Conservation laws and scattering for de Sitter classical particles
aa r X i v : . [ h e p - t h ] F e b Conservation laws and scattering for de Sitterclassical particles
S Cacciatori ‡ , V Gorini § , A Kamenshchik , k and UMoschella ¶ Dipartimento di Fisica e Matematica, Universit`a dell’InsubriaVia Valleggio 11, 22100 Como, Italy and INFN, Sez. di Milano, Italy Dipartimento di Fisica and INFN, via Irnerio 46, 40126 Bologna, Italy L D Landau Institute for Theoretical Physics of the RAS, Kosygin street 2, 119334Moscow, Russia
Abstract.
Starting from an intrinsic geometric characterization of de Sitter timelikeand lightlike geodesics we give a new description of the conserved quantities associatedwith classical free particles on the de Sitter manifold. These quantities allow fora natural discussion of classical pointlike scattering and decay processes. We alsoprovide an intrinsic definition of energy of a classical de Sitter particle and discuss itsdifferent expressions in various local coordinate systems and their relations with earlierdefinitions found in the literature. ‡ E-mail address: [email protected] § E-mail address: [email protected] k E-mail address: [email protected] ¶ E-mail address: [email protected] onservation laws and scattering for de Sitter classical particles
1. Introduction
Since the first pioneering observations of the luminosity-red shift relation of distanttype Ia supernovae [1, 2, 3, 4] it is by now accepted as an established fact that theexpansion of the universe is accelerated. This circumstance could be interpreted bysaying that there exists some kind of agent, dubbed dark energy, which exerts an overallrepulsive effect on ordinary matter (both visible and dark). This repulsion has longsince overcome the mutual attraction of the various parts of the latter, thereby beingresponsible for the present accelerated expansion. The nature of dark energy is to dateentirely mysterious. The only facts we know with reasonable certainty are that darkenergy contributes today in the amount of about 73 % (the exact figure depending onthe cosmological model adopted) to the total energy content of the universe, and thatits spatial distribution is compatible with perfect uniformity.The simplest possible explanation for dark energy which can be put forward is toassume that it is just a universal constant, the so called cosmological constant, denotedΛ. If we espouse this point of view, this would mean that the background arena for allnatural phenomena, once all physical matter-energy has been ideally removed, is notthe familiar flat Minkowski spacetime M (1 , . Instead, that it consists of the maximallysymmetric de Sitter spacetime dS whose radius R is related to Λ by the equation R = p / Λ. The actual value of Λ is extremely small in astrophysical and also ingalactic terms (Λ ≃ − cm − ), so that cosmic expansion has no significant effectsay on the structure of a typical galaxy, such structure being essentially controlled bythe material (in all its forms) composing the galaxy itself, by the mutual gravitationalattraction of the galaxy’s parts and by the galaxy’s angular momentum. On the otherhand, Λ has an essential effect on the distribution of matter on large cosmic scales,such as on the structure of the cobweb pattern of filaments and voids characterizing thearrangements of galaxies and galaxy clusters in the universe.It is not our purpose here to deal with the by now longstanding problem of thenature of dark energy and of why the dark energy content of the universe is, atthe present epoch, comparable with the universe’s ordinary matter content. See e.g.the reviews [5, 6, 7]. Instead, we adhere to the simple working hypothesis that thecosmological constant is a true universal constant, just like such are the speed of lightand Planck constant, say.In the approximation in which the effects of gravitation on the geometry ofspacetime can, at least locally, be neglected, the presence of a cosmological constantwould naturally lead to the problem of the formulation of the theory of special relativityin presence of a universal residual constant background curvature, namely of a de Sitterrelativity in place of the customary flat Minkowski one. Then, the symmetry group ofthe theory would be the de Sitter group SO (1 ,
4) (the Lorentz group in five dimensions)instead of the Poincar´e group, which is the contraction [8] of the latter arising in the limitΛ →
0. A considerable amount of work has already been performed in this direction, onservation laws and scattering for de Sitter classical particles M (1 , inwhich the de Sitter universe can be represented as an embedded four-dimensional one-sheeted hyperboloid. The relevant conserved quantities associated with free motion canthemselves be expressed in terms of the same lightlike five-vectors, as we do here. Then,it turns out that, in a given particle collision, the conservation of energy and momentumof ingoing and outgoing particles at the collision point can be expressed in terms of thecorresponding one particle conserved quantities before and after the collision.The structure of the paper is as follows. In Section 2 we first recall the expressionof the generators of the de Sitter symmetry group in terms of the flat coordinates of thefive-dimensional ambient Minkowski space. Then, by using Noether theorem appliedto the invariant action of a free massive particle we derive the set of the associatedconserved quantities K . Of such conserved quantities we give two different intrinsiccharacterizations. One in terms of the two lightlike vectors ξ and η of M (1 , that uniquelyidentify the given timelike geodesic. The other one in terms of either one of such vectorsand of a given point of the geodesic. These characterizations are independent of thechoice of any particular coordinate patch on the de Sitter manifold dS . We also findthe corresponding formulae for lightlike geodesics.In Section 3 we describe particle collisions and decays in terms of the conservedquantities introduced earlier. Precisely, we re-express the conservation of the totalenergy-momentum at the point of a collision as a conservation law for the total invariants K . In particular, the conservation equations can be given a perspicuous expression whichinvolves explicitly the collision point. The conservation of the invariants K allows us torelate the values of the energy and momentum at the point of collision to their valuesat any observation point. We do this by providing an explicit formula, valid both formassive and massless particles, which indeed relates the energy-momentum vector attwo arbitrary points on the geodesic. In particular, this formula applied to photonsyields the well-known frequency redshift relation.Section 4 is devoted to the definition of the energy of a free particle, both massiveand massless, by comparison of the corresponding geodesic to the reference geodesicassociated with a localized observer. This definition is itself intrinsic and does notmake reference to any particular coordinate patch. However, we also give the explicitexpression of the energy in terms of some specific coordinate choices on the de Sitter onservation laws and scattering for de Sitter classical particles
2. Conservation laws for de Sitter motion.
In what follows we will present our results by making reference to the (physical) four-dimensional de Sitter spacetime. However, as it will be evident from the discussion, ourformulae are completely general and valid in any dimension just by replacing 4 by d (and 5 by d + 1).The 4-dimensional de Sitter spacetime dS can be realized as the one-sheetedhyperboloid with equation dS = { X ∈ M (1 , , X = X · X = η AB X A X B = − R } (1)embedded in the 5-dimensional Minkowski spacetime M (1 , where a Lorentziancoordinate system has been chosen: X = X A ǫ A and whose metric is given by η AB = diag { , − , − , − , − } in any Lorentzian frame. The geometry of the deSitter spacetime is induced by restriction of the metric of the ambient spacetime tothe manifold: ds = ( η AB dX A dX B ) (cid:12)(cid:12) dS . (2)This is the maximally symmetric solution of the cosmological Einstein equations invacuo provided that R = p / Λ, with Λ >
0. The corresponding isometry group (therelativity group of dS ) is SO (1 , onservation laws and scattering for de Sitter classical particles M (1 , which is generated by the following ten Killing vector fields + L AB = (cid:18) X A ∂∂X B − X B ∂∂X A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dS . (3)Since the group acts transitively on the manifold dS it is useful to select a referencepoint (the origin) in dS as follows: X = (0 , , , , R ) . (4)Now consider a classical massive particle on the de Sitter universe. The usual actionfor geodesical (free) motion can be written by using the coordinates of the ambient five-dimensional spacetime as follows: S = − mc Z h ( V ) + a ( X + R ) i dλ ; (5)here λ → X ( λ ) is a parameterized timelike curve subject to the constraint X ( λ ) = − R as enforced by the Lagrange multiplier a ; V A ( λ ) = dX A /dλ is the corresponding velocity. V A ( λ ) is tangent to the curve X ( λ ) and therefore orthogonal to the vector X ( λ ) (inthe ambient space sense). The condition of tangentiality X · V = 0 has to be imposedalso on the initial conditions when solving the equations of motion. Consider now thegeneric infinitesimal isometry of dS X A X A + ω AB X B , (6)where ω AB are antisymmetric infinitesimal parameters. The action is invariant under (6)and using Noether theorem we find ten quantities that are conserved along the timelikegeodesics: K AB = m ( X A V B − X B V A ) R √ V = mR ( X A W B − X B W A ) == 1 R ( X A Π B − X B Π A ); (7) W A = dX A /dτ is the Minkowskian five-velocity relative to the proper time dτ = ds/c and Π A = m dX A dτ the corresponding Minkowskian five-momentum. Of these tenquantities only six are independent. Indeed, in order to specify a geodesic completelyone must for example assign the proper initial conditions, namely the initial point on dS and the initial velocity (at τ = 0, say). We note that the quantities (7) are of coursedefined also along particle trajectories which are not geodesics. However, in this casenot all of them (if any) will be constants of the motion. + The restriction of these operators to the de Sitter manifold is well-defined. This can be shown byintroducing the projection operator h and the tangential derivative D as follows: h AB = η AB + X A X B R , D A = h AB ∂ B = ∂ A + X A R X · ∂. It follows that L AB = X A ∂ B − X B ∂ A = X A D B − X B D A . onservation laws and scattering for de Sitter classical particles K AB which are independent on the choice of any particular coordinate patch on dS . Wedo this by exploiting an elementary way to describe the de Sitter timelike geodesics: incomplete analogy with the great circles of a sphere that are constructed by intersectingthe sphere with planes containing its center, the de Sitter timelike geodesics can beobtained as intersections between dS and two-planes containing the origin of M (1 , and having three independent spacelike normals. Each such two-plane also intersectsthe forward lightcone in M (1 , (the asymptotic cone) C + = { X ∈ M (1 , , X = 0 , X > } , (8)along two of its generatrices. Any two future directed null vectors ξ and η lying onsuch generatrices (see figure) can be used to parameterize the corresponding geodesic interms of the proper time as follows [22]: X ( τ ) = R ξ e cτR − η e − cτR √ ξ · η . (9) Figure 1.
Construction of a timelike geodesic of the de Sitter manifold. Theasymptotic future lightcone of the ambient spacetime; the vectors ξ , η belonging to C + play the role of momentum directions. Then, by inserting (9) into (7) we find that the conserved quantities have a verysimple expression, homogeneous of degree zero, in the components of the vectors ξ and η : K AB = mc ξ A η B − η A ξ B ξ · η . (10) onservation laws and scattering for de Sitter classical particles ∗ K = K ( ξ,η ) = mc ξ ∧ ηξ · η , (11)in the frame { ǫ A } that has been chosen in the ambient space. We normalize thedimensionless vectors ξ and η according with ξ · η = 2 m k , (12)where k is a constant with the dimensions of a mass whose value can be fixed accordingto the specific convenience. With this normalization, formulas (9), (10) and (11) writerespectively X ( τ ) = kR m (cid:0) ξe cτR − ηe − cτR (cid:1) , (13) K AB = k c m ( ξ A η B − ξ B η A ) , K ( ξ,η ) = k c m ( ξ ∧ η ) . (14)The replacements ξ −→ µξ , η −→ µ − η ( µ > τ variable. Therefore, the normalizations of ξ and η are fixed separately by equation (12) (in which a given choice has been made for thepositive constant k ) and by selecting the point on the geodesic corresponding to zeroproper time. With these qualifications it turns out that the pair ( ξ, η ) depends on sixindependent parameters. Then (14) shows once more that only six of the ten constantsof motion K AB are independent (in the appendix we illustrate this fact with an explicitexample).Formula (11) (or equivalently formula (14)) provides our first intrinsiccharacterizations of the constants K AB . The second characterization that we displaybrings about an arbitrary fixed point X ( τ ) on the geodesics. Indeed, from (13) one hasthe relation η = ξ − mkR ¯ X , (15)where ¯ X = X (0), which allows to rewrite the geodesic (13) in the alternative form X ( τ ) = ¯ Xe − cτR + kRξm sinh cτR . (16)Inserting (15) into (14) and using (16) gives K AB = kcR ( X A (0) ξ B − X B (0) ξ A ) = kcR e cτR ( X A ( τ ) ξ B − X B ( τ ) ξ A ) . (17)As before, we can introduce the tensor K = K ξ,X = kcR e cτR ( X ( τ ) ∧ ξ ) . (18)The normalization (12) and Eq. (15) imply ξ · ¯ X = − Rmk . (19) ∗ ξ and η denote here the covariant one-forms associated to the null vectors; we use the same symbolfor a vector and its dual. onservation laws and scattering for de Sitter classical particles K , in its two alternative expressions (11) and (18) will play an importantrole in the following.To perform the massless limit we set m = kǫ, τ = σc ǫ (20)and let ǫ −→ X ( σ ) = ¯ X + ξσ with ξ · ¯ X = 0 (21)where σ is an affine parameter. Therefore, a lightlike geodesic is characterized by onelightlike vector which is parallel to the geodesic and by the choice of an initial eventthat uniquely selects the particular geodesic among the infinitely many pointing in thatdirection. The conserved quantities are still given by formula (17)˜ K AB = kcR ( X A (0) ξ B − X B (0) ξ A ) = 1 R ( X A Π B − X B Π A ) , (22)where Π A = kc dX A dσ is the Minkowskian five-momentum of the zero mass particle. Thereis of course no analogue of formulas (11) and (14) because ξ and η coincide in the masslesslimit. An alternative standard way to arrive at formulas (22) starts from rewriting theaction for a massive particle in the first order formalism: S [ e, γ ] = k Z γ (cid:20) e V + a ( X + R ) (cid:21) dλ + 12 k m c Z γ edλ , (23)where e is a function of λ and k is once more a constant with the dimensions of a mass.The equations of motion are obtained by varying the action with respect to e and tothe curve γ . The action for massless particles is obtained by setting m = 0 in Eq. (23)and the corresponding equations of motion are V = 0 , ddλ (cid:18) e V A (cid:19) = 0 . (24)Introducing an affine parameter σ such that dσ = ce ( λ ) dλ , the general solution is (21).The conserved quantities can be determined as before by means of Noether theorem,giving (22). The setup that we have just described can also be employed to provide a fresh look to deSitter quantum mechanics and field theory. Indeed, the variables on the cone in M (1 , that we have been using to describe the geodesics can be employed to parameterize thephase space pertinent to elementary systems. Then one can invoke his favourite method,like geometric quantization [24, 25] or the method of coadjoint orbits [26] to obtain aquantum description of such elementary systems. In doing this, a substantial differencewill arise when quantization deals with massless particles and the method will fail toprovide a de Sitter covariant theory. This problem has been known for a long time and onservation laws and scattering for de Sitter classical particles L containinginitially a uniform distribution of a large number of identical point particles of mass m which are all at rest relative to the endpoints (walls) of the box. “Rigid” meanshere that we assume the internal forces holding the box together to prevent it fromparticipating unhindered to the de Sitter expansion, so that a local geodesic observer O comoving with the box sees that the spatial extension of the latter does not changein time. In other words, the walls of the box are not receding away during the cosmicexpansion. In particular, if we assume the observer O to sit at the midpoint of the box,the worldlines of the endpoints of the box will not be geodesics and the box itself willshrink compared to the comoving spatial coordinates. As to “initially at rest” it meansthat, at a given initial time, all the particles inside the box are assumed to move withzero velocity in the comoving frame. Then, because of the expansion of the universe,the observer O will see the particles move away from each others and eventually starthitting the walls of the box, bounce back and collide with each other. By virialization,the result is that (after a long time and in a nonrelativistic framework) they will reacha thermodynamic equilibrium at a temperature T c = H L m/ . k , where H is theHubble constant and k the Boltzmann constant (see appendix B).Now consider a quantum scalar field in dS , which we assume to be in its groundstate (the de Sitter vacuum [17, 31]). Due to the interaction with the spacetimecurvature the vacuum fluctuations generate real particles with a thermal spectrum attemperature T q = ~ c/ πRk ([32, 19]). It seems reasonable to assume that the lightestallowed mass (the dS mass) for such a field is the one corresponding to the Comptonwavelength of the order of the de Sitter radius R , giving m = m dS = h/Rc which wouldcorrespond to a quantum temperature T q = m dS c / π k ≃ m ds R H /k . This is of thesame order of magnitude of the classical temperature T c = H R m dS / . k calculatedbefore, for classical particles of mass m dS in a box extending to the cosmological horizon.This allows us to interpret the classical temperature T c = H L m/ . k as a classicalanalogue, in de Sitter spacetime, of the Hawking-Unruh effect.We have derived the above analogy for a two dimensional de Sitter spacetime.However, the above considerations can be easily extended to dS provided the classicalparticles are taken to be rigid spheres of some small but nonzero radius.In addition, the expression for T c has been derived under the assumption thatthe classical particles are non relativistic. Strictly speaking, this approximation is not onservation laws and scattering for de Sitter classical particles R ≃ cm , the quantum de Sitter temperature T q is of theorder of 10 − o K, whereas the de Sitter mass m dS is of the order 10 − g . Therefore, theaverage velocity of our de Sitter particles at the de Sitter temperature is comparablewith the speed of light, so that they are highly relativistic. This fact can be readilyunderstood by noting that, as the walls of the box approach the cosmological horizon,their speed relative to the comoving coordinates approaches c . Therefore, when thefirst of the de Sitter particle hits the wall it bounces back with a highly relativisticspeed, which is then transmitted to the other dS particles through particle collisions andfurther collisions with the walls themselves. Nonetheless, since we are only concernedwith orders of magnitude, we still claim that our crude estimates relating classical andquantum de Sitter temperatures are justified.Finally note that our effective vacuum particles can in some sense be viewed upon asthe lightest detectable particles in a de Sitter background. Indeed, in such a backgroundthe largest uncertainty in position is ∆ x ≃ R , whereas ∆ p ≃ mc , so that the Heisenbergprinciple gives m & h/Rc . It is not clear what this exactly means, though one may boldlysuggest that quantum effects in de Sitter forbid the existence of lighter particles. Morepresumably, it may mean that the semiclassical description of lightest particles is toona¨ıve and that strong quantum effects do come into play.
3. Collisions and decays.
We consider the collision of two ingoing particles which gives rise to the production ofa certain number of outgoing particles b + b −→ c + c + . . . + c N . (25)The particles b i , with masses m i , are described by geodesic curves ending at the collisionpoint ¯ X , which is also the starting point of the N geodesics describing the outgoingparticles c f with masses ˜ m f . We assume the collision point to be the common zero ofthe proper time of all particles involved in the process, namely ¯ X = X i (0) = X f (0).Denoting by ( χ i , ζ i ) and by ( ξ f , η f ) the pairs of normalized null vectors parameterizingthe ingoing and outgoing particles we have ζ i = χ i − m i k i R ¯ X , i = 1 , η f = ξ f − m f k f R ¯ X , f = 1 , , . . . , N , (26)and the quantities which are conserved along each geodesic are K i = k i cR ¯ X ∧ χ i , i = 1 , K f = k f cR ¯ X ∧ ξ f , f = 1 , , . . . , N . (27)Solving the collision problem amounts to finding the outgoing vectors ξ f given theingoing ones χ i . At the collision point the total covariant energy-momentum four-vectormust be conserved: π µ + π µ = M X f =1 π µf . (28) onservation laws and scattering for de Sitter classical particles x µ , µ = 0 , , ,
3, so that π µ = m dx µ dτ (respectively, π µ = kc dx µ dσ ) for any given massive particle of timelike (respectively,massless particle of lightlike) worldline x µ ( τ ) (respectively x µ ( σ )) on dS . In termsof the embedding in M (1 , , at the collision point ¯ X we have, for a given particle, K AB | X = ¯ X = 1 R (cid:18) X A ∂X B ∂x µ − X B ∂X A ∂x µ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =¯ x π µ , (29)where X A = X A ( x µ ( τ )) or X A = X A ( x µ ( σ )) depending on whether the particle ismassive or massless. By summing over all ingoing and outgoing particles and using (28)we find the simple relation K + K = N X f =1 K f . (30)Similarly, for the decay b −→ c + c + . . . + c N of a single particle K = N X f =1 K f . (31)Note that K AB K AB = − m c . This relation replaces in dS the Minkowskian one π µ π µ = m c . Then, choosing the normalization constants k i and k f equal for allparticles, Eq.(27) allows us to write the conservation equations (30) and (31) respectivelyas ( χ + χ − N X f =1 ξ f ) ∧ ¯ X = 0 , (32)( χ − N X f =1 ξ f ) ∧ ¯ X = 0 . (33)Though equations (30) and (32) are equivalent to equation (28) they have the advantageof being expressed in an intrinsic form. To further clarify their meaning it is interestingto find the explicit expressions of the null vectors χ i and ξ f corresponding to a particularchoice of the collision event ¯ X . For example, choosing ¯ X = X equation (32) becomesequivalent to χ µ + χ µ = N X f =1 ξ µf , µ = 0 , , , . (34)From Eq. (26) ζ µ = χ µ µ = 0 , , , , and ζ = χ − mk (35)(we have omitted the index i = 1 , χ and ζ are nullvectors, if m = 0 this relation implies χ = − ζ = mk . Therefore, we have χ = (cid:16) χ , ~χ, mk (cid:17) , ζ = (cid:16) χ , ~χ, − mk (cid:17) , (36) onservation laws and scattering for de Sitter classical particles χ ) − ( ~χ ) = m k . (37)By using the parametrization (13) it follows that m dX µ dτ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = kcχ µ = q µ , m dX dτ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = 0 , (38)with q = ( q ) − ( ~q ) = m c . (39)In a small neighborhood of X in dS we choose local coordinates x µ defined by x µ = X µ , µ = 0 , , ,
3. Since the plane X = R is tangent to dS at X we have ∂X /∂x µ | X = 0 so that, at X , the metric of dS , expressed in terms of the coordinates x µ , is given by ds | X = ( η AB dX A dX B ) | dS ,X = η µν dx µ dx ν . Then the x µ are locallyLorentzian at X and dX µ /dτ | τ =0 = dx µ /dτ | τ =0 where x µ ( τ ) is the parametrizationof the geodesic at X . Hence, equation (38) tells us that q µ can be interpreted as thecomponents (in the chosen frame) of the Lorentzian four-momentum at X of the particlemoving along the geodesic x µ ( τ ). This interpretation applies to zero mass particles aswell.Then, denoting by q µi and ˜ q µf respectively the four-momenta at the collision point¯ X = X of the incoming and outgoing particles relative to the coordinates x µ we have χ i = 1 kc ( q i , ~q i , m i c ) , i = 1 , , (40)and ξ f = 1 kc (˜ q f , ~ ˜ q f , ˜ m f c ) , f = 1 , , . . . , N (41)and the conservation equation (34) becomes q µ + q µ = N X f =1 ˜ q µf (42)expressing once more the equivalence of (32) to (28). Similar considerations apply tothe decay (31).The expressions of the incoming and outgoing null vectors χ i and ξ f in the generalcase, when the collision point ¯ X is arbitrary, can be obtained by applying to (40) and(41) an arbitrary five-dimensional Lorentz transformation.In conclusion, it is worthwhile noting that since any Lorentzian manifold is locallyinertial, at the classical level the conservation laws in de Sitter point particle collisionsexpress nothing more than the usual total energy-momentum conservation in theprocess, so that Λ plays no role here. The situation is drastically different in the quantumcase due essentially to the spread of wave packets. For example, in de Sitter particledecay the decay amplitude depends on Λ and the presence of curvature allows in somecases for a non-zero probability for an unstable particle of mass m to decay into particles onservation laws and scattering for de Sitter classical particles m , a process which is strictly forbidden in Minkowskispacetime due to energy-momentum conservation [20].Finally, as regards the geodesic motion of a single particle, it is important to remarkthat the explicit expressions ξ = 1 kc ( q , ~q, mc ) ,η = 1 kc ( q , ~q, − mc ) , (43)of the components of the pair of normalized null vectors ξ and η characterizing theparticle geodesic X ( τ ) when ¯ X is chosen at the origin (4) as well as their correspondingexpressions for arbitrary ¯ X , which are obtained by applying to (43) a suitable five-dimensional Lorentz transformation, depend solely on the choice of ¯ X and do not makereference to any particular local coordinate system on dS . Instead the introductionof one such suitable system about ¯ X is made necessary for the correct physicalinterpretation of the components of ξ and η . In a collision process any outgoing particle is not detected, and its properties measured,at a collision point ¯ X . Instead, the detection takes place at some other event far awayfrom ¯ X . In particular, if we measure the energy and the momentum of the particle,we need a formula which relates these quantities at the point of measurement to thesame quantities at the production point. To avoid being monotonous we illustrate theprocedure with a lively example. Consider the pp scattering p + p −→ p + p + a + b + c , and suppose that we are searching for an intermediate process p + p −→ p + p + Z −→ p + p + a + b + c , (44)where Z is a massive particle decaying into the triple a, b, c with a very short lifetime,so that it cannot be directly detected. Then, by (31) K Z = K a + K b + K c , so that 2 K Z := ( K aAB + K bAB + K cAB )( K ABa + K ABb + K ABc ) = − m Z c must hold. Assume we look at a large number of such processes and that we are ableto measure experimentally K aAB , K bAB and K cAB in each individual process. Then,plotting the number density of processes dn/dK as a function of the invariant mass Q Z := p − K Z , we should find a resonance at Q Z = m Z c . As a well-known example ofa reaction of the type (44) we may mention the process p + p −→ p + p + Z −→ p + p + π + + π − + π , onservation laws and scattering for de Sitter classical particles Z can either be one of the mesons η (547) or ω (782) or some broader resonance.See e.g. ref.[33].The experimental problem of measuring the quantities K a , K b , K c could be tackled asfollows. To fix ideas, consider just one particle, which we suppose to detect at an eventwhose local coordinates are x µ . Barring intrinsic indeterminacies the detection measuresthe position x µ and the momentum π µ ( τ ). Then K AB is determined by (29) as K AB = K AB | x = x = 1 R (cid:18) X A ∂X B ∂x µ − X B ∂X A ∂x µ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x π µ ( τ ) . (45)This formula can be used to relate the covariant momentum at the point of measurementto the one at the collision point. Indeed, if x µ are the local coordinates of the collisionevent and π µ (0) the covariant momentum of the particle at the same point, then K AB | x = x = 1 R (cid:18) X A ∂X B ∂x µ − X B ∂X A ∂x µ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x π µ (0) . (46)By multiplying both sides of this equation by1 R (cid:18) X A ∂X B ∂x ν − X B ∂X A ∂x ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x , and summing over A and B we find π µ (0) = G µν ( x , x ) π ν ( τ ) . (47)Here G µν ( x , x ) = − R g µρ ( x ) (cid:18) X A ∂X B ∂x ρ − X B ∂X A ∂x ρ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x (cid:18) X A ∂X B ∂x ν − X B ∂X A ∂x ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x , where g µν ( x ) = η AB ∂X A ∂x µ ∂X B ∂x ν , is the metric on dS in the given coordinates. Formulas (29), (30), (31) and (47) holdfor lightlike particles as well.As an example, choose the local coordinates x µ = { ct, x i } to be the flat ones: X ( t, x i ) = X = R sinh ctR + ~x R e ctR ,X i = e ctR x i ,X = R cosh ctR − ~x R e ctR . (48)If for simplicity we restrict ourselves to the two dimensional case in flat coordinates andchoose x = (0 ,
0) and x = ( ct, x ) we find G µν ( x , x ) = e ctR xRxR e ctR cosh ctR + e ctR x R ! . (49)In particular, consider the case of a photon transmitted from x to x . To expressthe momenta in terms of inertial frames at rest in each point with respect to the onservation laws and scattering for de Sitter classical particles e = cdt , e = e ctR dx . The inertialenergy-momentum ˆ π µ has components ˆ π = π and ˆ π = e ctR π . In particular, in x ,ˆ π µ (0) = π µ (0) andˆ π (0) = ˆ π + e ctR xR ˆ π , (50)ˆ π (0) = xR ˆ π + (cid:18) cosh ctR + e ctR x R (cid:19) ˆ π . (51)Obviously x cannot be any point, but must lie on a lightlike geodesic starting from x .It can be easily found puttingˆ π (0) = ˆ π (0) = hν c , ˆ π = ˆ π = hνc , in (50) and (51) and solving for x = x ( t ). This gives x ( t ) = R (1 − e − ctR ) . Using this in (50) we finally obtain ν = e − ctR ν . (52)This is the redshift measured by the observer at x : the photon emitted with frequency ν at x is perceived as a photon of frequency ν by the observer at x . onservation laws and scattering for de Sitter classical particles
4. Energy.
In Einstein’s special relativity the energy of a particle is defined (and measured) relativeto an arbitrary given Lorentz frame, it being the zero component of a four-vector. Inphysical terms, a Lorentz frame can be seen as an ideal global network of (free) particlesrelatively at rest and carrying clocks that stay forever synchronized. This picture doesnot extend to the de Sitter case where frames are defined only locally.However, the maximal symmetry of the de Sitter universe allows for the energyof a pointlike particle to be defined relative to just one reference massive free particleunderstood conventionally to be at rest (the sharply localized observer). Below we willcompare this definition with the ones obtained in various coordinate patches by a morestandard Lagrangian approach.The procedure amounts to fixing arbitrarily a timelike reference geodesic (thegeodesic of the particle “at rest”). Let us denote by u and v the future oriented nullvectors which identify such geodesic; the energy of the free particle (9) with respect tothe reference geodesic is defined as follows: E = E ( ξ,η ) ( u, v ) = − c K ( ξ,η ) ( u, v ) u · v . (53)We have that E ( u,v ) ( ξ, η ) = E ( ξ,η ) ( u, v ) which can be interpreted as the symmetrybetween the active and passive point of view. In particular, the proper energy is E ( ξ,η ) ( ξ, η ) = mc , as it should be. To further elaborate this definition let us choosean origin ¯ Y = Y (0) on the reference geodesic and denote by λ the scalar such that u · v = 2 λ , v = u − λ ¯ YR . (54)As before, fixing λ removes the scale arbitrariness in the choice of u and v and it followsthat Y ( τ ) = R u e cτR − v e − cτR λ = ¯ Y e − cτR + Ruλ sinh cτR (55)Proper times in (16) and (55) are of course not to be confused. Taking into accountEqs. (15) and (54) it follows that E = − mc ( ξ ∧ η )( u, v )( ξ · η )( u · v ) = − kc λR ( ξ ∧ ¯ X )( u, ¯ Y ) . (56)Finally, by inserting into this expression Eq. (19) and the analogous relation λ = − R ( u · ¯ Y ) (57)we get the expression E = mc ( u · ¯ X )( ξ · ¯ Y ) − ( ¯ X · ¯ Y )( ξ · u )( ξ · ¯ X )( u · ¯ Y ) , (58)which is an alternative form of (53). onservation laws and scattering for de Sitter classical particles E of the particle to a given coordinate patch.This can be done as follows. Suppose a local frame ( t, x i ) has been selected so that theembedding of dS in M (1 , is given by X A ( P ) = X A ( t, ~x ); to fix ideas let us performthis choice so that the event t = 0 , ~x = 0 is the ”origin” X of the de Sitter manifold.Then, we define the energy E of a particle relative to the given frame as the energyof the particle w.r.t. the particle (observer) at rest at the origin, i.e. w.r.t. the referencegeodesic passing through the origin with zero velocity ( ~x (0) = 0, d~xdt (0) = 0). We workout a few explicit examples. The flat coordinate system { t, x i } is defined by (48). In these coordinates the de Sittergeometry is that of a flat exponentially expanding Friedmann universe: ds = c dt − e ct/R δ ij dx i dx j = c dt − a ( t ) δ ij dx i dx j . (59)The reference geodesic (55) through the origin Y ( t = 0 , x i = 0) = (0 , , , , R ) with zerovelocity is uniquely associated to the choices ¯ Y = (0 , , , , R ) and u = λ (1 , , , , Y and u , Eq. (58) is explicitly written as follows: E = kc R ( ξ ¯ X − ξ ¯ X ) . (60)Noting that ξ = mkR (cid:18) ¯ X + Rc dX (0) dτ (cid:19) . (61)and using Eq. (48) we readily find E = mc dtdτ − cR x i p i = mc q − a ( t ) v c − cR x i p i (62)where we have set v i = dx i dt , p i = − me ct/R dx i dτ = − ma ( t ) v i q − a ( t ) v c . (63)In Section 5 we will show that p i = − a ( t ) p i = mv i q − a ( t ) v c (64)can be interpreted as the de Sitter version of the linear momentum (in flat coordinates).In the limit R −→ ∞ , (62) and (64) go over into the usual Minkowskian expressions ofthe energy and momentum.That (62) can be interpreted as the correct de Sitter energy of the particle isconfirmed by noting that it is the conserved quantity associated to the invariance of theparticle action (5) under time translation. Indeed, since in flat coordinates the spatial onservation laws and scattering for de Sitter classical particles e ctR , the expression of aninfinitesimal symmetry under time evolution is t −→ t + ǫ,x i −→ x i − cR x i ǫ. (65)The action S = − mc Z r − e ctR v i v j c δ ij dt (66)is invariant under (65) and, by Noether’s theorem, the corresponding constant quantityis precisely (62).For a massless particle, using Eq. (20) we find E = kc dtdσ − cR x i p i , (67)where we have set p i = − kce ctR ( dx i /dσ ) . (68)In particular, note that d~xdt · d~xdt = c e − ctR . To find the relation between t and σ , we takethe derivative with respect to σ of the relation defining the cosmic time X + X = Re ctR , and use (21) to obtain dtdσ = 1 c e − ctR ( ξ + ξ ) . (69)Inserting this into the expression of the energy we find E = kc ( ξ + ξ ) e − ctR − cR x i p i , p i = − k ( ξ + ξ ) e ctR dx i dt . (70)In the flat limit R −→ ∞ we have E −→ kc ( ξ + ξ ) so that, if we associate a frequencyto the de Sitter massless particle, we have hν = kc ( ξ + ξ ) and finally E = hνe − ctR − cR x i p i , p i = − hνc e ctR dx i dt . (71)In the limit R −→ ∞ we obtain the usual Minkowskian expressions E = hν , ~p = hνc ~n . (72) Let { t, ω α } , α = 1 , . . . ,
4, be such that ( X = R sinh ctR ,X α = R cosh ctR ω α , (73)where ω α ω β δ αβ = 1, that is the ω α is a vector on the sphere S of unit radius. Concretely ω = sin χ sin χ cos χ ω = sin χ sin χ sin χ ,ω = sin χ cos χ ,ω = cos χ (74) onservation laws and scattering for de Sitter classical particles ds = c dt − R cosh ctR n(cid:0) dχ (cid:1) + sin χ h(cid:0) dχ (cid:1) + sin χ (cid:0) dχ (cid:1) io . (75)The initial point and the lightlike vector identifying the reference geodesic are oncemore ¯ Y = X and u = λ (1 , , , , ξ in terms of X and dX/dτ it follows that E = mc ω − R c v sinh ctR q − R c cosh ctR v α v β δ αβ , (76)where α, β = 1 , , , v α = dω α /dt .Again, expression (76) can be recovered as the conserved quantity associated to a timetranslation plus a rescaling of the ω ’s that together leave invariant the action S = − mc Z r − R c cosh ctR Ω ij w i w j dt , (77)where d Ω = Ω ij dχ i dχ j and w i = dχ i dt . This is the coordinate system originally introduced by de Sitter in his 1917 paper [34].It describes a portion of the de Sitter manifold as follows: X = R q − r R sinh ctR ,X i = r i , i = 1 , , ,X = R q − r R cosh ctR , (78)where r = P i =1 r i r i . With these coordinates the metric exhibits a bifurcate Killinghorizon at r = R : ds = (cid:18) − r R (cid:19) c dt − dr (cid:0) − r R (cid:1) − r ( dθ + sin θdφ ) . (79)We choose the same origin and reference geodesic as before and find E = mc (cid:18) − r R (cid:19) q − r R − ( ~r · ˙ ~r ) ( R − r ) c − ˙ ~r · ˙ ~rc . (80)In these coordinates, the action for a massive free particle is S = − mc Z s − r R − ( ~r · ˙ ~r ) ( R − r ) c − ˙ ~r · ˙ ~rc dt . (81)Here the dot means derivation with respect to t . This action is invariant under timetranslations and the associated conserved energy coincides with (80).We leave it as an exercise to find the analogue of expressions (76) and (80) for masslessparticles. As mentioned in the introduction, a fourth example can be found in [16] wherethe expression of the energy is given in terms of stereographic coordinates. onservation laws and scattering for de Sitter classical particles
5. A possible definition of momentum.
Whereas, as shown in section 4, the energy of a particle can be defined relative to anarbitrary fixed reference geodesic, and therefore in a frame independent manner, nosimilar characterization can be given for the linear momentum. Instead, a definition ofmomentum for a de Sitter particle necessarily requires the selection of some coordinatesystem. In Minkowskian relativity energy and momentum are defined as the conservedquantities associated with the invariance of the action respectively under infinitesimalinertial time and space translations. In addition, in the de Sitter case, for a class ofreference frames which in the limit R −→ ∞ become inertial, we have seen that theparticle energy arises again as the conserved quantity associated to time translations(depending on the choice of the particular coordinate system, the latter may or maynot act on the space coordinates as well). Therefore, it is natural to attempt to definethe de Sitter momentum in any such frame as the conserved vector quantity which isassociated with infinitesimal space translations. This requires fixing the origin of thecoordinate system, since, due to the presence of curvature, changing the origin affectsthe definition of space translations. Of course, in order for such definition of momentumto be consistent, one must make sure to recover the usual Minkowskian momentum inthe limit R −→ ∞ . Then we search for Lorentz transformations in the embeddingspacetime M (1 ,
4) which generate spatial translations of the origin, once the latter hasbeen identified. Specifically, let ( t, x i ) be local coordinates, X A ( t, x i ) the embeddingfunctions and O ≡ X A (0) the origin. We consider the submanifold defined by t = 0.It defines an hypersurface of dS which identifies an osculating hyperplane in O . Anyinfinitesimal Lorentz transformation which leaves the osculating plane invariant definesan infinitesimal translation of O which is transverse to the reference geodesic definingthe energy. As stated above we use such transformations to define the momentum. Weillustrate again the construction for the coordinate systems of section 4. In this case, the t = 0 surface defines the osculating plane X + X = R . The Lorentztransformations leaving this hyperplane invariant are generated by the infinitesimalrotations of the form v ∧ v i , where v = (1 , , , , −
1) and v i = (0 , ~e i , ~e i is thestandard basis of R . Then the momentum is p i := K ( v, v i ) , (82)which expressed in coordinates gives p i = − me ctR dx i dτ . (83)This expression coincides with (63) and corresponds to the conserved quantitiesassociated to the invariance of the action under spatial translations x i x i + a i . Fora massless particle we should use ˜ K in place of K , finding the second of formulas (71) onservation laws and scattering for de Sitter classical particles v i , v . However such ambiguity can be fixed by requiring to obtain the usual Minkowskianexpression in the R −→ ∞ limit. In spherical coordinates the t = 0 slice correspond to X = 0. We have v = (0 , , , , v i as before. Then K ( v, v i ) = mc ξ η i − η ξ i ξ · η , (84)and, in term of the coordinates, p i = mR cosh ctR (cid:18) ω dω i dτ − ω i dω dτ (cid:19) . (85)The flat limit leaving the origin invariant can be easily performed and it gives the correctmomentum for Minkowski spacetime. Again the t = 0 slice defines the osculating hyperplane X = 0 and K ( v, v i ) is oncemore given by (84). Then p i = m r − r R dr i dτ cosh ctR − r − r R r i R c sinh ctR − q − r R r i R X k =1 r k dr k dτ cosh ctR . (86)
6. Conclusions.
In this paper we have taken the stance that in the absence of gravitation the spacetimearena in which all physical phenomena take place is the (maximally symmetric) fourdimensional de Sitter manifold dS . Then, barring discrete spacetime operationssuch as space reflection and time reversal, which are not exact symmetries of nature,the corresponding relativity group is the connected component of SO (1 , onservation laws and scattering for de Sitter classical particles c ) and a length scale (the cosmological constant Λ). The actual values of theseparameters in nature are of course not fixed by the basic symmetries and must be foundexperimentally. And, indeed, though limiting values of c and Λ (such as c = 0 , ∞ and/orΛ = 0 , ∞ ) cannot be excluded a priori, it comes as no surprise that the values of c andΛ determined by the observations are well defined and finite. It would be surprising ifit were otherwise! It is a different (and to some extent metaphysical) question why c and Λ have the values they have and not others. However we are not concerned withthis problem here.It is then clear that, if Minkowski space should be replaced by de Sitter space and,correspondingly, the Poincar´e group by the de Sitter group, one is naturally led to areformulation of the theory of special relativity on these grounds [10, 11, 12, 13, 14].However, compared to the Minkowski case, this task presents certain complicationswhich are essentially connected with the fact that in the de Sitter case there exists noclass of privileged reference frames as are the Minkowskian inertial ones. Indeed, theassociated coordinate systems of any such hypothetical class of equivalent frames shouldrespect the basic symmetries of the spacetime manifold. In particular, the homogeneityrequirement would imply the coordinate transformation between any two such equivalentframes to be affine [36], and this is impossible if the underlying manifold is curved.Therefore, the absence of a privileged class of equivalent frames suggests that, in deSitter relativity, it would be desirable, whenever possible, to characterize significantphysical quantities in an intrinsic way, namely in a manner independent of the choiceof any particular coordinate patch. In this paper we have accomplished this for anyset of independent conserved quantities along the geodesic motion of a free de Sitterparticle and for the overall conservation of the total constants of the motion in anyparticle collision. In particular, we have also been able to give an intrinsic definition ofenergy of a de Sitter particle, as the energy of any such particle relative to an arbitraryselected reference particle chosen conventionally to be at rest. In this respect, it isimportant to stress that in the same way as there is a unique Lorentzian (i.e. relativistic)generalization of the kinetic energy of a Galilean (i.e. nonrelativistic ) particle, the deSitter energy is the unique de Sitter generalization of the Lorentzian energy, which arisesfrom the appearance of an intrinsic residual curvature of the spacetime manifold.We remark that, due to the smallness of Λ, the actual corrections to Einstein’sspecial relativity which are brought about by the presence of the cosmological constant,such as for instance those embodied in formula (47), are utterly tiny at scales oflaboratory experiments performed on earth or in space. Specifically, formulas (50)and (51) tell us that to first order in 1 /R the fourmomentum of a particle traveling adistance x is altered by a relative amount of the order x/R . In particular, for example,since R ≃ cm, the order of magnitude of the fractional frequency shift of a photon onservation laws and scattering for de Sitter classical particles x in the de Sitter universe is∆ νν ≃ − for x ≃
10m (87)and ∆ νν ≃ − for x ≃ . (88)The figure ∆ π/π ≃ − would be relevant also for particles produced in a collision in aparticle accelerator such as the Tevatron or LHC since the distance between the collisionpoint and the detector is typically of the order of a several meters. By comparison, werecall that in the classical experiment by Pound and Rebka [37, 38, 39] devised tomeasure the gravitational shift of a photon falling in the earth’s gravitational field wehave ∆ νν ≃ − (89)whereas in experiments with atomic clocks which monitor the variation of thegravitational potential of the sun at the location of the earth between perihelion andaphelion [40] we have∆ νν ≃ − . (90)The precision with which the value of ∆ ν/ν has been measured in the Pound and Rebkaexperiment is of the order of one percent, whereas values of ∆ ν/ν in the range of thefigure of formula (90) in experiments with atomic clocks as are mentioned above are nowtested with a precision of the order of almost one part per million. This shows that,whereas there is no chance to realistically test formula (47) for Λ in the gravitationalfield of the earth, even for falls of thousands of kilometers, comparison of the ticks ofatomic clocks set in suitable eccentric orbits around the sun may in principle be able toreveal an effect due to Λ in a not too unforeseeable future. Indeed, in such a hypotheticalcase, due to the known periodicities it should be possible to filter out the effects fromall other contributions, gravitational and not.Finally, we remark that while at the present epoch the effects due to Λ aretangible only at cosmological scales, they might have been essential, even at microscopicdistances, during the period of inflationary expansion in the very early universe, whenthe effective de Sitter radius was extremely small ( ≃ − − − cm ), at which time,however, quantum effects are expected to have been dominant [41]. Acknowledgments
We are indebted to F. Hehl who stimulated our interest in the subject. onservation laws and scattering for de Sitter classical particles Appendix A
We show in this appendix by constructing an explicit example that the components ofthe future oriented lightlike fivevectors ξ and η which identify a given timelike geodesicare fixed by the assignement of the values of six independent parameters once ξ and η have been separately normalized according to equation (12) and the selection of thepoint of the geodesic corresponding to zero proper time. Precisely, let x µ = { ct, r i } bea set of local coordinates on dS and consider a timelike geodesic passing through ~r at time t = 0 with velocity ~v = d~r/dt | t =0 . Parameterizing the geodesic as in (9) andchoosing a suitable normalization for ξ and η , we can express this vectors in terms ofthe initial conditions ~r and ~v . Indeed X ( x µ ( t )) = R ξe cτ ( t ) R − ηe − cτ ( t ) R √ ξ · η . (91)Then, setting t = 0 and assuming τ (0) = 0, we find X ( x ) = R √ ξ · η ( ξ − η ) (92)and, taking the derivatives with respect to t at t = 0 we get dXdt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ∂ µ X ( x ) ˙ x µ = c √ ξ · η ( ξ + η ) ˙ τ . (93)Choosing k = m in (12), and solving with respect to ξ and η we find ξ = ∂ µ X ( x ) v µ c ˙ τ + X ( x ) R ,η = ∂ µ X ( x ) v µ c ˙ τ − X ( x ) R , (94)where v = c , v i = ˙ x i and c ˙ τ = r ds dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 . (95)The condition ξ · η = 2 and the choice of the initial point X ( x ) on the geodesic fixesthe separate normalizations of ξ and η and, as a consequence, formulas (94) express ξ and η as functions of ~r and ~v .For example, if we choose static coordinates we have explicitly ξ i = 1 c ˙ τ v i + r i R , (96) η i = 1 c ˙ τ v i − r i R , i = 1 , , ξ , η , ξ and η , where˙ τ = dτdt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = s − r R − v c − ( ~r · ~v ) c ( R − r ) = mc E (cid:18) − r R (cid:19) . (98) onservation laws and scattering for de Sitter classical particles ds is positive along timelike curves so that setting ~ρ = 1 R~r , ~β = 1 c ~v , (99)we have 1 − ρ − β − ρ β cos θ − ρ > , (100)where θ is the angle between ~r and ~v (and ~ρ , β and θ are functions of t ). Then β < − ρ p − ρ sin θ . (101)In particular for lightlike geodesics we find β = 1 − ρ p − ρ sin θ , (102)so that massless particles have velocities1 − ρ ≤ β ≤ p − ρ , (103)depending on the angle between the velocity and the position vector. Appendix B
In one dimension the de Sitter line element in flat coordinates is given by ds = c dt − e ctR dx = c dt − a ( t ) dx . (104)Then, denoting by y the spatial coordinate with respect to which the rigid box has fixedlength 2 L , we have y = xe ctR (105)so that the j-th particle of our classical comoving gas drifts towards one of the walls ofthe box according to the equation y j ( t ) = y j e ctR , y j = L jN , j = 0 , ± , ± , . . . , ± ( N − . (106)Hence the ± ( N − − th particle, which is the one initially closest to the right (left)wall, reaches the latter (and bounces back elastically) at the time t N − = Rc log NN − ± ( N − − th particle, thus starting thethermalization process through further collisions with the other particles. Now, theequation of motion of particle j is¨ y j − c R y j = 0 (108)corresponding to a potential V ( y ) = − c R y . (109) onservation laws and scattering for de Sitter classical particles T given by12 kT = h K i = m |h V i| = m R L − L ρ ( y ) dy Z L − L | V ( y ) | ρ ( y ) dy , (110)where K is the kinetic energy and ρ ( y ) is the mass density of the gas at equilibrium.Since our gas is an ideal one its equation of state is mp ( y ) = ρ ( y ) kT . (111)Then, eliminating the pressure from (111) and the Euler equation − dV ( y ) dy = 1 ρ ( y ) dp ( y ) dy (112)we get − dV ( y ) dy = kTm ddy log ρ ( y ) (113)from which kTm log ρ ( y ) ρ ( o ) = | V ( y ) | = c R y . (114)Hence ρ ( y ) = ρ (0) e mc kTR y (115)so that kT = mc R R L − L dyy e mc kTR y R L − L dye mc kTR y (116)which can be written as Z LR q mc kT dww e w = 12 Z LR q mc kT dwe w . (117)The solution of this equation is LR r mc kT = 1 .
063 (118)namely T ≃ mc L R k (1 . ≃ mH L . k . (119) onservation laws and scattering for de Sitter classical particles [1] S. Perlmutter et al. , “Measurements of omega and lambda from 42 high-redshift supernovae,” Astrophys. J. , vol. 517, pp. 565–586, 1999.[2] A. G. Riess et al. , “Observational evidence from supernovae for an accelerating universe and acosmological constant,”
Astron. J. , vol. 116, pp. 1009–1038, 1998.[3] A. G. Riess et al. , “The farthest known supernova: Support for an accelerating universe and aglimpse of the epoch of deceleration,”
Astrophys. J. , vol. 560, pp. 49–71, 2001.[4] D. N. Spergel et al. , “First year Wilkinson microwave anisotropy probe observations:Determination of cosmological parameters,”
Astrophys. J. Suppl. , vol. 148, p. 175, 2003.[5] V. Sahni and A. A. Starobinsky, “The case for a positive cosmological lambda-term,”
Int. J. Mod.Phys. , vol. D9, pp. 373–444, 2000.[6] T. Padmanabhan, “Cosmological constant: The weight of the vacuum,”
Phys. Rept. , vol. 380,pp. 235–320, 2003.[7] P. J. E. Peebles and B. Ratra, “The cosmological constant and dark energy,”
Rev. Mod. Phys. ,vol. 75, pp. 559–606, 2003.[8] E. In¨onu and E. P. Wigner, “On the contraction of groups and their represenations,”
Proc. Nat.Acad. Sci. , vol. 39, pp. 510–524, 1953.[9] L. F. Abbott and S. Deser, “Stability of gravity with a cosmological constant,”
Nucl. Phys. ,vol. B195, pp. 76–96, 1982.[10] R. Aldrovandi, J. P. Beltran Almeida, and J. G. Pereira, “Cosmological term and fundamentalphysics,”
Int. J. Mod. Phys. , vol. D13, pp. 2241–2248, 2004.[11] R. Aldrovandi, J. P. Beltran Almeida, and J. G. Pereira, “de Sitter special relativity,”
Class.Quant. Grav. , vol. 24, pp. 1385–1404, 2007.[12] H.-Y. Guo, C.-G. Huang, Z. Xu, and B. Zhou, “On beltrami model of de Sitter spacetime,”
Mod.Phys. Lett. , vol. A19, pp. 1701–1710, 2004.[13] H.-Y. Guo, C.-G. Huang, Z. Xu, and B. Zhou, “On special relativity with cosmological constant,”
Phys. Lett. , vol. A331, pp. 1–7, 2004.[14] H.-Y. Guo, B. Zhou, Y. Tian, and Z. Xu, “The triality of conformal extensions of three kinds ofspecial relativity,”
Phys. Rev. , vol. D75, p. 026006, 2007.[15] J. Kowalski-Glikman and S. Nowak, “Doubly special relativity and de Sitter space,”
Class. Quant.Grav. , vol. 20, pp. 4799–4816, 2003.[16] F. G. Gursey, “Introduction to the de Sitter group,” Group Theoretical Concepts and Methodsin Elementary Particle Physics edited by F. G. Gursey (Gordon and Breach, New York, 1965).[17] N. D. Birrell and P. C. W. Davies,
Quantum fields in curved space . Cambridge (UK): CambridgeUniversity Press, 1982.[18] J. Bros and U. Moschella, “Two-point functions and quantum fields in de Sitter universe,”
Rev.Math. Phys. , vol. 8, pp. 327–392, 1996.[19] J. Bros, H. Epstein, and U. Moschella, “Analyticity properties and thermal effects for generalquantum field theory on de Sitter space-time,”
Commun. Math. Phys. , vol. 196, pp. 535–570,1998.[20] J. Bros, H. Epstein, and U. Moschella, “Lifetime of a massive particle in a de Sitter universe,”
JCAP , 2008.[21] U. Moschella, “The de Sitter and anti-de Sitter sightseeing tour,” in
Einstein,1905-2005 (T. Damour, O. Darrigol, B. Duplantier, and V. Rivesseau, eds.), Progress in MathematicalPhysics, Vol. 47, Basel: Birkhauser, 2006.[22] U. Moschella, “Particles and fields on the de Sitter universe,”
AIP Conference Proceedings , vol. 910,pp. 396–411, 2007.[23] J. L. Synge,
Relativity: The General Theory . Amsterdam: North-Holland Publishing Company,1960.[24] J. M. Souriau,
Structure des Syst`emes Dynamiques . Paris: Dunod, 1970.[25] B. Konstant,
Quantization and Unitary Representations. Lecture Notes in Mathematics . Berlin: onservation laws and scattering for de Sitter classical particles Springer-Verlag, 1970.[26] A. A. Kirillov,
Elements of the theory of representations . Berlin: Springer-Verlag, 1970.[27] E. R. Harrison, “Fluctuations at the threshold of classical cosmology,”
Phys. Rev. D , vol. 1,pp. 2726–2730, May 1970.[28] Y. B. Zeldovich, “A hypothesis, unifying the structure and the entropy of the universe,”
Mon.Not. Roy. Astron. Soc. , vol. 160, pp. 1–3, 1972.[29] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations.part 1. classical perturbations. part 2. quantum theory of perturbations. part 3. extensions,”
Phys. Rept. , vol. 215, pp. 203–333, 1992.[30] B. Ratra, “Restoration of spontaneously broken continuous symmetries in de sitter space-time,”
Phys. Rev. , vol. D31, pp. 1931–1955, 1985.[31] T. S. Bunch and P. C. W. Davies, “Quantum field theory in de sitter space: Renormalization bypoint splitting,”
Proc. Roy. Soc. Lond. , vol. A360, pp. 117–134, 1978.[32] G. W. Gibbons and S. W. Hawking, “Cosmological event horizons, thermodynamics, and particlecreation,”
Phys. Rev. D , vol. 15, pp. 2738–2751, 1977.[33] D. Barberis et al. , “A study of the centrally produced π + π − π channel in pp interactions at450-gev/c,” Phys. Lett. , vol. B422, p. 399, 1998.[34] W. de Sitter, “On the curvature of space,”
Proc. Kon. Ned. Akad. Wet. , vol. 20, p. 229, 1917.[35] H. Bacry and J. M. Levy-Leblond, “Possible kinematics,”
J. Math. Phys. , vol. 9, pp. 1605–1614,1968.[36] J. M. L´evy-Leblond, “One more derivation of the Lorentz transformation,”
American Journal ofPhysics , vol. 44, p. 271, 1976.[37] R. V. Pound and G. A. J. Rebka, “Apparent weight of photons,”
Phys. Rev. Lett. , vol. 4, pp. 337–341, 1960.[38] R. V. Pound and J. L. Snider, “Effect of gravity on nuclear resonance,”
Phys. Rev. Lett. , vol. 13,pp. 539–540, 1964.[39] R. V. Pound and J. L. Snider, “Effect of gravity on gamma radiation,”
Phys. Rev. B , vol. 140,pp. 788–804, 1965.[40] N. Ashby et al. , “Testing local position invariance with four cesium-fountain primary frequencystandards and four nist hydrogen masers,”