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Hamiltonian of a many-electron atom in an externalmagnetic field and classical electrodynamics

C. Lhuillier, J.P. Faroux

To cite this version:C. Lhuillier, J.P. Faroux. Hamiltonian of a many-electron atom in an external mag-netic field and classical electrodynamics. Journal de Physique, 1977, 38 (7), pp.747-755.�10.1051/jphys:01977003807074700�. �jpa-00208635�

HAMILTONIAN OF A MANY-ELECTRON ATOMIN AN EXTERNAL MAGNETIC FIELD AND CLASSICAL ELECTRODYNAMICS

C. LHUILLIER and J. P. FAROUX

Laboratoire de Spectroscopie Hertzienne de 1’E.N.S. (*),24, rue Lhomond, 75231 Paris Cedex 05, France

(Reçu le 25janvier 1977, accepte le 14 mars 1977)

Résumé. 2014 La précision croissante des expériences concernant l’effet Zeeman des atomes à

plusieurs électrons necessite une connaissance toujours plus approfondie des effets relativistes etradiatifs. L’extension de l’ équation de Breit proposée par Hegstrom permet de rendre compte de ceseffets jusqu’à l’ordre 1/c3; a la limite non relativiste, le hamiltonien trouvé est d’une grandecomplexité. Nous montrons dans cet article qu’il est possible de rendre compte 2014 dans le cadre del’electrodynamique classique 2014 de la quasi-totalité de ses termes. Il apparait ainsi que l’équationde Breit et son extension traduisent en fait essentiellement l’existence de deux phénomènes physiquessimples et bien connus : i) l’interaction du moment magnétique de chaque particule avec le champqu’elle subit dans son référentiel propre, ii) la précession de Thomas des spins des particules accélérées.

Abstract. 2014 The increasing precision of experiments on the Zeeman effect of many electron atomsrequires an ever increasing knowledge of relativistic and radiative effects. The generalization of theBreit equation as proposed by Hegstrom includes all these effects up to order c2014 3. In the non rela-tivistic limit the resulting Hamiltonian is quite intricate, but we show in this paper that it is possibleto explain almost all of its terms within the frame of classical electrodynamics. In particular, picturingthe atoms as point-like particles with intrinsic magnetic moments, we show that the physical pheno-mena underlying the Breit equation and its recent extension are mainly two well-known classicalphenomena : i) the interaction of the magnetic moment of the constituent particles with the magneticfield they experience in their rest frame; ii) the Thomas precession of the spins of the acceleratedparticles.

Tome 38 N° 7 JUILLET 1977

LE JOURNAL DE PHYSIQUE

Classification

Physics Abstracts5.230 - 5.234

1. Introduction. - Increasing precision in the mea-surement of the Zeeman effect of atoms [1] has led toa revived interest in the theory of atomic systems.interacting with external electromagnetic fields. In

particular, the experimental accuracy of atomic gfactors requires an ever increasing knowledge of

relativistic, radiative and nuclear-mass corrections inan external magnetic field. In the absence of a deve-lopment from the fully covariant Quantum field

theory, we must use the generalized Breit equationincluding anomalous moment interactions, as first

proposed by Hegstrom [2]. This generalized Hamil-tonian, expected to be exact up to order C-3, displaysa rather striking feature : the anomalous part and theDirac part of the magnetic moment do not seem toplay a similar role, thus impeding a clear-cut inter-pretation of the various correcting terms. It happenshowever that the Hegstrom Hamiltonian can be

(*) Associ6 au C.N.R.S.

understood with the help of classical electrodynamicsand special relativity, and it is the object of thispaper to show that the generalized Breit equation ismainly a consequence of two phenomena : the

Thomas precession [3] of the spin of acceleratedparticles and the interaction of the magnetic momentof each particle with the magnetic field it experiencesin its rest-frame.The generalized Breit equation as proposed by

Hegstrom is presented in section 2. Section 3 dealswith the problem of the spin dynamics, and in particu-lar with the relativistic definition of the spin, theThomas precession, and the Hamiltonian suitablefor a satisfying representation of the spin motion.In section 4, we collect these results, with the wellknown Darwin Hamiltonian for particles withoutspin. We determine the various electromagnetic fieldsacting on charges and magnetic moments and weobtain the Hamiltonian describing a many-electronatom, including all terms up to order C-3 . Except

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003807074700

748

for the Zitterbewegung term, the resulting Hamilto-nian is the same as that deduced from the generalizedBreit equation. We then discuss the physical originof all the terms appearing in this Hamiltonian.

2. The generalized Breit equation in constant exter-nal magnetic field. - Within the formalism proposedby Hegstrom [2], the atom is considered as a systemof Dirac particles with anomalous magnetic moments.Its evolution is governed by a generalized Breit

equation and the corresponding Hamiltonian can bewritten as :

where :

and

This Hamiltonian is written in electrostatic units,with a choice of gauge such that :

where Bo is the constant external magnetic field, andthe ith particle is characterized by its mechanicalmomentum 1ti, mass mi, charge ei, position ri, andmomentum pi related to 1ti and Ai by :

Hi and Ei are the magnetic and electric fields expe-rienced by the particle i i.e.

Lastly xi is the anomalous part of the magneticmoment of the particle due to virtual radiative

processes (1) (the explicit value for electrons is givenin refs. [4] and [5]).For various reasons (and in particular because the

only good wave functions we know are non relativistic),it is desirable to reduce equation (2.1) to its nonrelativistic limit. The result is [2] :

(1) As far as radiative corrections are concerned, equation (2.1)assumes that the electron is a free one; in particular, this formalismdoes not account for the short-range modification of the Coulombpotential, responsible for the Lamb-shift. This effect manifests itselfas an overall shift of order Z03B13 Ry of the fine structure levels and for

each multiplet as a relative shift of order 03B15 log 03B1Ry [6]. In a magneticfield similar effects lead to corrections of order 03B13 03BCB B0 as discussed

by Grotch and Hegstrom [7]. Except for these corrections, theHamiltonian (2.1) is expected to be valid up to the order c20143, thatis to say up to order 03B13 03BCB B0 for the Zeeman effect.

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In these expressions poi represents the ratio eil2 mi ;the Lande factor gi of particle i is related to theanomalous part xt of the magnetic moment by :

Equation (2.6) is the starting point of the moreelaborate calculations on the Zeeman effects of many-electron atoms [8, 9], and it would be gratifyingto know the physics underlying the various termsappearing in this intricate Hamiltonian. There alreadyexists a traditional interpretation of all these terms :Jeo is the nonrelativistic Schrodinger Hamiltonian,JC, is the first relativistic correction to the kinetic

energy, JC2 the Darwin term due to the Zitterbewegungof the particles, JC3 describes the orbit-orbit coupling,JC4 the spin-orbit coupling, Jes the spin-other-orbitcoupling, Je6 the spin-spin interaction, and X7 theinteraction of the spin with the external magneticfield. Besides the great complexity of most of theseterms, let us notice that the anomalous magneticmoment appears sometimes as g;, sometimes as (gi -1)or ( gi - 2). That means we cannot merely replacein the Pauli Hamiltonian the Dirac magnetic moment2 itoi si by the anomalous moment gizoi si. We are thusfaced with a dilemma : either the anomalous part of amagnetic moment does not behave like the Dirac

part, or the traditional intepretation of equation (2.6)is erroneous. In trying to answer these questions, wefound it interesting to reconsider the problem of anatom in a magnetic field using a very simple relativisticbut classical description where atoms are thoughtof as systems of point-like particles with magneticmoment proportional to an inner angular momentum.This is the purpose of the next two sections.

3. Relativistic dynamics of particles with intrinsic

angular momentum. - 3.1 INTRINSIC ANGULAR

MOMENTUM AND INTRINSIC MAGNETIC MOMENT. - In

the framework of classical electrodynamics, a particlewith spin is defined by its intrinsic angular moment S,together with an associated magnetic moment Mrelated to S by :

a) Equation (3.1) is a relation between threedimensional axial vectors and when guessing its fourdimensional counterpart, two solutions are possible :the first one with four-vectors, the second one withantisymmetric four-tensors ; the first choice is the mostwidely used by people interested in polarized beams(Bargman, Michel, Telegdi [10], Hagedom [11] andothers [12]), it has the merit of mathematical simplicity,the covariant equation for the polarization motionbeing rather easy to get [10]. However, as stressed byHagedom [11], the physical meaning of the polariza-tion four-vector is ambiguous in any frame other than

the rest-frame. For this reason, we have chosen thesecond solution and our point of view is based on thefollowing hypothesis :The intrinsic angular momentum of a particle is

described by an antisymmetric 4-tensor E "’ ; in a rest-frame of the particle the purely spatial part of E JlVcoincides with S while the spatio-temporal part is null ;i.e. in a rest frame :

This definition implies that in any frame

where u, is the four velocity of the particle.b) In an inertial frame F where the particle moves

with velocity v = cp, the spatio-temporal part of EJlVis non zero, and appears to represent (except for afactor - gpofc) the electric dipolar moment associatedwith the moving magnetic moment. With the useof the Lorentz transformation of antisymmetrictensors, we easily get the components ( - cT, 8)of Zuv in F from the rest-frame components (0, S).The result is :

where :

and

is the Lorentz contraction tensor. The advantage ofthe E formalism is clear : in any frame, Z has a clear-cut physical interpretation, the associated vectors 8and S corresponding to usual quantities of electro-dynamics, a point already stressed and thoroughlyused by de Groot and Suttorp [13] (2).

3.2 THE SPIN EQUATION OF MOTION, THE THOMASPRECESSION. - We first consider the trivial case ofa particle moving with a constant velocity relative toan inertial frame Lo ; its motion is most simply

(2) The presence of the light velocity c in the tensor 03A3 03BCv(- c2014CP, S)arises from our choice of the electrostatic system of units

(defined by 4 03C003B50 = 1 and 03BC0/403C0 = 1/c2) in which the electromagnetictensor F03BCv is written as (2014 E/c, B), with E/c = (F10, F20, F30)and B = (F23, F31, F12), We shall later see that this choice is

particularly suitable whenever we want to develop the magneticinteraction terms in power of c-1.

750

described in its inertial rest frame where it satisfiesto the equation :

where B is the magnetic field experienced by theparticle and measured (like S, M and i) in the inertialrest frame.

In the general case where the particle is acceleratingrelative to Lo, we do not have one single inertial restframe, but rather a sequence of inertial rest frames,each member of the sequence being at a given propertime i an instantaneous rest frame of the particle.To define unambiguously this succession of rest

frames, we must further specify the orientation of theaxis of the successive rest frames. There are at leasttwo possible sequences :

a) we can use a sequence in which all the successiveinstantaneous rest frames R(i) have their axis parallelto an arbitrary inertial frame Lo (for example the labframe),

fl) we can also use another sequence P(i), whereP(i + di) is deduced from P(i) by a pure Lorentzboost of velocity w [bv being the velocity of theparticle at time r + dT as measured in P(T)] -

In the absence of any external forces (no magneticinteractions), it is expected that the spin does notprecess in the sequence P(i). But since the productof two pure Lorentz boosts is in general (for noncolinear velocities) the product of a pure Lorentzboost by a rotation [14], it happens that the sequenceP(i) is rotating relative to R(i) and Lo. If the spin ismotionless in P(i), it will thus appear to be rotatingin R(i) and Lo : a phenomenon known as the Thomasprecession [3].

In order to see the phenomenon from another pointof view, we shall now report another demonstrationof this precession. In this new demonstration we shallnot use the rest frames P(i) and the rather intuitiveassumptions we have just made; instead we shall useonly the well defined R(r) frames and the fact that inthese instantaneous rest frames, the spin remains apurely spatial quantity.

Thus we define the function S(i) which at eachtime &: measures the orientation of the spin in theinstantaneous rest frame R(1) :

the spatio-temporal part being in this succession offrames identically zero :

At a particular time ia there is a correspondinginertial frame R(Ta). At time i > ia we shall denote as

CPA(T) the velocity of the particle as judged fromR(Ta) [all the quantities measured in R(Ta) will be

affected with the subscript a]. At this time we havethe following relation between the components ofspin [0, S(r)] in R(-r) and in [ - c!fa( f), 8a(-r)] R(Ta) :

where A(P(T)) is the pure Lorentz boost whichtransforms coordinates of antisymmetric tensors fromthe laboratory frame Lo to the inertial frame R(i)moving with the velocity cp(T) with respect to Lo.With the help of equations (3.3) and (3.6), equa-tion (3. 7) can be written in the form :

Deriving equation (3.8) with respect to i and

taking the derivative at time T = Ta gives :

This equation shows us that the total rate of changeof spin in the frame R(T) is the sum of the rate of changedue to external interactions plus a purely kinematicterm which takes into account the Thomas pre-cession (3). More precisely we can write (3.9) in theform :

where :

BR is the magnetic field experienced by the spinas judged from R(i) and dfildt is the acceleration of theparticle viewed from Lo, given by the Lorentz forcelaw.

Equation (3. 10) introduces a good division betweenwhat we shall name magnetic interaction terms andpurely kinematic ones, and therefore gives us a simplephysical insight of the phenomena. On the other hand,the covariant character of the evolution law is totallyhidden due to the particular choice of sequence R(i)with respect to Lo. However, it is easy to derive

(3) It is possible to make a demonstration very similar in form forany purely spatial quantity of R(03C4) (vector or pseudo-vector);that is consistent with the geometrical interpretation we givepreviously.

751

equation (3.10) from the following covariant equa-tion [ 15] :

where Ffl represents the antisymmetric tensor of

electromagnetic fields. We have not followed this

approach as it was our purpose to give the mostpedestrian approach, with simple physical interpre-tation, but we draw attention to the fact that equa-tions (3.10)’ and (3.12) have exactly the same physicalcontent and the same limits, they are valid only forconstant or slowly varying fields : a hypothesis weshall make throughout all this paper. As a finalremark we must emphasize that equation (3.10)can equally well be derived from the B.M.T. equationwhich governs the evolution of four-vector Sa (seeby example refs. [14] and [16]).

3.3 HAMILTONIAN FORM OF EQUATION OF SPIN

MOTION. - Equation (3.10) with the Lorentz forcelaw provides a complete description of the motionof a charged particle with intrinsic magnetic momen-tum. From the Lorentz force law :

we easily obtain the acceleration of the particle in thelab frame as a function of the electro-magnetic fieldsacting on it :

and then the expression of the Thomas precessionby (3.11).From equation (3. 10), it is straightforward to

obtain the Hamiltonian of the motion in R(i) usingthe heuristic definition [17, 18] of Poisson brackets :

With the help of (3.14), (3.10) and (3.13) lead to theequation of motion of spin in R(i) with the form :

with

and

Finally, to obtain the equations of motion and thecorresponding Hamiltonian in the lab frame Lo we

have only to use the pure Lorentz boost given in

(3.3). By deriving equation (3.3), we obtain :

the first terms on the right hand sides represent therate of change of 8 and T due to the time-variationof the Lorentz boost which transforms Lo in R(T),they are just partial derivative of 8 and S ; usingmoreover equation (3.15), we can write equa-tions (3. 16) as :

Taking (3.14) into account, we then get :

which bring out the fact that the Hamiltonian whichgoverns the equation of motion of spin in the labframe is :

Remarks : 1) The factor 1 /7 introduced bet-ween (3.15) and (3.19) merely takes into accountthe change of time scale between the lab frame andthe rest frame of the particle (dt = y dT).

2) It is intentional that we keep in expression (3.19)the expression of spin components in the rest framesince they are the only intrinsic quantities with simplecommutation rules; we must also emphasize that thesame point of view prevails in quantum mechanics.Moreover note that in Blount’s quantum mechanicalpicture [19] we have an equation of motion verysimilar to (3.19) (see equally ref. [20], chap. 8).

3) As in the initial equation (3.10), we shall

distinguish in Hamiltonian (3.19) the magnetic inter-action from the kinematic effects. At this point it isinteresting to rewrite the magnetic part in terms ofquantities of the lab frame (81’ S, BL, EL) ; we then get :

752

This last form is physically interesting, it accounts forthe interaction in the lab frame of the dipolar magneticmoment with the magnetic field, and for that of thedipolar electric moment with the electric field. Butin fact, it will be simpler in following calculations touse the composite form :

4) We finally obtain the kinematic part due toThomas precession in the form :

In this last form, we can rediscover the well knowncontribution of the Thomas precession to the spin-orbit interaction, but we have also in addition a nonnegligible contribution to the Zeeman effect.The different origin of the terms of similar form

appearing in (3.21), and (3.22) now clearly explainsthe appearance of the various factors g, (g - 1),( g - 2) in (2.6) which was a priori so puzzling.

4. Classical Hamiltonian of a many-electron atomin a constant magnetic field. - In order to get thisHamiltonian we first recall the calculation of the

spin-independent terms (Darwin Hamiltonian).

4.1 THE DARWIN HAMILTONIAN. - Let us considera particle i without spin, moving in an electromagneticfield deriving from the potentials A, cpo Its Hamiltonianmay be written as :

where mi, ei and pi are respectively the mass, chargeand momentum of particle i. The expansion in powersof c-’ leads to the standard expression :

Ai and (pi are the potentials of the overall electroma-gnetic fields acting on particle i (i.e. the fields createdby other particles j :A i plus the external field Bo).To obtain the Darwin Hamiltonian, we must choosethe Coulomb gauge (div A = 0). Then, followinga method proposed by de Groot and Suttorp [20], weexpand the potential and fields in power of c - 1 andexpress their values at time t as a function of thevalues at the same time t of the dynamic variables of thesource particles. This method has the remarkable

advantage of implicitly taking the retardation effect

into account. The results are given in table I. We havekept in the expansion of A the terms up to the orderc - 3. The term A(3) leads to well known difficulties(radiation damping) connected with energy dissi-

pation within the atomic system [21]. Such a radiativeprocess cannot be coherently taken into account

unless we quantize the electromagnetic field, we shallhenceforth neglect it.When the potentials given in table I are introduced

into the Hamiltonian (4.2), we get, up to order c-’ :

with the mechanical moment now defined as :

and

Potentials (Coulomb gauge)

TABLE I

Potentials and fields created at point iby a particle j in motion

electrostatic units

Fields

753

It is noticeable that this Hamiltonian, where allterms up to order C-2 have been included, is exactlythe Darwin Hamiltonian [22]. Besides the relativistickinetic energy correction, it involves a correction tothe instantaneous electrostatic interaction often refer-red to in terms of a magnetic interaction betweenthe orbit of the moving particles. Let us stress that, ifwe can neglect the radiative terms A(3), this Hamil-tonian remains exact up to the order c- 3.The spin-independent part of the Hamiltonian of a

many electron atom can now be deduced by summingthe different terms (4.3) for each component of this

atom, provided we do not take the interaction termsinto account twice. (At this level of approximation,this Hamiltonian appears necessarily as a sum overthe individual Jei, a feature which may not persistat higher approximations because of three-bodyinteractions.)

4.2 SPIN-DEPENDENT PART OF THE HAMILTONIAN. -In section 3, we proved that the spin-dependent partof the interaction of a particle i with an electromagneticfield can be described by a Hamiltonian composedof a magnetic part Hmag and a kinetic part JC.

a) Calculation of Je:nag.Gathering the results of table I (fields of moving charges) and appendix (fields of moving magnetic dipoles)

and limiting ourselves to terms up to order c - 3, we obtain from the initial expression (3.21) :

and the final result :

The physical origin of all the above terms is now clear and we can successively distinguish the interactionof the magnetic moment of particle i with the external magnetic field and with the magnetic field of the movingcharges j, then the interaction between the motional induced electric moment with the electric fields producedby the j particles and finally the interaction between magnetic moments (4).

b) Calculation of the kinetic part HTh.To determine HTh from (3. 22) up to the order C-3, it is sufficient to consider the expressions of Ei and Bi

up to the order c-1 and we obtain :

We see that the Thomas precession plays an impor-tant role in the spin orbit interaction and also that itis responsible for a correction to the Zeeman effect.

4.3 COMPLETE HAMILTONIAN. - Gathering the

spin-independent Hamiltonian (4.3), the magneticHamiltonian JCmag (4.5) and the Thomas Hamilto-nian (4.6) and summing over the particles, we obtain

the atomic Hamiltonian. Except for the Zitterbewegungterm, the result is identical to the Hamiltoniandeduced from the generalized Breit equation andgiven in (2.6).The obvious advantage of our classical derivation

is to allow us to give a clear-cut interpretation of thevarious terms : the Schrodinger Hamiltonian Jeo andthe relativistic kinetic correction JCI are well known.Je2 appears as a purely quantum term that can beunderstood as the interaction with the mean potentialseen by delocalized electrons. 3C3 expresses the orbit-orbit interaction : it is not only due to simple interac-tion between the two orbits considered as magnetic

(4) It is possible to improve the calculation so as to get the contactpotential between magnetic moments. This is achieved by ascribinga tiny volume to the magnetic moment as can be seen for example inCohen-Tannoudji et al. [23].

754

moments but most exactly includes retardation effectsin the Coulomb interaction.The so-called spin-orbit term Je4 results from two

different effects : the first one has a magnetic originand appears as the interaction of the electric moment(originating in the moving magnetic moment) withthe electric field of the nucleus and electrons, whilethe second one has a purely kinematic origin, relatedto the Thomas precession of the intrinsic angularmomentum in the accelerated frame; this is at the

origin of the presence of the (g - 1) factor.The spin-other orbit interaction Jes is purely

magnetic and expresses the interaction of the magneticmoment with the magnetic field created by the othermoving electrons.As previously found JC6 is a purely magnetic

interaction between the magnetic moments of elec-trons.

Finally, JCy represents the interaction of the spinwith the external magnetic field; its previous expres-sion (2.6) can be advantageously replaced by themore explicit form :

where Botl and Sll represent the components of Boand Si perpendicular to the velocity of the particle i.

In this new expression, we see the interaction of thespin with the magnetic field as seen in the rest frameand a relativistic correction of purely kinematic

origin (which gives a contribution to the Margenaucorrection). This overall relativistic effect appearsas an anisotropic modification of the Thomas pre-cession in a magnetic field. (It has been shown in asomewhat different way in reference [24] that thiscorrection reduces for a Dirac electron to the so-calledrelativistic mass correction.)

It is thus remarkable that despite its intrinsic andwell known limitations (radiation damping), classicalelectrodynamics is able to offer a clear explanation ofall the relativistic terms up to order c-3 appearingin the Hamiltonian (2.6) derived from the generalizedBreit equation. ,

Appendix : Retarded potentials and fields created

by moving magnetic moments. - A moving magneticdipole reveals itself not only as a contracted magneticmoment :

but also as an electric dipole :

Consequently a set of particles with spin will bedescribed by a magnetization density :

together with a polarization density :

and in the presence of a charge density p and currentdensity j the Maxwell equations in vacuum may bewritten as :

As for the potentials, they satisfy the uncoupledequations :

When only p and J are present, the solution is wellknown (cf. ref. [20]). When P and M are non zero,we proceed as following. We look for the superpo-tentials 1t and 1t* satisfying :

Then it is easy to verify that the functions T and Adefined as :

and

are solutions of equations (A. 6) (A. 7) with the sourceterms P and M (they moreover satisfy to the Lorentzgauge condition).The superpotentials 1t and 1t* are readily calculated

since their generating equations (A. 8) and (A. 9)are identical to the well known equation :

755

Thus n and n* are given by the corresponding for-mulas :

and

where r(t’) = R - R’Following a method developed by de Groot and

Suttorp [20], we expand the delta function in a Taylorseries around (t - t’) and we obtain the expressionsfor the superpotentials created at point i by the setof particles j "# i, in the form of power series in c-1.(What is remarkable is that each term of the series issynchronous i.e. it involves the values of M and Pat the very time t the potentials 1t and n* are calculated.)The first terms of the series are :

from which we successively obtain the potentialscp and A :

and the fields E and B :.

References

[1] VAN DEN BOUT, P. A., AYGÜN, E., EHLERS, V. J., INCESU, T.,SAPLAKOGLU, A., SHUGART, H. A., Phys. Rev. 165 (1968)88.

LEWIS, S. A., PICHANICK, F. M. J., HUGHES, V. W., Phys. Rev.A 2 (1970) 86.

LEDUC, M., LALOË, F., BROSSEL, J., J. Physique 33 (1972) 49.AYGÜN, E., ZAK, B. D., SHUGART, H. A., Phys. Rev. Lett. 31

(1973) 803.LHUILLIER, C., FAROUX, J. P., BILLY, N., J. Physique 37 (1976)

335.

[2] HEGSTROM, R. A., Phys. Rev. A 7 (1973) 451.

[3] THOMAS, L. H., Phil. Mag. 3 (1927) 1 ; Nature 117 (1926) 514.[4] WESLEY, J. C., RICH, A., Phys. Rev. Lett. 24 (1970) 1320 ;

Phys. Rev. A 4 (1971) 1341.

[5] KINOSHITA, T., CVITANOVIC, P., Phys. Rev. Lett. 29 (1972)1534.

[6] DOUGLAS, M., KROLL, N. M., Annls. Phys. 82 (1974) 89.[7] GROTCH, H., HEGSTROM, R. A., Phys. Rev. A 8 (1973) 2771.[8] LEWIS, M. L., HUGHES, V. W., Phys. Rev. A 8 (1973) 2845.[9] LHUILLIER, C., These, Université de Paris VI (1976).

LHUILLIER, C., FAROUX, J. P., BILLY, N., J. Physique 37 (1976)335.

[10] BARGMANN, V., MICHEL, L., TELEGDI, V. L., Phys. Rev. Lett.2 (1959) 435.

[11] HAGEDORN, R., in Relativistic Kinematics (Benjamin, NewYork), chap. 9.

[12] DURAND, E., in Mécanique Quantique, tome II (Masson,Paris), 1976.

[13] SUTTORP, L. G. and DE GROOT, S. R., Nuovo Cimento 65A(1970)245.

[14] JACKSON, J. D., in Classical Electrodynamics (2nd edition,Wiley N. Y.) 1975, chap. XI.

[15] Equation (3.12) has been studied in great details by Suttorpand de Groot [13] who proved its equivalence with thespin covariant equations of motion deduced from theDirac equation.

[16] SARD, R. D., in Relativistic Mechanics (Benjamin, N. Y.)1970, section 5.3.

[17] DIRAC, P. A. M., in the Principles of Quantum Mechanics(Fourth Edition, Oxford) 1958, section 21.

[18] A more elaborate foundation of classical Hamiltonian for-malism has been developed by

BARUT, A. O., in Electrodynamics and Classical Theory ofFields and Particles (McMillan Comp., N. Y.) 1964,chap. II.

[19] BLOUNT, E. I., Phys. Rev. 126 (1962) 1636 ; Phys. Rev. 128(1962) 2454.

[20] DE GROOT, S. R., SUTTORP, L. G., Foundations of Electrody-namics (North Holland Publishing Comp., Amsterdam)1972, p. 138.

[21 LANDAU, L. D. and LIFSHITZ, E. M., in The Classical Theoryof Fields (Pergamon, Oxford) 1962, p. 231.

[22] DARWIN, C. G., Phil. Mag. 39 (1920) 537.[23] COHEN-TANNOUDJI, C., DIU, B., LALOË, F., in Quantum

Mechanics (Wiley, London-New York) 1977.

[24] HEGSTROM, R. A. and LHUILLIER, C., to be published in Phys.Rev. A (April 1977).