A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES

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1 A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES ADRIAAN DANIËL FOKKER ( ) A translation of: Ein invariantr Variationssatz für i Bwgung mhrrr lctrischr Massntilshn Z. Phys. 58, (199). ABSTRACT. A LAGRANGian for multipl, charg, point-masss that is invariant unr LORENTZ transformations is prsnt. It is istinguish by mploying both rtar an avanc potntials so that motion is fully tim symmtric. Also, th rsulting form of consrvation principls for momntum an nrgy ar givn an thir consquncs iscuss. 1. INTRODUCTION Quantum mchanics for singl particls was formulat, by mploying th corrsponnc principl, such that its structur parallls th HAMILTONian formalism of classical physics. With rspct to multipl intracting particls, howvr, no complt quantiz formulation has bn foun that is invariant unr LORENTZ transformations. For this task, in fact, vn a classical founation is absnt. It woul b, thrfor, of us to introuc an vlop a proposal for a pr-quantum ynamics for this purpos, 1 that coul srv as th founation for a quantum xtntion in th form of a HAMILTONian variational principl that taks account of mutual intraction of multipl particls without introucing fils.. RELATIVISTIC POINT-PARTICLE MECHANICS To rlativiz th usual formulation of a variational principl, namly: δ T U t 0 whr T is th kintic nrgy an U th potntial nrgy, as is wll known, in sta of T t, on uss: mc v 1 1 c t mcs whr m is th mass, v th vlocity of th particl, an c th sp of light. To convrt th xprssion Ut to an invariant scalar, on rcognizs that th potntial nrgy is just th tim-componnt of a 4-vctor whos spacial part is th ngativ of momntum P, which, analogously to nrgy componnts, on can not as potntial momntum. Thus, on may writ in plac of Ut: Ut P x Ut P x x P y y P z z Dat: 1 Oct FOKKER, A. D. Physica 9, 33 (199). A fil formulation was rcntly propos by HEISENBERG, W. an PAULI, W., Z. Phys (199). 1

2 A. D. FOKKER Whn consiring th motion of a charg, th covariant potntial nrgy-momntum vctor is givn by th prouct of th charg with th fil potntial, whr th 3-vctor potntial part is to b ivi by c. Lt us consir th potntial at th point of intrst X, with spac-tim coorinats x 0 x 1 x x 3 t x y z t x as affct by a charg, with spac-tim coorinats w i i as functions of a paramtr, u. To fin th form of th potntial, following LIÉNARD an WIECHERT, intify thos sourc or affctiv spac-tim points or vnts from which signals travling at th sp of light rach th point of application. Thy satisfy: c x 0 w 0 x w R 0 or x 0 w 0 r c r x w If th vlocity of th sourc charg on th lin irct towars th point of application is v r, thn th potntials ϕ an a for that charg can b writtn: ϕ 1 4π r 1 v r c a r 1 w w 0 v r c In orr to construct a covariant vctor from ths xprssions, w writ: ϕ an corrsponingly a c c w 0 c x 0 w 0 w 0 x 1 c x 0 w w 0 x 1 w 1 w 1 w 1 w 1 w 0 R w w R w whr, clarly, R w nots th 4-vctor scalar prouct of th intraction ray R with th isplacmnt iffrntial w. Th abov xprssion rprsnts a rtar potntial, which is u to th charg locat at point w as this charg woul affct anothr locat at th point x. Lt us consir now th trajctoris of chargs 3 as givn by thir coorinat functions y i u z i v, as functions of th paramtrs u v, rspctivly. To trmin th solution, x i, of th quation of motion for charg 1 as it is influnc by th othr chargs, th following LAGRANGian is to b us : 0 δ m 1 s 1 3 x z S z whr s x 1 an R S ar th intraction rays btwn th location of x 1 an y z of th influncing chargs at thir locations, whr, morovr, th magnitu or moulus of th intraction rays always is to qual zro. S Figur:.1. It is ssntial to rquir, that th iffrntial x 1 for th charg 1 is to corrspon to th motion of th othr chargs only unr th conition: R 0 S 0 tc As a consqunc of this stipulation, th following always hol: R x S z S x tc

3 A RELATIVISTIC LAGRANGIAN 3 FIGURE.1 Wr th motions of th chargs 1 3 givn a priori, an wr th task to trmin th motion of charg with mass m, thn th iffrntial y of this motion (with arclngth s ) woul corrspon via intraction rays, R! T " with iffrntial lmnts on th othr trajctoris x z ", so that th appropriat LAGRANGian woul b: 0 δ m s y # x Similar xprssions obtain for th rmaining chargs. 3 y $ z % T z 3. A SYSTEM LAGRANGIAN At this point, howvr, it sms auspicious to not that, rathr than having a variation for ach particl sparatly, it woul b sirabl to hav a singl unifi variation for an nsmbl as a systm. Th following fact, howvr, thwarts this siratum. Th intraction trm: x y R y accounts for th rtar action of on 1, but os not corrspon to rciprocal rtar influnc of 1 on. Nvrthlss, bcaus R y R x this sam intraction trm in th form y x accounts for xactly th avanc intraction of 1 on.

4 4 A. D. FOKKER On may wll assrt by no mans hr for th first tim that th habit of prfrring to work only with rtar potntials, is simply an arbitrary prjuic. In viw of th aim to hav a fully rciprocal intraction, it might b takn, that influncs 1 to on half via rtar, an on half via avanc intraction. Such woul hav as a consqunc, that th trm in th LAGRANGian accounting for th rtar action of th scon charg on th first, also accounts for avanc action of th first on th scon. This in turn, prmits using a singl LAGRANGian function to trmin trajctoris for all particls of a systm, i..: 0 δ m i cs i i j 3 y ' x & R x )( whr th first sum is ovr all particls in th nsmbl, an th scon is ovr th particl pairs for which mutual intraction is to b takn into account. This LAGRANGian is fully invariant unr LORENTZ transformations. It maks no us of th notion of fil whatsovr. Th rciprocity an symmtry of th intraction is complt. It can b sai, that it prtains mor to a systm of motions, than to a systm of particls. Th concpt of a systm of particls woul rquir a crtain orring of th iffrntials of motion. This orring can b uniquly spcifi, but thn it woul not b invariant; or, it can b ma invariant, but thn it woul not b unambiguous, i.., it is not possibl to fin a systm of particls that is simultanously both invariant an unambiguous. Thus, unavoiably, on must choos a systm of motions ovr a systm of particls. 4. CONSRVATION PRINCIPLES AND CONCLUSIONS Now w consir a variation of th trajctory of particl with charg 1 for which at ach spac-tim point x, an infinitsimal variation δx shall b introuc into th intraction: δ so as to calculat th ffct of on 1. By this variation, th stablish corrsponnc btwn th iffrntials x an y ar to b rtain: Th intraction ray, R, is to b a light signal (* R*$ 0 an th variation δx shoul imply intrinsically th variation: δy y R δx of th motion of. This motion changs nothing, its only ffct is to scur th corrsponnc btwn th variation iffrntials, i.., to assur that R δy R δx, an thrfor also δr 0. In aition, th corrsponnc of th quantitis: R x rmains intact, so that for th variation of th nominator on can tak th variation of R x. Thus, w may now writ: δ δ+ x δy δ R x R x On can intgrat this xprssion by parts an vrify that variation at th limits of th intgral, vanishs. This las to an intgran, kping th valu of δy in viw: δx i - y i y x R i i, m & R x ( y m R i x m x i R i R x R x R x.

5 A RELATIVISTIC LAGRANGIAN 5 Th xprssion in th parnthsis givs an incras in kintic nrgy an siz of th motions (for i 0 or i 1 3), in short, th forc xrt on th charg 1. On ss that, xcpt from th pnncs on vlocity that ar usually ignor, th forc pns partially on acclrations of th sourc chargs, an partially on trms inpnnt of acclrations. Th last trm corrspons to lctrostatic forcs an th so-call raiation ffcts (th accurant an subsqunt trm). In our formulations w hav also inclu th avanc influnc of in th intgral x y / R x x y / R y Excuting th variation in th sam way as abov whil taking account of th fact that δr0 δy # δx, thn rquirs th vanishing of th intgral: y x R i 0 m 1 c x i s 1 R i ym Ri y m R y - y i & R x ( yi x y / Ri & R x 1( x m x i R i R x R x R x x m x y / x y / x i R i R x R x R y R x R y. Within th larg parnthsis, which, brought to th lft si with minus signs woul b sn as a covariant vctor rprsnting transfr of nrgy an momntum from to 1, thr ar som total iffrntials. On coul intrprt thm as nrgy an momntum of 1 which is u to th prsnc an motion of. For th ffct of 1 on th scon particl, thr is a similar quation also compris of two componnts. If w writ this quation for th iffrntial y, (S Figur: 4.1), which via th intraction ray R 0 is boun with th prviously consir iffrntial x an with th intraction ray R 0 0 is boun with th iffrntial x, w gt: 0 m 1 c y i s Ri xm R x Ri x m R x - x i & R x xi & R x y m ym R y Ri Ri y # x R x R y 1( y x R x R y 1( yi y $ x R y R x yi y ' x / R y R x Ri Ri x y3 R y R x y ' x / R y R x. It is important to rcogniz hr, that th thir row in this xprssion togthr with th last row of th prvious xprssion, constitut a total rivativ. Both of thm rlat to th mutual intraction ray R btwn x an y. Togthr thy rprsnt & Ri x y3 R x R y (

6 6 A. D. FOKKER FIGURE 4.1 This xprssion, in turn, las th way to an unrstaning of how th principls of consrvation of nrgy an momntum prtain in this formulation. To mak things simpl, lt us consir only two particls. W must consir a sum of a progrssiv zig-zag chain of intraction rays (S: Fig. ) linking iffrntials of motion an thir quations. In this sum, ach vrtx of th zig-zag chain, pning on whthr it prtains to th action of 1 on, or visa-vrsa, corrspons to a contribution to th kintic nrgy of on or th othr particl: m 1 c x i s 1 or m c y i s whil for ach intraction-ray, R i btwn x i an y i thr corrspons a contribution to th potntial nrgy of: & x i R x yi R y Ri R x R y ( If th motion is prioic, thn th chain will also b prioic, or vry narly so, an th sum thn woul xtn only ovr a singl prio, as it closs back on itslf. In this cas it constituts a total iffrntial, so that thr is a quantity that rmains constant along th chain. If on ivis this constant quantity by th numbr of mutual intractions in a prio, on gts th nrgy, or th momntum, [of th systm]. This corraborats th rmark ma abov, that nrgy an momntum can not b fin for a systm of particls, but rathr only for a systm of mutual motions.

7 A RELATIVISTIC LAGRANGIAN 7 Ovr an abov this fact, whn th motion is not prioic, th sum os not clos on itslf, an thrfor, os not yil a total iffrntial. At th ns of th zig-zag chain thr will always rmain xclu contributions. If thy ar ignor nvrthlss, an th sum is ivi by th numbr of intraction rays, an vr mor xact finition of th total nrgy an momntum can b givn as th chain is takn longr an longr; but no mattr what, th nrgy an momntum at a particular momnt will rmain unfin. This is th pric for xcluing fils from th LAGRANGian; yt, this fatur os not conflict with quantum mchanics. Translator: A. F. 4 KRACKLAUER c 005 NATUURKUNDIG LABORATORIUM VAN TEYLER S STICHTING, HAARLEM