Conformal symmetry for relativistic point particles
aa r X i v : . [ h e p - t h ] J u l ICCUB-14-049
Conformal symmetry for relativistic point particles
Roberto Casalbuoni ∗ Department of Physics and Astronomy,University of Florence and INFN, 50019 Florence, Italy
Joaquim Gomis † Departament d’Estructura i Constituents de la Mat`eria and Institut de Ci`encies del Cosmos,Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
In this paper we consider conformal dynamics for a system of N interacting rel-ativistic massless particles. A detailed study is done for the case of two-particles,with a particular attention to the symmetries of the problem. In fact, we show thatthis analysis could be extended to the case of higher spin symmetries. Always in thetwo particle case a formulation in terms of bilocal fields is proposed. For a systemof N particles we consider two possible scenarios: i) the action is invariant underany permutation of the N particles. This case corresponds to completely democraticinteractions with each particle interacting with all the others. The action dependson N − PACS numbers: 11.25.Hf, 11.30.-j, 11.10.Ef, 03.30.+p ∗ Electronic address: casalbuoni@fi.infn.it † Electronic address: [email protected]
I. INTRODUCTION
The idea of conformal invariance in physics is more than one hundred years old (a very nicehistory of the conformal group can be found in H.A. Kastrup [1]). It started with H. Batemanthat in 1908 proved the invariance of the wave equation under inversion x µ → x µ /x . This isa discrete transformation that, as we shall see, will play a crucial role in the present paper. Afew months later, Biggs himself with two papers followed by one by E. Cunningham provedthat the Maxwell equations are invariant under the conformal group. The next step was byH. Weyl in 1918, who tried to unify the gravitational and the electromagnetic interactionsmaking use of conformal invariance. This approach was strongly criticized by Einstein. Asit is well known this idea led eventually to the phase invariance of the Schr¨odinger equationand to the gauge invariance of the Maxwell theory.The conformal invariance had a revival during the sixties and the seventies in two differentareas, particle physics and critical phenomena. The interest of scale invariance in criticalphenomena raised from the works of L.P. Kadanoff [2] and K.G. Wilson [3] In particlephysics the famous SLAC experiment on deep inelastic scattering aroused wide interest inscale symmetry and its extensions. The relevance of conformal symmetry in field theorywas outlined by Polyakov [4] and used in the Operator Product Expansion (OPE) by K.G.Wilson [5]. In the context of the OPE there was a revival of the idea of bootstrap (for areview see [6]).Since then, the attention to conformal symmetry has been always very high. In field the-ory we recall the already mentioned invariance of the Maxwell equations and of the masslessDirac equation. At the classical level the scalar theory with a φ self-interaction and thenon-abelian Yang-Mills theories are conformal invariant. In these cases the conformal invari-ance is broken by anomalies, but still it plays an important role. In condensed matter and instatistical field theory scale invariance at the critical points is a fundamental phenomenon.We also mention the importance of two-dimensional conformal symmetry in string theoryand in general in two-dimensional field theories, where the conformal group, contrarily tothe case of space-dimensions different from two, is infinite dimensional. As a last point wemention the AdS/CFT correspondence [7] [8] [9] which allows to define in a non-perturbativeway M/string theory in terms of a (superconformal) quantum field theory in flat space-time.This idea has opened the possibility to study strongly coupled field theories in terms ofgravitational theories.It should be underlined that one of the main reasons that makes conformal theories soattractive is that they do not depend on any dimensionfull coupling constant.More recently, conformal symmetry has become an important tool in the analysis of higherspin theories [10, 11], for a recent review see [12, 13]. About this point it is interesting tonotice that the higher spin symmetries of Vasiliev theory appear in the free massless KleinGordon equation [14]. At particle level these symmetries are all the symmetries of the actionof a relativistic massless particle and they generalize the well know conformal symmetriesof this action.In this paper we present an application of conformal invariance to classical interactingrelativistic particles. First of all, this problem, at the best of our knowledge, has beenconsidered only in the non-relativistic one-dimensional case. Examples are the Calogero-Moser rational model [15–17], describing N interacting particles via two body interactions.This model is very important in the context of integrable models. For an extension to thesupersymmetric case see, for example, [18]. The other example, always in one dimension,is in reference [19] where only one degree of freedom is considered but it contains a deepanalysis of the role of the conformal group. For the superconformal case see [20].The second point is that, after quantization, this theory is naturally connected with non-local field theories appearing in the context of higher spin theories, see for example [21].As we mentioned previously, the underlying physics of the latter theories is connected withmassless free particles. Since we are considering interacting massless particles, it would beworth to try to understand the possible connections.This paper is organized as follows: after the Introduction, in Section 2 we make use ofthe dilatation and translation invariance to show that a free massless particle cannot bedescribed in configuration space proving that the lagrangian vanishes identically. Therefore,in Section 3 we introduce lagrange multipliers (or einbeins) in order to impose the mass zerocondition. This is the description that we will use throughout all this paper. Furthermore,using conformal invariance we write down an action for two relativistic massless particles ina D dimensional space-time, with D = 2. In the case of two particles we further show thatit is indeed possible to write down a lagrangian using only the coordinate space. It turns outthat this lagrangian vanishes identically when turning off the coupling constant describingthe interaction, as expected from the previous considerations. In this Section we show alsothat there is a constraint in phase space involving the product of the momenta squared ofthe two particles.Section 4 is dedicated to the Hamiltonian analysis of the model. It turns out that there aretwo primary constraints and two secondary ones. We show that out of these four constraintstwo are first class and two second class. Eliminating the second class constraints throughthe use of the Dirac brackets we recover, in the reduced phase space, the constraint foundpreviously in the lagrangian analysis.By construction, our model is explicitly invariant under the conformal group acting uponthe coordinates of the two particles, but it is interesting to study in an explicit way theKilling vectors of the model. This is done in Section 5, where we show that both theKilling vectors associated to the two particles satisfy the conformal Killing equations withindependent infinitesimal parameters. However, due to the interaction, the two vectors mustsatisfy a further condition requiring that the infinitesimal parameters of the two Killingvectors coincide. This implies an explicit breaking of the symmetry group of the free case SO ( D, ⊗ SO ( D, to the diagonal subgroup SO ( D,
2) .The previous study is preliminary to what we do in Section 6, where we consider higherorder Killing tensors. This means to take powers of the generator defined in the previousSection. Obviously these powers are constant of motion, but there is some interest fromthe point of view of higher spin symmetries to study conformal Killing tensors [22], seealso the more recent papers [23, 24]. In the case of free massless relativistic particles thesesymmetries are the enveloping algebra of the relativistic conformal group [14, 25]. We studyin particular the case of a Killing tensor of rank 2, deriving the conditions that must besatisfied to provide the required invariance. The equations we get are obviously satisfiedwhen the Killing tensor is realized as the product of two Killing vectors, so the interest isto look for non factorized solutions. Is should also be noticed that in the case of higher spinthe interest is in Killing vectors corresponding to a single space-time variable, whereas inour case they depend on two space-time variables (corresponding to the fact that we arestudying a two-particle system).In Section 7 we construct a bilocal field theory, involving two bilocal fields, such toincorporate the constraint found in Section 2. This is also an interesting point since bilocalfields are naturally connected with higher spin symmetries.In Section 8 we extend the two-particle model to N massless particles interacting ina conformal invariant way. Here various possibilities open up according to the kind ofsymmetry we require under the exchange of the N -particles. In particular we will examinetwo models, in the first one we assume invariance with respect to any permutation amongthe N particles. This entails a completely democratic model in which each particle interactwith all the others. The model depends on N − N particles in two clusters one made upwith n and the other with m particles, and we send to infinity all the distances among theparticles of the first cluster and the particles of the second cluster, the original lagrangiangoes into the sum of two lagrangians of the same kind of the original one. In the secondmodel considered here, we associate the particle labels to the sites of a one-dimensionallattice, assuming nearest neighbor interactions. Therefore only two-body interactions areinvolved and the model is defined by a single dimensionless coupling. The asymptoticseparability holds also in this case. There are not symmetries related to the exchange ofparticles. However, for a closed lattice there is a symmetry under discrete translations.In Section 9 we draw some conclusions and give an outlook for further problems to bestudied. II. CONFORMAL INVARIANCE IN PARTICLE COORDINATES
We will discuss the requirements coming from conformal invariance on the lagrangian ofclassical relativistic point-particles. Let us start with one-particle. We will prove that aconformal invariant lagrangian for a single relativistic particle vanishes identically. Actuallyit will be enough to assume that the action is parametrization invariant and that it dependsonly on the coordinates of the particle. The generator of dilatations for a single particle isgiven by D = x µ p µ = − x µ ∂L∂ ˙ x µ . (1)Notice the minus sign in the definition of the canonical momentum. This follows from ourchoice of a mostly minus metric g µν = (+ , − , − , · · · , − ) in a D dimensional space-time. Werequire D in the previous equation to be a constant of motion and, furthermore, that theLagrangian is homogeneous of first degree in the time parameter. It follows0 = dDdτ = − ˙ x µ ∂L∂ ˙ x µ − x µ ∂L∂x µ = − L, (2)where we have used the Lagrange equations of motion ddτ ∂L∂ ˙ x µ = ∂L∂x µ = 0 (3)and the invariance under translations. Therefore, the only solution for D to be a constantin time, is that the lagrangian vanishes. It is obvious that this result applies to the caseof N non interacting particles (under the same assumptions). This is an important point,since, if we want to consider a conformal invariant theory for a given number of particles, wecannot describe the free case using only space-time variables. In fact, as it is well known, amassless particle is described using an einbein variable defined on the world line (in practicea Lagrange multiplier). Therefore, this is the description that we will adopt, although formore than one particle a conformal invariant lagrangian depending only on the coordinatescan be constructed. On the other hand this latter formulation is such that turning offthe interaction, the lagrangian vanishes identically, as it should be clear from the previousdiscussion. III. FORMULATION WITH THE EINBEINS
The lagrangian for a single free massless particle can be obtained through the use of aneinbein e : S = − Z dτ ˙ x e , (4)from which varying with respect to the einbein we get the equation ˙ x = 0 and evaluatingthe momentum p µ = ˙ x µ /e we obtain p = 0. The minus sign in front of the action is aconsequence of our choice of the space-time metric. Requiring that the einbein transformsas a time derivative, this action is invariant under reparametrization. It is also invariantunder Poincar´e transformations. As for dilatations, we require: x µ → λx µ , e → λ e. (5)Furthermore, we recall that a special conformal transformation can be obtained through thefollowing series of operations: (inversion) ⊗ (translation) ⊗ (inversion), therefore, to imposethe conformal symmetry it is enough to require the invariance under inversion x µ → x µ x . (6)The transformation properties of ˙ x is ˙ x → ˙ x x , (7)from which it follows e → ex . (8)Summarizing, the action (4) is invariant under conformal and reparametrization transfor-mations.Now let us discuss the case of two particles. We start at the free level with two masslessparticles S free = − Z dτ (cid:18) ˙ x e + ˙ x e (cid:19) . (9)In order to construct an interaction term depending on the relative coordinate r µ = x µ − x µ , (10)we notice that under inversion r → r x x . (11)Therefore a conformal invariant action for two relativistic particles is given by S = − Z dτ (cid:18) ˙ x e + ˙ x e + α √ e e r (cid:19) . (12)The variation with respect to the einbeins gives rise to the following equations ∂L∂e = ˙ x e − α r e e r = 0 ,∂L∂e = ˙ x e − α r e e r = 0 . (13)Resolving these two equations in the einbeins one finds1 e = α x (cid:18) ˙ x ˙ x r (cid:19) / , e = α x (cid:18) ˙ x ˙ x r (cid:19) / , (14)with α ≥
0. In extracting the square root we have chosen the minus sign, in order to havethe time component of the canonical momenta with the same sign of the time derivative ofthe coordinate times, x i . Substituting inside the action (12) we find S = − α Z dτ (cid:18) ˙ x ˙ x r (cid:19) / . (15)As we have discussed previously the conformal invariant action for two particles in configu-ration space vanishes when the interaction is turned off.Evaluating the momenta from (12) we get p µi = − ∂L∂ ˙ x iµ = ˙ x µi e i . (16)The equations (13) can be expressed in terms of the momenta obtaining p − α r e e r = 0 , p − α r e e r = 0 . (17)Finally, eliminating the ratio e /e from these two equations we get a constraint amongmomenta and coordinates p p − α r = 0 . (18)This relation can also be obtained as a primary constraint from the action (15).Notice that we have started with a flat metrics g µν , but we could have started witha conformal metrics as well, g µν → exp(2 γ ( x )) g µν . In fact, in the formulation (12), theconformal factor can be absorbed into the definition of the einbeins, whereas the formulation(15) is explicitly scale invariant. IV. HAMILTONIAN ANALYSIS
In our notations the Poisson brackets among coordinates and momenta are: { x µ , p ν } = − g µν , { e i , π j } = δ ij . (19)Once again, the sign in the first Poisson bracket is fixed by our choice of the mostly minusmetric. Notice also that there are two primary constraints π i = ∂L∂ ˙ e i = 0 , (20)therefore the canonical hamiltonian results to be H C = − p ˙ x − p ˙ x − L = − e p − e p + α √ e e r . (21)Following Dirac we define the Dirac Hamiltonian, H D , adding an arbitrary combination ofthe primary constraints π i = 0, in terms of two arbitrary functions λ i , H D = H C + λ π + λ π . (22)Requiring the stability of the primary constraints we get two secondary constraints { π , H D } = 12 (cid:18) p − α r e e r (cid:19) ≡ φ , { π , H D } = 12 (cid:18) p − α r e e r (cid:19) ≡ φ . (23)Notice that these two constraints are the same as the ones in (17). Then, we have to considerthe stability of the secondary constraints φ i , obtaining { φ , H D } = − α r √ e e p · r + s e e p · r + α r (cid:18) λ r e e − λ √ e e (cid:19) , { φ , H D } = + α r s e e p · r + √ e e p · r + α r (cid:18) λ r e e − λ √ e e (cid:19) . (24)These two constraints are not independent. In fact, the second equation can be obtained fromthe first one multiplying by − e /e . It follows that the stability of the secondary constraintscan be attained by eliminating one of the two parameters λ i , for instance, evaluating λ from the first equation (24). We find λ = e e λ + 4 r (cid:0) e p · r + e e p · r (cid:1) . (25)Correspondingly the Dirac hamiltonian becomes H D = H C + 4 r e ( e p · r + e p · r ) π + λ (cid:18) π + e e π (cid:19) . (26)It is convenient to redefine λ = ˜ λ e , then H D = H C + ˜ λ ( e π + e π ) + Cπ , (27)with C = 4 r e ( e p · r + e p · r ) . (28)This expression suggests that the coefficient of λ is a first class constraint. This can beverified by evaluating its Poisson bracket with H D { e π + e π , H D } = e φ + e φ − Cπ . (29)This shows that the constraint e π + e π is weakly stable. Then it is a simple algebra toprove that the two constraints e π + e π , e φ + e π − Cπ (30)0are weakly first class, that is their Poisson brackets with the other constraints π , π , φ φ are proportional to one of these constraints. In conclusion, the four constraints can bedivided as follows: first class e π + e π , e φ + e φ − Cπ , second class π , φ . (31)Then, introducing the Dirac parentheses one can put π and φ strongly to zero. In this way π and φ turn out to be strongly first class.The matrix of the second class constraints, χ ij , i, j = 1 ,
2, is quite simple χ = D − D , χ − = − /D /D , (32)where D = { φ , π } = α r e e r . (33)The Dirac brackets among any two dynamical variables are given by { O , O } ∗ = { O , O } + 1 D [ { O , φ }{ π , O } − { O , π }{ φ , O } ] . (34)In the reduced space using the second class constraints and the Dirac brackets the firstclass constraints become π = 0 , φ | φ =0 = 0 , implying (cid:18) p p − α r (cid:19) = 0 , (35)and we recover eq. (18) that was obtained previously by solving the equations for theeinbeins.Since among of the two first class constraints we have one that is primary, it is knownthat we should have one gauge transformation . The generator of this gauge transformationcan be constructed from a well know algorithm, see for example [26–31]. The generator G ,which is a constant of motion, is given by G = X i =1 (cid:18) ddτ ( ǫe i ) π i − ( ǫe i ) φ i (cid:19) , (36)where ǫ ( τ ) is an arbitrary function of the global parameter that parametrize the two worldlines.1The transformation generated by G is δe i = ddτ ( ǫe i ) , δx µi = ǫ ˙ x µi , (37)it is the global world line diffeomorphism (Diff). Note that the interaction breaks theindividual Diff invariance of the two world lines. V. ANALYSIS OF THE RIGID SYMMETRIES
We have constructed our lagrangian requiring conformal invariance, that is invarianceunder the group SO ( D, SO ( D, ⊗ SO ( D, acting on the variables of the particles 1 and 2 respectively. Whenthe interaction is introduced, the invariance is broken explicitly to the diagonal subgroup.It is interesting to analyze these symmetries by looking at the conditions the Killingvectors must satisfy in order our lagrangian is invariant under the symmetries generated bygeneric Killing vectors G = X i =1 ξ iµ ( x , x ) p µi . (38)In this Section we will make use of the lagrangian L , given in (15), in terms of which wehave p iµ = − ∂L∂ ˙ x µi = −
12 ˙ x iµ ˙ x i L (39)and the equations of motion˙ p µ = − ∂L∂x µ = + 12 r µ r L, ˙ p µ = − ∂L∂x µ = − r µ r L. (40)It is clear that the result will be that the Killing vectors are those of the conformal groupbut, the equations we will find here will be important for the analysis of the Killing tensorswe will do in the next Section. This analysis is relevant for the higher spin symmetries thathave been recently considered in the literature [14] [25].By taking the time derivative of G , using the expression of the momenta given in eq. (39)and the Lagrange equations of motion (40), we obtain˙ G = − X i,j =1 ( ∂ jµ ξ iν ( x , x )) ˙ x µj ˙ x νi ˙ x i L + ( ξ µ ( x , x ) − ξ µ ( x , x )) r µ Lr = 0 , (41)2where ∂ iµ = ∂/∂x iµ . Notice that the first term of this equation is not symmetric in j and i .A necessary condition to have a solution is ∂ jµ ξ iν ( x , x ) = 0 , j = i, (42)which implies that ξ iν = ξ iν ( x i ). If we use this information in (41) − X i = j ( ∂ iµ ξ iν ( x i )) ˙ x µi ˙ x νi ˙ x i L + ( ξ µ ( x , x ) − ξ µ ( x , x )) r µ Lr = 0 . (43)Now the first term is symmetric in µ, ν . The solution of this equation is12 ( ∂ iµ ξ iν ( x i ) + ∂ iν ξ iµ ( x i )) = g µν λ ( i ) ( x i ) , i = 1 , , (44)12 X i =1 λ ( i ) = ( ξ µ − ξ µ ) r µ r . (45)By contracting together the indices µ, ν in (44) we find λ ( i ) = 1 D ∂ ρi ξ iρ . (46)The two equations (44) tell us that ξ µ and ξ µ are the Killing vectors of two conformalgroups SO ( D, i acting on the two variables x and x respectively. This is the symmetrygroup of two massless non-interacting particles. However, it is easily proved that the secondcondition (45) is satisfied if and only if the infinitesimal parameters defining the two Killingvectors are identical. Therefore the symmetry SO ( D, ⊗ SO ( D, is broken down to thediagonal subgroup SO ( D,
2) due to the interaction between the two particles.
VI. HIGHER SPIN SYMMETRIES
In the previous Section we have shown that the quantity G (see (38)) is a constant ofmotion, if the parameters defining the two conformal Killing vectors, corresponding to thetwo particles, are the same. It is a trivial observation that the power G n is also a constantof motion dG n dτ = 0 . (47)The explicit expression for G n is G n = X i ,i , ··· ,i n =1 ξ i µ ξ i µ ξ i n µ n p µ i p µ i · · · p µ n i n . (48)3This defines a tensor of rank n constructed in terms of the n conformal Killing vectors in n variables x , x , · · · , x n .In principle, one could try to generalize the expression (48) to a generic Killing tensor[22] G ′ = X i ,i , ··· ,i n =1 ξ µ µ ··· µ n i i ··· i n p i µ p i µ · · · p i n µ n . (49)Notice that the tensor ξ µ µ ··· µ n i i ··· i n may depend on the variables x , x , · · · , x n . Requiring G ′ tobe a conserved quantity one gets0 = X k,i ,i , ··· ,i n =1 ∂ µk ξ µ µ ··· µ n i i ··· i n p i µ p i µ · · · p i n µ n ˙ x kµ ++ n X j =1 2 X i ,i ··· ,i n =1 ξ µ µ ··· µ n i i ··· i n p i µ p i j − µ j − (cid:16) ( − i j − r µ r L (cid:17) p i j +1 µ j +1 · · · p i n µ n , (50)where we have used the equations (40) in the form˙ p iµ = ( − i − r µ r L. (51)Then, using (39) p iµ = −
12 ˙ x iµ ˙ x i L, (52)0 = 12 X k,i , ··· ,i n =1 ∂ µk ξ µ µ ··· µ n i i ··· i n ˙ x i µ ˙ x i µ · · · ˙ x i n µ n ˙ x kµ ˙ x i x i · · · x i n − n X j =1 2 X i ,i ··· ,i n =1 ξ µ µ ··· µ n i i ··· i n ˙ x i µ ˙ x i j − µ j − ˙ x i x i · · · x i j − (cid:16) ( − i j − r µ r (cid:17) ˙ x i j +1 µ j +1 · · · ˙ x i n µ n ˙ x i j +1 ˙ x i n . (53)Proceeding as in the previous Section, one obtains equations for the tensor ξ µ µ ··· µ n i i ··· i n inde-pendent on ˙ x i . We know that these equations are satisfied when the tensor factorizes in n conformal Killing vectors.An interesting question remains open: is the factorized case the only solution to theprevious equations?Let us study in detail the case of n = 2. We have X ijk =1 (cid:18) ∂ ρk ξ µνij ˙ x kρ ˙ x iµ ˙ x jν ˙ x i ˙ x j (cid:19) − X ij =1 (cid:18) ξ µνij ( − i − ˙ x jν r µ ˙ x j r (cid:19) = 0 . (54)4Notice that this equation it is not symmetric in k, i, j . Proceeding as in the previous sectionthe necessary condition to have a solution is ∂ ρk ξ µνij = 0 , k = i, k = j, (55)using this condition we have X k = i,i = j (cid:18) ∂ ρi ξ µνij ˙ x iρ ˙ x iµ ˙ x jν ˙ x i ˙ x j (cid:19) + X k = i = j (cid:18) ∂ ρi ξ µνii ˙ x iρ ˙ x iµ ˙ x iν ˙ x i ˙ x j (cid:19) −− X i = j (cid:18) ξ µνij ( − i +1 ˙ x jν r µ ˙ x j r (cid:19) − X i = j (cid:18) ξ µνii ( − i +1 ˙ x iν r µ ˙ x i r (cid:19) = 0 . (56)We get a solution requiring12 ( ∂ ρi ξ µνij + ∂ µi ξ ρνij ) = g ρµ W ν ( ij ) , i = j, (57) ∂ ρi ξ µνii + ∂ µi ξ ρνii + ∂ νi ξ µρii = g ρµ V ν ( i ) + g µν V ρ ( i ) + g νρ V µ ( i ) , (58) X i =1 (cid:18) W ν ( ij ) − ξ µνij ( − i +1 r µ r (cid:19) = 0 , (59) X i =1 (cid:18) V ν ( i ) − ξ µνii ( − i +1 r µ r (cid:19) = 0 . (60)In these expressions W ν ( ij ) = 12 (cid:0) ∂ νi ξ µijµ + ∂ iµ ξ νµij (cid:1) , i = j, (61)and V ν ( i ) = 1 D + 2 (cid:0) ∂ νi ξ µiiµ + 2 ∂ iµ ξ µνii (cid:1) . (62)In the factorized case ξ µνij = ξ µi ( x i ) ξ νj ( x j ) , W νij = λ ( i ) ξ νj , V νi = 2 λ ( i ) ξ νi , (63)where the λ ( i ) ’s are defined in eq. (46). It is easily verified that the previous equations (57),(58), (59) and (60) are satisfied. On the other hand, in principle it is possible that theseequations have independent solutions, a fact that would be rather interesting.5 VII. A BILOCAL FIELD THEORY
Bilocal field theories have been considered recently in the framework of higher spin sym-metries, see for example [21]. These bilocal field equations are conformal invariant, thereforeit is of some interest to construct a bilocal conformal invariant field theory. This is madepossible by encoding the constraint equation given in eq. (18) in a bilocal field theory. Tothis end, let us introduce two bilocal fields, φ i ( x , x ) with i = 1 ,
2. Then consider the action S = R d x d x h
12 ( ∂ µ φ ( x , x ) ∂ µ φ ( x , x ) + ∂ µ φ ( x , x ) ∂ µ φ ( x , x )) − φ ( x , x ) V ( x , x ) φ ( x , x ) i , (64)where the potential V is given by V ( x − x ) = α x − x ) . (65)Varying with respect to φ and φ we get the equations of motion (cid:3) φ + V φ = 0 , (cid:3) φ + V φ = 0 . (66)Eliminating φ from the first equation φ = − V − (cid:3) φ , (67)and substituting inside the second one (cid:3) ( V − (cid:3) φ ) − V φ = 0 . (68)Then, multiplying by V we obtain V (cid:3) ( V − (cid:3) φ ) − V φ = 0 . (69)Now, let us look for solutions of the type φ i ( x , x ) = e i ( p x + p x ) ˜ φ i ( x − x ) . (70)Substituting inside the equations of motion (69) we find p ˜ φ − V ˜ φ = 0 , p ˜ φ − V ˜ φ = 0 . (71)6Eliminating again ˜ φ from the first one˜ φ = V − p ˜ φ , (72)and substituting inside the second one ( r = x − x )( p p − α r ) ˜ φ ( p , p , r ) = 0 , (73)and an analogous equation for ˜ φ . Notice that what we have done here is not to take theFourier transform of the bilocal fields, but we have simply looked at a particular solution ofthe field equations.As we have seen our theory gives rise to the constraint (18), therefore, in quantum theoryit would be natural to transform it in a wave equation of the type( (cid:3) (cid:3) − α r ) φ ( x , x ) = 0 , (74)which should be looked at as a generalization the conformal invariant massless Klein-Gordonequation to two conformal particles. On the other hand, this equation is fourth-order in thederivatives and it might produce problems in a related field theory. In fact, higher-ordertheories present , in general, ghosts in the spectrum. For this reason we prefer to start witha system of two fields each of them obeying a second order equation.An interesting point is to expand the equations of motion (69) in terms of a series ofhigher spin local fields, but we defer this problem to a future paper. VIII. CONFORMAL INVARIANT LAGRANGIANS FOR MANY PARTICLES
In this Section we would like to extend the case of a conformal invariant interactionbetween two particles to the case of N particles. The kind of model one obtains dependson the symmetries one assumes in the exchange of the particles. We will start assumingthe maximal symmetry, that is invariance under any permutation among the particles. Thisrequirement and conformal invariance fix completely the interaction among the N particlesup to N − N massless freeparticles L free = − N X i =1 ˙ x i e i . (75)7We recall, from Section III that under inversion:˙ x i → ˙ x i x i , r ij → r ij x i x j , i, j = 1 , , · · · N. (76)In order the free part to be invariant under inversion the einbeins must transform as e i → e i x i , (77)whereas under reparametrization they must transform as a time derivative. In order towrite down the invariant terms for the many particle case, let us notice that the two pointinteraction can be written in the following form: (cid:18) e i e j r ij r ji (cid:19) / . (78)This expression suggests that an invariant term for n particles is of the form e i e i · · · e i n r i i r i i · · · r i n − i n r i n i ! /n . (79)In fact, it is easily seen that this term is conformal invariant and transforms as a firstderivative with respect to time.Then, the most general conformal invariant lagrangian symmetric under the exchange ofany pair of particles has the following structure L = − N X i =1 ˙ x i e i − N X n =2 β n X i
In this paper we have studied what relativistic conformal symmetry can teach us aboutpossible interactions among N classical massless particles. The lagrangian considered heredepends on the symmetry we assume under the exchange of the particles. Assuming invari-ance under any permutation among the particles, the lagrangian is completely fixed up to N − n particles out of the totalset, it contains n -body interactions and it appears to be completely democratic. This is alsoshown by the number of terms in the lagrangian, which does not grow with a power of N but rather in an exponential way, namely like 2 N − . Another possibility that we haveconsidered is the one corresponding to nearest neighbor interactions. The interest of thiscase is mainly related to the possibility of getting a simple limit in the continuum, obtainingin this way a conformal string [32].We have analyzed the case N = 2 with a particular emphasis on the symmetries. In fact,it is known that the conformal symmetry of the free massless Klein-Gordon equation can beextended to the enveloping algebra of the conformal group, obtaining in this way higher spinsymmetries. In our case, we have two (or more) interacting particles preserving conformalsymmetry, so a natural question to investigate is the possibility to enlarge the higher spinsymmetries to interacting massless particles.An interesting point is the extension of the conformal models presented here to super-conformal ones [32].The N particle models could be considered in the case of D = 1, that is a pure quantummechanical case, in order to study their possible integrability. In particular, the nearestneighbor model looks close to the Calogero-type models [15–17].Another problem, to be investigated in the future, is the quantization of these models. A0promising possibility, in our opinion, would be to try the world-line quantization, along theway paved by string theory. We recall here that the world-line quantization can be extendedto the self-interactions of scalar particles [33]. Another option is field quantization using, inthe case of two particles, bilocal fields as introduced in Section 7. Acknowledgments
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