ONLY PARTICLES WITH SPIN 2 ARE MEDIATORS FOR FUNDAMENTAL FORCES: WHY?

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1 ONLY PARTICLES WITH SPIN 2 ARE MEDIATORS FOR FUNDAMENTAL FORCES: WHY? Vladik Kreinovich Computer Science Department University of Texas at El Paso El Paso, TX 79968, USA ABSTRACT According to modern physics, the mediators for all the fundamental forces of nature have spin S 2: photon for electrodynamics (S = 1), gluons for strong interactions (S = 1), bosons for the weak force (S = 1) and gravitons for the gravitational force (S = 2). Why? We show that the demand that the equations of a fundamental field are consistent in the classical limit (when quantum effects are negligible and the only sources of all the fields are classical point particles) leads to S 2. So this consistency demand explains the above observation. This result also leads to a qualitative conclusion that all the phenomena related to fields and particles of spin > 2 are purely quantum effects, and thus their intensity is much smaller than of fields and particles with S 2. This explains why the most stable and widely spread elementary particles are of spin 2 and why, in spite of the theoretical possibility that the torsion field (of spin 3) can also be responsible for gravitation, it has not yet been experimentally observed. Keywords and phrases: mediators of fundamental forces, large spin, torsion, equations of motion, classical approximation of a field theory. 1

2 1. ONLY PARTICLES WITH SPIN 2 ARE MEDIATORS FOR FUNDAMENTAL FORCES: A FACT TO EXPLAIN. A fact. According to modern physics (see, for example, Nachtmann, (1) or Griffiths (2) ) the mediators for all the fundamental forces of nature have spin S 2: photon γ for electrodynamics (S = 1), gluons g for strong interactions (S = 1), bosons Z, W for the weak force (S = 1) and gravitons for the gravitational force (S = 2). A natural question is: why? Why is this question important? The ultimate goal of particle physics is to develop a Final Theory that will explain all the existing phenomena. To develop a candidate theory is a complicated and time-consuming activity; to derive its experimental consequences is also extremely difficult. Theories that describe only existing fundamental forces are already very complicated, but physicists often propose to add extra forces. This addition further increases the complexity of a theory, and thus increases the time and efforts that are necessary to develop and check it. This additional complexity is caused by the fact that new variables are added to the system of equations of a theory: namely, the variables that describe the components of the new fields. If the mediator of a new field is a scalar particle, this field is scalar and we have to add only one component. If the mediator is a vector (spin 1) particle, this means that this fundamental field is a vector field, and so we have to add four components of this vector field A i. For spin 2 the field is a tensor field, and we add 10 components of a symmetric tensor field A ij, etc. The bigger the spin, the more components we have to add, and so the more complicated the theory becomes. So, in order to restrict the complexity of possible theories, it is desirable to find physically motivated limitations on the possible spin of mediators. Since for all the existing mediators the spin S is 2, a reasonable guess is that S 2 can be the desired physically motivated restriction. Why is this question non-trivial? The reason why this question is non-trivial is that in principle there is nothing wrong with particles with spins > 2: 1) such particles have been experimentally observed (up to spin 11/2) (see, for example, Gaziorowicz and Rozner, (3) Particle Data Group (4) ); 2) particles with arbitrarily high spins appear naturally (and consistently) in string theory (see, for example, Brink and Henneaux (5) ), one of the most promising approaches to particle physics; 2

3 3) a fundamental interaction that corresponds to the mediators of spin 3 (> 2) has actually been proposed as early as 1920s by Cartan (6,7). Namely, after Einstein proposed to use Riemann spaces to describe gravitation, Cartan proposed to use a natural generalization of Riemann spaces spaces with torsion. Unlike General Relativity, where one fundamental field g ij describes gravitation, in Cartan s approach the second fundamental filed is necessary, a torsion field S i jk. The correspondent quanta called torsions are of spin 3. This approach has been further developed by Trautmann (8). It is known to be consistent with all the existing experimental data (Misner, Thorne, and Wheeler (9) ), and it is still considered one of the possible approaches towards the Final Theory (see, for example, several publications in Pronin and Obukhov (10) ). How we are going to answer this question. Since we need an explanation that will work for future theories as well, we need an answer, that will not depend on the specific features of existing theories, we need a physically motivated explanation from first principles. For that purpose we ll consider the simplest possible case that we ll call a classical approximation, when quantum effects are negligible and thus the world consists of mutually interacting point particles. In this case the particles are the only source of all the fields. We are going to prove that only fields with S 2 are consistent in this classical limit. 2. AN EXAMPLE OF A CLASSICAL APPROXIMATION: DERIVATION OF EQUATIONS OF MOTION FROM FIELD EQUATIONS IN GENERAL RELATIVITY An example of classical approximation already exists. There already exists a successful example of a classical approximation in our sense: a so-called EIH (see, for example, Misner, Thorne, and Wheeler (11) ). So we ll describe it and then show how to generalize it and get a classical approximation of an arbitrary theory. Why point particles. This EIH approach was developed by Einstein, Infeld, and Hoffman (12) for General Relativity. This theory is formulated in terms of field equations; but the majority of the observations that it tends to explain are observations of celestial bodies. So we must be able to make conclusion about the motion of these bodies from the field equations. In other terms, we must be able to deduce equations of motion from the field equations. The size of celestial bodies is very small in comparison with their mutual distances, so representing them as point particles is a good approximation. If we want a better approximation, we can take into consideration dipole, quadrupole moments, etc; 3

4 and, for example, a dipole can also be represented as a pair of point particles, that are located close to each other. Perturbation techniques applied to General Relativity. The idea behind EIH approach is as follows. The fundamental field of General Relativity is a metric tensor field g ij. In vacuum (empty space-time) it is equal to the Minkowski metric tensor of Special Relativity η ij = diag(1, 1, 1, 1). So in celestial-mechanical problems, where the fields are weak, g ij η ij ; hence the difference h ij = g ij η ij is small. Therefore we can apply the techniques of the perturbations theory: namely, we can expand the Einstein equations into the series with respect to h ij. where As a result Einstein equations turn into Dh ij 1/2Dh k k η ij = t ij + Θ ij, Dh ij = h,k ij,k + h,ij h k i,jk h k j,ik,,k (as usual in tensor algebra) means derivative with respect to x k, repeated index means summation, h = h k k, t ij is an energy-momentum tensor and Θ ij is a sum of all the terms that are quadratic, cubic, etc in h ij and h ij,k. What is the source of the gravitational field? Here t ij is the source of the field h ij. We consider the case when the only source terms are those coming from classical point particles. In every moment of time a classical particle can be described by its location r(t), four-velocity u i (t) and mass m(t). For each of the particles the tensor t ij must be located in the same point r(t), so t ij ( r, t) must be proportional to a delta-function δ( r r(t)), i.e., t ij ( r, t) = A ij (t)δ( r r(t)), where A ij is a symmetric tensor. The only symmetric expression that we can get from u i is ku i u j for some constant k. Since m(t) describes the mass of the source, this constant k must be proportional to m(t), hence t ij = cm(t)u i u j δ( r r(t)). In case of several point sources the total tensor t ij is equal to the sum of these expressions: t ij = cm a (t)u a i ua j δ( r ra (t)), where different values of a correspond to different point sources. How to deduce the equations of motion. The possibility to deduce the equations of motion from these field equations stems from the fact (that is relatively easy to verify) that the divergence of the left-hand side of the field equations is identically 0: Dh j i,j 1/2Dh k k,jη ij = 0, 4

5 therefore the divergence of the right-hand side must also be identically 0: t j i,j + Θ j i,j = 0. If we substitute the above expression for t ij into these four equations, we get nonlinear equations for m(t), r(t) and u i (t) in terms of the field. Due to the fact that t ij is located only in finitely many points, its divergence is different from 0 also only in these points. The divergence in the point r a depends only on the mass, location and velocity of a-th particle. The above expression for t ij contains coordinates r a (t) and their first derivatives u a i, so the divergence of t ij (that is the sum of its first derivatives) includes at most the second derivative of the particle s coordinates with respect to time, i.e., its acceleration a = d 2 r(t)/dt 2. Likewise, since t ij contains the mass, its divergence must contain the mass s first derivative dm/dt. We can thus view this system as a system of four equations that relate three accelerations and one mass derivative (totally four values) to the coordinates, velocities and the values of mass. From these four equations we can express four unknowns ( a and dm/dt) explicitly in terms of the coordinates r, velocities u i and the values of the field h ij. Thus we get the first-order differential equation that describes how the mass changes and three second-order differential equations that describe how the coordinates change. In other words, we get the desired equations of motion. How to generalize the EIH approach to arbitrary field equations: harmonic coordinates approach. At first glance it looks like the above derivation is a trick that is possible because of the specific features of General Relativity. That this trick is not confined to General Relativity follows from the fact that similar derivations are possible for several other gravitational theories (see, for example, Misner, Thorne, and Wheeler, (11) Finkelstein and Kreinovich, (13) Finkelstein, Kreinovich, and Pandey, (14) Cannon, Lisewski, Finkelstein, and Kreinovich (15) ). As concerns General Relativity, a transformation proposed by Fock (16) makes the above-mentioned formulas much easier to generalize. Namely, since General Relativity is a covariant theory, there is a possibility to introduce special coordinates that would simplify the equations (it is similar to choosing a gauge for the electromagnetic field). 5

6 Fock proposed to use the so-called harmonic coordinates, in which h j i,j 1/2 h,i = 0. In these coordinates Einstein field equations are simpler: h,k ij,k 1/2 h,k,k η ij = t ij + Θ ij, If we introduce a new field ψ ij = h ij 1/2 hη ij, for which Fock s conditions imply ψ j i,j = 0, we get a still simpler equation ψ,k ij,k = t ij + Θ ij, that is a field equation for a spin 2 field with t ij as a source and Θ ij as a non-linear interaction term (that includes possible self-interaction). Now we are ready for the generalization. 3. CLASSICAL APPROXIMATION FOR AN ARBITRARY FIELD THEORY What does it mean that a field corresponds to spin- S particles. Let s first consider the case of integer spin. According to the field theory, a field that corresponds to the particles of an integer spin S is a symmetric rank-s tensor field a i...jk that satisfies the equation ai...j,k = 0. Comments. k 1) The demand that this tensor field must be symmetric stems from the idea that the correspondent representation of the Poincare group must be irreducible. An arbitrary tensor field can be represented as a sum of its symmetric part and parts with different other symmetries; these parts transform independently and so they correspond to different irreducible representations. For example, a rank-2 tensor p ij in a 3-D space can be represented as a sum of its symmetric part s ij = 1/2 (p ij + p ji ) and an antisymmetric part a ij = 1/2 (p ij p ji ). The antisymmetric part actually corresponds to a spin-1 field ε ijk a jk, where ε ijk is a permutation tensor. 2) The demand that the divergence should be equal to 0 is also motivated by the idea that a ij...k should represent a field of spin S, and not the combination of fields of different spins, because if the divergence is not equal to 0, then it is a new field of rank-(s 1) tensors, that corresponds to spin S 1. Field equations: first approximation. In the first approximation, when we neglect the interaction between this field and other fields (and also neglect non-linear terms that are due to its self-coupling), the field equations are,p ai...jk,p + m2 0a i...jk = ρ i...jk, 6

7 where m 0 is a constant called a rest mass of the correspondent particle and ρ i...jk is a source field. Field equations: general case. In general case, when we take interactions into consideration, the equations are,p ai...jk,p + m2 0a i...jk = ρ i...jk + Θ i...jk, where by Θ i...jk we denoted the sum of all non-linear terms. Source terms in the classical approximation. We assume that the only source terms are those coming from classical point particles. To describe the ability of a particle to generate this field we must use a scalar q that is usually called its charge. So in every moment of time a classical particle can be described by its location r(t), four-velocity u i (t) and charge q(t). For each of the particles located in the point r(t) the source tensor ρ i...jk must be located in the same point r(t), so ρ i...jk ( r, t) must be proportional to a delta-function δ( r r(t)). Therefore, ρ i...jk ( r, t) = A i...jk (t)δ( r r(t)). Since ρ i..jk is a symmetric tensor, A i...jk must also be symmetric. The only symmetric expression that we can get from the four-velocities u i is ku i...u j u k for some constant k. Since the charge q(t) describes the intensity of the source, this constant k must be proportional to q(t), hence ρ i...jk = cq(t)u i...u j u k δ( r r(t)). In case of several point sources the total tensor ρ i...jk is equal to the sum of these expressions: ρ i...jk = cq a (t)u a i...u a j u a kδ( r r a (t)), where different values of a correspond to different point sources. Resulting equations that control charges and coordinates. Since a k i...j,k = 0, the divergence of the left-hand side of the above field equations is identically 0. Therefore the divergence of the right-hand side must also be identically equal to 0, i.e., k ρi...j,k + Θi...j,k = 0. The left-hand side of this tensor equation is a rank-(s 1) symmetric tensor. Therefore this tensor equation actually consists of N(S) scalar equations, where N(S) is the total number of components of such a tensor. If we substitute the above expression for the point-particles source term ρ i...jk into these divergence equation, then, arguing like in the case of General Relativity, we conclude that each of N(S) scalar equations in the location of each particle r a turns into an 7 k

8 equation that relates the three accelerations a(t) of this point particles and the first time derivative of its charge dq/dt with r, u i and the value of the field. So we have totally N(S) equations for four unknowns: a and dq/dt. Whether this system has a solution or not, and is this solution unique, depends on the relationship between the number of equations N(S) and the number of unknowns (four). Since N(S) depends only on the spin S, this relationship, in its term, depends on the spin S of the field. Let s consider all possible values of S. If spin is 2, then everything is OK. If S = 0, we have no divergence equations at all, so there is no inconsistency. If S = 1, we have one equation ρ k,k + Θk,k = 0 for four unknowns: three accelerations and dq/dt. In particular, in case of an electromagnetic field, when there are no non-linear terms (Θ k = 0) this equation leads to dq/dt = 0, i.e., to the electric charge conservation law. If S = 2, then N(S) = 4 (the number of components of the rank-1 tensor in a 4- dimensional space), so we have four equations for four unknown components. Therefore, just like in General Relativity, we can deduce the equations of motion. Comment. If a spin-2 field is controlled by linear field equations, i.e. Θ ij = 0, then the resulting equations of motion t j i,j = 0 lead to dm/dt = 0 and a(t) = 0. In other words, the trajectories of all the point particles are straight, and there is no interaction with the field. Therefore a non-trivial spin-2 field theory must be non- linear (because else the particles would somply not interact with each other at all). This fact is a specific feature of spin-2 theories; it does not happen in spin-1 (vector) theories, where Maxwell equations for the electromagnetic field are linear, but interaction between charged particles is possible. This is the reason why General Relativity is a nonlinear theory, while electromagnetic field equations can be linear and still describe interactions. When spin is > 2, we get inconsistency. When S > 2, i.e., when S 3, then S 1 2, and therefore the number of components N(S) of a rank-(s 1) symmetric tensor is at least as big as the number of components of a symmetric rank-2 tensor, which is 10. So we have at least 10 different equations for four unknown functions. This means that the resulting system is over-determined. 8

9 In some cases when the number of equations is bigger than the number of unknown, there is a solution. For example, if a field theory is linear, the resulting equations lead to dq/dt = 0 and a = 0. However, in the general case, if the number of equations in an equations system exceeds the number of unknowns, this system has no solutions at all (i.e., it is inconsistent). In case of a classical approximation to a spin > 2 field theory we have at least 10 equations for four unknowns, so we can conclude that the classical approximation of a spin> 2 field theory is (in general case) inconsistent. For non-integer spin there is no classical approximation. Indeed, non-integer spins correspond to spinor and odd-rank spintensor fields, and there is no way to construct a source spintensor of odd rank from a scalar q(t) and a vector u i. 4. CONCLUSIONS 1) We showed that in the classical approximation only fields with spin 2 are consistent; this explains why the mediators of all the known fundamental forces are particles of spin 2. 2) As a corollary we conclude that all the phenomena related to fields and particles of spin > 2 are purely quantum effects. Therefore their intensity must be proportional to a Planck constant and thus must be much smaller than the intensity of fields and particles with S 2. So we get an explanation of the following two phenomena: a) we explain why the most stable and wide spread elementary particles are of spin 2 (see, for example, Gaziorowicz and Rozner (3) ); b) we explain why, in spite of the theoretical possibility that torsion field (of spin 3) can also be responsible for gravitation, its effects has not yet been experimentally observed (see, for example, Misner, Thorne, and Wheeler, (11), and Pronin and Obukhov (10) ). ACKNOWLEDGEMENTS. The author is greatly thankful to Thomas E. Phipps and to the anonymous referees for valuable comments and suggestions. This work was supported by a NSF Grant No. CDA , NASA Research Grant No and the Institute for Manufacturing and Materials Management grant. 9

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