Hamiltonian of a many- atom in an external magnetic 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￿

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tome 38 N° 7 JUILLET 1977 LE JOURNAL DE PHYSIQUE

Classification Physics Abstracts

5.230 - 5.234

HAMILTONIAN OF A MANY-ELECTRON ATOM IN 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 et radiatifs. L’extension de l’ équation de Breit proposée par Hegstrom permet de rendre compte de ces effets jusqu’à l’ordre 1/c3; a la limite non relativiste, le hamiltonien trouvé est d’une grande complexité. Nous montrons dans cet article qu’il est possible de rendre compte 2014 dans le cadre de l’electrodynamique classique 2014 de la quasi-totalité de ses termes. Il apparait ainsi que l’équation de Breit et son extension traduisent en fait essentiellement l’existence de deux phénomènes physiques simples et bien connus : i) l’interaction du moment magnétique de chaque particule avec le champ qu’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 atoms requires an ever increasing knowledge of relativistic and radiative effects. The generalization of the Breit 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 possible to explain almost all of its terms within the frame of classical electrodynamics. In particular, picturing the 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 classical phenomena : i) the interaction of the magnetic moment of the constituent particles with the magnetic field they experience in their rest frame; ii) the Thomas precession of the spins of the accelerated particles.

1. Introduction. - Increasing precision in the mea- understood with the help of classical electrodynamics surement of the Zeeman effect of atoms [1] has led to and special relativity, and it is the object of this a revived interest in the theory of atomic systems. paper to show that the generalized Breit equation is interacting with external electromagnetic fields. In mainly a consequence of two phenomena : the particular, the experimental accuracy of atomic g Thomas precession [3] of the of accelerated factors requires an ever increasing knowledge of particles and the interaction of the magnetic moment relativistic, radiative and nuclear-mass corrections in of each particle with the magnetic field it experiences an external magnetic field. In the absence of a deve- in its rest-frame. lopment from the fully covariant Quantum field The generalized Breit equation as proposed by theory, we must use the generalized Breit equation Hegstrom is presented in section 2. Section 3 deals including anomalous moment interactions, as first with the problem of the spin dynamics, and in particu- proposed by Hegstrom [2]. This generalized Hamil- lar with the relativistic definition of the spin, the tonian, expected to be exact up to order C-3, displays Thomas precession, and the Hamiltonian suitable a rather striking feature : the anomalous part and the for a satisfying representation of the spin motion. Dirac part of the magnetic moment do not seem to In section 4, we collect these results, with the well play a similar role, thus impeding a clear-cut inter- known Darwin Hamiltonian for particles without pretation of the various correcting terms. It happens spin. We determine the various electromagnetic fields however that the Hegstrom Hamiltonian can be acting on charges and magnetic moments and we obtain the Hamiltonian describing a many-electron (*) Associ6 au C.N.R.S. atom, 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- This Hamiltonian is written in electrostatic units, nian is the same as that deduced from the generalized with a choice of gauge such that : Breit equation. We then discuss the physical origin of all the terms appearing in this Hamiltonian. where Bo is the constant external magnetic field, and the ith particle is characterized by its mechanical 2. The Breit in constant exter- generalized equation momentum 1ti, mass mi, charge ei, position ri, and nal field. - Within the formalism magnetic proposed momentum pi related to 1ti and Ai by : by Hegstrom [2], the atom is considered as a system of Dirac particles with anomalous magnetic moments. Its evolution is governed by a generalized Breit and are the and electric fields equation and the corresponding Hamiltonian can be Hi Ei magnetic expe- written as : rienced by the particle i i.e.

where : and Lastly xi is the anomalous part of the magnetic moment of the particle due to virtual radiative processes (1) (the explicit value for is given in 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 non relativistic limit. The result is [2] :

(1) As far as radiative corrections are concerned, equation (2.1) each multiplet as a relative shift of order 03B15 log 03B1Ry [6]. In a magnetic assumes that the electron is a free one; in particular, this formalism field similar effects lead to corrections of order 03B13 03BCB B0 as discussed does not account for the short-range modification of the Coulomb by Grotch and Hegstrom [7]. Except for these corrections, the potential, responsible for the Lamb-shift. This effect manifests itself Hamiltonian (2.1) is expected to be valid up to the order c20143, that as an overall shift of order Z03B13 Ry of the levels and for is to say up to order 03B13 03BCB B0 for the Zeeman effect. 749

In these expressions poi represents the ratio eil2 mi ; the rest-frame. For this reason, we have chosen the the Lande factor gi of particle i is related to the second solution and our point of view is based on the anomalous part xt of the magnetic moment by : following 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 JlV is the of the more Equation (2.6) starting point coincides with S while the spatio-temporal part is null ; elaborate calculations on the Zeeman effects of many- i.e. in a rest frame : electron atoms [8, 9], and it would be gratifying to know the physics underlying the various terms appearing in this intricate Hamiltonian. There already exists a traditional interpretation of all these terms : is the nonrelativistic Jeo Schrodinger Hamiltonian, This definition implies that in any frame JC, is the first relativistic correction to the kinetic energy, JC2 the Darwin term due to the Zitterbewegung of the particles, JC3 describes the orbit-orbit coupling, JC4 the spin-orbit coupling, Jes the spin-other-orbit where u, is the four velocity of the particle. the and the coupling, Je6 spin-spin interaction, X7 b) In an inertial frame F where the particle moves interaction of the with the external spin magnetic with velocity v = cp, the spatio-temporal part of EJlV field. Besides the of most of these great complexity is non zero, and appears to represent (except for a terms, let us notice that the anomalous magnetic factor - gpofc) the electric dipolar moment associated moment sometimes as sometimes as appears g;, (gi -1) with the moving magnetic moment. With the use or ( gi - 2). That means we cannot merely replace of the of antisymmetric in the Pauli Hamiltonian the Dirac moment magnetic tensors, we easily get the components ( - cT, 8) 2 itoi si by the anomalous moment gizoi si. We are thus of Zuv in F from the rest-frame components (0, S). faced with a dilemma : either the anomalous part of a The result is : magnetic moment does not behave like the Dirac part, or the traditional intepretation of equation (2.6) is erroneous. In trying to answer these questions, we found it interesting to reconsider the problem of an where : atom in a magnetic field using a very simple relativistic but classical description where atoms are thought of as systems of point-like particles with magnetic moment proportional to an inner angular momentum. This is the purpose of the next two sections.

and 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 particle is the Lorentz contraction tensor. The advantage of with spin is defined by its intrinsic angular moment S, the E formalism is clear : in any frame, Z has a clear- together with an associated magnetic moment M cut physical interpretation, the associated vectors 8 related to S by : and S corresponding to usual quantities of electro- dynamics, a point already stressed and thoroughly used by de Groot and Suttorp [13] (2).

3.2 THE SPIN EQUATION OF MOTION, THE THOMAS a three a) Equation (3.1) is relation between PRECESSION. - We first consider the trivial case of dimensional axial vectors and when its four guessing a particle moving with a constant velocity relative to dimensional two solutions are counterpart, possible : an inertial frame Lo ; its motion is most simply the first one with four-vectors, the second one with antisymmetric four-tensors ; the first choice is the most widely used by people interested in polarized beams (2) The presence of the light velocity c in the tensor 03A3 03BCv(- c2014CP, S) (Bargman, Michel, Telegdi [10], Hagedom [11] and arises from our choice of the electrostatic system of units others it has the merit of mathematical [12]), simplicity, (defined by 4 03C003B50 = 1 and in which the electromagnetic the covariant equation for the polarization motion 03BC0/403C0 = 1/c2) tensor F03BCv is written as (2014 E/c, B), with E/c = (F10, F20, F30) rather to as stressed However, = being easy get [10]. by and B (F23, F31, F12), We shall later see that this choice is Hagedom [11], the physical meaning of the polariza- particularly suitable whenever we want to develop the magnetic tion four-vector is ambiguous in any frame other than interaction terms in power of c-1. 750 described in its inertial rest frame where it satisfies affected with the subscript a]. At this time we have to the equation : the following relation between the components of spin [0, S(r)] in R(-r) and in [ - c!fa( f), 8a(-r)] R(Ta) : where B is the magnetic field experienced by the particle and measured (like S, M and i) in the inertial rest frame. where A(P(T)) is the pure Lorentz boost which In the general case where the particle is accelerating transforms coordinates of antisymmetric tensors from relative to Lo, we do not have one single inertial rest the laboratory frame Lo to the inertial frame R(i) frame, but rather a sequence of inertial rest frames, moving with the velocity cp(T) with respect to Lo. each member of the sequence being at a given proper With the help of equations (3.3) and (3.6), equa- time i an instantaneous rest frame of the particle. tion (3. 7) can be written in the form : To define unambiguously this succession of rest frames, we must further specify the orientation of the axis of the successive rest frames. There are at least two possible sequences : a) we can use a sequence in which all the successive instantaneous rest frames R(i) have their axis parallel Deriving equation (3.8) with respect to i and to an arbitrary inertial frame Lo (for example the lab the derivative at time T = frame), taking Ta gives : fl) we can also use another sequence P(i), where P(i + di) is deduced from P(i) by a pure Lorentz boost of velocity w [bv being the velocity of the particle at time r + dT as measured in P(T)] - In the absence of any external forces (no magnetic interactions), it is expected that the spin does not in the But since the precess sequence P(i). product This equation shows us that the total rate of change of two Lorentz boosts is in non pure general (for of spin in the frame R(T) is the sum of the rate of change colinear the of a Lorentz velocities) product pure due to external interactions plus a purely kinematic boost by a rotation [14], it happens that the sequence term which takes into account the Thomas pre- P(i) is rotating relative to R(i) and Lo. If the spin is cession (3). More precisely we can write (3.9) in the motionless in P(i), it will thus appear to be rotating form : in R(i) and Lo : a phenomenon known as the Thomas precession [3]. In order to see the phenomenon from another point of view, we shall now report another demonstration where : of this precession. In this new demonstration we shall not use the rest frames P(i) and the rather intuitive assumptions we have just made; instead we shall use only the well defined R(r) frames and the fact that in these instantaneous rest frames, the spin remains a BR is the magnetic field experienced by the spin purely spatial quantity. as judged from R(i) and dfildt is the acceleration of the viewed from the Lorentz force Thus we define the function S(i) which at each particle Lo, given by law. time &: measures the orientation of the spin in the introduces a division between instantaneous rest frame R(1) : Equation (3. 10) good what we shall name magnetic interaction terms and purely kinematic ones, and therefore gives us a simple physical insight of the phenomena. On the other hand, the spatio-temporal part being in this succession of the covariant character of the evolution law is totally frames identically zero : hidden due to the particular choice of sequence R(i) with respect to Lo. However, it is easy to derive

At a particular time ia there is a corresponding (3) It is possible to make a demonstration very similar in form for inertial frame At time i > we shall denote as R(Ta). ia any purely spatial quantity of R(03C4) (vector or pseudo-vector); CPA(T) the velocity of the particle as judged from that is consistent with the geometrical interpretation we give R(Ta) [all the quantities measured in R(Ta) will be previously. 751 equation (3.10) from the following covariant equa- have only to use the pure Lorentz boost given in tion [ 15] : (3.3). By deriving equation (3.3), we obtain :

the first terms on the hand sides the where Ffl represents the antisymmetric tensor of right represent rate of of 8 and T due to the time-variation electromagnetic fields. We have not followed this change of the Lorentz boost which transforms in approach as it was our purpose to give the most Lo R(T), pedestrian approach, with simple physical interpre- they are just partial derivative of 8 and S ; using moreover we can write tation, but we draw attention to the fact that equa- equation (3.15), equa- tions as : tions (3.10)’ and (3.12) have exactly the same physical (3. 16) content and the same limits, they are valid only for constant or slowly varying fields : a hypothesis we shall make throughout all this paper. As a final remark we must emphasize that equation (3.10) can equally well be derived from the B.M.T. equation which governs the evolution of four-vector Sa (see by example refs. [14] and [16]). Taking (3.14) into account, we then get :

3.3 HAMILTONIAN FORM OF EQUATION OF SPIN MOTION. - Equation (3.10) with the Lorentz force law provides a complete description of the motion of a charged particle with intrinsic magnetic momen- tum. From the Lorentz force law : which bring out the fact that the Hamiltonian which governs the equation of motion of spin in the lab frame is : we easily obtain the acceleration of the particle in the lab frame as a function of the electro-magnetic fields acting on it :

Remarks : 1) The factor 1 /7 introduced bet- ween (3.15) and (3.19) merely takes into account the of time and then the expression of the Thomas precession change scale between the lab frame and the rest frame of = by (3.11). the particle (dt y dT). From equation (3. 10), it is straightforward to 2) It is intentional that we keep in expression (3.19) obtain the Hamiltonian of the motion in R(i) using the expression of spin components in the rest frame the heuristic definition [17, 18] of Poisson brackets : since they are the only intrinsic quantities with simple commutation rules; we must also emphasize that the same point of view prevails in . Moreover note that in Blount’s quantum mechanical With the of and lead to the help (3.14), (3.10) (3.13) picture [19] we have an equation of motion very equation of motion of spin in R(i) with the form : similar 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 it is with point interesting to rewrite the magnetic part in terms of we then and quantities of the lab frame (81’ S, BL, EL) ; get :

Finally, to obtain the equations of motion and the corresponding Hamiltonian in the lab frame Lo we 752

This last form is physically interesting, it accounts for into account. The results are given in table I. We have the interaction in the lab frame of the dipolar magnetic kept in the expansion of A the terms up to the order moment with the magnetic field, and for that of the c - 3. The term A(3) leads to well known difficulties dipolar electric moment with the electric field. But (radiation damping) connected with energy dissi- in fact, it will be simpler in following calculations to pation within the atomic system [21]. Such a radiative use the composite form : process cannot be coherently taken into account unless we quantize the electromagnetic field, we shall henceforth neglect it. When the potentials given in table I are introduced into the Hamiltonian (4.2), we get, up to order c-’ : 4) We finally obtain the kinematic part due to Thomas precession in the form :

with the mechanical moment now defined as : In this last form, we can rediscover the well known contribution of the Thomas to the precession spin- and orbit interaction, but we have also in addition a non negligible contribution to the Zeeman effect. The different origin of the terms of similar form appearing in (3.21), and (3.22) now clearly explains the appearance of the various factors g, (g - 1), TABLE I ( g - 2) in (2.6) which was a priori so puzzling. Potentials and fields created at point i by a particle j in motion 4. Classical Hamiltonian of a many-electron atom electrostatic units in a constant magnetic field. - In order to get this Hamiltonian we first recall the calculation of the spin-independent terms (Darwin Hamiltonian).

4.1 THE DARWIN HAMILTONIAN. - Let us consider Potentials (Coulomb gauge) a particle i without spin, moving in an electromagnetic field deriving from the potentials A, cpo Its Hamiltonian may be written as :

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

Fields

Ai and (pi are the potentials of the overall electroma- gnetic fields acting on particle i (i.e. the fields created by other particles j :A i plus the external field Bo). To obtain the Darwin Hamiltonian, we must choose the Coulomb gauge (div A = 0). Then, following a method proposed by de Groot and Suttorp [20], we expand the potential and fields in power of c - 1 and express their values at time t as a function of the values at the same time t of the dynamic variables of the source particles. This method has the remarkable advantage of implicitly taking the retardation effect 753

It is noticeable that this Hamiltonian, where all atom, provided we do not take the interaction terms terms up to order C-2 have been included, is exactly into account twice. (At this level of approximation, the Darwin Hamiltonian [22]. Besides the relativistic this Hamiltonian appears necessarily as a sum over kinetic energy correction, it involves a correction to the individual Jei, a feature which may not persist the instantaneous electrostatic interaction often refer- at higher approximations because of three-body red to in terms of a magnetic interaction between interactions.) the orbit of the moving particles. Let us stress that, if we can neglect the radiative terms A(3), this Hamil- 4.2 SPIN-DEPENDENT PART OF THE HAMILTONIAN. - remains exact to the order c- 3. tonian up In section 3, we proved that the spin-dependent part The spin-independent part of the Hamiltonian of a of the interaction of a particle i with an electromagnetic many electron atom can now be deduced by summing field can be described by a Hamiltonian composed the different terms (4.3) for each component of this of 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 interaction of the magnetic moment of particle i with the external magnetic field and with the magnetic field of the moving charges j, then the interaction between the motional induced electric moment with the electric fields produced by 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- the atomic Hamiltonian. Except for the Zitterbewegung tant role in the spin orbit interaction and also that it term, the result is identical to the Hamiltonian is responsible for a correction to the Zeeman effect. deduced from the generalized Breit equation and in 4.3 COMPLETE HAMILTONIAN. - the given (2.6). Gathering The obvious of our classical derivation spin-independent Hamiltonian (4.3), the magnetic advantage is to allow us to a clear-cut of the Hamiltonian (4.5) and the Thomas Hamilto- give interpretation JCmag various terms : the Schrodinger Hamiltonian Jeo and nian (4.6) and summing over the particles, we obtain the relativistic kinetic correction JCI are well known. Je2 appears as a purely quantum term that can be understood as the interaction with the mean potential (4) It is possible to improve the calculation so as to get the contact seen delocalized electrons. the orbit- potential between magnetic moments. This is achieved by ascribing by 3C3 expresses it a tiny volume to the magnetic moment as can be seen for example in orbit interaction : is not only due to simple interac- Cohen-Tannoudji et al. [23]. tion between the two orbits considered as magnetic 754 moments but most exactly includes retardation effects Consequently a set of particles with spin will be in the Coulomb interaction. described by a magnetization density : The so-called spin-orbit term Je4 results from two different effects : the first one has a magnetic origin and appears as the interaction of the electric moment (originating in the moving magnetic moment) with the electric field of the nucleus and electrons, while together with a polarization density : the second one has a purely kinematic origin, related to the Thomas precession of the intrinsic angular momentum 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 and the interaction of the magnetic expresses magnetic and in the presence of a charge density p and current moment with the field created the other magnetic by density j the Maxwell equations in vacuum may be electrons. moving written as : As previously found JC6 is a purely magnetic interaction between the magnetic moments of elec- trons. Finally, JCy represents the interaction of the spin with the external magnetic field; its previous expres- sion (2.6) can be advantageously replaced by the more explicit form :

As for the potentials, they satisfy the uncoupled equations : where Botl and Sll represent the components of Bo and Si perpendicular to the velocity of the particle i. In this new expression, we see the interaction of the spin with the magnetic field as seen in the rest frame and a relativistic correction of purely kinematic When only p and J are present, the solution is well origin (which gives a contribution to the Margenau known (cf. ref. [20]). When P and M are non zero, correction). This overall relativistic effect appears we proceed as following. We look for the superpo- as an anisotropic modification of the Thomas pre- tentials 1t and 1t* satisfying : cession in a magnetic field. (It has been shown in a somewhat different way in reference [24] that this correction reduces for a Dirac electron to the so-called relativistic mass correction.) It is thus remarkable that despite its intrinsic and well known limitations (radiation damping), classical Then it is easy to verify that the functions T and A electrodynamics is able to offer a clear explanation of defined as : all the relativistic terms up to order c-3 appearing in the Hamiltonian derived from the (2.6) generalized and Breit equation. ,

Appendix : Retarded potentials and fields created by moving magnetic moments. - A moving magnetic are solutions of 6) 7) with the source dipole reveals itself not only as a contracted magnetic equations (A. (A. terms P and M moreover to the Lorentz moment : (they satisfy gauge condition). The superpotentials 1t and 1t* are readily calculated since their generating equations (A. 8) and (A. 9) but also as an electric dipole : are identical to the well known equation : 755

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

and from which we successively obtain the potentials cp and A :

where r(t’) = R - R’ Following a method developed by de Groot and Suttorp [20], we expand the delta function in a Taylor series around (t - t’) and we obtain the expressions for the superpotentials created at point i by the set of particles j "# i, in the form of power series in c-1. (What is remarkable is that each term of the series is synchronous i.e. it involves the values of M and P and the fields E and B :. at the very time t the potentials 1t and n* are calculated.) The first terms of the series are :

References

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