Front form and point form formulation of predictive relativistic mechanics . Noninteraction theorems

Instantaneous relativistic dynamics of particles with direct interaction was initiated in a celebrated paper by Dirac, 1 entitled "Forms of relativistic dynamics." The line of thought set up there was further developed by Bakanjian, Thomas, and Foldy, 2 in the framework that Dirac had called "instant form." The subsequent development of the theory met the important drawback of the so-called "no-interaction theorem." 3 In general terms, it states that, if the position coordinates of the particles are to be canonical coordinates, and the particle worldlines must be Poincare invariant, then the only systems that are compatible with both requirements are those consisting of free particles. One attempt to circumvent this problem was initiated by Currie, and later on it has generated a rather wide stream of literature, which is known as predictive relativistic mechanics (and maybe, it should be called an instant form of PRM). It consists, first, in giving up the Hamiltonian formalism, which was taken for granted in former approaches, and starting from a more elementary level. The fundamental assumptions in predictive relativistic mechanics are (i) the equations of motion of the particles are Newton-like, that is, the acceleration of each particle is a given function of positions and velocities of all particles; and (ii) Poincare invariance, which is understood to mean two things: the acceleration functions must be formally the same in all inertial reference frames, and particle worldlines must be Poincare invariant. These requirements imply that some condition (the so-called Currie-Hill equations ) must be fulfilled by the acceleration functions. In addition, they also ensure the possibility of setting up a realization of the Poincare algebra on the system's tangent space (the one spanned by positions and velocities). Now, the no-interaction result can be obtained again 7 if one seeks for a Hamiltonian formalism such that the aforementioned realization of the Poincare algebra is canonical, and the position coordinates can be taken as canonical. As far as we know, all proofs of the no-interaction theorem hitherto derived share a common feature, namely, physical variables are assumed to be simultaneous in a given inertial frame. This is a specific trait of the "instant form" of relativistic dynamics. However, in the pioneering paper by Dirac,l two other possibilities were considered, namely, the "front form" and the "point form" (in fact, a later paper by Leutwyler and Stem 7 increases that number by two more "forms"). One then wonders whether the no-interaction theorem, or a related result, also holds in these two alternative forms of relativistic dynamics. Although this is, indeed, an interesting point to be elucidated, it seems not to have been proven yet. Indeed, in a relatively recent paper by Leutwyler and Stem 7 we can find the following sentence: "Although this no go theorem has been established only for theories of class (i) (i.e., the "instant form" of relativistic dynamics) it likely also holds for the remaining four forms of Hamiltonian dynamics." In the present paper we intend to give an answer to the question that is more or less implicit in the quoted sentence, and derive a no-interaction theorem in the front form as well as in the point form. The master lines of our proof are the same as those of the proof given by Hill for the no-interaction theorem in the "instant form." In a natural way, the paper is divided in two parts. The first one (Secs. II and III) is devoted to the front form, and the second one (Secs. IV and V) to the point form. Besides, each part is organized in two sections: one devoted to develop what could be called the front (resp. point) form of predictive relativistic mechanics, and the other to prove the nointeraction theorem.


I. INTRODUCTION
Instantaneous relativistic dynamics of particles with direct interaction was initiated in a celebrated paper by Dirac, 1 entitled "Forms of relativistic dynamics."The line of thought set up there was further developed by Bakanjian, Thomas, and Foldy, 2 in the framework that Dirac had called "instant form." The subsequent development of the theory met the important drawback of the so-called "no-interaction theorem." 3 In general terms, it states that, if the position coordinates of the particles are to be canonical coordinates, and the particle worldlines must be Poincare invariant, then the only systems that are compatible with both requirements are those consisting of free particles.
One attempt to circumvent this problem was initiated by Currie, 4 and later on it has generated a rather wide stream of literature, which is known as predictive relativistic mechanics (and maybe, it should be called an instant form of PRM).It consists, first, in giving up the Hamiltonian formalism, which was taken for granted in former approaches, and starting from a more elementary level.The fundamental assumptions in predictive relativistic mechanics are (i) the equations of motion of the particles are Newton-like, that is, the acceleration of each particle is a given function of positions and velocities of all particles; and (ii) Poincare invariance, which is understood to mean two things: the acceleration functions must be formally the same in all inertial reference frames, and particle worldlines must be Poincare invariant.These requirements imply that some condition (the so-called Currie-Hill equations 5 ) must be fulfilled by the acceleration functions.In addition, they also ensure the possibility of setting up a realization of the Poincare algebra 6 on the system's tangent space (the one spanned by positions and velocities).Now, the no-interaction result can be obtained again 7 if one seeks for a Hamiltonian formalism such that the aforementioned realization of the Poincare algebra is canonical, and the position coordinates can be taken as canonical.
As far as we know, all proofs of the no-interaction theorem hitherto derived share a common feature, namely, physical variables are assumed to be simultaneous in a given inertial frame.This is a specific trait of the "instant form" of relativistic dynamics.However, in the pioneering paper by Dirac,l two other possibilities were considered, namely, the "front form" and the "point form" (in fact, a later paper by Leutwyler and Stem 7 increases that number by two more "forms").
One then wonders whether the no-interaction theorem, or a related result, also holds in these two alternative forms of relativistic dynamics.Although this is, indeed, an interesting point to be elucidated, it seems not to have been proven yet.Indeed, in a relatively recent paper by Leutwyler and  Stem 7 we can find the following sentence: "Although this no go theorem has been established only for theories of class (i) (i.e., the "instant form" of relativistic dynamics) it likely also holds for the remaining four forms of Hamiltonian dynamics." In the present paper we intend to give an answer to the question that is more or less implicit in the quoted sentence, and derive a no-interaction theorem in the front form as well as in the point form.The master lines of our proof are the same as those of the proof given by Hill 5 for the no-interaction theorem in the "instant form." In a natural way, the paper is divided in two parts.The first one (Secs.II and III) is devoted to the front form, and the second one (Secs.IV and V) to the point form.Besides, each part is organized in two sections: one devoted to develop what could be called the front (resp.point) form of predictive relativistic mechanics, and the other to prove the nointeraction theorem.

MECHANICS
In the instant form of predictive relativistic mechanics 8 (which has been its only formulation up to now), the extended configuration space of N spinless particles is spanned by the 3N + 1 variables: t, x~, b = 1, ... , N, i = 1,2,3; where the evolution parameter is the time ,coordinate as measured in a given inertial frame, and the x~ are the space coordinates of the event determined by the intersection of the worldline of particle b and the space hyperplane x 4 = t.
The equations of motion are then required to be secondorder differential equations, that is, (2.1) Thus the space of initial data is spanned by the following 6N + 1 variables: t, xL u~, b,e = 1, ... , N, i,k = 1,2,3.
If the space hyperplanes x 4 = t characterize the instant fonn, likewise the null hyperplanes x 3 + X 4 = A will playa central role in the front fonn of relativistic dynamics (here t and A are two real parameters).So, the extended configuration space in the front fonn will be coordinated by the 3N + 1 variables: A, x~, b = 1, ... , N, i = 1,2,3; where A is the evolution parameter and xL i = 1,2,3, are the space coordinates of the event where the worldline of particle b meets the null hyperplane or, using the notation introduced in (AS), (the same value of A for all particles).For convenience, our configuration space coordinates will be (see Appendix A) x;: , a = 1, ... , N, A = 1,2, -, rather than the Cartesian x~, i = 1,2,3.
We now require the motion to be governed by a secondorder differential system, For every given solution of (2.3), we have a set of N worldlines describing the history of the system.Indeed, if lP:(x:,zf; A), A,B,D = 1,2, -, a,b,e = 1, ... ,N, is the solution of (2.3) corresponding to the initial data then, according to (2.2) and (AS), the worldlinexb(A) of particle b will be taken as which in the adapted coordinates (A4) reads (2.5) Similarly to the instant fonn description, the principle of relativity will be used at two different levels.First, the "acceleration" functions at on the right of Eq. ( 2.3) must have the same fonn in every inertial frame.And second, the dynamic system must be worldline invariant.The latter requirement means the same as in the instant fonn case, namely, that if Sand S I are two inertial frames related to each other by a Poincare transfonnation (2'j,..ra1 D ), A,Ii)) = 1,2, -, + -see the Appendix-and X:(A ), b = 1, ... ,N, A = 1,2, -,+ are the worldlines of the particles in the frame S, when the system starts from a given set of initial data Zo=(X;:,u~), a,e = 1, ... ,N, A,D = 1,2, -; then the Poincare-transfonned worldlines must be obtained in the frame S I, when the system starts from the transfonned set of initial data Zo =(X~A ,U;D).
Thus, as happens in the instant fonn, the mapping Zo --Zo defines the induced action of Poincare transfonnation (2'j,..ra1 D ) on the space of initial data.
The infinitesimal generators are then obtained in the usual way: (2.9) a~Fax;: a~F au: To obtain the specific expressions for these generators, we shall work out the condition of worldline invariance (2.7), together with the worldline equations (2.6).Introducing the latter into both sides of (2.7), we obtain lP~(zo, Aa (zo It should be noticed that, since the "acceleration" functions have the same fonn in frame S and in S I, the same general solution lP ~ has been substituted into both sides of Eq. (2.7).However, whereas in the right-hand side we take the initial data Zo = (x;: ,v:), which correspond to the frame S, in the left-hand side we have to put the transfonned initial data Zo = (X~A,UbB) which correspond to the worldlines as viewed from the frame S I.Moreover, the value of the evolution parameter in the left-hand side ofEq.(2.10), which we have written as Aa (zo, A), will be presumed different from the parameter A in the right-hand side.This is due to the fact that the worldline invariance only ensures that each worldline transfonns into another one as a whole, no matter how the respective parametrizations are related to each other.
Equations (2.10)-(2.14),which hold for every value of A and for every Poincare transformation (2' ~ ,.9/ D), actually determine the functions!:,gg in (2.8).Although, apart from a few trivial cases, it would be impossible to derive explicit expressions for such functions, the above equations permit us to obtain the infinitesimal generators in a rather straightforward way.Indeed, introducing the infinitesimal expression (AI2) for the Poincare transformation (2'~,.9/D)into Eqs.
Upon substitution into (3.9),this finally yields {a!,V:} =0, a#b.(3.17) That is, the acceleration a:, B = 1,2, -, of each particle b does not depend on the positions and velocities of the remaining ones, but only on its own position xt and velocity vt.This conclusion would be enough to consider that the nointeraction result is proven, since the motion of each particle is not affected by the presence of the others.However, in the case we are considering (i.e., front form) a little bit deeper analysis reveals that the accelerations actually vanish.Indeed, from (2. 15b) and (3.17In the instant and front forms of dynamics, the construction of the configuration space was somehow linked to the choice of either the space hyperplanes X4 = t or the null ones x 3 + X4 = A., respectively.In the point form, the hyperboloids xP xp = -A. 2 will be assigned a similar role. Each point in the extended configuration space will be characterized by 3N + 1 coordinates (x~ , A. ), a = 1, ... , N, i = 1,2, 3, where A. is taken as an evolution parameter and the x~ are the spacelike coordinates of the event where the worldline of the ath particle intersects the hyperboloid As in the earlier two cases, the equations of motion are second-order differential equations  (4.6) As is easily seen from this equation, and also from (4.4), the correspondence between A and the time coordinate qJ ~ is not one-to-one.In order to avoid the nondifferentiability in the branch point A = 0, we shall take hereafter ..10 and A positive.
Moreover, the translation parametersAIt will be assumed to be small enough for A ~ (zo, A ) on the left-hand side of Eq. (4.6) to remain positive.By differentiating (4.5) with respect to A we obtain the transformation formula for the velocities Lit v 4; ~ (zo, A ) = 4;: (z~, Aa (zo, A )) Aa (zo, A) ,   is the infinitesimal generator of A evolution.Equations (4.13) play a similar role as Currie-Hill conditions in the instant form of dynamics.It can be easily proven that they are equivalent to the following requirements.
(i) The generators Pit' J ltV given by (4.12) generate a re- alization of the Poincare algebra on the space of initial data (i.e., their commutation relations are the suitable ones).