A model for a relativistic, many-particle Lagrangian ...

A model for a relativistic, many-particle Lagrangian with electromagnetic interactions R. A. MOOREA N D T . C. SCOTT Guelph-Waterloo program for graduate work in physics, Waterloo Campus, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 AND

M. B. MONAGAN

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Department of Computer Science, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 Received April 20, 1987 It is shown that the Fokker-Wheeler-Feynman theory for the relativistic modelling of a system of massive, charged, point particles interacting via action-at-a-distance forces, electromagnetic in origin, can be reformulated and reinterpreted so that it retains all of its required physical attributes but is devoid of the absurdities originally ascribed to it. That is, Lorentz covariance, time-reversal symmetry, and particle-interchange symmetry are maintained, whereas lack of causality and the paradox of "discontinuous" forces are removed. The reformulated theory yields a physically acceptable relativistic, many-particle Lagrangian. The Euler-Lagrange equations of motion can be written down for either closed or open systems. For closed systems, a generalized Hamiltonian, linear momentum, and angular momentum are constants of the motion. The concept of an open system is used to show that radiation reaction follows straightforwardly from the Euler-Lagrange equations of motion and their past and present time solutions. It is concluded that the basis for this type of modelling of such systems is now established. On montre que la thCorie Fokker-Wheeler-Feynman pour la modClisation relativiste d'un systkme de particules ponctuelles, massives et chargCes, en interaction B distance par des forces d'origine ClectromagnCtique, peut Ctre reformulCe et rCinterprCtCe de f a ~ o nB garder tous les attributs physiques requis, tout en Climinant les absurditks qu'on lui a originellement imputCes. La covariance lorentzienne, la symCtrie de renversement du temps et la symCtrie d'Cchange des particules sont maintenues, alors que la non-causalit6 et le paradoxe des forces "discontinues" sont enlevCs. La theorie reformulCe donne pour un systkme de plusieurs particules un lagrangien relativiste acceptable. Les Cquations Euler-Lagrange du mouvement peuvent &tre Ccrites pour des systkmes fermCs ou pour des systkmes ouverts. Pour les systkmes fermCs, un hamiltonien gCnCralisC, l'impulsion et le moment cinttique sont des constantes du rnouvement. Le concept de systkme ouvert est utilisC pour montrer que la rkaction de rayonnement dCcoule de f a ~ o ndirecte des Cquations Euler-Lagrange et de leurs solutions temporelles pour le pass6 et le present. On conclut que la base de ce type de rnodClisation est maintenant Ctablie pour de tels systkmes. [Traduit par la revue] Can. J. Phys. 66,206 (1988)

1. Introduction The purpose of the present work is twofold. The first objective is the relativistic modelling of a system of charged, massive, point particles interacting via action-at-a-distance (AAD) forces, electromagnetic in origin, without self-interactions, in complete analogue to the nonrelativistic case. Herein, a theory based on the Fokker-Wheeler-Feynman (FWF) model (1, 2) is presented that yields a relativistic many-particle Lagrangian expressed in terms of the particles' variables and the observer's time. Such a Lagrangian is one basis on which to build a description of these systems. The second objective is to determine the fundamental properties of the theory and to show that they are physically acceptable and are relativistic generalizations of the nonrelativistic case. This is achieved by combining a number of previous works and by reformulating and reinterpreting the FWF theory. There are a number of reasons for presenting the current work. First, although one might expect that the electromagnetic problem is completely understood, this is, in fact, not the case. The necessity for further work on some relativistic aspects of this problem is discussed in the next paragraph and in the next section. Second, such a modelling procedure is relatively straightforward, is open to a direct physical interpretation, and has been extremely useful for a wide range of nonrelativistic physical problems, in particular, many-particle systems. In addition, it has proven sufficient for many purposes, namely, phenomena not involving quantum-field corrections. Therefore, it would appear desirable to have the relativistic generalization. Third, only two brief initial reports (3, 4) exist and details need to be presented. Fourth, and final, similar, (see, e.g., refs. 5-14) and related, (see, refs. 15-19) modelling, with

the electromagnetic case being used as a prototype, is being done for systems with other interactions. However, in the former the interactions are, in general, included in either plausible or ad hoc ways. Thus it is of some interest and importance to establish criteria for physically acceptable interactions from the electromagnetic case. In fact, it is now known that the present theory can be generalized (20) to cover a class of interactions with the same properties. Although the above-mentioned type of modelling has been used extensively for electromagnetic interactions, current models appear to be incomplete for a description of relativistic corrections to applied-field effects in many-electron atoms; see e.g., refs. 21-23. Of course, the one-particle Dirac (24) equation has been extremely valuable; however, not surprisingly, it is inadequate for these systems. Perhaps the best available model is that given by the Breit (25) Hamiltonian. In this case, in its derivation, the particle velocities are replaced by their Dirac velocities in the 1/ c 2 contribution to the interparticle interaction energy, rather than by their position and momentum dependences. Thus, in applied fields, the standard substitution p - + p - ( e / c ) Acannot be made in this interaction and terms of the desired order are lost. Now, because the Breit Hamiltonian has been very successful in the description of zero-field situations, modifications must be considered carefully. Hence, the origins of the Breit Hamiltonian are reconsidered, namely, the Darwin Lagrangian (26). Now, difficulties (see, e.g., refs. 27 and 28) with the Darwin approach are known to exist. Thus, the Darwin approach is reviewed and examined in Sect. 2, and the conclusion is made that it is incomplete for a relativistic description of physical systems. In this way, we justify reviving the FWF theory (1, 2).

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A considerable amount of effort has been expended in the search for an acceptable relativistic AAD theory that is the generalization of the nonrelativistic theory, and the problem has been examined from different aspects (see e.g., Kerner (29) and Llosa (30) for reviews and references). Nevertheless, in spite of it being rejected as an alternate theory of electromagnetism, the FWF model appears to be the only other existing model for the present problem, and it has all of the ingredients necessary for the type of modelling that is being sought. It is this aspect of the theory that we exploit. The FWF theory is reviewed, reformulated, and reinterpreted in Sect. 3. It is based on a Lorentzinvariant action integral, which guarantees the proper Lorentz covariance of the theory, and has built in both time-reversal symmetry and particle-interchange symmetry. However, as interpreted by Wheeler and Feynman (1, 2), it yields an acausal particle Lagrangian and, hence, contains the paradox of "discontinuous" forces. Both of these features are physically unacceptable for the type of model being sought, and this seems to be why it has been rejected as a model for the present problem, as deterministic (causal) models are sought. It is shown that the FWF theory can be recast in a form that is causal, that the paradox of "discontinuous" forces is then nonexistent, and that the theory can be reinterpreted in a physically acceptable way. Furthermore, the resultant Lagrangian can be used to define a closed system, with the constants of motion as noted in Sect. 3, or an open system, with interactions with its surroundings. Thus, it is concluded that the reformulated FWF theory provides a suitable, physically acceptable particle Lagrangian for electromagnetic interactions, and that it can be used as the basis for the description of the type of phenomena mentioned above. The reformulated FWF theory is tested in Sect. 4, where the force of radiation reaction on a charged particle is derived directly from the equations of motion and the known features of the solutions. In this respect, the model is restored solely to a system of interacting particles and their motions. No absorber theory, no absorber conditions, no fields, and no special properties of the universe are required. Finally, Sect. 5 contains further discussion.

2. Rejection of the Darwin approach Darwin (26) starts by making the physically plausible assumption that the one-particle Lagrangian for particle one, say, is its free-particle Lagrangian (see, e.g., Goldstein (31)) plus its interaction energy with the retarded Litnard-Wiechert fields of particle two. That is, where mlo is the rest mass of particle one; c is the speed of light; and

with ul (= drl/dt) being the usual spatial three velocity, rl being the spatial position vector, and t being the observer's time. Finally, vY2 is given by, in a convenient form for our purposes,

Here the 4;'s are the particles' charges; the tilde denotes a four vector; and hence, Yi = (ri, ict,) and p i = mioyi(ui, ic) are the position and momentum four vectors. The a denotes that either the retarded time, tR, a = R , s~ = 1, or the advanced time, tA, a = A , SA = - 1, as given by [2.4], is the argument; otherwise, it is the current time, t. For many particles the 2 is replaced by j, which is summed over. Next, Darwin converts vY2, [2.4], to its l / c power-series expansion. The (l/c)O term is the instantaneous Coulomb interaction between the two charges. The (1 /c)l term is zero. The (l/c)' term, originally asymmetric in the two-particles' variables because vY2 is asymmetric, is symmetrized by adding a total-time derivative, a divergence, to this term. It is this form that was used to produce the Breit Hamiltonian. The (1 / c ) ~term is the first term known (see, e.g. refs. 27 and 28) to cause difficulties and is proportional to

+

Here the dot signifies a time differentiation. Simply put, VY2,, cannot be written as a total-time derivative nor can it be symmetrized with respect to particle interchange by adding another divergence. Thus it cannot be used to define a two-particle interaction. Using the Maple symbolic-algebra package (32) available on the University of Waterloo computing systems, we have to 1/el0. We have verified extended the 1/c expansion of that each of the next odd-powered terms in the series has the same difficulties as the 1/c3 term. Thus, in spite of its intuitive correctness, one can only conclude that the Darwin approach is incomplete for the proper description of the relativistic twobody problem with electromagnetic interactions. This leads to the consideration of the FWF model (1, 2).

e2

3. Reformulation of the Fokker-Wheeler-Feynman theory The attractiveness of the FWF theory lies in the original formulation, namely, a Lorentz-invariant action integral, J, that has particle symmetry and satisfies time reversal.

Here -ri (= d t l y j ) is the proper time of the ith particle; the delta is the usual Dirac delta function; and since j i , there are no self-interactions. The fixed end points have been adequately discussed elsewhere (1 3, 14, 33). All one needs to know is that the end points are such that the variable of integration can be taken as the observer's time and that the Euler-Lagrange equations are meaningful. Because J is a Lorentz scalar, the theory satisfies Lorentz covariance. To extract the Lagrangian, we must write J in standard form. The free-particle contributions are

+

and yield the standard free-particle Lagrangians (3 1); namely, the quantity in parentheses. The interaction terms are obtained by using (34) subject to

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and

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to carry out one of the time integrations and to obtain

shown that the l l c power-series expansion of [3.6] can be exactly particle symmetrized, a possible key in understanding this system as suggested by our example, by the addition of a divergence. However, the physical interpretation is completely obscured by the power series. Therefore, we look for an equivalent divergence in closed form. This divergence turns out to be quite elusive until one recognizes that vT2 (t = t) is closely related to VFI (t = g).Use [2.3] to define

Recognize from [2.4] with tl = t that where the V",.'s are defined by [2.3] and [2.4]. Extracting the Lagrangian for, say, particle one interacting with, say, particle two yields [3.6]

L I = -mloc2/yl - +(Vy2 + Vf2)

The extension to many particles is straightforward. L I is in exactly the correct form (35, 36) to ensure that the EulerLagrange equations of motion satisfy the correct Lorentz covariance. Now, because particle one appears to interact with both the retarded and advanced fields of particle two, Wheeler and Feynman interpreted this result as a violation of causality and simply postulated causality away. This leads to their paradox of discontinuous forces or, equivently, lack of free choice, which they again postulated away. Such an interpretation is simply unacceptable for a physical model. To resolve this situation, one must recall some perhaps peculiar features of Lagrangians. At this point, we digress to a well-known problem; namely, a pair of particles constrained to move along the x axis and connected by an ideal spring of equilibrium length toand force constant k. The Lagrangian is Each term is well defined and has a precise physical meaning. However, an arbitrary total-time derivative can be added to L. For example, one can write

Now, L' yields exactly the same Euler-Lagrange equations of motion as does L. However, the interaction in L' is devoid of physical meaning, and one can imagine the absurdities that would result if a physical interpretation were attempted. This example illustrates a curious feature of Lagrangians and the need to use extreme caution in extracting a physical picture of the system from a particular Lagrangian. It is exactly this dilemna that we are faced with in intepreting L , , [3.6], in spite of its simple form. In this case we have no a priori reasons for making a decision. To gain some insight into Lagrangians for the present problem, we note two points. First, a direct interpretation of [3.6] attributes physically unacceptable properties to the model. Second, although J is initially symmetric under particle interchange, the interaction in L1 is not. These imply that L, as written in [3.6] is not in an appropriate form for a direct physical interpretation. Now, Kerner (37) (see also Kennedy (38)) has

where we have put it in a particularly convenient form. Finally, note that It follows immediately that

Hence,

Now, it must be recognized that a distinction between L 1 ,[3.6], and Li, [3.13], cannot be made using the Euler-Lagrange equations; they are identical because the divergence satisfies them identically. This means that there is an equivalence between VFl and Vf2. Looking elsewhere, note that as required, the interaction in Li is symmetric under particle interchange and, in fact, has precisely the form that one would intuitively guess. Further, only the present and past times appear; hence, in this form, the system can be interpreted as being causal. In other words, it was unnecessary for Wheeler and Feynman to postulate lack of causality on the basis of [3.6]. Ihe paradox of discontinuous forces is now nonexistent, and additional interactions and forces can be added at any time in any physically acceptable way. It is true that our concepts of interactions of electromagnetic origin must be revised; however, once the Darwin approach has been rejected, this is unavoidable. Ndice that Thus, the interaction energy of particle one with particle two is just the usual contribution plus a correction. From the discussion of Sect. 2, this correction is of order 1/c3; it must represent the need for the model to have an internal self-consistency and must be an expression of the mutual self-consistent coexistence of the two particles, or the universe, from all past time to the present. This completes the reformulation and reinterpretation of the FWF model, and one is left with examining its properties. The constants of motion for closed systems are most readily (29, 37, 38) ascertained using the l / c power-series expansion of the interaction. Because of the occurrence of past times and the constraints, [2.4], all orders of the time derivatives of position appear and generalized mechanics (see, for example, ref. 39) applies. Note that [2.4] is a constraint that must be used

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in the Lagrangian at the beginning and removes any explicit time dependence from the Lagrangian. As noted earlier, there are three constants of the motion. Define

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where j is the particle identifier. Define the generalized momenta by

The Euler-Lagrange equations of motion are

For the FWF model, the system satisfies spatial translational invariance and hence the total first order linear momentum is a constant of the motion, that is, [3.18]

lated FWF model has all of the properties required for the type of modelling of physical systems being sought.

4. Radiation reaction For the reformulated FWF model to be acceptable, it must give a direct explanation of radiation reaction; that is, it must be based solely on the basis of point particles, their action-at-adistance forces, their equations of motion, and the past and present time solutions. It should be devoid of fields and special properties ascribed to the universe, that is, absorber theory. It is now shown that the reformulated FWF model satisfies all these conditions. We use [3.13] and [3.14] to write the Lagrangian for particle one interacting with the rest of the universe:

where the prime means j + 1. The 1l cpower-series expansion V?, is

2 pj') = constant j

A generalized Hamiltonian function is defined by [3.19] H = I

2

with all quantities being evaluated now at time t . Setting Kl, = q l q j / ~ , ,r, l j = r l - rj, and rlj = Irl - r j l

r,("+').p,(")-~

; n=1

Because the Lagrangian has no explicit time dependence, this function is a constant of the motion. Finally, the generalized angular momentum defined by

is also a constant of the motion. This follows because the FWF model satisfies spatial rotational invariance. Thus the reformu-

one has [4.3a]

V t v o= K1,/rl,

[4.3b]

V;,~=O

[4.3d]

v;,~ = Klj

rU a, . u1- a, uj + -. a j 3

etc. Similar expressions exist for v>.It is convenient to write each of these terms, as much as possible, with a part that is symmetric in particle interchange and a part that is a total-time derivative. Thus set

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where

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Combining the above results reduces L 1 ,[4.1], to the form

Here df/dt represents all the total-time derivatives contained in the terms given by [4.4a]-[4.4c] and O(1 /c5) represents all contributions from the l / c power-series expansions of order 1/c5 and higher. The Euler-Lagrange equation of motion is

Here F Ris the standard retarded Lorentz force on particle one due to the rest of the universe. The remaining force arises from the self-consistency of the universe and, as noted earlier, is small. Both terms are instantaneous as the information has already come from the past and particle one simply responds to it. Clearly, no information from the future is involved nor are there any response-time delays; that is, it is particle one that responds to the rest of the universe and it is not the rest of the universe responding to particle one. However, to solve for the motion of particle one, we need to know the motions of all of the other particles in the universe, in principle. This is to be expected for this model. Not only is this impossible but this model would not yield observed results, because the real universe is not described solely by electromagnetic forces. As a direct solution is not possible, we look for another approach; namely, we infer sufficient information from observation. First, the universe appears to be neutral. This aspect must be used to define the system and yields no information about the motions. Second, in spite of large variations in distribution, on average, any finite volume of the universe is also neutral. This we call local charge neutrality, and we quantify it by

Here the sum over j can be over any finite volume, including the entire universe. Another interpretation of [4.4] is that on average, any finite volume of the universe does not have a net dipole moment. Any deviations appear as fluctuations or corrections. Notice that [4.8] is not a constraint on the system and, hence, cannot be used in the Lagrangian. Rather, it describes the present solution (state) of the motion of the particles in the universe. If the universe were in a different state, this term would be different. Letting the sum over j in [4.8] go over the entire universe, including particle one; differentiating thrice; and substituting this in [4.7] yields

The last term is exactly the standard radiation-reaction force (see, e.g., Jackson (35) eq. [17.8]). The reformulated FWF model permits a straightforward physical explanation of the standard form for radiation reaction

using only the dynamic properties of the model and known properties of the universe. Because we have started with an exact Lagrangian, [4.1], and derive only the lowest order approximation to the radiative-reaction force in [4.9], a few additional remarks may be useful. First, note that the exact perfect-absorber condition is given by Rohrlich (ref. 40), eq. [7.25]). This condition depends upon the difference (v?,Vt). Now from Kerner (37, 41) and Kennedy (38) this difference is known to be an odd function of c, and hence, because of [4.3b], the first nonzero contribution comes from the l / c 3 term, that is, [4.3d]. Evaluating this l / c 3 contribution for the perfect-absorber condition yields

Thus, the present state of the universe, [4.8], satisfies the perfect-absorber condition to the same accuracy as it yields the radiation-reaction force in [4.9]. Second, the validity of [4.9] depends upon the convergence of the l / c power series. For example, the expansion of the right-hand side of [2.4] converges only if the distance moved by particle two in the time Itl - tyl is less than its separation with particle one. This is equivalent to saying that the particle's speed is less than c. However, from [4.3a]-[4.3e] it is clear that this is simply a low-order statement and that a general convergence condition may not be possible. To simplify this problem, we have considered classically the circular motion of two oppositely charged equal-mass particles. Now there is a single expansion parameter u/c, where u is the common speed of the particles. In this case, the power series of the time delay, the interaction energies, and the forces all have the same radius of convergence, namely, u / c < 0.6627 ..., which is quite small. However, it is anticipated that in general, the nearby charges, for which the power series are nearly valid, will dominate the radiative-reaction force, whereas the effects of distance charges will tend to go to zero and cancel. Thus [4.9] should be approximately valid. Finally, we note that it is known (see, e.g., ref 42), how to incorporate some of the higher order corrections into [4.9].

5. Discussion It has been shown that the FWF theory can be reformulated and reinterpreted to yield a physically acceptable relativistic model, through a Lagrangian, for a system of massive, charged point particles interacting via action-at-a-distance forces of electromagnetic origin. The theory starts from a Lorentzinvariant action integral, and this ensures the proper Lorentz covariance to the theory, in particular, to the equations of motion. The theory is constructed to satisfy time-reversal symmetry and particle-interchange symmetry. However, both

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MOORE ET AL.

of these symmetries cannot b e displayed in the multiparticle Lagrangian simultaneously. The theory can b e cast in a form in which causality is seen to be satisfied. Thus, physically acceptable additional interactions or forces can be applied at any time. The model can be used to describe a closed system or an open system. In the case of a closed system, the model yields a generalized Hamiltonian, a generalized linear momentum, and a generalized angular momentum that are constants of the motion. In the case of an open system, the interaction with the surroundings, e.g., with the rest of the universe, can be unambiguously defined. This permits a straightforward description of the so-called radiation-reaction force. At this point, the foundations have been laid for further examination of relativistic corrections to the instantaneous approximations, both classically and quantum mechanically. Further, there exists a considerable amount of literature on a variety of aspects of electromagnetic problems, and the proposed model has rather wide ranging implications on many of them. Also as noted earlier, the present model with electromagnetic interactions can be used as a prototype and can be generalized (20) to specify a class of interactions, all having the same properties; it is anticipated that this work will have a number of uses. Incidently, this work shows why the nointeraction theorems, e.g., see Chelkowksi (43) for a recent review and references, exist. In these cases the assumed forms of the interactions are incomplete and hence ultimately yield contradictions.

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