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The classical description of charged particles F. Rohrlich Citation: Phys. Today 15(3), 19 (1962); doi: 10.1063/1.3058058 View online: http://dx.doi.org/10.1063/1.3058058 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v15/i3 Published by the American Institute of Physics. Additional resources for Physics Today Homepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition Downloaded 05 May 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://www.physicstoday.org/about_us/terms

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Page 1: The Classical Description of CHARGED PARTICLES

The classical description of charged particlesF. Rohrlich Citation: Phys. Today 15(3), 19 (1962); doi: 10.1063/1.3058058 View online: http://dx.doi.org/10.1063/1.3058058 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v15/i3 Published by the American Institute of Physics. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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TheClassicalDescription of

CHARGED PARTICLESBy F. Rohrlich

OF the approximately thirty elementary particlesknown or assumed to exist today about half arecharged. The rest are electrically neutral, though

some of these undoubtedly consist of positive and nega-tive charge distributions, as does for example the neu-tron. A complete understanding of these particles willinvolve quantum mechanics, quantum field theory, orperhaps a not-yet-dreamed-of theory. But under cer-tain conditions a classical description is adequate.

In fact, charged elementary particles are studied bymeans of particle accelerators, i.e., by means of elec-tric and magnetic fields; their motion is almost en-tirely described by means of classical particle electro-dynamics. No physicist bats an eyelash about that. ButI know a number of them who would not readily agreethat quantum electrodynamics has a classical limit.

Is there a consistent classical description of a chargedparticle? To be sure, it would have a limited domain ofvalidity, viz., outside the domain of quantum mechan-ics. It would be related to a quantum description, justas classical mechanics is related to quantum mechanics.Classical mechanics, although having only a limiteddomain of validity, is a consistent theory which standson its own. Is there an equally consistent classical par-ticle electrodynamics?

Background

AT first, we should be quite content with a descrip-tion of spinless particles such as IT mesons. All

the essential difficulties are already contained in sucha theory.

A spinless elementary particle is spherically sym-metric. Its size is given in order of magnitude byits Compton wavelength; the largest such particle istherefore the electron (X « 4 X 10"11 cm). It followsthat the size of elementary charged particles is toosmall to be seen by classical observations. They arepoint particles. This is the more reasonable, becauseany structure that these particles might have is cer-tainly of a quantum-mechanical nature and not mean-ingfully described in a classical theory. And as far aselectrons are concerned, even quantum electrodynamics

F. Rohrlich, a theoretical physicist, holds a professorship in theDepartment of Physics and Astronomy at the State University ofIowa in Iowa City.

pictures them as point particles in the sense that theyhave no structure. All experiments so far have con-firmed this.

But here is also the first and most important diffi-culty: while a sphere of radius r and charge e has anelectrostatic energy (self-energy) kez/r, where A is anumber which depends on the charge distribution in-side the sphere, a point charge (r —•> 0) has infinite self-energy.

This difficulty was already encountered by Lorentzand Abraham who, after the discovery of the electronat the turn of the century, constructed the first impor-tant classical theory of a charged particle. Another diffi-culty which is closely connected with this one is theinstability of a classical charge: a (nonelectromagnetic)attractive force is apparently necessary to hold togetherthe individual parts of a charged particle. This force isoften referred to as the "Poincare stress".

Going from statics to kinematics we find more diffi-culties: the Coulomb field surrounding the electron hasenergy and, correspondingly, a mass, according to spe-cial relativity. Since an electron always carries its Cou-lomb field with itself, this field must satisfy the kine-matics of classical mechanics. If me is the mass of thatfield and v its velocity, then its momentum must begiven by p = mev. Lorentz showed that this is not thecase. A factor -£ on the right-hand side of this equationspoils the theory. The momentum and the velocity donot keep in step with each other even for a free par-ticle. The descriptions of the particle and of the sur-rounding Coulomb field cannot be combined into aunified whole, contrary to our observational evidence.

Finally, when we inquire about the dynamics of amoving charge, we are given the well-known Lorentzforce equation

= e(E + v X B) (1)

discussed in every text, but we are warned not to takethis equation too seriously: the radiation reaction isneglected. The radiation emitted by an acceleratedcharge causes a recoil effect and an energy loss, quanti-ties which are not properly included in the Lorentzforce equation.

For a point particle, it was shown by Lorentz thatthe radiation reaction can be included by adding a term

March 1962

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proportional to the time derivative of acceleration.Abraham generalized this to a relativistically validterm. Dirac, in a famous paper1 published in 1938, re-derived the equation of motion, including the Abrahamterm, from first principles by a very different methodthan was used by Lorentz and Abraham. This Lorentz-Dirac equation thus seems to be the answer: It is arelativistically correct equation and it includes properlythe radiation-reaction effects.

There is, however, one trouble with this equation.It yields, in addition to the physical solution of a givenproblem, unphysical solutions in which the charged par-ticle characteristically accelerates indefinitely as timegoes on, even if the force ceases to act, and, in fact,even when there is no force at all. These "runaway"solutions prevented the Lorentz-Dirac equation frombeing generally accepted as a fundamental equation.But our theory must consist of Maxwell's equations forthe fields associated with moving charges and fieldequations for the motion of charges caused by fields,the latter being established only in the form of Eq. (1),viz., when radiation reaction is neglected. Thus, we donot have a completely consistent classical particle elec-trodynamics.

The above difficulties of self-energy and self-stress,of kinematics and of dynamics for a classical chargedparticle, are in striking contrast to relativistic quantumelectrodynamics. Though this theory is by no means inperfect shape or in its final form, it has been developedsufficiently far to cope with self-energy problems byrenormalization, to make the self-stress vanish, and totreat the dynamics of electrons, including radiation re-action in agreement with the most accurate experimentsin physics today, involving confirmation to nine figuresand more. Yet we still consider classical electrody-namics, which is in much worse shape, to be a specialcase of quantum electrodynamics.

This paradoxical situation can be explained to alarge extent by the fact that relatively very few theo-retical physicists are nowadays working on the funda-mentals of classical electrodynamics, many of them be-ing carried away by the wealth of new discoveries inexperimental physics, especially in high-energy and ele-mentary-particle physics. It is deplorable that the logi-cal consistency of various branches of physics, and thebeauty and harmony resulting from an understandingof their relationship with each other, is taking a backseat in favor of the necessarily speculative approachdemanded from the latest "theory" about the most re-cent results from the biggest accelerator.

Recent Progress—Dated 1922

THE infinite self-energy of a point charge is oftenblamed for all the difficulties of the classical the-

ory- In particular, it has been stated repeatedly thatthe unsatisfactory relation p = ^ mrv arises becauseboth p and me are infinite. At the same time, this re-lation is not consistent with relativity, because the en-ergy and momentum of a particle constitute a four-

(0)

Fig. 1

vector, so that the relation mec- — E for the rest en-ergy is inconsistent with the factor-|. Thus, one arrivesat the conclusion that the transformation properties ofp are incorrect (i.e., it does not arise from the non-relativistic limit of a four-vector) because the theorydiverges.

But this conclusion is wrong. We know the exampleof quantum electrodynamics. Here is a relativistic the-ory which diverges. The self-energy is infinite, but allquantities transform correctly. Why. then, should thisnot also be the case in its classical counterpart?

The solution of this dilemma is very simple. It wasfirst pointed out in 1922 (!) by Fermi.- but was thencompletely forgotten. It never found its way into thetextbooks. It was rediscovered by Kwal3 in 1949, butagain without receiving any attention. Ten years laterI also rediscovered the solution to this problem andpublished it4 before learning of the work of these twoauthors.

Lorentz knew how to compute the energy and themomentum of radiation from the energy density andthe Poynting vector. He then assumed that the sameformulas can be used to compute the energy and themomentum of the Coulomb field surrounding a freecharged particle. He did this nonrelativistically as wellas relativistically, and he took great pains to accountfor the relativistic deformation of a moving sphere.But he did not use covariant notation and consequentlyhe did not notice that he made a mistake. He did nottake into account that the three-dimensional volumeelement in his integrals is one component of a four-vector and must be transformed accordingly. He didnot realize that it is absolutely essential in a relativistictheory that the observer in the rest system of themoving particle should always see the same thing, no

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matter what the velocity of the uniformly movingcharge might be. This is not a trivial matter, becausea Coulomb field extends over all space; it is a non-local object. And the point particle, being the limit ofan extended particle, must be "rigid". This means ex-actly that the comoving observer sees the particle andits field in its usual state of rest. In a Minkowski space-time diagram this implies that the plane which de-scribes three-dimensional space must be tilted when oneintegrates over a moving charge. The tilt must be suchthat the velocity and the normal to the plane are paral-lel at every instant. (Fig. 1.)

When this is taken into account, the nonrelativisticlimit shows that Lorentz' formula for E was correct,but that the one for p lacked a term. That missingterm involves the Maxwell stress tensor (to no one'ssurprise after the above remarks about rigidity). Ityields — J m,.v, so that one finally obtains the ex-pected result p = mcv without the factor | .

One concludes that in the classical theory also thedivergent nature of certain quantities cannot be heldresponsible for incorrect transformation properties. Wehave a relativistically correct theory, though it is stilldivergent.

In classical electrodynamics we can now do just aswell as in quantum electrodynamics. There the methodof renormalization removed all divergences and madethe self-stress vanish. Renormalization can also be ap-plied to classical charges. In fact, it was used in theclassical case by Kramers ° and Dirac 1 long before itsuse in quantum electrodynamics. Today, with the ex-perience of quantum field theory, this can be donemore generally and more elegantly.

You Pay For What You Get

H P HE classical and the quantum description of-*- charges are now at par. The only remaining stum-

bling block is the equation of motion. The Lorentz-Dirac equation would be quite satisfactory if it werenot for the runaway solutions.

There is, however, another feature of the Lorentz-Dirac equation. As mentioned previously, it contains aradiation-reaction term which is the time derivativeof acceleration, i.e., the third derivative of position.The differential equation is therefore of third order.Such an equation requires three initial conditions. TheNewtonian equation of motion of classical mechanics isof second order, requiring initial position and velocity.Here, we need also the initial acceleration. This is verybad, because we don't know how to choose this initialacceleration. Only for a very special choice of it doesone obtain the physical solution. The smallest error inthis choice is sufficient to assure a runaway solution.

This situation shows clearly that, in agreement withone's physical intuition, the equation should not be ofthird order. What should be specified is not the initialacceleration (which we don't know how to choose) butthe final acceleration. We do know something about theacceleration in the distant future: it should not be infi-

nite, i.e., it should not be a runaway solution. Thus,what we have here is an asymptotic condition on theacceleration, a condition which must be an integralpart of the equation of motion, so that only physicalsolutions will arise.

This asymptotic condition can indeed be combinedwith the Lorentz-Dirac differential equation. The resultis an integro-differential equation." This equation mustnow be regarded as the equation of motion and as oneol the fundamental equations of classical particle elec-trodynamics. It is not equivalent to the Lorentz-Diracequation, but contains, in addition, the restriction tophysical solutions only.

This new equation of motion is of second order, sothat it poses an initial value problem of the Newtoniantype, requiring only initial position and velocity. Thus,both difficulties of the Lorentz-Dirac equation (run-away solutions and third-order character) are elimi-nated simultaneously, as was in fact to be expected.

Do we have to pay for this gain? Yes, indeed. Thenew equation of motion exhibits a new feature notknown in classical mechanics. From the latter we be-came accustomed to an instantaneous relationship be-tween the onset of a force and the onset of accelera-tion. And, similarly, when the force ceases, the ac-celeration ceases too, at once. This instantaneity nowno longer holds for a charged particle subject to aforce. The onset of acceleration can in fact occur priorto the onset of the force. An example of this is shownin Fig. 2.

Thus, according to the new equation of motion, it ismeaningless to talk about a causal relationship be-

Fig. 2

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tween force and acceleration. But this does not meanthat a specific force does not give rise to a precise,unique orbit of the particle. The particle's position atany time is completely and exactly predictable fromthe given force and the initial conditions.

This, then, is the price we must pay for having aconsistent, complete particle electrodynamics. But isn'tthis price too high? Haven't we bartered the non-physical runaway solutions for equally nonphysicalacausality? 1 do not think so, and the reason is asfollows:

The term which involves the third time derivative ofposition in the Lorentz-Dirac equation, and which nolonger occurs in the new equation of motion, containsa factor T(l = jj e-/(m c:i). This is a constant for a givenparticle of mass m and charge e; it is the time it takesa light ray in vacuum to traverse a distance 5 e'-/(m c'-).Of all charged particles known the electron has thelargest TU. It is 0.62 X 10~*3 sec, a very short time in-deed. But it is just this time TO which determines theorder of magnitude of the noninstant, "acausal" rela-tion between force and acceleration. It can easily beseen that there is no classical measurement by whichsuch a small acausality can ever be observed. And if aquantum mechanical measurement is made, the presenttheory is not valid. The unphysical nature exhibited bythe new equations of motion is therefore not observ-able. It should disturb us no more than the computa-tions in nonrelativistic mechanics which predict thatwith sufficient power a space ship can go to the nearestfixed star, a Centauri, and back again in only 90 days,while a light ray takes nine years for the same roundtrip. The validity limits of the classical theory with re-spect to small time intervals is here an essential con-sideration. Taking these limits into account, the pricewe pay is small indeed.

Galileo and Maxwell vs. Energy Conservation

TS the classical theory now put in order? Not so•*- quickly. We have not yet looked into its consistencywith the theory of gravitation. The central questionhere is whether the principle of equivalence also holdsfor electromagnetic processes. We would certainly likea positive answer on this. Is the new equation of mo-tion consistent with it?

Consider the following thought experiment. Galileoclimbs up the tower and drops one neutron and oneproton side by side onto the ground below in a suitablyevacuated Pisa (Fig. 3). Will they fall equally fast?

If the principle of equivalence is to be valid, thenan observer falling with the particles should see a neu-tron and a proton at rest side by side throughout thetrip from the top of the tower to the ground. Theymust consequently fall equally fast and reach theground simultaneously.

If Maxwell's equations are valid (and we certainlybelieve that they are), the proton, being acceleratedgravitationally, will emit radiation. Conservation of en-ergy then implies that it must slow down and reach the

ground later than the neutron. Thus, Maxwell's equa-tions and the principle of equivalence are apparentlynot consistent with each other.

This is an incorrect conclusion. Does the emissionof radiation necessarily imply a slowing down of thecharge? Energy conservation seems to require it, butthe correct answer must be found from the equation ofmotion of the charge for the case of acceleration in aconstant force field.

We see now that the integro-differential equation ofmotion is put to a test here. Is the particle electrody-namics based on this equation together with Maxwell'sequations consistent with the principle of equivalence?

The motion we are concerned with here is known ashyperbolic or uniformly accelerated motion. A chargedparticle undergoing such motion emits radiation at aconstant rate.7 The equation of motion can be solvedexactly. It yields a trajectory which is identical withthat of a neutral particle and is therefore consistentwith the principle of equivalence.

This leaves the question of energy conservation. Weare now in agreement with Galileo and Maxwell, butwe seem to be running head-on into a violation of en-ergy conservation.

Let us again consult the equation of motion. Beingan equation for the acceleration four-vector, one com-ponent of it is the energy conservation law per unittime. It becomes, for this case,

,IFM)dt

= F(t + TO)V(1 + T,,) - <R(t + ro) (2)

The left hand side is the rate of change of kinetic en-ergy at time /. The right hand side is the work doneby the gravitational force per unit time less the rateof radiation energy emission, all taken at time t + T0.This is exactly the conservation law which we expectedand on the basis of which the proton should go slower

• O

Fig. 3

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than the neutron; but the two sides of the equationrefer to different times! If this were not so, a compari-son with the corresponding equation for the neutron,

dE(t)dt

= F(t)v(t), (3)

would indeed result in the predicted lag of the proton.As it stands, equation (3) is also valid for the protonand so is equation (2). They do not contradict eachother.

Furthermore, in a classical experiment no differencebetween the conservation laws (2) and (3) can ever beseen, because both T0 and tR in (2) are much, muchtoo small to be observed, even if we had the strongestgravitational fields available.

This fact, and the validity of both equations (2) and(3), indicates the subtleties involved in energy con-servation on the basis of the integro-differential equa-tion of motion. The concepts are new and thereforeappear strange. But it would be difficult to contest themathematical consistency of this theory or its agree-ment with experiment.

At some future time we shall (hopefully) have suc-ceeded in unifying the theories of gravitational andelectromagnetic forces. Given this very general clas-sical theory, it will then be interesting to consider thelimit in which gravitational forces vanish. One will thusobtain a particle electrodynamics containing Maxwell'sequations and an equation of motion. Again hopefully,this equation of motion will be the Lorentz-Diracequation. In our present, nonunified theory this is in-deed the case.8

The internal consistency of fundamental physicaltheory will then be tested in this indirect fashion: bothquantum electrodynamics and general relativity (in-cluding electromagnetic phenomena) must have suit-able limits in which the same classical description ofcharged particles emerges.

Some Semiphilosophical Remarks

TV/I" ANY questions remain to be answered: the rela-•*•*•*- tivistic two-body and many-body problem; theassociated initial-value problem; the problem of a for-mulation free from the need for renormalization, andothers. But the basic equations and the characteristicfeatures of this classical theory are now pretty well

understood. And, in any case, classical electrodynamicsis no longer in worse shape than quantum electrody-namics.

One important problem which remains to be solvedconcerns the gap that still exists between the classicaland the quantum description of charged particles. Asuitable limiting procedure, explicitly carried through,must yield the classical theory from quantum electro-dynamics. The logical structure of physical theory re-quires that such a limit exist and that it yield exactlythe classical equations. The situation is completelyanalogous to the small velocity limit of relativisticmechanics: The equations of Newtonian mechanics areobtained in the limit. A crucial test of the new equationof motion will therefore be to prove whether it indeedresults from quantum electrodynamics in the classicallimit.

Pending the execution of such a program, one mightquestion whether there are not alternative equations ofmotion, yielding an equally consistent theory mathe-matically, and showing strange features which are toosmall to be seen classically. Many papers have, in fact,been written suggesting cut-off functions, shape factors,etc., in order to cope with the difficulties. Here a basicphilosophical principle comes into play: the demandfor simplicity. The introduction of form factors consti-tutes a considerable complication and involves greatarbitrariness. It should also be noted that quantumelectrodynamics predicts no structure for the electronand, consequently, its classical limit must also describea point particle. On the other hand, if a particle withstructure is to be described, like, for example, the pro-ton, it would remain to be shown that its charge andmagnetic moment distribution will survive on takingthe classical limit.

In conclusion, it might be worthwhile to point outthat the classical theory discussed here has several fea-tures of interest to field theory, including quantum fieldtheory in general. There is the nonlocal behavior intime; there is the characteristic time ("fundamentallength") governing this behavior; there is the asymp-totic condition which has become an essential part ofthe fundamental equations of the theory; and there aremany interesting consequences if one assumes that TOis so large that it can actually be measured. It mightwell serve as an interesting model of a nonlocal non-microcausal field theory.

References1. P. A. M. Dirac, Proc. Roy. Soc. (London) A167, 148 (1938).2. E. Fermi, Phys. Zeitschrift. 23, 340 (1922).3. B. Kwal, J. Phys. Rad. 10, 103 (1949).4. F. Rohrlich, Am. J. Phys. 28, 639 (1960).5. H. A. Kramers, Collected Scientific Papers, North-Holland 1956

(Ned. T. Natuurk. 11, 134 (1944)).6. F. Rohrlich, Ann. Phys. 13. 93 (1961).7. T. Fulton and F. Rohrlich, Ann. Phys. 9, 499 (1960); F. Rohrlich,

Nuovo Cim. 21, 811 (1961).8. B. S. DeWitt and R. W. Brehme, Ann. Phys. 9, 220 (1960).

March 1962

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