Nonlinear Theory of Elementary Particles Part VII: Classical Nonlinear Electron Theories and Their Connection with QED

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1 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi 6 Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Alxand G. Kyiakos * Aticl Abstact In this aticl of nonlina thoy of lmntay paticls (NTEP) a viw of th nonlina fild thois in th famwok of classical lctodynamics is psntd. It is shown that th sults found within ths thois can b tansfd to quantum thoy. Ths sults can also hlp us to undstand many aspcts of th quantum dsciption of lmntay paticls. In paticula, thy xplain why lcton can b intptd as a point and non-point paticl simultanously. Kywods: lcton thoy, Mi nonlina thoy, Schöding nonlina thoy... Intoduction In th pvious aticls w showd that a qustion about th fomation of lmntay paticls and th appaanc of its chaactistics, in paticula, mass and chag, is inspaably connctd with th slf-action of fild of a paticl. Th slf-action quis nonlina dsciption. Th physicists aivd not immdiatly at ths conclusions. Th thoy of chag, mass and oth chaactistics of lcton has aisn oiginally on th basis of classical lctodynamics and was dvlopd by W. Klvin, J. Lamo, H. Lontz, M. Abaham, A. Poinca, and many oths (s th viws: (Pauli, 958; Ivannko and Sokolov, 99)). Th fist xampl of a thoy (Coll. of aticls, 959) which unifid lctodynamics and mchanics was H. A. Lonz's attmpt to xplain th intia of an lcton on th basis of classical (lina) lctodynamics. H, th lcton was psntd as a clot of lctomagntic fild. Th pupos of th thoy was to show that th quation of an lcton s motion follows fom th quation which dscibs th fild of th lcton.. Impotant sults hav bn achivd within th famwok of this thoy. Howv, th dsciption of an lcton quid th intoduction of an additional non-lctomagntic fild. Altnativly, slf-action filds insid th lcton could b intoducd, but this quis th cation of a nonlina thoy of th lcton. Th ida of nonlinaity appad as answ to th difficultis, mgnt in th lina thoy of lcton (which was th fist lmntay paticl)... Th gnal sults and difficultis of th classical lcton thoy Accoding to th hypothsis, which has bn put fowad in th nd of th 9 th cntuy by J.J. Thomson and advancd by H. Lontz, M. Abaham, A. Poinca, tc. (Lontz, 96; Ivannko and Sokolov, 99), th lcton s own ngy (o its mass) is compltly causd by th ngy of th lctomagntic fild of lcton. In th sam way it is supposd that th lcton momntum is obligd to th momntum of th fild. Sinc lcton, as any mchanical paticl, posssss th momntum and ngy, which a togth th -vcto of th gnalizd momntum, th * Cospondnc: AlxandG.Kyiakos, Saint-Ptsbug Stat Institut of Tchnology, St.-Ptsbug, Russia. Psnt addss: Athns, Gc, a.g.kyiak@hotmail.com ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

2 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi ncssay condition of succss of th thoy will b th poof that th gnalizd momntum of an lctomagntic fild is a -vcto. Thus, fo th succss of th fild mass thoy th following conditions should b satisfid at last: Fistly, it is ncssay to civ final valu of th fild ngy, gnatd by a paticl, which could b pcisly quatd to final ngy of a paticl (i.. poduct of th mass by th squa of th light spd). Scondly, th valu of a momntum of th fild, gnatd by a paticl, must not only b final, but also has th pop colation with ngy, foming with th last a fou-dimnsional vcto. Thidly, th thoy should manag to dduc th quation of movmnt of lcton. Fouthly, it is ncssay to obtain of lcton spin, as a spin of a fild (that nds th quantum gnalization of th thoy of fild mass, sinc a spin is quantum ffct). All th paamts in classical lctodynamics can b xpssd though th symmtical ngy-momntum tnso of lctomagntic fild (Tonnlat, 959; Ivannko and Sokolov, 99) is dtmind by th following xpssions: i j i j ij i S i u, (7..) ij, (7..), (7..3) w, indics,,,3,, i, j,, 3 ; ij, whn i j and ij fo i j. Moov, a x xi x, x x, y, z, ict -vcto of th spac-tim has th fom,. Th analysis shows, that th a two conditions, by which th gnalizd fild momntum is a -vcto. In cas of spac without chags th siz i G ( d) c, (7...) will psnt a -vcto if divgnc of ngy tnso of a fild tuns into zo: x, (7..5.) Fo xampl, th lctomagntic fild, which is locatd in a spac without chags, satisfis simila conditions. In paticula, du to this fact, in th photon thoy, EM fild is chaactizd not only by ngy, but also by momntum. Th condition, by which th ngy and momntum of an lctomagntic fild fom a -vcto at th psnc of chags, is fomulatd by th Lau thom. Accoding to th last, at th psnc of chags th siz G is a -vcto only in th cas whn in th coodinat systm, lativly to which lcton is in st, fo all th ngy tnso componnts th following paity is obsvd ( d ), (7..6.) o xcpt fo th componnt T, th intgal of which is a constant and is qual to full ngy of th fild, gnatd by paticl (h d ) is lmntay volum in fnc systm, in which th ( o 63 G ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

3 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi lcton is in st). Th quality (.3) xpsss a ncssay condition, by which th whol paticl chag should b in balanc. W can quat this fild ngy to th paticl s own ngy, xpssing in this way th basic ida of a fild hypothsis. Accoding to th last: m ( d ) o c c, (7..7.) Thus, th mass of a paticl fom th fild point of viw can b dfind in two ways: ) pocding fom EM momntum of a fild G it is possibl to dfin mass as facto of popotionality btwn a fild momntum and th-dimnsional spd of a paticl. ) if w consid th lcton s own ngy as qual o contminous to th ngy of a fild, c and mass as th atio of a fild ngy G, to a squa of light spd (i.. as th fouth i componnt of a gnalizd momntum). Th attmpts to xcut this pogam, pocding fom classical lina Maxwll thoy, hav ld to difficultis. In paticula, it was not possibl to pov th Lau thom (Tonnlat, 959). In th classical thoy th dynamics (mchanics) and lctodynamics a compltly indpndnt fom ach oth. Elctomagntic actions a chaactizd by componnt T of an ngymomntum tnso of an lctomagntic fild. It dos not includ th ngy and momntum of th substanc, which should b subsquntly instd. Th attmpts of Lontz and Poinca to coodinat th thoy on th basis of th assumption that ngy of substanc has an lctomagntic oigin, hav not ld to a positiv sult. In Lontz lcton thoy (lina in ssnc) xistnc of chags it is possibl to xplain only by intoduction of focs of nonlctomagntic oigin. Nvthlss (Sokolov and Ivannko, 99), th w also a numb of succsss, which caid a hop to solv this poblm by som chang of th thoy. Th most pspctiv chang of Maxwll-Lontz thoy appad to b its non-lina gnalization by Gustav Mi Th Gustav Mi non-lina lcton thoy Within th famwok of classical physics, th fist compltly succssful nonlina thoy was catd by Gustav Mi (Mi, 9a, 9b, 93, 7; Pauli, 958; Tonnla, 959; Sommfld, 96). Th most widly known vaiant of this thoy was obtaind by M. Bon and L. Infld (Bon and Infld, 93b). Simila vaiant was also obtaind by E. Schöding and oths. Gustav Mi took th fist stp in th diction of th gnalization of Maxwll's quations and to th thoy of th lmntay paticls in 9 in his famous paps' "Foundations of a Thoy of Matt". Thi goal is no lss than th gnalisation of th Maxwll quations so that thy includ th xistnc of th lcton. This gnalisation is subjctd fom th stat to th pincipl of lativity and divd fom a "wold function" (Lagang function), which may dpnd only on Lontz-invaiant quantitis. H a distinction is mad possibly fo th fist tim in a consistnt fashion - btwn ntitis of intnsity and ntitis of quantity, i.. th diffnc btwn F B, ie and A A, i, on th on hand, and f H, id and j, i on th oth. Mi tstd th invaiants, which may b st up with th ntitis of intnsity and th ntitis of quantity spctivly. ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

4 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Mi st himslf th task to gnaliz th fild quations and th ngy-momntum tnso in th Maxwll-Lontz thoy in such a way that th Coulomb pulsiv focs in th intio of th lctical lmntay paticls a hld in quilibium by oth, qually lctical, focs, whas th dviations fom odinay lctodynamics main undtctabl in gions outsid th paticls. In oth wods Mi intoducd a unifom viw of th fild and substanc, and got id of Poinca Lontz focs, which hav a non-lctomagntic oigin Th G. Mi thoy Lagangian Mi was th fist who suggstd that th thoy should b constuctd on th basis of a Lagangian that dpnds on fundamntal invaiants. It is possibl to mak som gnal statmnts about th fom of th Lagangian L, which is oftn calld th wold function. Indpndnt invaiants, which can b fomd fom th bivcto F (wh F a th tnso componnts of EM fild stngths) and th vcto-potntial A i A i, A A, of an EM fild, a th following:, i A i. A squa of th bivcto F : I F F ;. A squa of th psudo-vcto I FF 3. A squa of a -vcto of EM potntial A :. A squa of th vcto 5. A squa of th vcto A * (wh I A A 3 ; F : I F A F A F * A : ; I * * 5 FA F A. * F is dual lctomagntic tnso). Thfo, L can dpnd only on ths fiv invaiants. If L is qual to th fist of ths, th fild quations a th odinay quations of EM thoy fo a spac without chags. Thus, L can diff substantially fom I only insid matial paticls. Invaiant can b includd into L only whn squad, in od not to bak th invaianc latd to spatial flctions. Invaiants 3-5 bak th gaug invaianc. W cannot mak futh statmnts about th wold function L. Thus, th is an infinit numb of ways in which w can slct L. Mi supposd that only th invaiants () and (3) nd to b considd fo th dsciption of quasistationay pocsss and constuct a wold function (Lagangian), such that at lag distanc fom th lcton th odinay Maxwll quations apply, whas th quations a modifid at th lcton and in its immdiat nighbouhood. Th ntitis of intnsity a obtaind fom Mi s wold function by diffntiation with spct to th ntitis of quantity. Th Lagangian is to b intgatd ov an abitay gion of th fou-dimnsional wold and to b vaid in suitabl mann. G. Mi accptd th following Lagangian as th initial on: o L Mi L Mi F 8 F f A A E H f A A, (7.. ), (7.. ) ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

5 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi wh f is som function, and spctivly. E, H a th stngth vctos of th lctic and magntic filds, Using this Lagangian, Gustav Mi obtaind th final ngy (o mass) of a chagd paticl as a valu compltly dtmind by th ngy of th paticl s fild. In this thoy, Lau s thom of paticl stability is satisfid, and th pop colation btwn th ngy and momntum of th paticl is achivd. An attmpt by Wyl should b mntiond h, in which h tis to mak th asymmty btwn th two kinds of lcticity undstandabl fom th point of viw of Mi s thoy. If wold function L is not ational function of A A, w can put L Mi FF f A A, (7.. ) L Mi FF f A A, (7.. ) wh f dnots any function which is not vn. Fo th statical cas th fild quations will not main invaiant fo an intchang of lctostatic potntial with (positiv and ngativ lcticity). Thus, if L is a multipl-valud function of th invaiants, mntiond abov, thn it is possibl to choos vaious singl banchs of this function as wold function fo positiv lcticity, and anoth fo ngativ lcticity Th Mi thoy difficultis Mi assums (Pauli, 958) th fild of th stationay lcton to b static and sphically symmty. But th latt assumption is admittdly not justifid by ou xpimntal knowldg alon, but commnds itslf fo its simplicity. W will thn hav to look fo thos solutions of th fild quations which a gula vywh fo as wll as fo. A much mo sious difficulty is causd by a fact alady noticd by Mi. Onc w hav found a solution fo th lctostatic potntial of a matial paticl of th quid kind, +const. will not b anoth solution, bcaus th fild quations of Mi's thoy contain th absolut valu of th potntial. A matial paticl will thfo not b abl to xist in a constant xtnal potntial fild. This sms to constitut a vy wighty agumnt against Mi's thoy. In th thois which w a going to discuss in th following sctions, this kind of difficulty dos not ais. 3.. Bon-Infld nonlina thoy Th Bon-Infld fild (Bon and Infld, 93a, 93b; Pauli, 958; Tonnla, 959; Ivannko and Sokolov, 99) thoy can b considd as a vival of th old ida of th lctomagntic oigin of mass. This non-lina fild thoy (Bon, 953) is a modification of Maxwll's lctodynamics in which th slf-ngy of th lcton is finit. Mi had shown alady in 9 that th quations of th lctomagntic fild can b fomally gnalizd by placing th lina lations btwn th two pais of fild vctos E, B and D, H by non-lina ons. Yt h did not spcify ths lations, and thus his fomalism maind mpty. Th ida which Bon applid to Mi plan is, as h not, a spcial cas of what Whittak has calld th pincipl of impotnc. If sach lads to an obstacl which in spit of all ffots cannot b movd, thoy dclas it as insumountabl in pincipl Exampls a ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

6 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi lativity, wh th impossibility of matial and signal vlocitis lag than th vlocity of light is dclad, and th unctainty lations of quantum mchanics, which fobid th simultanous dtmination of position and vlocity and of simila pais. In th cas of th lctomagntic fild th slf-ngy can b mad finit by pohibiting th incas of E th lctic vcto byond a ctain limit, th absolut fild. This can b don by imitating lativity wh th classical Lagangian of a f paticl L m is placd by L mc c, (7.3.) fom which c follows. In a simila way th Lagangian dnsity of Maxwll's lctodynamics can b placd by a squa oot xpssion. Thus a finit slf-ngy of a point chag is obtaind which psnts not only th intial mass but also, as Schöding has shown, th gavitational mass. To obtain th laws of natu Bon and Infld us a vaiation pincipl of last action of th fom 3 L d ', d ' dx dx dx dx, (7.3.) and postulat that th action intgal has to b an invaiant. Th poblm to find th fom of L satisfying this condition aiss h. Bon and Infld consid a covaiant tnso fild a kl. Th qustion is to dfin L to b such a function of a kl that (7.3.) is invaiant. Th wll-known answ is that L must hav th fom L a kl ; ( a kl = dtminant of a kl ), (7.3.3) If th fild is dtmind by sval tnsos of th scond od, L can b any homognous function of th dtminants of th covaiant tnsos of th od ½. Each abitay tnso a kl can b split up into a symmtical and anti-symmtical pat: a kl gkl fkl ; kl glk g ; fkl flk, (7.3.) Th simplst simultanous dsciption of th mtical and lctomagntic fild is th intoduction of on abitay (unsymmtical) tnso a kl ; it can b idntifid its symmtical pat g kl with th mtical fild, its antisymmtical pat with th lctomagntic fild. Th quotint of th fild stngth xpssd in th convntional units dividd by th fild stngth in th natual units may b dnotd by b. This constant of a dimnsion of a fild stngth may b calld th absolut fild. Th Bon-Infld fild quations a fomally idntical with Maxwll's quations fo a substanc which has a dilctic constant and a suscptibility, bing ctain functions of th fild stngth, but without a spatial distibution of chag and cunt. As Bon and Infld in th summay wit (Bon-Infld, 93b), Th nw fild thoy can b considd as a vival of th old ida of th lctomagntic oigin of mass. Th fild quations div fom th postulat that th xists an "absolut fild" b which is th natual unit fo all fild componnts and th upp limit of a puly lctic fild. Th fild quations hav th fom of Maxwll's quations fo a polaizabl mdium fo which th dilctic constant and th 67 ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

7 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi magntic suscptibility a spcial functions of th fild componnts. Th consvation laws of ngy and momntum can b divd. Th static solution with sphical symmty cosponds to an lcton with finit ngy (o mass); th tu chag can b considd as concntatd in a point, but it is also possibl to intoduc a f chag with a spatial distibution law. Th motion of th lcton in an xtnal fild obys a law of th Lontz typ wh th foc is th intgal of th poduct of th fild and th f chag dnsity Bon-Infld fild and Lagangian As in Maxwll's lctomagntism th Bon-Infld fild F A A of di magntic fild B : F is divd fom a potntial. This condition cancls th oto of th lctic fild E and th divgnc F F F, (7.3.5) Th Bon-Infld fild diffs fom th Maxwll fild in th dynamic quations, which a wittn in tms of th tnso I * F F ~ F b, (7.3.6) I I b b wh S and P a th scala and psudoscala fild invaiants: I F F B E, (7.3.7) I * F F E B, (7.3.8) * ( F is th dual fild tnso, i.. th tnso sulting fom xchanging th ols of E and Bon-Infld dynamical quations a which is obtaind fom th Bon-lnfld Lagangian o: L BI F, (7.3.9) b I I L b b b E B E B b b, (7.3.), (7.3.) B ). Th constant b in quations (7.3.6), (7.3.) and (7.3.) is a nw univsal constant with units of hld that contols th scal fo passing fom Maxwll's thoy to th nonlina Bon-Infld thoy, in th sam way as th light spd c is th vlocity scal that indicats th ang of validity of Nwtonian mchanics. Th Maxwll Lagangian and its latd dynamical quations a covd in th limit b, o in gions wh th hld is small compad with b. Bsids. Bon-Infld solutions having S = = P ('f wavs') also solv Maxwll's quations. As it follows fom Bon-Infld solution th constant b is a maximum lctic fild of an lcton ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

8 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi b E. Thus, if w consid E and B as wav functions, thn th atios E b and B b can b considd as th nomalizd wav functions Th point and non-point solutions of Bon-Infld s nonlina thoy Th sults psntd abov ld thm to th following Lagangian considd as a function of invaiants I and I : E E H E H L BI E, (7.3.) E wh E is a maximum lctic fild of an lcton. Using H, E gad, ( x ) ( ) ( t s), wh is th Diac fuction, w can find th following Lagangian fom: E E Ln ( ) E Thn, with th hlp of th vaiation pincipl, w can obtain: D L E wh D is an lctical induction vcto (D-fild): L E D E E E which cosponds to quation: div D ( ) Solution of this quation, which cosponds to lina Maxwll thoy, is as follows: D, (7.3.3) 3 As w can s, fom th point of viw of D-fild, th lcton should b considd as a point paticl. Fo th lctic fild (E-fild), w obtain: D E D E, (7.3.) wh point paticl. E. Thus, fom th point of viw of lctic fild (E-fild), th lcton is not th ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

9 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi In compaison to a lina thoy, ths sults psnt vy impotant spcifics of a nonlina thoy. This can xplain why xpimnts on dispsion of lcton can b intptd so that th non-point lcton can look as a point paticl. W can obtain th potntial s valu at th cnt of a paticl as follows: E d Th chag s dnsity distibution of th non-point lcton can b found in th following way: dive, (7.3.5) ( ) 3/ 7 Fo,, thfo diminishing vy apidly as incass. Fo,, thfo, but. It asy to vify that th spac intgal of is qual to. Fo a full chag w hav: d dived Eds lim E Th chag can b considd as distibutd in a sph of adius, sinc, bcaus of th condition, th dnsity will quickly go to zo. Thfo, th siz can b considd as an ffctiv adius of an lcton. Th valu of an lctomagntic mass of an lcton can b found basd on th quality condition: d m EM c 3 c c Using xpimntal valus fo th mass and th chag of an lcton, it is possibl to obtain fo 3 an ffctiv lcton adius th valu of,8 cm, which is pactically qual to th classical adius of an lcton. In this cas, w hav fo th lcton ngy d mem c. Also, it is asy to find th maximal fild of an lcton, which is in th cnt of an lcton at : 5 V E 9,8 CGS,75. m As is known two typs of filds and two dfinitions fo th chag dnsity, cosponding to thm, psnt in thoy of dilctics. Th atio of D to E can b considd as a dilctic constant : d D d, (7.3.6) E which in this cas is th function of a position. On lag distancs fom th chag, whn, is qual to on, th sam as in convntional lctodynamics. W can say that instad of th d 7 ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

10 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi 7 ngy xpssion th valu of th incas of d, so that th full ngy mains final. d is usd, and that th duction of is compnsatd by Thus, w povd that it is possibl to obtain th final slf-ngy (o th mass) of th chagd paticl as a valu compltly conditiond by th ngy of th fild of this paticl, whn on uss a ctain fomally ipoachabl hypothtical nonlina gnalization of lctodynamics as th basis. Futhmo, th Lau thoy of stability is valid within this thoy, so that a coct lationship btwn th ngy and th paticl s momntum is stablishd. Thus, basing on som fomal hypothtical nonlina gnalization of lctodynamics, it appad possibl (Ivannko and Sokolov, 99):. to pov th thom of stability, i.. to pov, that in th nonlina thoy th lcton is stabl without intoduction of focs of non-lctomagntic oigin;. to civ th final ngy (mass) of a paticl; 3. to civ th final siz of an lctic chag;. to civ th final siz of an lctomagntic fild... Schöding vaiant of Bon-Infld thoy without oot As E. Schöding has notd (Schöding, 935), Bon's thoy stats fom dscibing th fild by two vctos (o a "six-vcto"), B, E, th magntic induction and lctic fild-stngth spctivly. A scond pai of vctos (o a scond six-vcto) H, D, is intoducd, mly an abbviation, if you plas, fo th patial divativs of th Lagang function with spct to th componnts of B and E spctivly (though with th ngativ sign fo E ). H is calld magntic fild and D dilctic displacmnt. It was pointd out by Bon that it is possibl to choos th indpndnt vctos in diffnt ways. Fou diffnt and, to a ctain xtnt, quivalnt and symmtical psntations of th thoy can b givn by combining ach of th two "magntic" vctos with ach of th two "lctic" vctos to fom th st of six indpndnt vaiabls. Evy on of ths fou psntations can h divd fom a vaiation pincipl, using, of cous, ntily diffnt Lagang functions physically diffnt, that is, though thi analytic xpssions by th spctiv vaiabls a ith idntical o vy simila to ach oth. In studying Bon's thoy E. Schöding cams acoss a futh psntation, which is ntily diffnt fom all th afomntiond, and psnts cuious analytical aspcts. Th ida is to us two complx combinations of B, E, H, D as indpndnt vaiabls, but in such a way that thi "conjugats," i.., th patial divativs of L ~, qual thi complx conjugats. Choosing th following pai of indpndnt vaiabls ~ ~ F B id, G E ih, (7..) (which fom a tu six-vcto) th appopiat Lagangian woks out ~ F ~ G L ~ ~, (7..) FG and on has ~ * L ~ * L F ~, G ~, (7..3) G F ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

11 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Th * indicats th complx conjugat. Th divativ with spct to a vcto is shot fo: a vcto, of which th componnts a th th divativs with spct to th componnts of that vcto. Th units a "natual" units, Bon's constant b bing qualld to (in oth units L ~ would tak th facto b ). What is so vy supising with E. E. Schöding s ida (Schöding, 935) is that th squa oot, which is so chaactistic fo Bon's thoy, has disappad. Th Lagangian is not only ational, but homognous of th zoth dg. As Schöding show, th tatmnt of th fild by th Lagangian (7..) is ntily quivalnt to Bon's thoy. Thfo it cannot yild any nw insight which could not, vitually, b divd fom Bon's oiginal tatmnt as wll. Moov, fo actual calculation th us of imaginay vctos will hadly pov usful. Yt fo ctain thotical considations of a gnal kind I am inclind to consid th psnt tatmnt as th standad fom on account of its xtm simplicity, th Lagangian bing simply th atio of th two invaiants, whas in Maxwll's thoy it was qual to on of thm. It is not difficult to obsv that th componnts of (s abov) a idntical in fom with th componnts of Maxwll s vacuum tnso, F ~, G ~ bing substitutd fo H, E. By invstigating th tansfomations of th al tnso, it is asy to find a fam of fnc, in which th physical maning of ou "condition of conjugatnss" is adily disclosd. What distinguishs a Maxwll tnso fom th gnal symmtical tnso is only that its oots o ignvalus hav th fom this typ. At that th valu woks out to, ach doubl. Th fist pat of i, viz., ~ FG ~ 7 is pcisly of i ~ L, (7..) by considing that in Maxwll's cas is known to b H E HE (s th poof in (Lightman, A.P., t al., 975))... Oth Lagangians of nonlina thois It was notd (Ivannko and Sokolov, 99), that vaious and abitay vaiants of fomal nonlina lctodynamics lad to clos valus of cofficints, if w tak into account that th lcton adius is qual to a classical adius of an lcton. Fo xampl, in this way E. Schöding obtaind simila sults using th following Lagangian: E E H L Sch ln, (7..5) 8 E Th only sious dficincy in ths nonlina thois is that thy a not quantum. ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

12 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Th vacuum polaization as caus of fomation of lcton As is known (Wikipdia, Vacuum polaization; Pskin and Schod, 995) in quantum fild thoy, and spcifically quantum lctodynamics, vacuum polaization dscibs a pocss in which a backgound lctomagntic fild poducs vitual lcton-positon pais that chang th distibution of chags and cunts that gnatd th oiginal lctomagntic fild. It is also somtims fd to as th slf ngy of th gaug boson (photon). Vacuum polaization was obsvd xpimntally in 997 using th TRISTAN paticl acclato in Japan. Accoding to quantum fild thoy, th gound stat of a thoy with intacting paticls is not simply mpty spac. Rath, it contains shot-livd "vitual" paticl-antipaticl pais which a catd out of th vacuum and thn annihilat ach oth. Som of ths paticl-antipaticl pais a chagd;.g., vitual lcton-positon pais. Such chagd pais act as an lctic dipol. In th psnc of an lctic fild,.g., th lctomagntic fild aound an lcton, ths paticl-antipaticl pais position thmslvs, thus patially countacting th fild (a patial scning ffct, a dilctic ffct). Th fild thfo will b wak than would b xpctd if th vacuum w compltly mpty. This ointation of th shot-livd paticl-antipaticl pais is fd to as vacuum polaization. Th vacuum polaisation fistly was discussd 9 by G. Mi in his classical nonlina lctomagntic thoy of lcton. On of th fist a studnt of Wn Hisnbg and Anold. Sommfld - D. Eich Bagg - connctd th task of th foming th lmntay paticl (in paticula, lcton) with th polaization of physical vacuum In th simplst cas th fomulation of poblm of fomation of an (sphical) lcton looks as follows (Bagg, 95). Th fact of th fomation of th lcton pais by high ngy light quanta can b considd as dp polaization of vacuum that lads to th baking of th lctic dipol und th action of EM fild of light wav. This maks it possibl to xamin th psnc of this polaization also in th cas, whn fild ngtically cannot caus a pai fomation. This lads to intaction of light by light, as this showd Hisnbg, Eul and Kockl (Eul.and Kockl, 935; Eul, 936; Hisnbg and Eul, 936) in quantum fild thoy. Mayb Pt Dby (Dby, 93) was fist, who not, that this appoach can b dvlopd also in th famwok of Diac s thoy: If Diac's pictu of pai poduction in 'hol thoy' is coct, thn at low ngis a kind of vacuum polaisation should b ppad simila to th polaisation ffct in a dilctic mdium shotly bfo an lctic dischag occus." E. Bagg has shown (Bagg, 95; 988; 99, 993) that with th asonabl choic of th function V, t it is possibl to dscib th st sphical lcton within th famwok of Diac s lativistic thoy of paticls as lcto-magntic stabl, sphoidal paticls. Thi poptis, spcially thi spins and thi magntic momntum, a xactly thos, which hav bn masud at fist and lat on divd by Diac. This concpt thotically can b alisd by th hlp of th fomula fo th mdia dispsion t of constants of optical sonanc ffcts, allowing th calculation a polaisation function V, ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

13 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi th vacuum. In this cas th atio of D D (pmability) of vacuum : V, t. E 7 to E can b considd as a dilctic constant It is natual thfo to psnt in this cas th quations of fild in that fom, which thy hav fo th polaization mdium. This mans that in paticula th dilctic constant V must b considd as th function of spac. This fact natually constains th fild in th nvionmnt of lcton, which automatically lads to nd valu fo th slf-ngy of lcton. Considing th dilctic constant of vacuum with paticls V () not as constant, but as th function of th-dimnsional vaiabls and fild (in simplst cas, of scala potntial ), w must plac th known Laplac quation fo th mpty spac, (7.5.) by th quation, which consids th inconstancy of dilctic constant: V V V, (7.5.) x x y y z z This quation alady can not hav th infinit solution fo th fild of sphical chag. Actually, if V V ( ), ( ), thn th diffntial quation (7.5.) taks th fom: d d d(ln V ) d, (7.5.) d d d d If a dilctic constant is constant, th quation (7.5.) has a solution in th fom of th Coulomb potntial: C. It is obvious that w must obtain this xpssion at a gat distanc fom th lcton, whas in th limits of th basic volum, which contains th basic ngy (mass) of lcton, th solution must not giv th divgnt sult. By analogy with th dispsion fomulas of optical thoy Bagg poposd th following xpssion fo th dilctic constant (without taking into account a damping): C V, (7.5.5) wh C is a constant, which must b dtmind; (~ ) is th sonanc ngy ( m c ), which is dtmind by photon ngy; (~ ) is ctain ngy of th filds (which can b idntifid with th ngy of lcton-positon, which compos th dipol of intmdiat photon (s Kiyakos, a). It is natual to assum ; thn fo th dilctic constant w obtain th xpssion: C V, (7.5.6) ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

14 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Using (7.5.6), w will obtain fom (7.5.) th following xpssion: d d d d d d, (7.5.7) 75 Considing that th stngth of th fild of lcton has an finit maximum quantity, w must to assum that th sonanc ngy, which cosponds to this stngth, is th constant, which can b xpssd in th fom:, so that if mc, thn is a adius of th mc chagd sph, whos chag is distibutd unifomly by th volum. Thn fom (7.5.7) w obtain th nonlina diffntial quation: d d d d d d, (7.5.8) Th gnal solution of this nonlina diffntial quation of th scond od, as it is not difficult to vify (Bagg, 95), taks th fom: A B, (7.5.9) A B wh A and B a constants of th intgations, which can b found fom th limiting conditions: a) with, which givs s B, b) with, which givs A, Thus, th potntial of sphically symmtical lcton upon considation of slf-action can b codd in th fom:, (7.5.) As w s, potntial at point dos not go to infinity, but is takn th spcific valu:, which cosponds to th maximum valu of fild. Solution (7.5.) maks it possibl to calculat also stngth E and lctical displacmnt D of fild, and also ngy of sphical chag. Fom (7.5.), accoding to th connction of scala potntial and stngth of lctic fild, w obtain fo th valu: E gad, (7.5.) Fom this th maximum valu of fild will b qual to: E gad, (7.5. ) Using (7.5.), fom (7.5.6) w obtain fo th dilctic constant: C V, (7.5.) and fo th displacmnt: ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

15 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi C D V E, (7.5.3) 3 3 Equation (7.5.3) givs th possibility to stablish th connction btwn C and. Fom on sid: Fom th oth sid 8 Co DdS 3 EDd m It follows fom (7.5.) and (7.5.5): c C mc, C m c,, (7.5.), (7.5.5) Substituting, w obtain complt xpssion fo th dilctic constant th xpssion: V m c m c, (7.5.6) which compltly satisfis to th physical conditions of th lcton-positon pai poduction: taking into account w obtain V, whil th ngy mc, which is ncssay fo th appaanc of pai, givs fo th dilctic constant th infinit valu, which cosponds to th baking of photon to two pats. Both conclusions cospond pcisly to physical statmnt of poblms. Following E. Bagg (Bagg, 95), lt us not also that in th gnal cas th poblm must b placd and b solvd, taking into account all filds and thi pcis configuation. In this cas th dilctic constant will b not th simpl function of potntial, but th tnso, dtmind by th lctical and magntic filds of th intmdiat photon and nuclus. It was notd (Ivannko and Sokolov, 99), that vaious and, as was outlind, fom th physical point of viw, abitay vaiants of fomal nonlina lctodynamics lad to clos valus of cofficints, if to tak into account, that th lcton adius is qual to classical adius of lcton. It was also notd, that th basic dfct of ths thois, as wll as of Mi thoy, was th abitay choic of Lagangian, which had no connction with th quantum thoy, in paticula, with Diac thoy, and did not tak into account poptis of lcton, vald by th last. W will show that ths thois can b considd as appoach of th NTEP and that thy a mathmatically connctd to th Diac lcton thoy Nonlina classical thois as appoximations of NEPT 6.. A connction of G. Mi s thoy with Bon-Infld thoy and NEPT Lt us show that th Mi Lagangian, aft som additions, can b psntd as a Lagangian that is simila to th Lagangian of NEPT (and, consquntly, QED). Rcall th Mi Lagangian (7..): ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

16 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi o L E H f A Mi A L Mi FF f A A 8 As w know (Pauli, 958; Sommfld, 96), th chag dnsity is not an invaiant with spct to Lontz tansfomations. Howv, th chag is an absolut invaiant with spct to Lontz tansfomations. It is also known that th squa of th -potntial, i.. I A A 3, is an invaiant with spct to Lontz tansfomations. Howv, it is not an invaiant with spct to gaug tansfomations. It appas that th poduct of th squa of chag and I 3 will b invaiant with spct to both Lontz and gaug tansfomations Lamo Schwazschild s invaiant Accoding to (Pauli, 958) and (Sommfld, 96), R. Schwazschild (Schwazschild, 93), intoducd th valu S w A, (7.6.) c which h calld "lctokintic potntial". H showd that this valu, whn multiplid by th dnsity of chag, foms a lativistic invaiant: L' ( A) j A S w, (7.6.) c c ic, A i, A is a -potntial. Futh, Schwazschild wh j is a -cunt dnsity, foms th Lagang function by intgation with spct to spac L H E dv ( A) dv (7.6.3) c H thn obtains th action function by intgation with spct to tim. Thus, in fou dimnsions, th Lagang function dnsity (o Lagangian) can b wittn as follows: L FF j A, (7.6.) c whil th Lagang function will b: L F F d c j A d, (7.6.5) (In not to his book (Pauli, 958) Pauli notd that bfo Schwazschild, th sam Lagangian has bn suggstd by J.J. Lamo (Lamo, 9)) Th Mi vaiant of gaug-invaiant thoy W will now consid th adicand function in Mi s Lagangian: A A A A i, (7.6.6) Multiplying it by th squa of an lcton s chag, w obtain: A ( ) ( A), (7.6.7) Sinc th valu:, (7.6.8) ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

17 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi is th ngy of lcton intaction, and th valu: p A i i, (7.6.9) c is th momntum of lcton intaction, w obtain fom (7.6.9): A cp, (7.6.) ˆ ˆ Taking into account that o, p c p ˆ c p ˆ p, ths xpssions can also b wittn as: A i i, (7.6.) 78 Using th abov sults, w will accpt th following xpssion fo th nonlina pat of Mi s Lagangian L f A A N Mi L N Mi : c A, (7.6.) Using th poptis of Diac matics, it is asy to obtain th following dcomposition: A c ˆ, (7.6.3) ˆ p which sults in th following xpssion fo th nonlina pat of th Lagangian: N L c ˆ p, (7.6.) Taking into account that Mi ˆ mc ˆ p ˆ c ˆ, (7.6.7) w s that w can put th mass tm of Diac s quation into Mi s Lagangian. Th usag of ths xpssions lads to Diac s quations of lcton and positon, and givs th basis to Wyl s attmpt to intpt an asymmty of both typs of lcticity not with gad to a mass, but with spct to th diffnc btwn th paticl and antipaticl. Also, this way w can lat th Mi s Lagangian to Bon-Infld s Lagangian. W can wit using (7.6.) th following: A ud c gd ( u cg) d ( u cg) d, (7.6.8) o Thn, using th appoximation in th pvious aticl, w hav: A u c g, (7.6.9) Thus, in appoximat fom, th Mi s Lagangian can b wittn in th following fom: L Mi u c g, (7.6.) 8 Rcalling lctomagntic psntation of th Fiz idntity: 8 u c g 8 u S, (7.6.) E H E H E H E H w can wit an appoximation of th Mi Lagangian as ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

18 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi L Mi 8 8 E H E H, (7.6.) 79 As it is not difficult to s this appoximation is simila to Bon-Infld Lagangian. Moov, in th following aticl w will show that Mi s Lagangian can b tansfomd into th fom of th Lagangian of nonlina fild thoy, which cosponds to th lcton thoy of NEPT (s th following aticl NTEP. 8. Nonlina quantum lcton thoy ). 7.. Nonlina classical thois as appoximations of NTEP. Gnal statmnt of poblm As w has shown alady, th condition of quantization in NTEP is causd by th intoduction of lationships of th ngy (accoding to Planck) and momntum (accoding to d Bogli) quantization. Thfo, th classical nonlina thois, in which th quimnt of noncommutativity is valid bcaus of th spcific nonlinaity (causd by cyclic tanspot of fild vctos), can b considd as quantum thois if thy a supplmntd with this condition. Obviously, th NTEP can includ all possibl invaiants of lctomagntic fild. Thfo, its Lagangian can b wittn as som function of th fild invaiants I E H and I E H : L f I, I ), (7.7.) L ( Appantly, th function f L can hav a spcific fom fo ach paticula poblm. Howv, an xpansion of th function f L ( I, I) in Taylo Mac-Launt pow sis must xist in vy cas. In gnal cas, ths xpansions must contain th sam st of tms that will diff only by constant cofficints, som of which can b qual to zo (s xampls of such xpansions in (Akhiz and Bsttskii, 965; Wisskopf, 936; Schwing, 95). Thfo, in gnal, th xpansion will look as follows: L E B M L', (7.7.) 8 wh L' E B E B E B E B 3, (7.7.3) E B E B E B... is th pat sponsibl fo nonlina intaction (h,,,,,,... a constant cofficints). On th oth hand, th Bon-Infld Lagangian can b xpandd into a sis with small paamts a E and a B, wh a : E a L BI E B E B E B O( E, H ), (7.7.) 8 3 wh O( E, H ) is th sis maind with th tms, containing vctos of lctomagntic fild in pows gat than fou. Obviously, at a lag distanc fom th cnt of a paticl (wh th maximal fild is), und th conditions a E and a B, th tms of ths ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

19 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi sis quickly convg. Howv, th tms with high pows must b takn into account at a small distanc fom th paticl cnt. In th following aticl w will show that th fist appoximation of Lagangian of NTEP can b psntd in EM fom as: L E B A E B E B N, (7.7.5) 8 wh A is som constant. Thus, taking into account (7.7.), w can wit: LN L BI, (7.7.6) Thn, within th famwok of NTEP, an appoximat solution of th lcton quation will b simila to th solution of th Bon-Infld thoy. Obviously, this asstion will b also coct fo oth nonlina thois. Having this, it is not difficult to answ th qustion, why th vaious vaiants of fomal nonlina lctodynamics lad to clos valus of cofficints: th xpansions of nonlina Lagangian (7.7.3) a appoximatly simila fo vaious vaiants and consquntly thy poduc clos sults. At th sam tim, sinc th Lagangian and quations of NTEP compltly coincid with th Lagangian and quations of quantum lctodynamics, th Mi thoy and its vaiant - th Bon- Infld thoy, a closly latd to Diac s lcton thoy. In th following aticl w will show that th Diac lcton quation can b xpssd in fom of nonlina quation, which in th lctomagntic fom is simila to classical nonlina quations, considd abov, and in th quantum fom is simila to known nonlina quantum fild quations. Rfncs Akhiz, A.I. and Bsttskii,. W.B. (965). Quantum lctodynamics. Moscow, Intscinc publ., Nw Yok. Bagg, Eich. (95) Di Polaisation ds Vakuums als Usach in Fldbgnzung bim Elcton. Zits. F. Physik, Bd. 3, S (95) Bagg, E. (988). Das Kuglschalnmodl ds Elkton als Altnativ zu Punktladung, Fusion 9, 39(988) Bagg, E. (99). Wlt und Antiwlt als physikalisch Ralitat, Babl Mnd Vlag, Kil (99) Bagg, E.R. (993). Elcton, ponton, nuton as sphoidical paticls. Fotsch. Phys. (993) 6, Bon, M. (93a). On th quantum thoy of th lctomagntic fild. Pocdings of th Royal Socity of London A, 3: 37 Bon, M. and Infld, L. (93b). Foundations of th nw Fild Thoy, Poc. Roy. Soc. A, 5 (93). Bon, M. (953). Th Concptual Situation in Physics and th Pospcts of its Futu Dvlopmnt. Poc. Phys. Soc., 66, А, 5 (953) Coll. of th aticls (959). Nonlina quantum fild thoy. Moscow, Foign Litatu Publishing Hous. Dby, Pt. (93). Publishd in P. Du, Quantum und Fld, Viwg, Baunschwig (97), pag 3. Eul, H. and Kockl, B.. (935). Üb di Stuung von Licht an Licht nach d Diacschn Thoi. Natuwissnschaftn 5 (935), p. 6 Eul, H. (936). Üb di Stuung von Licht an Licht nach d Diacschn Thoi Ann. d. Phys. 6 (936), 398 Hisnbg, W. (93). Bmkungn zu Diacschn Thoi ds Positons, Zitschift fü Physik Volum 9, Numbs 3-, 9-3 (93). Hisnbg, W. and Eul, H. (936). Folgungn aus d Diacschn Thoi ds Positons, "Z. Phys.", 936. Bd 98, S. 7; Ivannko, D. and Sokolov, A. (99). Th classical fild thoy (in Russian). Moscow-Lningad Kyiakos, A.G. (a). Nonlina Thoy of Elmntay Paticls: IV. Th Intmdiat Bosons & Mass Gnation Thoy. 8 ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.

20 Pspactim Jounal Mach Vol. Issu 3 pp. 6-8 Kyiakos A. G. Nonlina Thoy of Elmntay Paticls Pat VII: Classical Nonlina Elcton Thois and Thi Kyiakos, A.G. (b). Nonlina Thoy of Elmntay Paticls: V. Th Elcton & Positon Equations (Lina Appoach). Kyiakos, A.G. (c). Nonlina Thoy of Elmntay Paticls: VI. Elctodynamic Sns of th Quantum Foms of Diac Elcton Thoy. Lamo, J.J. (9). Ath and Matt. Cambidg. Ch. 6. Lightman, A.P., t al. (975). Poblm book in lativity and gavitation. Pinston Univsity Pss, Piston, 975. Lontz, H.A. (96). Th thoy of lctons. Lipzig. B.G. Tubn. Mi, G. (9a). Gundlagn in Thoi d Mati. Ann. d Physik, 37, 5. Mi, G. (9b). Gundlagn in Thoi d Mati. Ann. d Physik, 39,. Mi, G. (93). Gundlagn in Thoi d Mati. Ann. d Physik,,. Pauli, W. (958).Thoy of Rlativit.y. Pgamon, London. Pskin, M.E.and Schod, D.V (995). An Intoduction to Quantum Fild Thoy, Addison-Wsly, 995; sction 7.5 Schöding, E. (935) Contibutions to Bon s nw thoy of th lctomagntic fild. Pocdings of th Royal Socity of London. Sis A, Mathmatical and Physical Scincs, Volum 5, Issu 87, pp , 935 Schwazschild, R. (93). Got. Nach.math.-natuw. Kl., S.5. Schwing, J. (95). Phys. Rv., 8, 66. Sokolov, A. and Ivannko, D. (95) Th quantum fild thoy (in Russian), Moscow-Lningad. Sommfld, A. (96). А. Elctodynamics: Lctus on Thotical Physics. Nw Yok: Acadmic Pss, 96.. Tonnlat M.-A., (959). Ls Pincips d la Thoi Elctomagntiqu t d la Rlativit. Masson t C., Pais. Wisskopf, V. (936). Kgl. Dansk Vidn 8 ISSN: Pspactim Jounal Publishd by QuantumDam, Inc.