particles there is no operational way to set of coordinates: The Riemannian normal coordinates. In give an order of the combination rule if we have Bose statis- this coordinates the distance from 0 is given by the usual flat tics, therefore taken strongly, that is if we do not allow for any space formula hence the mass shell condition in this coordi- modification of the statistics of identical particles, this princi- nates is simply? ple also implies that the product is symmetric and hence the connection is torsionless. C(k) = 4-4-m2. 0. (43) The strong equivalence principle relates the metric and the connection of P by imposing that the connection is metric The Lorentz transformations that preserves the zero momenta compatible. The metric determines distance between two and the metric therefore acts in the usual manner in this coor- points in momentum space, and hence governs the mass shell dinate systems. relations of single particles, while the?connection determine Moreover under the hypothesis of homogeneity these coor- what is the straightest path between two points, and?hence is dinates can be extended to cover almost all the manifold P. determined by interactions which combine momenta. Since The Lorentz generators therefore satisfy the usual algebra. If they are given by different physics, they are in principle in- we assume in turn that the Lorentz transformations are canon- dependent. However, there are indications, to be discussed ical transformations preserving the Poisson bracket, they also in 131 that, at least in some cases, the non-metricity of the satisfy the usual Poisson algebra. That is given the boost and connection is related to violations of the equivalence between rotation generators Ni and j.k = 1, 2,3, we have relativistic energy and mass. It is intriguing that Einstein's ob- servation that in a relativistic theory E = ?ma appears to relate = Eijk MA, { Mi,NA = Eijk Nk, {114,Ni} = —eijk NA the metric and the connection of momentum space. (44) The first three principles impose therefore that the geome- and the generators act on the momentum space P through try of momentum space is entirely fixed by a Lorentzian met- these brackets. Moreover, as in special relativity, we assume ric. The connection is then the unique connection which is that the momentum space P splits into collections of orbits of torsionless and compatible with the metric. the Lorentz group: the zero momentum point, which is left in- The fourth principle of maximal symmetry is the most re- variant by Lorentz transformations; the positive and negative strictive. This could be called the principle of "special relative energy mass shells of massive particles; the light cone corre- locality" since it essentially implies a unique fixed dual geom- sponding to massless particles; and mass shells of tachyons of etry on momentum space. What it means in a nutshell is that imaginary mass. It is a direct consequence of this assumption the space of Killing vector fields of the metric form a ten di- that the function C(k) in the mass shell condition (18) must be mensional Lie algebra. This symmetry algebra also preserve a Lorentz scalar, so that all Lorentz observers agree what the the connection. This implies that there exists Lorentz trans- value of the invariant mass in2 is. It follows then that vectors formations A acting on P fixing the identity 0 such that corresponding to the infinitesimal Lorentz transformations are Killing vectors of the metric (4), so that the surfaces of con- Ath = A(p)e A(q). (40) stant distance from the origin (the point in P corresponding to This implies that the conservation law transform covariantly zero momentum) are orbits of action of Lorentz group. under Lorentz transformations Combining the assumptions of Lorentz symmetry with translation invariance or, equivalently, homogeneity then