Magnetic Moment of the Proton

Transcription

1 SB/F/ Magtic Mot of th Proto G. Gozálz-Martí*, I.Taboada Dpartato d Física, Uivrsidad Sió Bolívar, Apartado 89, Caracas 18-A, Vzula. ad J. Gozálz Physics Dpartt, Northatr Uivrsity, Bosto, U.S.A. *Wbpag: Th agtic ot of th proto is calculatd usig a gotric uifid thory. Th gotry dtris a gralizd Pauli quatio showig aoalous agtic trs du to th triplt proto structur. Th thortical rsult givs a bar aoalous Ladé gyroagtic g-factor clos to th xprital valu. Th cssary radiativ corrctios should b icludd i th actual thortical drssd valu. Th first ordr corrctio raiss th valu to 2( Siilarly w obtai for th utro gyroagtic g-factor th valu 2( PACS :4.5.+h; 14.2.Dh; 2.2.+h

2 Proto Magtic Mot 2 1. Itroductio. Th proto tripl structur ay b rlatd to a o-liar gotric thory that uifis gravitatio ad lctroagtis that offrs th possibility of rprstig othr itractios by a sctor of th coctio [1,2,3 ]. It has b show that w lctroagtic cosqucs of th thory lad to quata of lctric charg ad agtic flux, providig a plausibl xplaatio to th fractioal quatu Hall ffct [4,5 ]. O th othr had w also hav cosidrd a approxiatio to this gotric o liar thory [6 ] whr th icroscopic physical objcts (gotric particls ar ralizd as liar gotric xcitatios, gotrically dscribd i a jt budl foralis show to lad to th stadard quatu fild thory tchiqus. Ths gotric xcitatios ar sstially prturbatios aroud a o-liar gotric backgroud spac solutio, whr th xcitatios ay b cosidrd to volv with ti. I this frawork, a gotric particl is actd upo by th backgroud coctio ad is vr rally fr xcpt i absolut pty backgroud spac (zro backgroud curvatur. Th backgroud spac carris th uivrsal irtial proprtis which should b cosistt with th idas of Mach [7 ] ad Eisti [8 ] that assig fudatal iportac of far-away attr i dtriig th irtial proprtis of local attr icludig th irtial ass. W ay itrprt th gotric xcitatios as gotric particls ad th backgroud as th particl vacuu. This is a gralizatio of what is orally do i quatu fild thory wh particls ar itrprtd as vacuu xcitatios. Th vacuu is rplacd by a backgroud solutio that w call th substratu. Th fild quatio of th thory adits a costat coctio ad currt substratu solutio dtriig a gotric sytric curvd spac [9,1 ]. Th gotry is rlatd to a coctio Γ i a pricipal fibr budl (E,M,G. Th structur group G is SL(4,R ad th v subgroup G + is SL 1 (2,C. Th subgroup L (Lortz is th subgroup of G + with ral dtriat, i othr words, SL(2,C. Thr is oly aothr subgroup P i th possibl group chais G H L, which is Sp(4,R. Th holooy groups of th coctio ay b usd gotrically to classify th itractios cotaid i th thory. Th subgroup chai SL(4,R Sp(4,R SL(2,C charactrizs a chai of subitractios with rducig algbraic sctors. Elctroagtis is associatd to a SU(2 Q subgroup. Mattr is rprstd by a currt J xprsd i trs of a pricipal fibr budl sctio, which is rlatd by charts (coordiats to atrics of th group SL(4,R. This structur dfis a fra of SL(4,R spiors ovr spac-ti. Th coctio is a sl(4,r 1- for that acts aturally o th fra (sctios. Thr ar crtai iportat probls i th thory that should b discussd, as follows: 1- To fid th rol playd by th xtra grators or coctio copots i th thory as o classical itractios, possibly uclar itractios. I particular, sic oly o of th thr SU(2 Q grators is obsrvabl as a log rag lctroagtic pottial 4-vctor A, th sigificac of th othr 2 uobsrvabl short rag coctio copots should b clarifid; 2- To furthr discuss th quatios of otio for a coplt G-syst which should display additioal ffcts i copariso with th siplr quatios for a P-syst. Hr w addrss ths probls. Wh w writ th quatios of otio for a coplt G-syst, w fid that w ar abl to calculat th bar agtic ot of th fudatal (hadro particl associatd to th strog uclar itractios, i a o rlativistic approxiatio. 2. Gral quatios of Motio. Th o liar quatios for th coctio hav a solutio that w hav calld substratu [9, 1] which ay b xprssd, i a particular rfrc syst, as a tsor valud for Λ i trs of a costat solutio Λ = J. ( 1 4 Usig th dfiitio of ass fro th rgy J.Γ [11 ] th calculatd ass corrspodig to this substratu solutio is = 4tr( J Γ = 4 tr ( J Λ = 4 tr ( κ κ = 4 tr( I =. ( 2 4

3 Proto Magtic Mot 3 W ay dfi a scod coctio by subtractig th tsor valud for Λ fro th gotric coctio! Γ Γ Λ = Γ J. ( 3 4 Usig th w coctio th quatio of otio xplicitly displays a substratu ass tr!! κ = κ ( Γ = κ Γ J = κ = ( 4 4 ad th quatios for th v ad odd parts of a G-syst ar, usig th corrspodig xprssios i th appdix,! ξ = ηγ η, ( 5! η = ξ Γ ξ. ( 6 W ow dsigat diffrt vctor pottials i th coctio with th followig otatio: A is th lctroagtic pottial corrspodig to th v part of th SU(2 Q sctor of th coctio; o A is th copltary odd part of th SU(2 Q sctor; Γ is th SL(2,C sctor of th coctio which dtris a L covariat drivativ; ϒ is th copltary part. Both ο Α ad ϒ ar coplx lts with ral part alog κ ad iagiary part alog κ κ 5. Th first quatio ay b writt as ( o ξ iaξ ξγ = Aη + η ϒ η. ( 7 Isrtig th uit -i 2 i th lft sid of this quatio o ( ( ( ( ( i iξ + A iξ iξ iγ = A η + η ϒ η ( 8 ad usig η = η, ( 9 ξ = ξ, ( 1 w gt o ( ( i + A iξ = A η + ηϒ + η. ( 11 Siilarly for th scod quatio w gt aftr ultiplyig by i o ( i A ( i A ( i ( i ( i η+ η η Γ = ξ + ξ ϒ ξ, ( 12 o ( i A ( A ( i ( i ( i η ξ ξ ϒ ξ + = + +. ( 13 If w dfi ( i 1 ϕ η + ξ (, ( 14 χ η i ξ, ( 15 w ca writ th quatios i th followig for o o ( i A ϕ ( i A χ Aχ Aϕ + + = + + χ ϒ ϕ ϒ ϕ + + +, ( 16

4 Proto Magtic Mot 4 o o ( i A χ ( i A ϕ Aϕ Aχ + + = + ϕ ϒ χ ϒ χ. ( No Rlativistic Approxiatio W ow tak th quatios as liarizd quatios aroud th substratu solutio with a coctio fluctuatio rprstig slf itractios. For a o rlativistic approxiatio, th low vlocitis ad corrspodig boosts ar of ordr v/c. W ay glct th trs of ordr v/c, which ar th boost ϒ ad th hritia parts of ad lavig thir atihritia parts η ξ = η, ( 18 = ξ, ( 19 ad w obtai ( i ( i 1 1 χ = η + ξ = η ξ = ϕ 2 2 ( i ( i 1 1 ϕ = η ξ = η + ξ = χ 2 2 ϕ, ( 2. ( 21 As usually do i rlativistic quatu chaics [12 ] i a o rlativistic approxiatio, w lt it ϕ, ( 22 χ χ it, ( 23 obtaiig slowly varyig fuctios of ti with quatios ( ( i + A ϕ i + A χ = Aϕ A χ + o o ϕϒ χϒ ( ( i + A χ i + A ϕ = A χ + A ϕ +, ( 24 o o o + χϒ + ϕϒ 2 χ. ( 25 Th quatios bco aftr rcogizig th spac copot o A as a odd lctroagtic vctor pottial. ( i A o A ( i o A A + + ϕ + χ =, ( 26 ( ( o o i + A i + A + A = + A χ ϕ χ χ 2, ( 27 whr χ is th sall copot rlativ to th larg copot ϕ. W glct th sall trs, as usual th χ trs i th last quatio ulss ultiplid by, ad substitut th xprssio for i quatio (26. Th rsult is o ( i + A + A ϕ o o ( i + A A ( i + A + A ϕ = 2. ( 28 Sic w hav th wll kow rlatio. a. b= a. b+ i.( a b, ( 29 substitutio i quatio (28 givs

5 Proto Magtic Mot 5 i ϕ o o ( i + A A( i i + A + A o ( A + A + 2 ϕ i( ( A + A i( A ia A = o o o. ( 3 which is Pauli s quatio [13 ] with xtra trs dpdig o th v ad odd vctor pottials A, o A. 4. Magtic Mot Tr Equatio (3 displays th rarkabl gotrical structur of triplts. Accordig to physical gotry [14] a G-solutio should iclud th thr SU(2 Q grators. W ay associat th ffct of a cobiatio of thr SU(2 Q coctio copots, o for ach possibl P i G, as thr classically quivalt lctroagtic pottials A s [2, 3]. It has b show [14], i this gotry, that all log rag filds corrspod to filds associatd to th fibr budl obtaid by cotractig th structur group SL(4,R to its v subgroup SL 1 (2,C which i tur corrspod to classical filds. Thrfor th log rag copot of th sl(4,r-coctio coicids with th log rag copot of a sl 1 (2,C-coctio corrspodig to a v subalgbra sl(2,c/u(1 of sl(4,r [14] rlatd to gravitatioal ad lctroagtic filds. I fact ay dirctio i th tridisioal gotric lctroagtic su(2 Q subalgbra ay b idtifid as a valid dirctio corrspodig to this raiig log rag classical lctroagtic u(1. Thr is o prfrrd dirctio i su(2 Q. If w obsrv a log rag lctroagtic fild, w ay always alig th classical fild A with ay of th 3 gotric lctroagtic κ grators i su(2 Q, or a liar cobiatio, by prforig SU(2 Q trasforatio. Nvrthlss th two xtra A s should ak additioal cotributios to th agtic rgy of th short rag G-syst, as show i quatio (3, ad thrfor to its corrspodig agtic ot. Th lctroagtic subgroup is SU(2 Q, siilar to th spi subgroup SU(2 S. Th group itslf, as a fibr budl (SU(2,S 2,U(1 carris its ow rprstatios. Th bas spac is th cost SU(2/U(1 which is th bidisioal sphr S 2. Th fibr is a arbitrary v subgroup U(1. Th actio of this lctroagtic SU(2 Q is a ultiplicatio o th fibr by a lt of th U(1 subgroup ad a traslatio o th bas spac S 2 by th actio of th SU(2 Q group Casiir oprator, rprstig a squard total SU(2 Q rotatio. This actio is ot as sipl as traslatios i flat spacs, but rathr has coplicatios siilar to thos associatd with agular otu du to th SU(2 group gotry. Th oritatio of dirctios i SU(2 is quatizd. I particular oly o copot of th lctroagtic rotatio grator E couts with th group Casiir oprator E 2. This oprator acts o th sytric cost S 2 bcoig th Laplac-Bltrai oprator o th cost. Its igvalus ar gotrically rlatd to traslatios o S 2 i th sa ar as th igvalus of th usual Laplac oprator ar rlatd to traslatios o th pla. Thr ar dfiit siultaous igvalus, with th Casiir oprator, oly alog ay arbitrary sigl su(2 dirctio, which w hav tak as th dirctio of th v grator E. Th grator E ay b dcoposd i trs of th copltary odd copot o E. W ca ot dcopos th A coctio ito copots with dfiit xpctatio valus. Th splittig of A ito a v part A ad a odd part o A rprsts, rspctivly, th splittig of th group actio ito its vrtical v actio o th fibr ad a copltary odd traslatio o th bas S 2. Cosidr that th xpotial fuctios k.r for a rprstatio of th traslatio group o th pla. Th agitud of th traslatio k is dtrid by th igvalu of th Laplac oprator <, kix kix 2 kix = λ = k, (31 whr th absolut valu of k is k ( δ k 12 k =. (32 Th odd subspac of su(2 Q, spad by th two copact odd lctroagtic grators i o E, is isoorphic to th odd subspac of th quatrio algbra, spad by its orthooral st q a. Associatd to this orthooral st w hav th Dirac oprator q$= o a curvd bidisioal spac. This Dirac oprator is th rprstatio of o E o th vctor fuctios o th sphr. W obtai for this actio of o E, if w sparat th wav fuctio ϕ ito th SU(2 S igvctor φ ad th SU(2 Q igvctor ψ, ad us th fact that th Lvi-Civita coctio is sytric,

6 ( a 2 a b ( a b [ a b] a a b a b a b ab = a b = = i( i+ 1 q ψ = q q ψ = q q ψ + q q ψ Proto Magtic Mot 6 ψ ψ ψ g. (33 This quatio shows that th squard curvd Dirac oprator is th Laplac-Bltrai or Casiir oprator, with igvalus qual to th squard traslatio agitud o th odd sphr S 2. Thrfor th odd lctroagtic grator o E is rlatd to traslatios o th lctroagtic S 2. I gral quatrio o E squard is th Casiir quatrio C 2 which acts o th cost SU(2/U(1. Th o E dirctio i th odd tagt pla is idtriabl bcaus thr ar o odd igvctors coo with E ad E 2 ad thrfor o dfiit valus for th grators i this pla. Nvrthlss th absolut valu of this quatrio ust b th squar root of th absolut valu of th Casiir quatrio, which also corrspods to th squard total rotatio E 2. I th orthooral bas of th coo igvctors of E ad E 2, th absolut valus of quatrios E ad E 2 ar th rspctiv igvalus. Th absolut valus of E ad # E dfi a polar agl θ i th su(2 algbra. W coclud that th lctroagtic grator has a idfiit aziuthal dirctio but a quatizd polar dirctio dtrid by th possibl traslatio valus. Thrfor w obtai 1 o 2 2 ( + E E i i 1 = = taθ. (34 E E Th itral dirctio of th pottial A ust b alog th possibl dirctios of th lctroagtic grator E i su(2 Q. Th A copots ust b proportioal to th possibl v ad odd traslatios. I cosquc th total A vctor ust li i a co dfid by a quatizd polar agl θ rlativ to a axis i th v dirctio ad a arbitrary aziuthal agl. Th rgy tr i quatio (3 bcos ( i( i o U = i( ( A + A ϕ = 1+ i ( A ϕ. ( Th fudatal stat rprstig a proto is th SU(2 Q stat with charg +1, corrspodig to lctroagtic rotatio igvalus ½, ½. I trs of th v agtic fild th rgy bcos, ( ( + 1 i B ( U = 1+ ϕ = + Bϕ i. (36 Associatd to th scod ª3 tr i th parthsis w hav itroducd th ffctiv dirctio agl θ of th total grator, for a G-syst, i th su(2 algbra. Th valu of θ is π/3. Statistically this dirctio corrspods to th xpctd (a valu of th projctd copot, of a rado classical dirctio, alog th chos v dirctio. Th first tr i th parthsis is rlatd to th P-syst which has oly a U(1 lctroagtic subgroup ad th coplicatios du to SU(2 ar ot prst. Th copltary odd subspac corrspodig to S 2 dos o xist. Th oritatio of th coplt lctroagtic coctio A ca always b tak alog th dfiit v dirctio dfid by th physical u(1 algbra. Th agl θ ay b tak qual to zro. It corrspods to a P-syst, associatd to th lctro, with oly a κ lctroagtic copot. I this cas th rgy rducs to 1 ( 1 U = i A = i B. ( If it wr possibl to ak a trasforatio that aligs th itral dirctio θ alog th v dirctio vrywhr, w rally would b dalig with a P-syst bcaus w actually would hav rstrictd th coctio to a P subgroup. A P-syst provids a prfrrd dirctio i th subgroup SU(2 Q of G, th oly lctroagtic grator κ of th associatd P group. If w us a P-tst-particl (a lctro to itract with a xtral agtic fild B, w actually alig th log rag copot A with this prfrrd dirctio. I this ar w ay xplai th log-rag physical xprits usig a ablia lctroagtic fild quatio with a rot sourc ad a Dirac quatio. If w us a G-tst-particl (a proto to itract with a xtral fild, th ost w ca do is to alig th classical lograg copot A with a rado dirctio i su(2. Part of th total itral fild i B is oly obsrvabl i a short

7 Proto Magtic Mot 7 rag rgio. To ach B dirctio rlatd by a SU(2 Q trasforatio, thr corrspods a partr spi dirctio, dfiig associatd scalar products.b. Th additioal scalar products, rlativ to a P-syst, aris fro th two additioal gotric filds, ad/or, quivaltly, fro th two additioal copis of Pauli atrics κ κ i th uivrsal Clifford algbra, dtrid by th additioal spi-agtic dirctios κ, which origially wr alog κ 1 κ 2 κ 3 ad κ 5. Th additioal spi oprators, itroducd by th short-rag o-classical lctroagtic pottials, rprst xtra itral currt loops that grat a aoalous icras of th itrisic agtic rgy. W hav calculatd th agtic rgy of a fr G-syst, with zro xtral fild B, i trs of th obsrvabl v itral fild B whos itral dirctio coicids with th vtual itral dirctio of a xtral fild B. I this gdk xprit, th agtic ot is dfid as th partial drivativ of th agtic rgy, producd by th total itral gralizd agtic fild i B, with rspct to th v copot B, i whos itral dirctio would alig th itral dirctio of th xtral B fild, U. (38 B Th stadard physical thods of agtic ot asurt ar uclar paraagtic rsoac [15, 16 ], olcular bas ad optical spctroscopy [17 ]. I a ral physical xprit, wh th sapl is placd i th xtral fild, thr is a chag i th agtic rgy of ithr a G-syst or a P-syst. For both systs, our tst particl will rspod to a xtral fild, ssig a variatio of th lctroagtic fild likd to th tst particl, i th itral dirctio of th xtral agtic B fild which coicids with th itral dirctio of th v B fild. Th chag i th agtic rgy, aftr rstorig i th quatio th fudatal physical costats,, c which ar all qual to 1 i our gotric uits, is ( 1+ taθ du U i $ U = B = B = i B db B 2 c $ SiB SiB = 21 ( + taθ ι = gi i 2c i $ $ i (39 whr th variatio s by th tst-particl is qual to th xtral fild B, th itral agl θ ι is zro for th lctro or π/3 for th proto ad i is th rspctiv ass. This xprssio dfis th agto (atoic or uclar i ad th aoalous Ladé gyroagtic factor g i. Th bar agtic ot is giv i gral by 21 ( taθ ι $ S = + = gi i 2c i 2 $ i trs of th charg, ass ad agl corrspodig to th proto or lctro. For th proto w obtai (4 21 ( 3 $ = + = g 2 p c p 2 N S $ dtriig th proto aoalous gyroagtic factor (2.732 ad th uclar agto. For th lctro w obtai 2 $ S = = g B 2 c 2 $ dtriig th lctro aoalous gyroagtic factor (-2 ad th Bohr agto. Ths idtificatios giv th bar agtic ots, fro th agtic rgy i th gotric gralizd Dirac quatio, for th oly two stabl chargd fudatal frioic xcitatios i th gotric thory, th lctro ad th proto. Thy rquir radiativ corrctios to obtai th xprital valus. I QED th calculatd valus of th zro ordr agtic ot for th lctro ad th proto, giv by thir xtral Coulob fild scattrig diagras, ar sigular. For both cass, th itractio hailtoia for th xtral Coulob fild diffrs by th additioal aoalous cofficit g, dtrid by quatio (4. Th ffct of this diffrc is to adjoi th cofficit g to th xtral vrtx i th diagra. Thus aftr roralizatio th zro (41 (42

8 Proto Magtic Mot 8 ordr agtic ot valus for th proto ad lctro ar proportioal by th factor of ordr zro g. Th radiativ corrctios for th lctro hav b calculatd by Schwigr [18, 19, 2 ]. For both, lctro ad proto, th first ordr corrctios dtrid xclusivly by th vrtx part of th Coulob scattrig diagra ar proportioal to th corrspodig zro ordr trs. For th lctro th vrtx corrctio diagra, ford by a itral photo li btw th two itral frio lis, givs Schwigr s corrctio factor α/2π. For th proto th tripl lctroagtic U(1 structur prst i th SU(2 sctor of th gotric itractio oprator J*Γ, dtris thr stadard j.a couplig trs. Th tripl structur is also prst i th ocopact sctor of th algbra [21]. This idicats that a full proto xcitatio dscriptio rquirs thr boost ota k i. Cosqutly a full dscriptio of th xtral Coulob fild scattrig diagra rquirs a pair of itral frio boost otu triplts, istad of siply th pair of lctro ota. Thr ar additioal U(1 radiativ procsss btw th six itral frio boost lis corrspodig to th two triplts. Thrfor, thr ar ultipl additioal itral vrtx diagras obtaid by prutatio of th itral photo li aog th six cssary itral frio lis, all cotributig to th first ordr corrctio. Th ultiplicity of th vrtx corrctios of th 6 frio boost lis tak 2 at a ti is! 6! M = = = 15. (43 p!( p! 24!! Thr should b 15 vrtx corrctios, ach qual to Schwigr s valu, α/2π, bcaus of th SU(2 group quivalc. Th total first ordr radiativ cotributio to th proto agtic ot tr givs g1 g 15α = 1+ = ( = ( π which rducs to.5 th discrpacy with th xprital valu, qual to I a siilar ar w ay calculat th gotric agtic ot of a cobiatio of G, P ad L xcitatios. Th total lctroagtic pottial fild of this xcitatio syst is th su of A, th SU(2 Q pottial of th G- xcitatio, ad A U, th diffrt U(1 pottial of th P-xcitatio. Lt ϕ b th coo fra (½ igfuctio of th pottials A ad A U. Thus, w ay writ th xprssio for th total agtic rgy, quatio (35, i th followig for, ( i( i o U = i( ( AU + A + A ϕ = 1+ i ( ( AU + A ϕ, ( U 2 which bcos, i trs of th total v agtic fild B, 1 1 ( ( ( 2 2 ( i B U = ϕ = i Bϕ. ( Th ass corrspods to a G-xcitatio ass or proto ass p. W assu th rsultat charg distributio prsrvs th doiat agtic ffct of th lss assiv E-xcitatio (lctro. Th corrspodig xprssio for th agtic ot, i trs of th uclar agto, is 21 ( 32 $ = + = g 2 c p 2 N S $. ( 47 Th radiativ corrctio of this valu should accout for th prsc of th additioal P-xcitatio i th syst. This idicats that th xcitatio dscriptio rquirs four lctroic boost ota istad of th thr ota rquird for th proto xcitatio. Th ultiplicity of th vrtx corrctios of th 8 frio boost lis tak 2 at a ti is 8! M = = 28. ( 48 26!! Icludig th total first ordr radiativ cotributio, th gotric agtic ot tr givs

9 Proto Magtic Mot 9 g1 g 28α = 1+ = ( = ( π This rsult has a.5 discrpacy with th xprital valu of th utro agtic ot, qual to W ay cosidr th utro as a (p,,ν cobiatio. 5.Coclusio. Th gotrical agtic ot of th proto ad th utro, du to thir spis ad lctroagtic grators, hav a thortical first ordr aoalous gyroagtic Ladé g-factor which is clos to th xprital valu with a.5 discrpacy. Thr ay b othr corrctios i th gotric thory, icludig cssary highr ordr trs, which could liiat th sall discrpacy btw th thortical drssd valu ad xprital valus. 6. Appdix. 6.1 Algbra Rprstatio. Ay lt of th gotric algbra [2] ay b writt i trs of its v ad odd parts as a = a + κ a + ad ay b rprstd by a atrix of twic disios with v copots, as follows ( 5 a a a a + a +. ( 51 Usig this tchiqu w rprst th various objcts as follows η ξ ξ η, ( 52 Γ Γ Γ ad sic Γ Γ + + κ = κ κ κ = κ ( 53, ( 54 κ. ( Equatios of Motio for a Mattr Fra. Th xplicit xprssio for th quatios of otio of a fra [1 ] is ( 2 ν α 1 α ν κ Γ + κ u = which has for v ad odd parts, ( α ν ξ ξγ η 1 Γ + ξ =, ( ν α ( Γ ν α u, ( 57 1 α ν η ηγ + ξ u η =, ( 58

10 Proto Magtic Mot 1 ad ay b writt, νu α 1 α ν ξ+ ξ = η νu α Γ, ( 59 1 α ν η η = ξ Γ. ( 6 Rfrcs. 1 G. Gozálz-Martí, Phys. Rv. D 35, 1225 ( G. Gozálz-Martí, G. Rl. ad Grav. 22, 481 ( I. Taboada, Thsis, Uivrsidad Sió Bolívar ( G. Gozálz-Martí, G. Rl. ad Grav. 23, 827 ( G. Gozálz-Martí, Rport SB/F/273-99, Uiv. Sió Bolívar (1999, Lal archiv cod-at/ G. Gozálz-Martí, G. Rl. Grav. 24, 51 ( E. Mach, Th Scic of Mchaics, 5 th Eglish d. (Op Court, LaSall, ch. 1 ( A. Eisti Th Maig of Rlativity, 5 th d. (Pricto Uiv. Prss, Pricto, p.55 ( J. G. Gozálz-T., Thsis, Uivrsidad Sió Bolívar ( G. Gozálz-Martí, Rport SB/F/274-99, Uiv. Sió Bolívar (1999, Lal archiv physics/ G. Gozálz-Martí, G. Rl. Grav. 26, 1177 ( J. D. Bjork, S. D. Drll, Rlativistic Quatu Mchaics (Mc Graw-Hill, Nw York, ch 1, ( L. D. Ladau, E. M. Lifshitz, Mécaiqu Quatiqu, Théori o Rlativist (Ed. Mir, Moscow, 2d. Ed. p. 496 ( G. Gozálz-Martí, Physical Gotry, (Uivrsidad Sió Bolívar, Caracas (2 15 E. M. Purcll, H. C. Torry, R. V. Poud, Phys. Rv. 69, p.37 ( F. Bloch, W. W. Has, M. Packard, Phys. Rv., 7, p. 474 ( N. F. Rasy. Nuclar Mots, (Wily, Nw York, ( J. Schwigr, Phys. Rv. 73, 416 ( J. Schwigr, Phys. Rv. 76, 79 ( J. M. Jauch, F. Rohrlich, Th Thory of Photos ad Elctros, (Sprigr-Vrlag, Nw York, Scod Ed., p342 ( G. Gozálz-Martí, Rport SB/F/279-, Uiv. Sió Bolívar (2, Lal archiv physics/952