Article Nonlinear Theory of Elementary Particles: VI. Electrodynamic Sense of the Quantum Forms of Dirac Electron Theory. Alexander G.

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1 58 Arl Nonlnar Thor o lnar Parls: VI. lrodna Sns o h Quanu Fors o Dra Alandr G. Krakos * Absra In h prsn papr s shown ha a ull orrspondn bwn h quanu and h lroagn ors o h Dra lron hor ss so ha ah ln o h Dra hor has h known lrodnas anng and v-vrsa. K-words: lron quaon blnar or sasal nrpraon Lagrangan alonan anonal ransoraon... Inroduon. Th spnor and bspnor quaons o phoon On h bass o h prvous paprs w wll show hr ha all h ahaal parulars o h Dra lron hor hav h known lrodnas anng. As s known Akhr and rssk 965 hr ar an dnal ahaal rprsnaons o h quaon o lron. Sn aordng o h nonlnar hor o lnar parls NTP an lron has lroagn orgn an b assud ha all hs rprsnaons hav a bas n nonlnar lrodnas. In parular h us b basd on h lnar quaons o phoon Krakos a and nonlnar quaon o nrda assv phoon Krakos b. L us an h spal aurs o hs quaons whh us also b rld n h quaons o lron. As w nod Krakos a h quanu quaon o phoon an b rordd n h or o Mawll-Lorn quaons akng no aoun h quanaon o nrg aordng o Plank. In parular hs quaons an b wrn down n h spnor and bspnor or. Th Mawll-Lorn quaons or a phoon as quanu o lroagn wav n h spnor or looks as ollows: p 6.. p whr and p ar h opraors o nrg and onu; ar h spn ars o Paul: ; and ar h wav unons o an phoon n h spnor or dsrbd b ans o h ollowng ars: 6.. * Corrspondn: AlandrG.Krakos San-Prsburg Sa Insu o Thnolog S.-Prsburg Russa. Prsn addrss: Ahns Gr -al: a.g.krak@hoal.o

2 L us an h phoons whh ov n h d oordna ss X Y Z along h -as. For suh phoons aordng o lrodnas h rlaonshp aks pla whh s nvaran rlav o h Lorn ransoraons. Thror h ar dsrbd onl b wo vors prpndular o -as. I w onn h Frns rhdron n b o h lral agn and Ponng vors S rspvl hn h lar wll b ollnar wh h -as hr n s a noral un vor o h raor o h phoon b s h un vor o bnoral and s h un vor o angn o h raor o phoon wh h rlnar oon hs vor onds wh h raor o h oon o phoon and wh h urvlnar s drd angnall oward h urv. Addonall h lral and agn vor an b urnd rlavl o -as o an angl n h ls whou hangng h phsal hararss o phoon. Takng no aoun hs on should onlud ha n h gnral as hr s an nn nubr o suh phoons. Obvousl an lnarl polard phoon an b rprsnd as h su o wo phoons wh uuall prpndular vors and wh a drn absolu valu. In ha as vors ar haron unons hs su s h llpall-polard or rularl polard phoon. Sn aordng o h os-nsn hor h onohroa lroagn wav s os-ondnsa o h phoons o on rqun w an prsn phoons as M wav and v vrsa. Fgur 6. dps a hang n h lr ld o phoon whh ovs along h -as h polaraon plan o whh oposs h angl wh h plan ZOY. Th gur also shows h proons o h oon o lr vor on h plan ZOY and XOY whh llusra h phoons wh s ld oponns and : 59 Рис. 6.. Furhr w wll hav n nd h gnral as o wo spara phoons on rularl polard phoon oposd o hs par o M ld vors Fg. 6.: and or

3 5 Рис. 6.. Inrodung spnors n h or: 6.. and akng no aoun ha w wll oban h Mawll quaons o wo lroagn wavs or phoons n h as o h quanaon o hr nrg: d b a 6.. I s no dul o s ha wo quaons 6.. an b rordd n h or o on quaon. Auall nrodung a wav unon alld bspnor b ans o h ollowng ar: 6..5 wo spnor quaons an b rwrn as on quaon: p 6..6 whr ar h Dra ars:. I s no dul o prov ha usng bspnor w wll oban h sa M quaons 6... Th quaons 6.. a and d orrspond o h polard n h plan XOY phoon. quaons 6.. b and orrspond o h polard n h plan ZOY phoon. Phsall hs phoons ar dnal n vw o unor and sorops o p spa. u h ahaal rord us onsdr h spal aurs o hr propagaon dpndng on h ho o oordnas. Th

4 dron o propagaon o lroagn wav s drnd b h Ponng vor and n h Gauss ss uns aks h or: SP 6..7 For h wav along h -as w hav: S P 6..8 whr s h un vor o -as. As w s h phoon ovs agans h dron o -as and h quaon o ld oponns n hs as us b wrn n h or: k o 6..9 k o Phoon ovs along h dron o -as and h quaon o s oponns us b wrn n h or: k o 6.. k o As ollows ro h prvous papr Krakos b or h oraon o lron h phoon us rs o all prn h ransoraon o roaon n h plan n. Obvousl h vnual rsul o ransoraon dos no dpnd on h urnng o h oordna ss. Ths ans ha h ransoraons whh dsrb a passag ro on Carsan ss o an ohr us s whou a hang n h rsul o h roaor ransoraon o phoon. wsng h phoon an or a rng hr n h plan XOY or n h plan YOZ. In h pross o roaon ransoraon h rng urrns ar ord: and lnar ass-r phoon s onvrd no h assv nrda boson whh w ondonall all nrda assv phoon. Th lr harg o hs phoon s qual o ro sn onans alrnang urrn. A quson arss o how ar lds drd n h nrda assv phoon? Th roaon o phoon an b aoplshd lokws and ounr-lokws lookng ro h nd o h agn vor Fg. 6.: 5 Рис. 6. Th rsul wll b sohow drn boh phoons or a rularl polard phoon. W wll an hs as n h papr ddad o nurno.

5 I ollows ro abov ha boh h nrda phoons and lrons and posrons us hav a s o ors o rprsnaons whh do no hang h phsal sns o quaons and parls hararss. 5. Th lrodna and quanu ors o Dras quaon As w showd n h prvous papr Krakos assv phoon an prn sponanous brakng o sr and an b dvdd no wo hal-prods wo s-phoons: lron and posron. Aordngl w oband h quaons o hs parls. Th ar known as spnor and bspnor ors... Th spnor lron quaon For h apparan o h hargd parl - lron s nssar ha h roaon o phoon ourd n h plan whh onans h lr vor and h vor o Ponng. Ar h brakng o nrda phoon no s-phoons h an or rngs hr n h plan XOY or n h plan YOZ. In hs as agn urrns ar no ord. u or h sr w wll an h gnral as o sn o boh lr and agn urrns: akng no aoun ha or h lron and posron h agn urrns ar qual o ro. Inrodung and as h wav unons o an lron n h spnor or b ans o h ollowng ars: 6.. w oban Dras quaon n h spnor or: p 6.. p Takng no aoun ha w wll oban Mawlls quaons n h opl or wh lral and agn urrns: 6.. whr. In h quanu or hs ss s or onvnnl wrn n h or o on bspnor quaon o Dra... Th bspnor lron quaon Dras lron quaon

6 Dras quaon or rqunl s wrn n a bspnor or. Inrodung a wav unon b ans o h ollowng ar: 6.. wo spnor quaons o lron an b rwrn as on Dra s lron quaon: p Quanu ors o Dra s lron and posron quaons Thr ar wo bspnor Dra quaons Akhr and rssk 965; h 96; Sh 955; Fr 96: o p o p whh orrspond o wo sgns o rlavs prsson o an lron nrg: p 6..8 Morovr or ah sgn o prsson 6..8 hr ar wo ran-onuga Dra quaons. Thus hr ar wo ran-onuga quaons orrspondng o a nus sgn o h prsson 6..8: o p p o and wo quaons ha orrspond o plus sgns o 6..8: o p 6.. p 6.. o W wll urhr us h wav unon o a rular-polard M wav ha s ovng as n h prvous paprs along h - as: M ors o Dra s lron quaon L us onsdr rs wo ran-onuga quaons orrspondng o a nus sgn o prsson 6..8: Usng 6.. w oban ro 6..9 and 6..9 : 5

7 whr r C 6.. r C 6.. ar opl urrns. Thus h quaons 6..9 and 6..9 ar Mawll s quaons wh opl urrns whr ran-onuga quaons 6.. and 6.. dr b urrn drons. L us onsdr now quaons ha orrspond o a plus sgns o An lroagn or o quaon 6.. s: 6.. Obvousl an lroagn or o quaon 6.. wll hav h oppos sgns o urrns wh rgard o 6... Coparng 6.. and 6.. w an onsdr quaon 6.. as h Mawll s quaon o h rardd wav n rlaon o Mawll s quaon o advand wav 6... So w do no wan o us h rardd wav w an ransor h wav unon o h rardd wav o h ollowng or: r 6..5 Thn onrar o h ss 6.. w g h ss 6... Th ransoraon ro h unon r o h unon adv s alld a harg onugaon opraon. Thus w an sa ha h lron and posron wav unons an b onsdrd as rardd and advand wavs.

8 No ha h abov rsul rlas o h hor o advand wavs o Whlr and Fnan Whlr and Fnan 95; Whlr 957. S also h Dra s work on -sr lassal lrodnas Dra 98 and Konopnsk s book on h sa op Konopnsk lrodnas anng o blnar ors Ths s wll known ha hr ar 6 Dra ars o dnsons. W plo h sa s o ars usd b Dra and na as -s. Th valus O whr s an o h Dra s ars ar alld blnar ors o Dras lron hor. I an b shown ha h nsor dnson o a blnar or ollows ro h nsor s nonlnar lrodnas ors. L us nura h Dra s ars as ollows Akhr and rssk 965; h 96; Sh 955: A r w hav: salar -vor psudosalar -psudovor 5 ansral nsor o a sond rank. L us alula lrodnas valus orrspondng o h abov ars usng aordng o I whr I s h rs salar nvaran o Mawll hor; s also Lagrangan o an lroagn ld n vauu; o 8u whr u s h nrg dns o h lroagn ld; 8 SP 8 g whr g s a onu dns o an lroagn wav s ld ovng along h Y-as. As s wll known h valu u g s a -vor o h nrg-onu. s a psudosalar o lroagn ld and 5 I s h sond salar nvaran o lroagn ld hor. W wll show subsqunl ha hs blnar or s rlad o spral o parls

9 56 5. In quanu hans hr-dnsonal oponns o h spn s nsor ar prssd hrough hs ars Sokolov and Ivannko Tnsor an b prsnd n a opa or as ollows: In quanu hans nsor Lvh 969 dsrbs agn and lr ons o an lron. In our hor h blnar or o hs nsor dsrbs also h Lorn s or ang o an lron... On sasal nrpraon o h wav unon In quanu hor h r wav unon rrs gnrall o a opl n-oponns unon as an ln n a opl lbr spa. In non-rlavs hor a wav unon rrs o a on-oponn opl unon. In rlavs hor a wav unon s a vor wh n oponns whh s usuall dsrbd as h ollowng ar: n In h quanu hor a wav unon sl dos no hav a phsal sns and s onsdrd as a ran ahaal odl appng o a ral lnar parl. Th ral unon oband ro h wav unon va s quadraur has hr a phsal sns. In h nonrlavs as h quadraur nsus ro ulplaon o wav unon b s opl onuga unon. In h rlavs as produ o h ran-onugad wav unons s usd. In h quanu hor s apd ha h squar n h sns ndad abov o norald b un wav unon dsrbs h probabl dns o ndng h parl n h gvn pla n gvn. In h as o nonlnar hor s prvous paprs Krakos ab h oponns o h non-norald wav unon ar h proon on h sld ss o oordnas o h vors o nonlnar lral and agn ld o lnar parl phoon lron and ohrs. Nonlnar s dsrbd n hs as wh opl unons. Th bas asurabl parars o h lnar parl ar s nrg onu angular onu and so orh whh ar prssd n lrodnas hrough h squars o h oponns o h vors o h M ld s abov. Thror or obanng suh ral valus s nssar o g rd o h opl and oban a ran ral unon or nubr. al or hs as h produ o h ran-onugad ars s nssar. In h non-norald or hs produ s qual o h nrg dns o parl. Noralaon n rgard o h oal nrg o parl or s ass urns h nrg dns no a raon n h rang ro ro o on whh an b rad as h probabl dns o nd h parl n h gvn pla o spa a h gvn nsan.

10 57 vrhng whh was sad n h papr Krakos a abou h norald and nonnorald rprsnaon o h wav unon o phoon s n parular paragraph.. Norald and non-norald rprsnaon o h wav unon o h phoon s vald or h wav unon o lron. In ohr words h wav unon o lron n h nonnorald or s h proon o h srngh vor o h nonlnar lroagn ld a a ran and poson. Aordngl h squar o h non-norald unon o lron s h nrg dns o lron. In h norald or hs squar an b onsdrd as h dns o h probabl o nd h lron a a ran and poson. For onraon o hs assron w wll addonall an h probabl onnu quaon o lron hor. As s wll known h probabl onnu quaon an b oband ro h Dra s quaon Akhr and rssk 965; h 96; Sh 955; Fr 96: Ppr r dv S pr r 6.. r r s h probabl dns and S pr r s h probabl P pr lu dns. Usng h abov rsuls or non-norald wav unon w an oban: r 8 and g 8 S. Thn an lroagn or o quaon 6..5 an P pr u S pr b prsnd n h ollowng or: u dv S 6.. whh s h or o law o nrg onsrvaon o lroagn ld. 5.. Th lrodnaal anng o ars ho Aordng o Fr Fr 96 " an prov ha all h phsal onsquns o Dra s quaon do no dpnd on h spal ho o Dra s ars In parular s possbl o nrhang h rols o h our ars b unar ransoraon. So hr drns ar onl apparn". Th ar squn agrs wh an lroagn wav ha has -dron. Th quson arss: how o dsrb h wavs ha hav and - drons? Inrodung h as nds ha nda h dron o an lroagn wav w an wr hr groups o ars ah o whh orrsponds o on and onl on wav dron:. L us hoos now h wav unon ors whh gv orr Mawll quaons or and - drons. Takng no aoun 6.. as an nal or o h - dron w wll g ohr ors ro b ans o nds ransposon around h rl s. Fg. 6.. Fg. 6..

11 58 Sn n hs as h Ponng vor has h nus sgn w an assu ha h ransposon us b ounrlokws. L us an hs supposon hkng h Ponng vor s valus: Th ss orrspond o wav unons and o non-ro Ponng vors rspvl. As w an s w oband a orr rsul. W an assu now ha h wav unons wll dsrb lroagn wavs b a lokws nds ransposon. Ths wav unons ov n a posv dron along h drn oordna as. L us prov hs: Th ss orrspond o wav unons and o non-ro Ponng vors rspvl. As w an s on agan w g orr rsuls. Now w wll prov ha h abov ho o ars and wav unons gvs orr lroagn quaon ors. I w us or nsan quaon 6.. and ranspos h nds lokws hn w orrspondngl oban or h posv dron o lroagn wav h ollowng rsuls or -drons: 6.5. Thror w hav oband hr quaon groups ah o whh onans our quaons as hs s nssar or dsrpon o all lroagn wav s drons. In h sa wa analogous rsuls an b oband or all ohr ors o Dra quaon. Obvousl s also possbl usng anonal ransoraons o hoos h Dra ars n suh a wa ha lroagn wav wll hav an dron. L us show hs.

12 59.. M anng o anonal ransoraons o Dras ars and bspnors Th ho o h -s ars s no unqu Akhr and rssk 965; Sh 955; Fok 9. As s wll known hr s a r ransoraon o a knd o S S whr S s a unar ar alld a anonal ransoraon opraor o whh h wav ransoraon o unons orrsponds: S. Ths dos no hang h rsuls o h hor. I w hoos ars as 6.5. hn h unons wll b assoad wh unons aordng o h rlaonshps: 6.5. A unar ar S whh orrsponds o hs ransoraon s qual o: S 6.5. I s no dul o vr ha b ans o hs ransoraon w wll also oban quaons o Mawll hor. In a s as o g h ollowng usng 6.. and 6.5.: whn: Subsung hs unons no h Dra s quaon w wll oban orr Mawll quaons or lroagn wavs n doubl quan. I s possbl o assu ha unons orrspond o an lroagn wav ovng a angl o 5 dgrs o boh oordna as. Thus ollows ro h abov rsul ha vr ho o Dra ars dns onl h dron o an nal lroagn wav. Obvousl hs s a phsal rason wh h phsal onsquns o h Dra s quaon do no dpnd on h spal ho o Dra s ars Fr An lroagn or o h lron hor s Lagrangan Th Lagrangan o h Dra hor an hav h ollowng or Sh 955: p L 6.6.

13 I an lroagn wav s ovng along h -as hn quaon 6.6. an b rwrn as ollows: L 6.6. Transorng ah r n 6.6. o lrodnas or w oban an lroagn or o Lagrangan o Dra s hor: u L dv S 6.6. No ha n a as o varaon produr w us dsngush h opl onuga ld vors * * and. Usng opl lral and "agn" urrns 6.. and 6.. w hav: u L dv S 6.6. I s nrsng ha sn L du o 6.. w an ak h ollowng quaon: u dv S whh has a or o h law o onsrvaon o nrg-onu or Mawll quaon wh urrn An prsson o Lorn or n M rprsnaon Aordng o our hor h or ha s prpndular o h raor o oon o M lds us appar o provd sabl o M parls. A h sa h angnal or b our ho along h -as us absn n hs as sn would provok a angnal alraon o lr harg. As s known ro Jakson 999 an prsson o Lorns or n a vor or s dsrbd b h prsson: F whr s h harg vlo. Inrodung h harg dns d d s possbl o rwr hs prsson n a or: F d whr s h volu oupd b harg. Th prsson n df braks s alld a Lorn or s dns whh as o an par o h d harg lron sl. Sn whr n as o NPT nsd h lron hn w an rwr hs prsson as. I a phoon undrgos h roaon ransoraon around h OZ as w oban: OZ I a phoon undrgos h roaon ransoraon around h OX as w oban:

14 OX h uppr l nd shows h spnnng as OZ or OX. 5 Th Lorn s or dns n lassal lrodnas an b prssd hrough h sral nrg-onu nsor o lroagn ld Tonnla 959; Ivannko and Sokolov 99: 6.7. whr s drnd b h ollowng prssons: S u r nds ; whn and or. Morovr a r. -vor o h spa- has h or Usng 6.7. h or oponns an b wrn as: g grad U 6.7. u dv g 6.7. r rs hr oponns dsrb h Lorn or dns vor and h ourh oponn orrsponds o law o nrg onsrvaon. As w an s w us a sral nrg-onu nsor hn w do no oban h ndd oponns o h or. Auall n hs as h oponns and ar qual o ro bu h oponn no. I appars ha h rgh rsul an b oband w us ansral nsor Thn w hav: 6.7. Or usng h nsor oponns: Usng 6.. and 6..5 w oban oponns o h Lorn s or:

15 No ha w oband hr h doubl nubr o braks sn bspnor 6.. onans wo plan polard wavs o h sa dron whh urn around h drn as. For a lnar phoon all prssons n braks n ar qual o ro aordng o Mawll quaon. I ans ha no ors appar n h lnar M wav quanu. Whn phoon roas around an as whh ar prpndular o -as w wll g addonal urrn rs whr h agnar un ndas ha h angnal urrn s prpndular o lr vor o wav. For h ransord phoon h or oponns ar: OZ For h ransord phoon w hav: OX As w an s h rsuls orrspond o our rprsnaons o dnas o a s-phoon. 8.. M and QM rprsnaon o nraon Lagrangan and alonan o nonlnar hor Th alonan and Lagrangan o NPT onsdrd as a nonlnar hor us onan all possbl nvarans o nonlnar lroagn ld hor. Thus w an assu ha h Lagrangan us b so unon o ld nvarans: I I L L 6.8. whr I I. alonan s ull dnd b h Lagrangan. Thus unon 6.8. s known hn s as o alula h alonan usng orulas. whh wll b now unons o varous powrs o lroagn ld vors: 6.8.

16 5 Apparnl h unons L and us hav s spal or or ah probl. Ths or s unknown bor h probl s soluon. As s known an approa or o unon an b ound on h bass o Shrodngr s or Dra s wav quaons usng h so-alld prurbaon hod. r w assu ha hr s an panson o unon n h Talor MaLaurn powr srs wh unknown panson on. Thn h probl s rdud o alulaon o hs ons. Th soluon s sarhd or ah r o h panson sparal sarng ro h rs. Usuall hs s a probl or a r parl whos soluon s alrad known. Thn usng an quaon wh wo rs rs w nd h on o h sond r. Usng urhr an quaon or h rs hr rs w nd h on or h hrd r o panson. In an ass s possbl o oban h soluon b hs hod wh an dsrabl aura. In as o unon o wo varabls h Talor MaLaurn powr srs n h vn o h pon s: n k n k O k! 6.8. whr In as whn w oban h MaLaurn srs. Obvousl or h os ps o unons I I L h panson onans approal h sa s o rs ha dr onl b onsan ons an o whh an b qual o ro as an apl s pansons o quanu lrodnas Lagrangan or a parl n h prsn o phsal vauu Akhr and rssk. 965; Shwngr 95; Wsskop 96. In gnral h panson wll look lk: 8 L L M whr... L s h par whh s rsponsbl or h nonlnar nraon hr... ar onsans. Th orrspondng alonan wll b dnd as ollows: 8 L L whr h alonan par rsponsbl or h nonlnar nraon s:

17 I s no dul o oban a quanu rprsnaon o h alonan o nonlnar hor. Rplang vors o lroagn wav ld b quanu wav unon w wll oban h srs o h ollowng p: k whr k ar Dra s ars ar ons o panson. As w an s h rs o Lagrangan and alonan srs onan h sa lns suh as and so ohrs. I s possbl o assu ha ah ln o srs has so parular phsal anng. In hs as s possbl o s analog wh h panson o lds o lroagn ons. and also wh h doposon o S- ar on h lns Akhr and rssk 965 ah o whh orrsponds o parulars o nraon o spara parls. 5 Rrns Akhr A.I. and rssk. W Quanu lrodnas. Mosow Inrsn publ. Nw York. h. A. 96 Inrda Quanu Mhans. W. A. nan In. Nw York - Asrda Par II Chap. 7. Dra P.A. M. 98. Pro. Roal So. London A67 8 Fr. 96. Nos on quanu hans. Th Unvrs o Chkago Prss Fok V. 9. Th prnpls o h quanu hans. Lnngrad Par III Chap. I 5 n Russan Ivannko D. and Sokolov A. 99. Th lassal ld hor n Russan. Mosow-Lnngrad Jakson J.D Classal lrodnas rd d. 88 pp. Wl Nw York. Konopnsk.J. 98. lroagn Flds and Rlavs Parls Papr XIV MGraw-ll Nw York. Krakos A.G. a. Nonlnar Thor o lnar Parls:. Th Phoon Thor. hp://vra.org/abs/6.9 Krakos A.G. b. Nonlnar Thor o lnar Parls:. Th Inrda osons and Mass Gnraon Thor. hp://vra.org/abs/9.66 Krakos A.G.. Nonlnar Thor o lnar Parls: 5. Th lron and posron quaons lnar approah Sh L.T Quanu Mhans nd don MGraw-ll ook Copan Jn Nw York. Shwngr J. 95. Phs. Rv Sokolov A. and Ivannko D. 95 Th quanu ld hor n Russan Mosow-Lnngrad. Tonnla M.-A Ls Prnps d la Thor lroagnqu d la Rlav. Masson C. Pars. Wsskop V. 96. Kgl. Dansk Vdnsk. Slskab. Ma.-s. Mdd. 6. Whlr J.A. and Fnan R.P. 95. Rvws o Modrn Phss Whlr J.A Rvws o Modrn Phss 9 6.