PhysicsLettersA 159 (1991) 193—197North-Holland PHYSICSLETTERS A

On the motion of chargedrelativistic particleswith spin

David Hartley 1,2 and Robin W. Tucker 1,3

DepartmentofPhysicsand MathematicalPhysics,UniversityofAdelaide,GPOBox498, Adelaide5001, Australia

Received30 July 1991; acceptedfor publication 14 August 1991Communicatedby J.P. Vigier

We derivean involutive systemof equationsdescribingtheclassicalrelativistic motionof a chargedparticle with spin in anexternalelectromagneticfield. The method is basedon a geometricvariational principlefor an integral manifold of anexteriordifferentialsystemonanextendedframebrundleandthedynamicsis motivatedby thetheoryofan idealisedradiatinggyroscope.

1. Introduction the spin degreesof freedomare describedin termsof anti-commutingvariables[14—16].The analysis

The classicalbehaviourof a relativistic charged of theoriesbased on supersymmetricactions forparticle with spin in an externallyprescribedelec- spinningparticleshasbeenoneof themajorreasonstromagneticfield hasbeena subjectof considerable for the continuedinterestin the subject.investigationandcontroversyin thepast [1—5].Such It is thepurposeofthis noteto offer an alternativea topic continuesto be of interestandin our opinion formulationbasedon ageometricalvariationalprin-no entirely satisfactory treatmentexists in the lit- ciple that is generalenoughto accommodateall theerature.The existingformulationsfall into two broad relevantinteractionsof a spinningpoint particleincategories.In thefirst categoryonefindstheoriesthat an externallyprescribedelectromagneticfield. Weare generatedfrom a reparameterisation-invariant shall supposethat the particlesuffersa radiationre-actionprinciple. Othertheoriesare basedon partic- actionwhich in the absenceof spin is describedbyularchoicesof covariantequationsfor the spin and the Lorentz—Diracequation [17]. It is notthe pur-position degreesof freedomfor the particle. In both poseof this articleto discussthe meritsofthis equa-casesconstraintsariseandthe natureandinterpre- tion or its generalisationsbut ratherto demonstratetation of such constraintsdiffer considerably.Typ- how radiativeinteractionscan be naturallyaccom-ical of theoriesin the first categoryonemaycite the modated along with the other interactions ofworkof Barducciet al. [6,71andHansonet al. [8,9]. relevance.Such theoriesdo not howeverinclude theforcesdue Ourapproachis basedon recentlydevelopedvan-to radiationreaction.Theseeffectsmaybe found in ational techniquesand the methodsof exterior dif-the work of Corben [10] and Bhabhaand Corben ferential systems.Thereaderisdirectedto refs. [18—[11] who, like Anderson[121 (generalisingthework 20] for furtherdetailsoftheapproach.Thebasicideaof Bargmannet al. [131), postulatecovariantequa- is to setup a variationalproblemon a spaceofframestionsof motion compatiblewith conservationlaws, thatextendsa field of DarbouxbodyframesassignedIn additionto thesepurely classicalaccounts,there to the world line of the particle.Sucha field offersexist numeroussemi-classicaltreatmentsin which a naturalinterpretationof classicalspin and the dy-

namicsof thesystemwill be motivatedby thetheoryPermanentaddress:Schoolof PhysicsandMaterials,Univer- of the idealisedgyroscope.sity of Lancaster,LancasterLAI 4YB, UK.

2 E-mail: D.Hartley@uk.ac.lancaster.E-mail: R.W.Tucker@uk.ac.lancaster.

0375-9601/9l/$03.50 © 1991 ElsevierSciencePublishersBy. All rights reserved. 193

Volume 159, number4.5 PHYSICSLETTERSA 14 October1991

2. The relativistic rigid body port law of thebody framealong C( I) accordingtothe matrix equation

We are concernedwith the dynamicsofamassive V( pT~ ~ (6)point particle describedby a propertime parame-tensedtime-like curvein a four-dimensionalpseudo- whereRiemannianspace—timeM with a prescribedmetric / o ~, ~. ~,

with it a framethat arisesfrom its structure.Thuswe i> = I (7)tensorg. Wesupposethat theparticlehasassociated 0 —s-, ~,It’3 53 0 —s I

assignto theworld line aone—oneinertiatensorfield /t5~

3 —S2 ~1 /

®dx”. (I) andT denotestransposition.

wherein arbitrary local coordinates{x°~a portionof the world line is given by the map 3. The exterior system

C’:I—*M . TF.V°=X°(i)}. (2) We adopta variationalapproachusingthemeth-

ods of exterior differential systems. These tech-We supposethat there exists a local oriented g-orthonormalbody frameF= {F~,}with dual coframe niquesare admirablysuitedto problemsof this typewhere extendedframe bundlesarise naturally. Wef={fu} along C(l) in which ~ takesthe form

wish to determinea one-dimensionalintegralman-ifold of an exteriorsystemthat also providesa local

= ~ a1F1®f’, (3) extremumfor a variationalproblem.To establisha1=1

suitableexteriorsystemwe extendthecoframef(andwith the a,} positive constantsand J’() tangent to its dual) to a tabularregionU~aboutthe world line(‘(I). and introduceon this region connectionforms Q”,,

This definesthe point particleto be a relativistic of V:rigid body. Foreacheventon C( I) theconfigurationspaceoftherigid body degreesof freedomis thespace VF,,= Q’~,,®F~,. U, Ii = 0, 1, 2, 3 . (8)ofall orientedbodyframeswith onebasisvectortan- Theg-orthonormalcoframe~t~”}is said to beadaptedgent to C( I) and is isomorphicto thegroup SO(3). to C, or a Liarboux cofratne for C’ if C* ~f1) fk =

The dynamicsof the rigid bodymay bespecifiedin ~di, O}, k=l, 2,3.Comparing(8) with the transportterms of a space-like(angularvelocity or classical law (6) we observethat in anadaptedbody coframe“spin”) one-formSdefinedalong (‘(1) which may the connectionforms satisfybe written C~*QOAIV~dT (9)

3

S= ~ s~(t)J’ (4) (~*�,jkQ,J=skdi, L], k= 1,2.3. (10)1=

Workingwith suchan adaptedcoframewe seekthenin termsof thebody frameand thespin components an integralmanifold of the exterior system

Wedenotethecomponentsof the (space-like)ac- Li ~ ~,1kQ,1~Skf

0. QoA — Wkf°] . (II)celerationof the world line by ~ Let X be the orthonormal frame bundle over U~.

3 which hascoframe{f’~,Qai,= —Q,,~,}.We seekan cx-~ w

1(r)F~. (5) tremum, (~), of the integral J(.(J)A which is an in-

I tegralmanifoldof ~Oflsuchthat C’~f~~0sothatwewhereC=10 is thetime-like unit tangentvectorto may project to a solution on space—time.To obtaintheworld line, andV is theChristoffel connectionof sucha manifold we extendX to Y = X >< ~ x t~wherespace—time.Thevariablesis,, Wk} describethetrans- tR/~ hascoordinates{s&, w~and ER

9 hascoordinates

194

Volume 159, number4,5 PHYSICSLETTERSA 14 October1991

{Pi}. The local Euler—Lagrangesystemon Y is given putationof the radiationreaction from an acceler-by the set of all one-forms atingpointcharge.Wedo notconsidercontributions

to the radiationreactionfrom higherorder electro-i~clA~=0, (12) magneticmultipoles.We havehowever includedawhereA~=A+~?~..1p10A andVisanarbitraryvari- Pauli coupling to an intrinsic gyromagneticratio qyational vectorfield on Y. Before contractiononeuses of the particle. In termsof our adaptedbody frame,the structureequations

*(S~.fl=s.Bf0+f(EXs) , (19)df”=—Q”,,~f”, (13)

whereE, ‘F, ~!Fo .~ andB = ~rkJ

1F

1

1Fk .~ aretheelectricdQa,,=—Q~C,

2Q’,,+R’~h, (14) andmagneticfield componentsof .~ in the adapted

where{R”,,} are (thelifts of) thecurvaturetwo-forms body frame.TheEuler—Lagrangeequationsgeneratedby vary-correspondingto the curvaturetensorof V in the

adaptedbody frame. In the following we shall ne- ing p, it andp set to zero the exterior system(11)glectcurvatureeffectsby taking space—timeto be flat. andtheseequationswill be usedto simplify the re-

mainingEuler—Lagrangeequations.By varyingfweobtain

4. Dynamical equations dp+{sxp+ qE+ [s~ir+ ~~p+ j~— ~s~I(sflw

Fromthestructureof theLorentz—Diracequation, +q2/3sx w+ qy [s X (sXE) — (s~B)w] }f

0the theoryof the non-relativisticrigid bodyand the +q

2fldw+qyd(sxE)=0. (20)expressionfor the torqueon a non-relativisticmag-neticdipole in a magneticfield we are led to the ac- From the variation of the connectioncomponentstion one-form we obtain

A=[u0—~g(S,.i(Sfl]f0+qA+qy*(S~ ~). (15) dp+(p+sxp—~vxir+qysXE)fo=0, (21)

In this expressionp0is the constantrestmassand q dir—(~vXp+sxir).f~0, (22)

the constantchargeof the particle.In termsof spin while from thevariationof thespinandaccelerationcomponents

components3

g(S,.~~(Sfl=~ a1s~, (16) [ir.—I(s)—qyB]f0=0, (23)1=1

pf0=0. (24)Theone-formA = .ci+ qflx~~1çwhere.~=zddis anex-ternally prescribedelectromagnetictwo-form and The tangentialequationobtainedby varyingf° is

~eIfd~eIf is the Abrahamself-field that gives rise not an independentequation.to a radiationreactionforce on the particle. Foran involutive system(j’0�0) it follows that all

It is convenientto introducea Euclideanthree- nine momenta {P~i} can be eliminated using thevector algebraand definep= ~ P2, P3), ~ (p,, equations

—p5, pc),p= (P7, P8’ P9), s= (si, s2,s3),w= (w1, w2,w3), I(s) = (a1s, a2s2, a3s3) andf= (~f~f3). pO’ (25)

Fromthecovariantform of theAbrahamone-form ir=I(s)+qyB, (26)

r= —f3i~(C~V~V~~t) (17) p=wXI(s)+qywxB—qysxE. (27)

we find in termsof our adaptedbody coframe: From (27) we may computedp and substitutein

~eIfdWk,.f+.fO~ [f (sxw)J , (18) (20) anduse(26)10substitutein (22) to obtain asystemof first order coupledordinary differential

with fl l/67t~9in MKS units. This accommodates equationsfor the vectorsw(t) ands(t). The equa-dynamicaleffectsfollowing from Dirac’s [17] corn- lion for the spin vectors is simply

195

Volume 159,number4,5 PHYSICSLETTERSA 1 October1991

provide a set of coupleddifferential equationsindI(s)—sxl(s)f0—qysxBf0+qydB=0, (28) termsof~v(r)ands(i) for the parameterisationof

or, with dB=Bdr, f~=—di, and dI(s)=I(i) di, A~”(r).The adaptedbody frame componentsof theexternal electromagneticfield are related to its in-

I(i)+sxl(s)+qysxB+qyB=0. (29)ertial components• ~ ~ by the relations

From (22) and (25) we observethat(33)

ird,r=0.B, = f,~,.4j,”A”~ (34)I I”’

Henceirir is a constantof the motion.It is worth pointingoutthat in theabsenceof both Eq. (32) togetherwith the equationsobtained by

the radiationreaction (/3=0) andthemagneticmo- substituting(33) and (34) into (20) and(28) pro-ment coupling (y=0) the abovesystemis no longer vide a coupledsystemfor the determinationof 0’, sin involution. In this casetheconditions(25), (26), and Aa

1’. In global Minkowski coordinates{x~weand (27) togetherwith (22) imply may choose{N

11=a/axU} so that thetangentvectorto the world line is given by the equation

dp=dwxl(s)+wx[sxl(s)]f0. (30)

Inserting this into (20) and taking the Euclidean ~ =A0~’(r)~ X~’ ciox” (r)~—~. (35)

scalarproduct with I(s) yields the further algebraicconstraint Thus we may solvethe differential equations

I(s){~vx [sxI(s) ]}+I(s). (sXB) X”(t) =A01’(r) (36)

+qI(s) E+ Ns’I(s) + Po]I(~) w= 0. (31) to determinethe world line of the spinningparticle.

Genericallythis providesa linear equationfor one In theinertial framethespin vectoron C( I) is givenof thecomponentsof w andin conjunctionwith (20), by(30) and (22) yields a deterministicsystemfor w 3 aand s. ~‘= j=I s

1(’r)F3= ~ . (37)

We believe the above analysisoffers a straightfor-5. Comments wardwayto generatea constraint-freesystemof dy-

namical equations.The method is powerful andWe havederivedan involutive systemof equa- readilygeneralisedto accommodategravitationaland

lions describingthedynamicsof a radiatingrelativ- other interactions,and doesnot require the use ofistic spinningparticle in an arbitrary externalelec- anti-commutingvariables to describe the spin de-isomagneticfield. The evolutionof the spin vector greesof freedom.Throughoutwe havemaintainedwith propertimeis describedin termsof theadapted a proper time parameterisationof the dynamical

coframeandthe externallyappliedfield appearsin evolutionandseeno reasonto arguethat the elapsethe equationsof motion with componentsspecified of proper time for spinning particlesdependsonin the sameframe. It is customaryto refer the be- anythingotherthan the length of the world line de-haviourof thespin vectorto a “fixed” global inertial terminedby theambientmetricof space—time[211.frame {N,,} in flat space—timein which the compo-nentsof the externalfield will normally be defined.In sucha frame theconnectioncomponentsof V arezero. If we relate {N9} and {Fa} by a matrix AcknowledgementAa

9ESO(3,1) such that Fa=Aa0N,, then theequations The authorsare grateful for the hospitality pro-

vided by the Departmentof Physics and Mathe-Q”h( t) ((I) = (A-’) ~ ~4~” (32) rnatical Physicsat the University of Adelaide,and

196

Volume 159, number4,5 PHYSICSLETTERSA 14 October1991

for thesupportof a BP VentureResearchFellowship. [lii H.J.BhabhaandH.C. Corben,Proc. R.Soc. A 178 (1941)243.

[l2]J.L. Anderson,Principlesof relativity physics (AcademicPress,NewYork, 1967).

References [13] V. Bargrnann,L. Michel andV.L. Telegdi, Phys.Rev. Lett.

[141 L. Brink, S. Deser,B. Zumino, P. deVecchioandP. Howe,[1] T. Takabayashiand J.P. Vigier, Prog. Theor. Phys. 18 Phys.Lett. B 64 (1976) 435.

(1957)573. [IS] F. BerezinandM.S. Marinov, Ann. Phys. (NY) 104 (1977)[2] D. BohmandJ.P.Vigier, Phys.Rev. 109 (1958)1882. 336.[3] H.C. Corben,Phys.Rev. 121 (1961)1833. [161 P.A. Collins andR.W. Tucker, NucI. Phys.B 121 (1977)l4]W.G. Dixon, Proc. R.Soc.A314 (1970)499. 307.[5] W.G. Dixon, Proc. R.Soc. A 319 (1970)509. [17] P.A.M. Dirac,Proc. R. Soc.A 167 (1938)148.16] A. Barducci,L. LusannaandE.Sorace,NuovoCimento46B [18] P.A. Griffiths, Exterior differential systemsandthecalculus

(1978)287. of variations(Birkhäuser,Basel,1983).[7] A. BarducciandL. Lusanna,Nuovo Cimento47B (1978) [191 D.H. HartleyandR.W. Tucker,in: LMS lecturenoteseries,

54. Vol. 150. Geometry of low-dimensional manifolds[8] A.J. Hansonand T. Regge, Ann. Phys. (NY) 87 (1974) (CambridgeUniv. Press,Cambridge,1990).

498. [20] D.H. Hartley, R.W. Tuckerand P.A. Tuckey,Constrained[9] A.J. Hanson,T. Reggeand C. Teitelboim, Constrained dynamics and exterior differential systems, Lancaster

Hamiltonian systems (Academia Nazionale dei Liucei, UniversityPreprint (1991).Rome, 1976). [21] J.W.van Holten,Nuci. Phys.B 356 (1991)3.

[10] H.C. Corben,Classicaland quantumtheoriesof spinningparticles(Holden-Day,SanFrancisco,1968).

197