The non-linear wave theory, adequate of Standard Model

Transcription

1 Th non-lnar wav hor adqua of Sandard Modl Alandr G. Krakos S. Prsburg Russa. -al: Absra A solar sabl wav solon - s dfnd as a saall onfnd (loald) non-dsrsv and non-sngular soluon of a non-lnar wav hor. For an non-lnar wav hor h solons ar h sa fundanal soluons as h usual wavs ar h fundanal soluons of h lnar quaons. For lnar arl hss a loald and sabl wav s a rf odl for lnar arls onng u n a non-lnar fld hor h ossbl of wha would hav o b a wav ak n a lnar on. As s known h nwr fundanal non-ablan gaug hors ar non-lnar and hav h solon soluons. In h frawork of h quanu fld hor s no dfful o fnd h rlaons bwn solons and lnar arls ha go vr d and ar nrl und fro h lassal on of vw. In h rsn ar s offrd h hor of h wavs wh nw of non-lnar rgn hanks o h ransforaons of h gaug. Th ular solons ar h onsuns of hs hor whh ar dnal wh h obs of Sandard Modl. In arular h hav asss whh aar du o h sonanous brakdown of sr of h nal wavs; h an b onl n wo sas boson and fron; h an hav osv and ngav hargs. Th ahaal dsron of h rsn hor an b nrrd whn h frawork of h Conhagn radon bu as s shown blow non-lnar hor nrraon s uh or onsuv and roduv. A abl of onns Inroduon o h hor () Char. Th hor aoas (5) Char. Th lron hor (7) Char 3. Quanu and lroagn fors of lron hor () Char. Pon and non-on soluons of lron quaons (9) Char 5. Th assv nurno hor (36) Char 6. On h hadrons hor (6) Char 7. On lnar arls sra (53) Bblograh (6) Inroduon o h hor Th Sandard Modl hor (SM) s h gaug hor or h hor of asslss vor flds n whh h arls aqur h asss hanks o h sonanous brakdown of h hsal vauu sr. To undrsand h onnon of h rsn hor wh SM w an on sl al whh wll b dsrbd n dal and gnrald n followng hars of hs book. As on of h als of h raon of assv fld (.. arls) fro h asslss vor fld an srv h as of h of lron-osron ar hooroduon: γ N N () Auall h gaa-quanu (for sl sak hoon) γ s h asslss boson whos fld s onvrd as vor. Th nulus fld N nas ransforaon of hs hoon no wo assv arls - lron and osron whos flds ar snors whh ar onvrd dffrnl n oarson wh h vor fld. Thus an sa ha quaon () dsrbs h ross of h sr brakng of nal asslss vor fld wh h rsul - h raon of assv snor arls. Sulanousl quaon () shows ha h ranson fro boson o snor fld s onnd wh so sf ransforaon. To fnd h las on w onsdr hs ross as h Fnan dagra (fg. ):

2 Fg. W know nohng abou wha hand n h brhon of h ar. W onl s hr h bgnnng and h nd of h ross. Bu wha ransforaon ould ak la wh asslss hoon n a fld of an ao nulus whh has ld o h aaran of wo rlavl oonlss arls boh wh ass and sn whh ar qual o half of nrg and sn of a hoon and also wh uuall oos lr hargs? L s r o answr hs quson. Th ral hoon sruur s unknown o us. Bu a hoon s h quanu of lroagn (M) wav and hrfor h bndng of a raor of M wav n h srong M fld us hav la n hs as. So w an suggs ha undr abov ondons h hoon M wav as so srng an sar o ov along h losd urvlnar raor forng h obs whh as a sngl whol us hav ass and hrfor anno ov wh h lgh sd. Blow w show ha suh obs dndng on h knd of raor wll hav all h hararss and arars of ral lnar arls and hav dsrbd b known rlavs quaons of Dra and Yang-Mlls (h dals s n h nond hars). Aordng o abov suggson n sls as of Fg. a hoon as M wav srng should o b wrld no a rng and hn should b dvdd no ohr wo rngs whh an ov now wh a sd lss han h sd of lgh. Thus a lnar hoon -.. hoon obs h lnar wav quaon - us ba h non-lnar hoon and hn rodus wo non-lnar wavs whh us ob ohr quaon. Obvousl a wrld hoon wll hav h ass ha s qual o nrg of a lnar hoon dvdd on a squar of lgh sd and as s as o show h sn qual o on. Aarnl afr hoon dvson w rv wo arls wh rs ass qual o half of ass of a wrld hoon and wh sn qual o half of h sn of a hoon. L s r o fnd h bass of hor dsron of hs ross (s n dal h har ). L us rurn o Fg. and rall h dsron of h orrsondng arls. As w know hoon s dsrbd b ans of h quand quaon of Mawll. Th lron and osron ar dsrbd b h quaons of Dra. Thus ondonall sakng w s how o h lf h quaon of Mawll of h quand lroagn wavs fls no h srong lroagn nular fld of ao. Thn w s how o h rgh wo quaons of Dra (on for h lron and anohr for h osron) dar. Thus aordng o our sh follows ha h Dra quaons ar h M fld quaons of wo ars of h wrld quand M wav. W an sa also ha n h nrson on of h Fnan dagra lns a roaon ransforaon s rald. W an assu ha hs ransforaon s dnal o h gaug ransforaon baus h las on s (s.g. (Rdr 985)) h ransforaon of fld roaon n h nnr sr sa of arl. Dos hs assuon onrad o h odrn quanu fld hor? A frs w know ha (Gsonr ) h onvnonal vw s ha sn and sn ½ arls blong o dsn rrdubl rrsnaons of h Ponar grou so ha hr should b no onnon bwn h Mawll and Dra quaons dsrbng h dnas of hs arls. ow hs s n agrn wh h fa ha h offrd lnar arl hor s lroagn? I s of ours ru ha h Dra quaon s no quvaln o lassal Mawll quaons (alhough for a long s sablshd ha h an b forall rsnd n h dnal ahaal for). Bu n h rsn hor hr s no a quson abou Mawll hor bu abou h sal non-lnar lroagn wav hor. Th wav funon of h Dra quaon aars hr as a rsul of ransforaon and brakng of hoon vor wav funon o h snor wav funons n full aordan wh quaon (). L s rbr n hs onnon ha h Dra quaon n h ffh ars of h rvous nur was nad h s-vor quaon and hr wav funon was nad h s-vor (s for al (Gonnr )) baus h las on s onnd wh h vor fld b ran ahaal rlaons (s.g. (Rdr 985)). Sondl ss hr s a dfful: h Mawll dndng quaons onan s vors and s quaons (h sour quaons ar ossbl o onsdr as h nal ondons); a h sa

3 3 h snor Dra lron quaon onans wo wav funons and wo quaons and h bsnor - aordngl four. Bu w us rall ha n h rsn hor h quson s abou h lroagn wavs no abou M fld gnrall. As s known M wav dos no onan h longudnal fld oonns and orovr hs ror s Lorn-nvaran. Du o h las fa w an lan hr wh Dra quaon onns alwas wo or four oonns wav funons. Auall hr ar onl wo ossbls: ) on lan olard M wav onans wo wav fld oonns and an gnra on snor onl; ) n h gnral as on rular olard M wav onans four fld oonns and so an gnra wo snors.. on bsnor. Obvousl o adus hs rqurns s nssar h dvson of h wrld hoon o b a sal ross. Bu how an h wrld hoon srng b dvdd so ha wo ansral b harg arls wh sn half aar? Unqu oorun of suh ross s h dvson of h wrld hoon no wo wrld half-rods of hoon aordng o followng sh (fg. ): Fg. r h lnar hoon s wrld n non-lnar on whh for ons ar braks n wo non-lnar half-rods. In ohr words w an sa ha fro on vor arl w rv wo s-vor arls (wo snors) whh aordng o fgur ar full ansr (no hr ha h robl of arl s don ss; s furhr h sond son of nroduon) Blow (s h har ) s shown onssnl fro ahas on of vw how an lroagn quaon of h wrld wav (no h lassal Mawll-Lorn quaons bu so non-lnar quaon of M fld!) s drvd fro h lnar M wav quaon. Thn fro h las h quaons for wo - rardd and advand - wrld half-rod ar ddud whh n h ar for ar h Dra lron and osron quaons. Usng hs rsuls w an lan hr also wh on h Fnan dagra (fg.) h osron rrsns as hough ovs bak n h : sn an lron and osron ar h advand and rgardd wrld wavs rsvl hrs ol dsron dffrs b sgn of has. Durng h wav wrlng hr vors lr agn and Ponng vor ors h rhdron orrsondng o rhdron of h urv un vors noral bnoral and angnal whh ar known n h dffrnal gor as Frn-Srr rhdron. In h rsarh s shown ha h lral urrn of lron (osron) s an addonal ar of h Mawll dslan urrn whh aars du o h ransor of lral wav vor along h urvlnar raor. I also aars ha hs addonal r orrsonds o onnon offns of R (n as of lons) or of Crsoffl (n as of hadrons) whh harar h urns of fld vors a hr oon n urvlnar sa. No also ha n h gnral as of rularl olard wav h agn urrn aars as hs was rdd b Dra (bu n an as h agn harg s qual o ro). Furhr s also shown (s h har 3) ha all quanu-hanal valus and hararss (nludng sasal nrraon of wav funon blnar fors..) n lrodnas of urvlnar M wavs hav sl hsal sns whh howvr don onrad o h quanu fld hor nrraon. Sn lron and osron orrsond o wo wrld half-rod wavs of on hoon follows fro hs fa ha n Unvrs h nubrs of osv and ngav hargs us b alwas ff-ff (hs lads o h harg onsrvaon law and h nural of Unvrs). In h frawork of CWD n h lron quaon h r of h nraon aong arls aars auoaall n h on of brakng of h nural wrld hoon no wo hargd arls. I orrsonds o h rsson of h nal nraon. Ar hr sll h bass o a hs aroah? Ys hr ar and vr srous ons.. In hs as h oal-hanal analog of alon fro whh h quanu hor bgan fnds s subsanaon. Auall h offrd hor s h os of urvlnar wavs whh sulanousl an dsrb h oon of h ar obs.

4 . Th aaran n h Dra lron quaon of Pauls ars whh dsrb h roaon n lassal hans n D sa rvs an lanaon as wll as h aaran of Gll-Mann ars n h Yang-Mlls quaons whh dsrb h roaon n 3D sa. 3. Th nss of a nulus lroagn fld rvs an lanaon: srvs as h du wh h bg rfraon nubr lanng on whh h M wav bns (obvousl hs rqurn s dnal o h rqurn of onsrvaon of ss onu).. Th ford M arls ar sulanousl boh wavs and arls;.. h wav-arl duals s nhrn o h. 5. Sn h wrld hoon has ngr sn ( s a boson) bu h wrld s-hoons hav sn half ( s a fron) w auoaall rv an lanaon of dvson of all lnar arls no bosons and frons. 6. I s as o s ha h fg. rfls h ross of sonanous sr brakdown of an nal hoon and aaran of ass of lnar arls whh hav la n rsn of a nulus fld as so suor and aals of h raon (lang hr h rol of ggs boson). 7. In h hor of h sa shral lron of h Lorn lassal hor hr ar no h lroagn fors aabl o onsran h rulson of lron ars fro ah ohr and s nssar o nr Ponars fors of non lroagn orgn. I s as o s ha hr owng o rsn of a urrn hr s h agn ar of full Lorn for drd agans lrosa fors of rulson and ounrbalanng h. Thus suh lron dos no dand h nroduon of ranous fors of an unknown orgn and s sabl. Abou so ohr onsquns whh follow fro h suggson abou hoon wrlng w wll brfl alk blow. In h rsarh s also shown (s h har 5) ha for h nal lnar rular olard hoon whh wrls a lan s dvson an rodu h nural assv lons of h sa as nurno and annurno whh ar also dsrbd b Dra quaons. Thanks o h rular olaraon of nal hoon wo wrld half-rods of suh hoons hav h nnr hl and dffr n h hral. In hs as nurno as wrld hlods rrsns Mobuss sr: s fld vor a nd of on ol (36 o ) has h oos dron n rlaon o h nal vor sa and onl a wo ols (7 o ) os bak o h sarng oson. Ths ror of h M-lon vor orrsonds o h sa ror of wav funon of Dra lon hor. I s nrsng ha aordng o R. Fnan (Fnan 987) h arl whh has h Mobus sr oolog us ob h Paul luson rnl. Thus n h frawork of h rsn hor h wrld s-hoon arls us ob o Fr-Dra sas. Suh lon has ass bu dosn hav lral harg. Rall h ass of a arl s dfnd b ngral fro dns of nrg whh s rooronal o h sond dgr of fld srngh. In hs as h ngral s alwas dsn fro ro f h fld srngh s dsn fro ro. A h sa h arl harg s dfnd b ngral fro dns of a urrn whh s rooronal o h frs dgr of fld srngh. Obvousl hr s a han whn h sub-ngral rsson s no qual o ro bu h ngral s qual o ro. I s as o hk ha w wll rv suh rsul hr sn h sub-ngral funon hangs undr h haronous law. Furhr (s h har 6) n rsarh s dsrbd h aaran of saal arls as h suroson of h wrld s-hoons. Th quaons of suh arls ond wh Yang-Mlls quaons for hadrons so h D suroson of wo wrld s-hoon srngs gnras h sons and 3D suroson of hr wrld s-hoon srngs lads o aaran of barons.g. roon. In hs as a Frn-Srr rhdron ovs n hr-dnsonal sa urnng and wsng onnuousl. Thrfor h urrn of ah loo of abov obs wll no or b onsan as ook la for a rular raor and wll hang s valu. n h harg of ah loo wll b lss han h harg of lron. If o dnf h sara lns of suroson (.. h knos) wh quarks w an rv an lanaon of h rnal fas nlabl n fra of SM. Frs hr s a lar rlaonsh bwn quarks and lons. Sondl bos undrsandabl h onfnn of quarks and gluons. Thrdl h dsnon of lnar arls no hr grous - lons sons and barons rvs h sl lanaon. Fourhl h fraonal of hargs of quarks rvs an lanaon oo as an ohrs. In h rsarh s also shown h ossbl of ohr or ol arl foraon (har 7) as wll as h arl arars alulaon.

5 5 Abou arl s and hddn varabls n quanu hor Whn h frawork of h rsn hor h lron s h lroagn fld of a sal onfguraon onnrad n sall volu wh harars s of Coon wavlngh. Dos h rsn of h lron s n frawork of non-lnar hor onrad o s absn n h Dra hor? No sn n boh ass hs s h sa quaon - h Dra lron quaon. Bu how h sa quaon an onan and sulanousl no onan a s? r w aroah o vr nrsng rsul of h rsn hor whh solvs nurous dsus doubs and qusons onnung an ars: ar hr n h quanu hans hddn arars; s ossbl o nr h no dsrong h quanu hans. I aars hr ha von Nuann was arall rgh who has rovd ha s ossbl h hddn arars o nr no h gvn sh of QM bu also d Brogl D. Boh and ohrs ar rgh whh hav shown ha h Nuanns roof s ld b frawork of sng nrraon. Th non-lnar wav hor shows ha nohng or us b nrd no h sng quaons of Sandard Modl baus vrhng wha s nssar hr alrad ss. In h Dra lron quaon alrad hr s a s of lron bu s hddn no b h faurs of h quanu hor bu b h for n whh w rrsn and nrr. L s lan hs san. Th urrn r of h non-lnar lron quaon s onnd wh aralll ransor of a fld vors along a urvlnar raor. I s dfnd b h urvaur of a raor (or n ohr rrsnaon h R offn of roaons) whh ar rssd b Coon lron wavlngh: λ C! (whr s h lron ass and s h lgh sd). As follows fro h rsarh for h urvaur of a raor w hav r r! whh dfns n h sa h fr ass r of Dra lron quaon! whh as w s dosn onn an s. (No ha h olarsaon of hsal vauu odfs or rnorals h lron bar s b followng wa: r r r whr r s h lassal radus! s lroagn onsan whh n h frawork of non-lnar wav hor s onnd wh h olarsaon of hsal vauu). Thus unl w do no know ha h Dra quaon h lron radus onans rall s h hddn arar. Bu on h ohr hand s hddn onl baus w us h anond for of h Dra quaon. So h sng of radus dos no onrad o h quanu hans n an wa. Thrfor as shown n h har an lron an b dsrbd boh as on (wh us of rnoralsaon) and non-on arl whou volaon an hor. In hs wa w an lan h ohr hddn arars of lron - for al h arars of so-alld Zrbwgung - rblng or or orrl osllaor oon of rlavs lron found ou b. Shrodngr. I s no dfful o undrsand ha h orgn of Zrbwgung s h roaon of lron flds (sn h roaon s hr h su of wo rndular osllaons). Thus an sa ha h rsn hor nluds quanu hans as h foral lnar ahaal sruur and ranl dos no anl an of s rsuls bu onl lans h and lds addonal rsuls. I rans onl o no ha hs work s dvod anl o dsron of h sngl fr arls. I s roosd ha h sss of h dnal arls and h rlad robls would b onsdrd n ohr book. Char. Th hor aoas Th aoa bass of h rsn hor nluds 6 osulas h frs of whh ar h osulas of h odrn fld hor. Th osulas 5 and 6 rss h sf of h rsn hor and do no onrad o odrn hss.. A osula of fundanal of an lroagn fld: h slf-onssn Mawll- Lorn roso quaons ar h ndndn fundanal fld quaons. Th Mawll-Lorn quaons ar followng four dffrnal (or quvaln ngral) quaons for an lroagn du (Jakson 999; Tonnla 959):

6 6 π ro ( ) ro dv π ρ ρ (.) (.) ( ) (.3) dv (.) whr F { } s h lroagn fld vors ρ s h harg dns; s h urrn dns; s h sd of lgh. Valus and ρ (or n -vor for { } whr 3 funonals) of srngh and ( ρ ) n hs quaons should b onsdrd as funons (or rsl of h sa flds whh hs hargs and urrns subsanall dfn: ). Th ar of hargs and urrns an b ausd b rnal n rlaon o h gvn ss rasons. Suh hargs and urrns whh do no dndn on and of an nal sour sos ar rfrrd o as rnal and ar dsgnad as and. (W shall rnd ha gnrall h flds and urrns of hs ρ quaons ar nrdndn and h quaons ar hrof non-lnar). As s known h Mawll-Lorn hor rd h sn of lroagn wavs. In rlaon o hs h followng osula has la.. Plank s-nsns osula of quanaon of lroagn wavs: h lroagn wavs ar h suroson of h lnar wav flds nad hoons havng h ran nrg onu and ro rs ass. In hs osula ar sulanousl akn no aoun boh quanaon of lroagn wavs and h blongng of hoons o bosons. 3. A osula of duals of hoons: hoons s as ral ndndn obs whh hav a) h wav rors dsrbd b Mawll-Lorn quaons (.) - (.) and b h wav quaon followng fro h: F (.5) b) quanu rors whh ar no ddud fro Mawll-Lorn hor bu also no onradd o. Th nural hararss of hoons ar drnd b a blow osula of Plank.. Plank osula: Connon bwn nrg frqun and wavlngh of a hoon s s b h followng forulas: h ν (.6) whr ν λ ε! h h λ ε (.7) ε ar h nrg lnar frqun rular frqun wavlngh onu and sd of a hoon and h! ar h usual and bar Planks onsans aordngl. Th followng osula s nhr rovd nor dsrovd b rns bu dosn onrad o h horal dsron of hoons: 5. Th osula of M srng. Sn hs osula s nral n our hor dands a srous subsanaon of s onssn o odrn rsuls. As s known n frawork of QD (Ahr and Brssk 969) o oban h hoon wav funon h sond ordr wav quaons for M fld vors and (.5) ar usd. Faorng h wav quaon o h quaons for rardd and advand wavs w rv wo quaons of h frs dgr rgardng h funon f whh adqua o a wav vor k and s so k

7 7 gnralaon of h M fld vors. Th quaon for hs funon s quvaln o h Mawll- Lorn quaons. Bu h funon f k an b nrrd as wav funon of a hoon onl n h onu sa. I dos no allow o dsrb an nraon of a hoon n h loal on of oordna sa sn for hs a h wav funon n h oordna rrsnaon s rqurd. Unforunal h a o nr h hoon funon n h oordna rrsnaon has srk on an nsurabl dfful. Aordng o analss of Landau L.D. and Prls R. (Landau and Prls 93) and lar of Cook R.J. (Cook 98a;98b) and Inagak T. (Inagak 99) h hoon wav funon s nonloal. Auall havng ad h nvrs Four ransforaon of abov f k funon kr f k d k f 3 3 ( r ) s ossbl o dfn f ( r ) as h hoon wav funon n ( π ) oordna rrsnaon. Bu h f ( r ) funon s no dfnd b h valu of h fld ( r ) n h sa on; dnds on h fld dsrbuon n so ara whh ss ar of h ordr of h hoon wavlngh. Ths ans ha h loalaon of a hoon n a sallr ara s ossbl and hn h valu f ( r ) wll no hav h sns of robabl dns o fnd a hoon n h gvn on of oordna sa. Th lnar ob whh on h on hand obs h wav quaon and on h ohr hand has so s s rfrrd o as a srng. Thus whn h frawork of CWD s adssbl o dsrb a hoon as an ln of lroagn wav - an lroagn srng (no forgng of ours ha hs suoson anno hav an rlaonsh o h ral sruur of a hoon). Ths allows us o forula h followng osula: Whn h frawork of h rsn hor h fundanal arl of an M fld - h hoon - an b dsrbd as a rlavs M srng of on wavlngh s whh orrsonds o s nrg aordng o Planks forula. Th an roof of vald of hs osula s h oorun o onsru on s bass h hor whh onds oll wh h sng quanu fld hor. 6. A osula of foraon of assv arls: Whn h frawork of h rsn hor undr h ran rnal ondons h M-srng an sar o ov along h losd urvlnar raor forng h lnar arls. As s known h bndng of a raor of an M wav n h srong M fld follows alrad fro h Mawll-Lorn hor. Thus srl sakng h oorun of an M wav roagaon along a urvlnar raor dos no dand a sal osula. A h sa s obvous ha du o h quanu naur of a hoon (M-srng) h ford arls should ossss a las a rs ass and h angular onu (sn). Morovr h dal analss shows ha suh lnar arls an hav lr harg hl and all ohr hararss and arars of ral lnar arls. I would b vr dfful o fnd sulanousl h adqua dsron of all M lnar arls. Th os logal wa of onsruon of h gnral hor - o bgn wh h os sl and good sudd - horall and rnall arl: h lron. Chars - of h book ar dvod o hs. Thn w shall r o gnral h rvd rsuls for h dsron of or o and ol arls. Furhr for brv hs hor s rfrrd o as CWD - Curvlnar Wavs lrodnas. Char. Th lron hor.. Inroduon Th ossbl of h foral rrsnaons of h Shrudngr and h Dra lron quaons n h for of h lnar Mawll quaons was nond n svral arls and books (Arhbald 955; Akhr and Brssk 965; Koga 975; Caolaoro 98; Rodrgus ). Aordng o osula 6 an lroagn wav whh ov along h losd urvlnar raor us ra h sabl obs ha orrsond o lnar arls of dffrn knd. L s now ransla hs suoson no h ahas languag and show ha n h sls as h ar for of h quaons of suh urvlnar wavs ahaall full onds wh

8 8 quanu quaons of vor and snor (s-vor) arls and gvs an nrsng onsquns whh suln h quanu fld hor rsuls... Lnar M wav quaon n h ar for W dfn as lnar wav h soluon of h lnar wav quaon. L us onsdr h lan-olard lnar lroagn (M) wav ovng for al on - as h lr and agn flds of whh an b wrn n h ol for as: o o ( ± k) (.) ( ± k) Th lroagn wav of an dron has wo lan olaraons and onans onl four fld vors; for al n h as of -dron w hav: Φ ( ) { } (.) and for all ransforaons. No n hs onnon ha h Dra bsnor has also four oonns. Th M wav quaon has h followng known vw (Jakson 999): whr Φ () Φ( ) (.3) s an of h abov lroagn wav fld vors (.). In ohr words hs quaon rrsns four quaons: on for ah vors of h lroagn fld. W an also wr hs quaon n h followng oraor for:!! ( ) Φ( ) ε (.) whr ε ar h oraors of h nrg and onu orrsondngl and Φ s so ar whh onsss four oonns of Φ (). Takng no aoun ha ( ) ( ) σ oε ε whr ; σ σ ; σ β ar Dras ars and σ σ ar Paul ars h σ σ quaon (.) an also b rrsnd n h ar for of h Kln-Gordon-lk quaon whou ass: ( ) ( ε ) Φ o (.5) Takng also no aoun ha n as of hoon w hav ε and k fro (.5)!! ε as s has la for a hoon. Thrfor w an onsdr h Φ - usng (.) w oban wav funon of h quaon (.5) boh as M wav and as a hoon. Faorng (.5) and ullng fro lf on h ran-onuga funon Φ ( ) ( ε ε ) Φ (.6) o o Th quaon (.6) a b dsngrad on wo Dra-lk quaons whou ass: ( Φ ) oε ( ε ) Φ o (.7 ) (.7 ) Φ w g: I s no dfful o show ha onl n h as whn w hoos h Φ -ar n h followng for:

9 9 Φ ( ) Φ (.8) h quaons (.7) ar h rgh Mawll quaons of h lroagn wavs: rardd and advand. Auall usng (.8) and ung n (.7) w oban: (.9 ) (.9 ) For wavs of an ohr dron h sa rsuls an b oband b h l ransoson of h nds and b h anonal ransforaon of ars and wav funons (s har 3). W wll furhr ondonall na ah of (.7) quaons h lnar s-hoon quaons rbrng ha was oband b dvson of on wav quaon of a hoon no wo quaons of h lroagn wavs: rardd and advand. 3.. Twrl ransforaon of lroagn wav Th ransforaon of h lnar wav o h urvlnar (brfl wrl ransforaon ) an b ondonall rrsnd as followng rsson: Φ Ψ R (3.) whr R s h oraor of raor ransforaon of M wav fro lnar o urvlnar h Φ s h wav funon dfnd b ar (.8) whh sasfs h quaons (.5) and (.7) and Ψ s so wav funon: Ψ (3.) whh aars afr non-lnar ransforaon; hr { } ) ( Ψ ar lroagn fld vors afr wrl ransforaon. As s known h dsron of vor ranson fro lnar o urvlnar raor s full dsrbd b dffrnal gor (snhar 96). No also ha ahaall hs ranson s quvaln o h vor ranson fro fla sa o h urvlnar sa whh s dsrbd b Rann gor. In onnon o hs l us rnd ha h Paul ars as wll as h hoon ars ar h sa roaon oraors -D and 3-D orrsondngl (Rdr 987). 3.. Th wrl ransforaon dsron n dffrnal gor L h lan-olard wav whh has h fld vors ) ( b wrld wh so radus r n h lan ) ( Y O X of a fd o-ordna ss ) ( O Z Y X so ha s aralll o h lan ) ( Y O X and s rndular o (fgs ):

10 Fg. Aordng o Mawll (Jakson 999) h dslan urrn n h quaon (.9) s dfnd b h rsson: ds π. (3.3) Th abov lral fld vor whh ovs along h urvlnar raor (l hav dron fro h nr) an b wrn n h for: n (3.) whr and n s h noral un-vor of h urv (havng dron o h nr). Th drvav of an b rrsnd as: n n (3.5) r h frs r has h sa dron as. Th sn of h sond r shows ha a h wrlng of h wav h addonal dslan urrn aars. I s no dfful o show ha has dron angnal o h rng: whr τ s h angnal un-vor n υ κτ (3.6) υ s h lroagn wav vlo κ s h urvaur of h raor and r s h urvaur radus. Thus h dslan urrn of h lan wav ovng along h rng an b wrn n h for: whr hoon nrg τ! τ π n τ π π ds υ r κ (3.7) w wll na h urvaur angular vlo s so ass orrsondng o h nrg ε r ε s n n π ar h noral and angn oonns of h urrn of h wrld lroagn wav orrsondngl. Thus: τ (3.8) ds n Th urrns n and τ ar alwas uuall rndular so ha w an wr h n h ol for: ds n τ (3.8 ) and

11 whr τ π. Thus h angn urrn aaran auss h aaran of agnar un. Fro h abov w an also assu ha h aaran of agnar un n h quanu hans s d wh h angn urrn aaran. 3.. Th wrl ransforaon dsron n Ran gor W an onsdr ondonall h Mawll wav quaons (.7) wh wav funon (.8) as Dra quaon whou ass. Th gnralaon of h Dra quaon on h urvlnar (Rann) gor s onnd wh h aralll ransor of h snor n h urvlnar sa (Fok 99ab; Fok and Ivannko 99; Van dr Wardn 99; Shrodngr 93; Infld und Van dr Wardn 933; Gonnr ). For h gnralaon of h Dra (whou ass) quaon on h Rann gor s nough o rla h usual drvav / (whr ar h o-ordnas n h -sa) wh h ovaran drvav: D (3.9) Γ whr 3 ar h sung nds and Γ s h analogu of Chrsoffls sbols n h as of h snor hor alld R sbols (or onnon offns). In h hor shown ha Γ whr and ar h ral valus. I s no dfful o s ha h angn urrn τ orrsonds o h R onnon offns (sbols) Γ. Whn a snor ovs along h sragh ln all h sbols Γ and w hav a usual drvav. Bu f a snor ovs along h urvlnar raor no all h Γ ar qual o ro and a sulnar r aars. Tall h las on s no h drvav bu s qual o h rodu of h snor slf wh so offn Γ whh s nrn n snor. Sn aordng o h gnral hor (Sokolov and Ivannko 95) h nrn n snor Γ has h for and h dnson of h nrg-onu -vor s logal o dnf Γ wh -vor of nrg-onu of h hoon lroagn fld: whr ε and { ε } Γ (3.) s h hoon nrg and onu (no h oraors). In ohr words w hav: ε Γ (3.) Takng no aoun ha aordng o nrg onsrvaon law no dfful o s ha h sulnar r onans a wrld wav ass... Th quaons of wrld lroagn wav ε ± β.. Kln-Gordon-lk quaon of wrlng hoon As s follows fro rvous sons du o h urvlnar oon of h lroagn wav so addonal rs K β orrsondng o h angn oonns of h dslan urrn wll aar n h quaon (.6) so ha fro (.6) w hav: ( ) ( ε K ε K) Ψ (.) o o Thus n h as of h urvlnar oon of h lroagn flds of hoon nsad of h quaon (.6) w oban h Kln-Gordon-lk quaon wh ass (Shff 955): ( ) Ψ ε (.) As w s h Ψ -funon whh aars afr lroagn wav wrlng and sasfs h quaon (.) s no dnal o h Φ -funon bfor wrlng. Th Φ -funon s h lassal s

12 lnar lroagn wav fld whh sasfs h wav quaon (.7); n h sa h Ψ - funon s h non-lassal urvlnar lroagn wav fld whh sasfs h Kln-Gordonlk quaon (.). As s known n quanu hss h Kln-Gordon quaon s onsdrd as h salar fld quaon. Bu obvousl h Kln-Gordon-lk quaon (.) whos wav funon s - ar wh lroagn fld oonns anno hav h sns of h salar fld quaon. Auall l us anal h obs whh hs quaon dsrbs. Fro h Mawll quaons follows ha ah of h oonns of vors of an lroagn fld subs o h sa for of h salar wav quaons. In h as of h lnar wav all fld oonns ar ndndn. r b sud of on of h vor s oonns w an onsdr h vor fld as salar. Bu afr h wrl ransforaon.. n h frawork CWD whn a angnal urrn aars w anno rod o h salar hor sn h oonns of a vor π as follows fro h ondon (Mawll law) o (whr s h un vor of wav vlo) ar no ndndn funons. Thrfor alhough h Kln - Gordon quaon for salar wav funon dsrbs a assv arl wh sn ro (snlss boson) h quaons (.) onrnng lroagn wav funons (3.) whh aars afr urvlnar ransforaon rrsns h quaon of h vor arl wh rs ass and wh sn on. In hs sns hs quaon la h rol of h Proa quaon. To hs dfful wh h na w wll na as h wrld hoon quaon... Th quaon of h wrld s-hoon Usng h dsngraon (.) w an oban fro wrld hoon quaon (.) h quaons of h wrld lroagn wav advand and rardd: whr { } [( ) oε β ] [( ) ] oε β (.3 ) (.3 ) s so M wav funon whh w wll b na furhr h wrld s-hoon quaons. And h abov ranson fro (.) o (.3) w an ondonall na a sr brakng ransforaon. Now w wll anals h arulars of h quaons (.3). I s no dfful o s ha h lass ar slar o Dra lron quaons. Bu no ha nsad of lron ass quaons (.3) onan h wrld hoon ass. Th quson arss wha of M arls h quaons (.3) dsrb? In h as of lron-osron ar roduon us b so ha fro (.3) w hav: [( ) oε β ] [( ) ] oε β (. ) (. ) Obvousl afr h wrld hoon brakng.. afr h harglss wrld hoon s dvdd no wo hargd s-hoon h lus and nus hargd arls aqur h lr flds and ah arl bgns o ov n h fld of anohr. In ordr o bo h ndndn (.. fr) arls h us b drawn awa on fro h ohr on gra dsan Thrfor h quaons whh ars afr h wrld hoon dvson anno b h fr osv and ngav (lron and osron) arl quaons bu h arl quaons wh h rnal fld. In hs as durng h hargd arls ov awa on fro anohr h nrg whh orrsond o h nrg of h lr fld raon us b ndd. In fa bng h arls obnd h ss dosn hav an fld (fg. 3). A vr sall dsan h ra h dol fld. And a a dsan uh or han h arl radus h lus and nus arls aqur h full lr flds. As s known (Jakson 999) h onal V P of boh lus and nus hargs n h on P s dfnd as:

13 3 π r r d osθ V P (.5) whr ± ar h dol hargs d s h dsan bwn h hargs and θ s h angl bwn as and radus-vor of lus arl. For d w hav V. For d w oban as h l as h Coulob onal for ah fr arls: V P d l π r (.6) Thus durng h brakng ross h arl hargs aar. For h arl rovd o nfn h work agans h arav fors ndd o b fulflld: ε rl V P dυ π dυ r (.7) Obvousl h rnal arls fld dfns hs work so ha h rlas nrg s h fld roduon nrg and n h sa s h annhlaon nrg. Thrfor du o nrg onsrvaon law hs nrg valu for ah arl us b qual ε rl. So h quaons (.3) w an wr n h followng for: [( ) ε β β ] [( ) oε β β ] (.8 ) (.8 ) Usng h lnar quaon for dsron of h nrg onsrvaon law w an wr: β ε ϕ A ± (.9) whr ans rnal. Pung (.9) n (.8) w oban h Dra quaon wh rnal fld: [ ( ) ( ) ε ε # β ] # (.) whh a d gv h Dra fr lus and nus arl quaons: [( ) oε β ] [( ) ] oε β (. ) (. ) Fro abov so nrsng onsquns follow:. bfor brakng h wrld hoon s no an absolul nural arl bu a dol; hrfor us hav h dol on.. h forula (.9) shows ha n CWD h ass s no quvaln o h nrg bu o h - vor of h nrg-onu; fro hs follows ha n CWD h nrg has h kn orgn. 3. n frawork of CWD for fr r of arl quaon h followng rsson ak la: β ε ϕ A n ± n n n n (.) whr n ans nrnal. In ohr words h valus ( ε n ) dsrb h nnr fld and h valus ( ε ) h rnal fld of lron-osron arls. Whn w onsdr h lron arl fro gra dsan h fld ( ε n n ) works as h ass and h r ( ε ) dsrbs h rnal lroagn fld (and w hav lnar Dra quaons of arls). Insd h lron h r ε ) s ndd for h dald dsron of h nnr fld of arl whh ( n n harars h arl ars nraon (as s shown blow hs r arrs o non-lnar quaon of arl). Usng (3.) w oban lroagn for of h quaons (.):

14 (.3 ) (.3 ) whr r r π π π π (.) ar h agnar urrns n whh! and r! s h radus of wrlng of M wav (and s also h half of Coon wavlngh of h lron). As w s h quaons (.3 ) and (.3 ) ar Mawll quaons wh agnar lr and agn urrns. As s known h sn of h agn urrn dosn onrad o h quanu hor (s h Dra hor of h agn onool (Dra 93)). In our as of h lan olard wav (s fgs. and 3) h agn urrns ar qual o ro. Thus h quaons (.) ar Mawll quaons wh agnar angnal urrns and sulanousl h ar h Dra quaon of lron. 5.. Analss of h fr lron quaon soluon fro M on of vw In aordan wh h abov rsuls h lroagn for of h soluon of h Dra fr lron quaon us b a wrld lroagn wav. If hs suoson s auall orr for h -dron hoon wo soluons us s: ) for h wav wrld around h OZ -as o (5.) and ) for h wav wrld around h OX -as 3 o (5.) Th - funons (5.) and (5.) as h soluons of h quaons (.) us hav h sa rssons as h Dra lron hor soluons (Shff 955). L us anal h Dra lron hor soluons fro CWD on of vw. I s known (Shff 955) ha h soluon of h Dra fr lron quaon (.) has h for of h lan wav: ) ( r B! ε (5.3)

15 5 φ whr 3 ; B b ; h aluds b ar h nubrs and φ s h nal wav has. Th funons (5.3) ar h gnfunons of h nrg-onu oraors whr ε and ar h nrg-onu gnvalus. r for ah h nrg ε has hr osv or ngav valus aordng o h nrg-onu onsrvaon law quaon ε ±. For ε w hav wo lnar-ndndn s of four orhogonal noralng aluds: ) ) B ( ) B B3 B ε ε B ( ) B B3 B ε ε and aordngl for ε : 3) ) B B ± ( ) B B3 B ε ε ( ) B B3 B ε ε ah of hs four soluons (Shff 955) an b norald b s ullaon b noralaon faor: κ ( ε ) whh gvs Ls dsuss hs rsuls. ) Th sng of wo lnar ndndn soluons orrsonds wh wo ndndn ornaons of h lroagn wav vors and gvs h unqu log lanaon for hs fa. ) Sn () w hav and for h fld vors w oban: fro (.) and (.5) for osv nrg B b b 3 b b (5.) (5.5) (5.6) (5.7) () φ ( ) φ (5.8) B and fro (.6) and (.7) for ngav nrg: B b b b b 3 () φ ( ) φ (5.9) B whh al orrsond o (5.) and (5.). 3) Calula h orrlaons bwn h oonns of h fld vors. Pung φ ε and ε w oban orrsondngl: () ( ) B B (5.) π for

16 6 () ( ) B B (5.) and ar Obvousl h agnar un n hs soluons ndas ha h fld vors uuall orhogonal. Also w s ha h lr fld alud s wo s lss han h agn fld alud. Ths fa shows ha h lroagn fld s valus whh orrsond o h Dra quaon soluon. (I an b ar dffrn onrar o flds of h lnar wav of h Mawll hor whr shown ha hs rsul rovds h lron sabl). ) I s as o show ha h lroagn for of h soluon of h Dra quaon s h sandng wav. Rall n as of h rl-wrld wav w hav r and hrfor r ; hn nsad (.3) w oban: b ε! (5.) ϕ 5) Aordng wh h ulr forula osϕ snϕ h soluon of h Dra quaon (5.) dsrbs a rl as orrsonds o our hor. 6) L s alula h noralaon faor subsung: κ 5 ε : (5.3) and oar wh noralaon faor whh s rvd fro h lroagn rrsnaon of h hor. In vw of ha h lr fld s w lss of agn fld h nrg dns of wrld s-hoon wll b qual: 5 ( ) s h s h s h s h s h s h W 8π 8π 8π (5.) Usng non-norald rsson for h wav funon: ( kr ) ( kr ) B B B (5.5) (whr B s so onsan gnrall dnsonal) and usng also h ran-onuga funon: ( kr ) ( kr) BB B (5.6) for fld nrg w wll rv h followng rsson: 5 W B 8π 8π (5.7) whh rsl orrsonds o h quanu hor rsul. 6.. Parulars of wav funon of lron quaon As s known h flds of a hoon ar vors ransforng as lns of grou (O3). Th snor flds of h Dra quaon ar ransford as lns of grou (SU). As s shown b L.. Rdr (Rdr 987) and ohrs wo snor ransforaons orrsond o on ransforaon of a vor. For hs rason h snors ar also nad s-vors or nsors of half rank (Gonnr ; Sokolov & Ivannko 95).

17 7 Fro abov followng ha h wrlng and brakng of h wrld hoon wavs orrsonds o ranson fro usual lnar Mawll quaon o h M urvlnar wav quaon wh an agnar angnal urrns (.. o h M Dra quaon). Obvousl h ransforaon rors of lroagn flds a hs ranson hang. Jus as h wav funons of h Dra quaon (.. snors) sub o ransforaons of grou (SU) h s-hoon flds us sub o h sa ransforaons. L us r now o sf h dffrns bwn lroagn flds { } h Ψ -funon of wrld hoon and lroagn flds { } of of -funon of wrld s-hoon. Takng no aoun ha w hav h sa ahaal quaons boh for h CWD Dra quaon and h Dra lron quaon w an affr ha hs ransforaon faurs ond wh h sa faurs of h snor (Rdr 987; Gofrd & Wsskof 98). Th snor nvaran ransforaon has h for: U (6.) whr h oraor of ransforaon s nrd as follows: U ( nθ ) os θ n σ sn θ (6.) whr n s h un vor of an as θ s a roaon angl around hs as and σ ( σ σ σ ) s h sn vor. Th roaon ar (6.) osssss a rarkabl ror. If h roaon ours on h angl θ π around an as (hrfor ours h rurnng o h nal ss of rfrn) w fnd ha U nsad of U as was ossbl o. In ohr words h sa vor of ss wh sn half n usual hr-dnsonal sa has wo-valudnss and asss o slf onl afr urn o h angl π. Ths rsul an b land onl f w suos ha h M lron s h wrld half-rod of a wrld hoon arl and hrfor nds o b road w o rurn o h nal sa. In ohr words h wrld s-hoon s h wrld half-rod of h hoon. Takng no aoun h abov rsuls h soluon of h M lron quaon (.. Dra quaon n h M for) w an na lroagn snor. In ohr words h lroagn snor s h s-rod of wrlng M wav. Thus h ransforaon of h lnar lroagn wav no urvlnar wav and s sr brakng rodus h lroagn snors. 7.. lroagn Non-lnar lron quaon and s Lagrangan 7.. Th M nonlnar lron quaon Obvousl h urvlnar of hoon or s-hoon oon us b dsrbd b non-lnar quaon. Fro hs follows ha h CWD quaon of M lron us b h non-lnar fld quaon. L us fnd lroagn and quanu fors. Th sabl of wrld s-hoon s ossbl onl b h s-hoon ar s slf-aon. Usng (.) fro (.) w wll oban h followng non-lnar quaon: [ ( ) ( ε ε n n )] (7.) whr h nnr nrg and onu an b rssd usng h nnr nrg dns U and onu dns g (or Ponng vor S ) of M wav: ε n τ U dτ π 8 τ ( ) dτ τ τ n g dτ S dτ π τ [ ] (7.) dτ (7.3) ung h ur l τ o b varabl. Subsung of h rsson (7.) and (7.3) o h M lron quaon w oban h non-lnar ngral-dffrnal quaon whh s as w suos h sarhd for of h non-lnar quaon whh dsrbs h M-lron n boh lroagn and onurrn quanu fors.

18 8 To show ha h quaon (7.) an auall rnds o h rol of h quaon of non-lnar lrodnas of h lron M arl w fnd s aroa quanu for. Usng M for of - funon s as o rov ha h quanu fors of U and S ar: S 8π g 8π U (7.) (7.5) Takng no aoun ha h fr lron Dra quaon soluon s h lan wav: [ ( k) ] (7.6) w an wr (7.) and (7.3) n h n aroa for: τ ε U τ 8π τ g τ S τ 8π (7.7) (7.8) τ s h volu whh onan h an ar of h wrld s-hoon nrg. Thn h whr aroa for of h quaon (7.3) wll b followng: ( ) τ 8π (7.9) I s no dfful o s ha h quaon (7.9) s h non-lnar quaon of h sa as nonlnar snbrg quaon(snbrg 966; Par ranslaon ollon 959): [ γ ( γ ) γ γ ( γ γ )] γ l 5 5 (7.) f nsad of -s Dra ars w wll us γ -s ars (hr l s so osv onsan). Th non-lnar quaon (7.) was osulad and nvsgad b snbrg. al. as h unar quanu fld hor quaon. Conrar o h las on h quaon (7.9) s oband b logal and orr wa and h slf-aon onsan l aard n (7.9) auoaall. As s known n h frawork of hs non-lnar unar fld hor so subsanal ahvns wr ad. 7.. Th Lagrangan of h nonlnar lron hor Th Lagrangan of h Dra lron hor of lnar n quanu for s (Shff 955): L ( ε β ) D I s no dfful o fnd s lroagn for: U L D dv S ( ) 8π (7.) (7.) (No ha n h as of h varaon rodur w us dsngush h ol onuga fld vors * * and ). Th Lagrangan of non-lnar hor s no dfful o oban fro h Lagrangan (7.) usng h hod b whh w found h nonlnar quaon. B subsung (5.) w oban: L N ( ε ) ( ε ) (7.3) n W suos ha h rsson (7.3) rrsns h oon for of h Lagrangan of h nonlnar lron hor. In ordr o oar (7.3) wh h known rsuls of lassal and quanu hss l us fnd h aroa lroagn and quanu fors of hs Lagrangan. Usng (7.7) and (7.8) w an rrsn (7.) n h followng quanu for: ( ) ( ) ( ) ( ) τ L N! dv (7.) 8π n

19 9 To oban h M for of (7.) w nall ass on o norald -funon usng h rsson 8π L N L. Thn w ransfor (7.3) usng quaons (7.) and (7.5) and oban fro (7.) h followng aroa lroagn for: L N! N U τ dv g ( U g ) (7.5) I s no dfful o ransfor h sond r usng h known dn of lrodnas: ( 8 ) ( U g ) ( ) ( ) ( ) π ( ) (7.6) Thus akng no aoun ha L D and usng (7.) and (7.6) w oban fro (7.5) h followng rsson: L N 8π ( ) ( 8π ) τ [( ) ( ) ] (7.7) As w s h aroa for of h Lagrangan of h nonlnar quaon of h wrld lroagn wav onans onl h nvarans of h Mawll hor and s slar o h known Lagrangan of h hoon-hoon nraon (Akhr and Brssk 965). L us now anal h quanu for of h Lagrangan dns (7.7). Th quaon (7.) an b wrn n h for: Q τ 8π ( ) ( ) L (7.8) I s no dfful o s ha h lrodnas orrlaon (7.6) n quanu for has h for of h known Fr dn (Chng and L 98; ): ( ) ( ) ( $ $ $ ) ( $ 5) Usng (7.9) fro (7.8) w oban: Q τ 8π [( ) ( ) ] 5 (7.9) L (7.) Th Lagrangan (7.) onds wh h Nabu and Jona-Lasno Lagrangan (Nabu and Jona-Lasno 96; 96a) whh s h Lagrangan dns of h rlavs suronduv hor. As s known hs Lagrangan dns s usd for h soluon of h robl of h lnar arls ass aaran b h hans of h vauu sr sonanous brakdown ( orrsonds also o h Coor s ar roduon ross n h suronduv hor). No agan ha n our hor h brakdown of sr also aks la whn a ass of arls aars whn wrlng and brakng of hoon. 8.. Abou ulars of CWD as h non-lnar hor I s no dfful o s ha CWD dslos wo s of non-lnar. Th frs s onnd wh osula 6 of CWD abou h oon of M wav along urvlnar losd raors. Th urvlnar as a dvaon fro lnar s ossbl o onsdr as on of knds of non-lnar. Bu n our as hs non-lnar raors onrn o onr knd: h ar rad and dsrbd b haron funons and b hr surosons. I allows o dsrb hs non-lnar b h lnar quaons. Rall h oon along a rl an b rsnd as h su of wo lnar haron osllaons. Th su of grar nubr of osllaons lads o h ulfor (nludng saal) urvlnar raors known as Lssaous fgurs. Aarnl n hs onnon all hs non-lnars ar onvnnl and sl dsrbd b ol funons (or dal s har 8). I s ossbl o assu ha h Fourr aaraus of h analss and snhss of funons rfls suh oorun of h lnar dsron of urvs whh an b dsrbd b h su of h lnar haron osllaons. In hs as s ossbl n h sn of Fourr hor o s h rflon of h ral dsrbd b CWD. Sn h Fourr hor an b usd onl n h as of lnar funons obvousl hs haron urvlnar allows n hs ondons o onsdr h CWD o b h

20 lnar hor.. h hor n whh as wll as n h quanu fld hor h rnl of suroson s srl arrd ou. Bu on h ohr hand as w saw h wrlng of M wavs rsuls also n ohr of nonlnar. Rall w dal hr no onl wh raors bu wh h flds whh ar aahd o hs raor b srl dfnd annr. Durng foraon of M arls.. as a rsul of bndng of raor of an M wav nsd of s volu h fld onfguraon vars. Ths nrs no h quaons h non-lnar rs whh ar rsnd nhr n lassal lrodnas nor n h lnar quanu fld hor. Th slng u of h wrld hoon no wo wrld half-rod vn or olas hs ur. Thus srl sakng nsd of a arl oras h non-lnar fld hor and aarnl h rnl of suroson should hr no hav la. Char 3. Th quanu and lroagn fors of lron hor.. Inroduon In h har w hav shown ha h Dra lron quaon s h quaon of M wav ovng along a rng raor. Thus h dffrn bwn wo fors quanu and lroagn - onsss onl n h ahaal for of rord: h ol for of h M quaons orrsonds o h oraonall-ar for of h quanu quaons. Th Dra lron hor has a lo of arulars. In h odrn nrraon hs arulars ar onsdrd as ahaal faurs ha do no hav a hsal anng. Th lroagn for of ar of h w hav onsdrd n h har. On h bass of h har w wll show also ha all ohr ahaal arulars of h Dra lron hor hav h known lrodnas sns... lrodnas anng of h fors of h Dra quaons.. Th quanu Dra quaon fors wh ass Thr ar wo bsnor Dra quaons (Akhr and Brssk 965; Bh 96; Shff 955; Fr 96) (h dsron of h quaon hararss and arars s n h har ): [( ) oε β ] [( ) oε β ] (.) (.) whh orrsond o wo sgns of h rlavs rsson of h lron nrg: ε (.3) ± bu for ah sgn of h ron (.3) hr ar wo ran-onuga Dra quaons. Thus hr ar wo ran-onuga quaons orrsondng o h nus sgn of h rsson (.3): [( ) oε β ] [( ) ] oε β and wo quaons ha orrsond o lus sgns of (.3): [( ) oε β ] [( ) ] oε β (. ) (. ) (.5 ) (.5 ) W wll us furhr h wav funon n h ar for of h lan M wav ovng as n h har along - as:

21 ( ) (.6) whh wh h followng ho of h Dra ars σ σ σ σ σ σ β (.7) whr $ σ ar Paul sn ars gv h orr lrodnas rssons... Th M Dra quaon fors L us onsdr frs wo ran-onuga quaons orrsondng o h nus sgn of h rsson (.3). Usng (.6) fro (. ) and (. ) w oban: (.7 ) (.7 ) whr r π π (.8 ) r π π (.8 ) ar h ol urrns n whh! and r!. Thus h quaons (. ) and (. ) ar Mawll quaons wh ol urrns. As w s h ran-onuga quaons (.7) and (.8) dffr b h urrn drons. L us onsdr now h quaons ha orrsond o lus sgns of (.3). Th lroagn for of h quaon (.5 ) s: (.9) Obvousl h lroagn for of h quaon (.5 ) wll hav h oos sgns of h urrns oaravl o (.9).

22 Coarng (.9) and (.7) w an s ha h quaon (.9) an b onsdrd as h Mawll quaon of h rardd wav. If w don wan o us h rardd wav w an ransfor h wav funon of h rardd wav o h for: r (.) Thn onrar o h ss (.9) w g h ss (.8). Th ransforaon of h funon r o h funon adv s alld h harg onugaon oraon. No ha h lron and osron wav funons an b onsdrd as h rardd and advand wavs. So h abov rsul lnks also wh h hor of advand wavs of Whlr and Fnan (Whlr and Fnan 95; Whlr 957). (S also Dra s work on -sr lassal lrodnas (Dra 938) and abou hs h - Kononsk s book (Kononsk 98). 3.. lrodnas anng of h bsnor fors I s known ha hr ar 6 Dra ars of dnsons. W us h s of ars whh usd Dra hslf and w wll na -s (.). I an b shown ha h nsor dnson of blnar for follows fro s nonlnar lrodnas fors. nura orrsondng Dra s ars (Akhr and Brssk 965; Bh 96; Shff 955): ) β (3. ) { } { 3 } ) 3) 5 3 ) 5 5) (3. ) (3. ) A ν ν (3. ) β ν ν ν (3.) whr ) salar ) -vor 3) sudosalar ) -sudovor 5) ansral nsor of sond rank ar. Ls alula lrodnas valus orrsondng o hs ars: O whr s gvn b (.6): π I ) ( ) ( ) 8 whr I s h frs salar (nvaran) of Mawll hor.. h Lagrangan of lroagn fld n vauu; ) 8π U whr U s h nrg dns of lroagn fld; o 8 π S P 8π g whr g s lroagn wav fld ovd along h Y-as. As s known h valu of nrg-onu. 3) 5 ( ) ( ) h onu dns of h U g s -vor whh s h sudosalar of s h sond salar (nvaran) of lroagn fld I lroagn fld and ( ) hor. ) ( ) ( ) 5 5 ( )

23 3 5 ( 5 3 As w wll show n h har 5 h -sudovor s onnd wh sral of arls. 5) Tnsor w an wr n oa for: ( ) ν ν ( ) ( ) ). ( ) ( ) ( ) ( ) ( ) As w wll show blow hs hnsor dfns h Lorn for... Abou sasal nrraon of h wav funon As s known fro h Dra quaon h robabl onnu quaon an b oband (Akhr and Brssk 965; Bh 96; Shff 955; Fr 96): P r r P ( r ) dv S r r ( r ) r ( ) s h robabl dns and ( r ) robabl flu dns. Usng h abov rsuls w an oban: ( r ) 8π U S r g 8π S. Thn h lroagn for of h quaon (3.5) s: U dv S (.) S r s h P r (.) whh s h for of nrg-onu onsrvaon law of h M fld. 5.. Th lrodnas anng of h ars ho Aordng o Fr (Fr 96) an rov ha all h hsal onsquns of Dra s quaon do no dnd on h sal ho of Dra s ars In arular s ossbl o nrhang h rols of h four ars b unar ransforaon. So hr dffrns ar onl aarn. Th ar squn ) agrs wh h lroagn wav whh has - ( 3 dron. A quson arss: how o dsrb h wavs whh hav and - drons? Inrodung h as nds whh nda h lroagn wav dron w an wr hr grous of h ars ah of whh orrsonds o on and onl on wav dron: ) ) ). ( 3 ( 3 ( 3 L us hoos now h wav funon fors whh gv h orr Mawll quaons for h and - drons. Takng no aoun (.6) as h nal for of h - dron fro b ans of h nds ransoson around h rl w wll g ohr fors. Sn n hs as h Ponng vor has h nus sgn w an suos ha h ransoson us b ounrlokws. L us an hs suoson hkng h Ponng vor valus: Th ss ) ) ) orrsond o h wav ( 3 ( 3 ( 3 funons ( ) () () and o non-ro Ponng vors [ ] [ ] [ ] rsvl. As w s w ook h orr rsul. W an suos now ha b h lokws nds ransoson of h wav funons wll dsrb h lroagn wavs whh ov n a osv dron along h o-ordna as. L us rov hs: and

24 Th ss ) ( 3 ) ( 3 ) ( 3 orrsond o h wav funons: ( ) () () and o followng non-ro Ponng vors [ ] [ ] [ ] rsvl. As w s on agan w g h orr rsuls. Now w wll rov ha h abov ho of h ars and wav funons gvs h orr lroagn quaon fors. Usng for al quaon (.5 ) and ransosng h nds lokws w oban for h osv dron of h lroagn wav h followng rsuls for -drons orrsondngl: (5.) As w an s w hav oband hr quaon grous ah of whh onans four quaons as s nssar for h dsron of all lroagn wav drons. In h sa wa for all ohr fors of h Dra quaon analogu rsuls an b oband. Obvousl s also ossbl va anonal ransforaons o hoos h Dra ars n suh a wa ha h lroagn wav wll hav an dron. L us show. 5.. Th M anng of anonal ransforaons of Dras ars and bsnors Th ho (.7) of h ars s no unqu (Akhr and Brssk 965; Shff 955; Fok 93). As s known hr s a fr ransforaon of a knd: S a S whr S s a unar ar alld h anonal ransforaon oraor and also h wav funons ransforaon S whh dos no hang h rsuls of h hor. If w hoos ars as: σ σ σ σ σ σ 3 σ σ (5.) hn h funons wll b onnd o funons aordng o h rlaonshs: (5.3) Th unar ar S whh orrsonds o hs ransforaon s qual o: S (5.) I s no dfful o hk ha b ans of hs ransforaon w wll also rv h quaons of h Mawll hor. Auall usng (.6) and (5.3) s as o rv: 3 3 (5.5)

25 5 whn: (5.6) Subsung hs funons n h Dra quaon w wll rv h orr Mawll quaons for h lroagn wavs n doubl quan. I s ossbl o assu ha h funons orrsond o h lroagn wav ovng undr h angl of 5 dgrs o boh oordna as. Thus fro abov follows ha vr ho of h Dra ars dfns onl h dron of h nal lroagn wav. Obvousl hs s a hsal orgn wh h hsal onsquns of Dra s quaon do no dnd on h sal ho of Dra s ars (Fr96). 6.. Th lroagn for of h M lron hor Lagrangan As a Lagrangan of h Dra hor an ak h rsson (Shff 955): ( ε β ) L D (6.) For h lroagn wav ovng along h -as h quaon (6.) an b wrn: L β! D (6.) Transfrrng ah r of (6.) n h lrodnas for w oban h lroagn for of h Dra hor Lagrangan: U L DM dv S ( ) π (No ha n h as of h varaon rodur w us dsngush h ol onuga fld vors * * and ). Usng h ol lral and agn urrns (.8 ) and (.8 ) w ak: L DM U dv S ( ) (6.3) (6.) I s nrsng ha sn L hanks o (.6) w an ak h quaon: U s ( ) dv S (6.5) whh has h for of h nrg-onu onsrvaon law for h Mawll quaon wh urrn. 7.. Th Lorn for rsson n M rrsnaon Aordng o our hor for h M arls sabl n h wrld wavs (.. no h M arls) h for us aar whh s rndular o h raor of oon of h M flds. Bu n hs as h angnal for (b our hos along h -as) us absn sn would rovok h angnal alraon of h lron flds. ν Th rsson of Lorn s for b h nrg-onu nsor of lroagn fld τ s wll known (Tonnla 959; Ivannko and Sokolov 99). Ths nsor s sral and has h followng oonns: τ π ν ν ν ντ π f (7.) r frs hr oonns dsrb h Lorn for dns vor and fourh oonn orrsonds o h nrg onsrvaon law. Usng (7.) an b wrn:

26 6 U grad g f f f (7.) g dv U f (7.3) As w s b usng of h sral nrg-onu nsor w don oban h ndd oonns of h for sn hr ) ( f. Th rgh rsul an oban usng ansral sn nsor ν (3.). Thn w hav: ν ν ν ν π π f (7.) or: 3 3 f f f f (7.5) Usng (.6) and (3.) w oban of Lorn s for oonns: f π f (7.6) f π f For h lnar hoon all h braks n (7.6) ar qual o ro aordng o Mawlls quaon. I ans ha aar no fors n lnar hoon. Whn hoon rolls u around an of h as whh ar rndular o h Y-as w wll g h addonal urrn rs. If o ak ha h fld vor of r F F ) ( dsrbs gorall h vors roaon w an for h wrld s-hoon wr: (7.7) (7.7) For snnng hoon ( ) h for oonns ar (h ur lf nd shows h snnng as OZ or OX). ( ) ( ) f τ π (7.8)

27 7 for snnng hoon ( ): f π ( ) ( ) (7.9) τ f (7.) f (7.) wha orrsonds o our rrsnaons abou h dnas of wrld s-hoon. 8.. Th quaon of h rng M wav fld oon W an suos ha -vor-onal of lroagn fld ulld o h lron harg ϕ A s h -vor of h nrg-onu of h urvlnar wav fld { ε }(s har ). Thrfor h wll-known analss of Dras lron quaon n h rnal fld an b usd for h analss of h quaons of h nnr wrld hoon fld b h hangs: A ϕ ε (8.) As s known (Akhr and Brssk 965; Shff 955) h quaon of h lron oon n h rnal fld an b found fro h n oraor quaon havng h Posson braks do O ( O O ) (8.) d! whr Ô s h hsal valu oraor whos varaon w wan o fnd and s h alon oraor of Dra s quaon. Th alons oraor of h Dra quaon s qual (Shff 955; Akhr and Brssk 965): P β ε (8.3) whr P s full onu of wrld hoon. For O P fro (8.3) w hav: $ dp A grad ( ϕ) [ υ ro A] (8.) d or subsu υ $ whr υ - vlo of h lron ar w oban: dp [ υ ] f d L (8.5) dp Sn for h oonlss lron h oon quaon s: d grad ε [ ] υ ro (8.6) Passng o h nrg and onu dnss g U ε (8.7) τ τ w oban h quaon of ar oon of wrld hoon:

28 8 g grad U [ υ ro g ] (8.8) L us anals h hsal anng of (8.8) onsdrng h oon quaon of dal lqud n for of Lab s-groks quaon (Lab 93). In hs as whn h rnal fors ar absn hs quaon s: whr U l g l grad U l [ υ ro gl] (8.9) g -nrg and onu dns of dal lqud. l Coarng (8.8) and (8.9) s no dfful o s hr ahaal dn. Fro hs follows h nrsng onluson: h nnr arls quaon a b nrrd as h oon quaon of dal lqud. Aordng o ( ) fro (8.9) w hav g grad U f L (8.) whr f L s h Lorn for. As s known h r [ ro ] υ g n (8.9) s rsonsbl for nral alraon. Probabl w hav h sa n (8.8). If h hoon lqud ov along h rng of r radus hn h angular oon vlo s d wh ro υ b rsson: and nral alraon s ro υ (8.) υ an υ ro υ r r (8.) r whr r s un radus-vor - s un vor of OZ-as. As a rsul h quaon (5.5) has h for of Nwons law: ρ an f L (8.3) Ths rsul an b sn as h lroagn rrsnaon of h rnfs hor (Shff955). Conluson Th abov rsuls roof ha h non-lnar M rrsnaon of h Dra hor gv h lassal lanaons of all arulars of h Dra lron hor whh nvrhlss don onrad o h quanu nrraon. Char. Pon and non-on soluons of lron quaons.. Inroduon.. San of h robl Th hor of alulaon of harg ass and ohr hararss of lron on h bass of h fld quaons has arsn orgnall n lassal lrodnas and was dvlod b W. Klvn J. Laror. Lorn M. Abraha A. Ponar. (Paul 958; Ivannko and Sokolov 99). I s basd on hohss of h fld ass and fld harg aordng o whh h arls own nrg or ass s oblgd o nrg of flds and h harg of arls s dfnd b h arls own flds. Ths das afrwards wr ransfrrd o quanu hans. Bu nhr lassal nor quanu hors ould lan onssnl h naur of ass and harg of lnar arls alhough for h lron so onsuv hors hav bn onsrud... Th gnral rqurns o h lassal lron ass hor A frs w wll addrss o h hohss of lron fld ass whn h frawork of lassal lrodnas (Lorn 96; Ivannko and Sokolov 99).

29 9 Aordng o h hohss whh has bn u forward n h nd of h 9 h nur b J.J. Thoson and advand b. Lorn M. Abraha A. Ponar. h lron s own nrg (or s ass) s oll ausd b h nrg of h lroagn fld of lron. In h sa wa s suosd ha h lron onu s oblgd o h onu of h fld. Sn lron as an hanal arl osssss h onu and nrg whh ar oghr h -vor of h gnrald onu h nssar ondon of suss of h hor wll b h roof ha h gnrald onu of an lroagn fld s a -vor. Thus for h suss of h fld ass hor h followng ondons should b sasfd a las: A frs s nssar o rv fnal valu of h fld nrg gnrad b a arl whh ould b rsl quad o fnal nrg of a arl (.. rodu of h ass b h squar of h lgh sd). A sond h valu of a onu of h fld gnrad b a arl us no onl b fnal bu also has h ror orrlaon wh nrg forng wh h las a four-dnsonal vor. Thrdl h hor should anag o ddu h quaon of ovn of lron. Fourhl s nssar o oban of lron sn as a sn of a fld (ha nds h quanu gnralaon of h hor of fld ass sn a sn s quanu ff). Th analss shows ha hr ar wo ondons b whh h gnrald fld onu G s a -vor. In as of sa whou hargs h s G T ( dr) (.) wll rrsn a -vor f dvrgn of nrg nsor of a fld urns no ro: T λ λ (.) For al h lroagn fld whh s load n a sa whou hargs sasfs slar ondons. In arular du o hs fa n h hoon hor M fld s harard no onl b nrg bu also b onu. ) Th ondon b whh h nrg and onu of an lroagn fld for a -vor a h rsn of hargs s forulad b h Lau hor. Aordng o h las a h rsn of hargs h s G s a -vor onl n h as whn n h oordna ss rlavl o whh lron s n rs for all h nrg nsor oonns h followng ar s obsrvd Tν ( dro ) (.3) T ( o for h oonn h ngral of whh s a onsan and s qual o full nrg of h fld gnrad b arl (hr d r ) s lnar volu n rfrn ss n whh h lron s n rs). Th qual (.3) rsss a nssar ondon b whh h whol arl harg should b n balan. W an qua hs fld nrg o h arl s own nrg rssng n hs wa h bas da of a fld hohss. Aordng o h las: ε T ( dr ) o (.) Thus h ass of a arl fro h fld on of vw an b dfnd n wo was: ) rodng fro M onu of a fld G s ossbl o dfn ass as faor of rooronal bwn a fld onu and hr-dnsonal sd of a arl. ) f w onsdr h lron s own nrg as qual or onrnous o h nrg of a fld and ass as h rao of a fld nrg G o a squar of lgh sd (.. as h fourh oonn of a gnrald onu). Th as o u hs rogra rodng fro lassal lnar Mawll hor hav ld o dffuls. In arular was no ossbl o rov h Lau hor (Tonnla 959). In h lassal hor h dnas (hans) and lrodnas ar oll ndndn fro ah ohr. lroagn aons ar harard b oonn T of an nrg-onu nsor of

30 3 an lroagn fld. I dos no nlud h nrg and onu of h subsan whh should b subsqunl nsrd. Th as of Lorn and Ponar o oordna h hor on h bass of h assuon ha nrg of subsan has an lroagn orgn hav no ld o a osv rsul. In Lorn lron hor (lnar n ssn) sn of hargs s ossbl o lan onl b nroduon of fors of non-lroagn orgn. Nvrhlss (Sokolov and Ivannko 99) hr wr also a nubr of susss whh arrd a ho o solv hs robl b an hang of h hor. Th os rsv hang of Mawll- Lorn hor aard o b s non-lnar gnralaon..3. Non-lnar lassal lrodnas In h har whn h frawork of CWD w hav rvd h non-lnar quaon for h lroagn (M) lron and hav shown ha on suffn dsan fro a arl onds wh h lnar Dra lron quaon. Bu unforunal h soluon of h non-lnar quaon of h urvlnar lroagn wav s no rvd. Is frs aroah h non-lnar snbrg quaon - also dd no anag o b solvd (alhough hr h nouragng rsuls hav bn rvd). W wll show h known lassal non-lnar hors of Gusav M M. Born - L. Infld. Shrodngr. rrsn h aroa soluons of non-lnar quaon of CWD whh nabl us o sa h ss of a arl and dsrbuon of a fld n aroah of shral lron. Bsds h nonlnar hors fnd ou an oorun of dsron of M lron as on or no on dndng on h usd ahas... Gusav M aroah o h lron hor.. Pror hsor Gusav M ad h frs a o onsru a url lroagn hor of hargd arls. (M 9a; 9b; 93; Paul 958; Tonnla 959)). Prodng fro so forall rrroahabl hohal non-lnar gnralaon of lrodnas h anagd o onsru a hor whh has ovro all dffuls of h lassal hor. As w hav sad abov n h hor of h lron bfor G. M (Balnk-Brula 983) h lron was no rad as a url lroagn n bu was also ad of ohr suff lk for al Ponar srsss and h hanal ass. M wand onl h lroagn fld o b rsonsbl for all h rors of h lron. In arular h wand h lroagn urrn o b ad of lroagns. In ordr o ahv hs goal M assud ha h onal four-vor nrs drl no h Lagrangan and no onl hrough h fld srngh. Th gnraon of h urrn has bn ahvd n hs annr bu h r was vr hgh. Th onals aqurd a hsal anng and h gaug nvaran was los. Ths ror has bn found unaabl b ohr hsss and h hor of M has bn shlvd for an long ars... G. M hor In hs hor M has ad wo ssnal ss (Paul 958; Tonnla 959). A frs M was h frs who suggsd n h onsruon of h hor lanng on a Lagrangan dndn on fundanal nvarans. A sond o g rd of Ponar Lorn fors ha hav non-lroagn orgn M nrd a unfor sgh a a fld and subsan. s a robl n ordr o gnral h fld quaons and h nrg-onu nsor of Mawll-Lorn hor n a wa ha nsd h lnar hargd arls h rulson Coulob fors would ounrbaland b ohr fors of orgn also and ousd of arls h dvaon fro ordnar lrodnas would rbl. assud ha an nrg and subsan has an lroagn orgn and ss as h uros o ddu rors and hararss of hargs fro rors of a fld. Abou h knd of Lagrangan L whh s frqunl alld a world funon n non-lnar lrodnas s ossbl o ak so gnral sans. Th ndndn nvarans of an lroagn fld whh an b ford fro bvor F ν (whr F ν ar h nsor oonns of lroagn fld srnghs) and a vor A ( ϕ A) ( ϕ A ) ( A ) followng: I ) Th squar of bvor F ν : F ν F * ν (whr A ar h ν I F ν F ; ) h squar of a sudo-vor *ν F s h dual lroagn nsor). 3) Th squar of a -vor of

31 3 I A A lroagn onal A : 3 ; ) Th squar of a vor: I Fν A ; 5) Th squar * of a vor: I 5 Fν A. Thrfor L an dnd onl on hs fv nvarans. If L s qual o h frs of h sfd nvarans h fld quaons ar dgnrad no ordnar quaons of h lron hor for sa ν whou hargs. Thus L an noabl dffr fro F ν F onl nsd h aral arls. Invaran an b nludd n L onl as a squar n ordr no o brak h nvaran onrnng saal rflons. Invarans 3-5 brak h gaug nvaran. Furhr sans abou h world funon L anno b ad. Thus for h slon of L hr ar an nfn nubr of ooruns. Gusav M ad as nal h followng Lagrangan: or L M F ± ( A ) ν ν F f A ( ) f ( A A ) L M ± 8π (. ) (. ) whr f s so funon. Usng hs Lagrangan (Tonnla 959) Gusav M anagd o rv h fnal nrg (or ass) of h hargd arl as a valu oll ausd b h nrg of h fld of hs arl. Bsds n hs hor h Lau hor of sabl s arrd ou and h ror orrlaon bwn nrg and onu of a arl s rahd. For furhr analss s also usful o non h a of. Wl (Paul 958) o nrr on h bass of M hor h asr (wh rs o dsnon of asss) of boh sors of lr. If L s no a raonal funon of A A L M F L M F s ossbl o u: ( A ) ( A ) ν ν F f A ν ν F f A (. ) (. ) Thus f L s a ull-valud funon of h nvarans nond abov s obvousl ossbl o hoos as world funons for osv and ngav hargs varous unquvoal branhs of hs funon..3. Connon h M hor wh h CWD Ls show ha M Lagrangan afr so addons an b subd as Lagrangan slar o Lagrangan of CWD (and onsqunl of QD). As w know (Paul 958; Sorfld 958) h harg dns s no nvaran onrnng Lorn ransforaon bu a harg s. Also s known ha h squar of -onal.. I A A 3 s nvaran onrnng Lorn ransforaon bu s no nvaran rlavl o gaug ransforaons. Bu aars ha h rodu of a squar of a harg on I 3 wll b an nvaran onrnng boh Lorn and gaug ransforaons. L s show hs..3.. Laror - Shwarshld nvaran Aordng o (Paul 958) and (Sorfld 958) R. Shwarshld (Shwarshld 93) nrd h valu υ S w ϕ A (.3) whh h alld lrokn onal and has shown ha hs valu bng ulld b dns of a harg fors h rlavs nvaran: υ L ρs ρ ϕ A w ( ) A (.)

32 whr { ρ ρυ} 3 s -urrn dns A { ϕ A} s -onal. Usng (.) Shwarshld has ford h followng Lagrang funon: υ L ( ) dv ρ ( ϕ A) dv (.5) and b -ngraon (.5) h has rvd h funon of aon. Thus n -dnsonal dsgnaons h Shwarshld Lagrang funon dns (or Lagrangan) wll b wrn down as follows: L and h Lagrang funon wll b: ν L Fν F dτ A dτ (.7) ν Fν F A (.6) (In h no o h book (Paul 958) Paul arkd ha bfor Shwarshld h sa Lagrangan has bn suggsd b J.J. Laror (Laror 9)). L s onsdr now h radand n h Ми Lagranganе: ϕ A A A (.8) Mullng on h squars of dns of a harg and a squar of a harg w shall rv aordngl: A ) ( ) ( A ϕ A (.9) W wll nr h valus of dns of nrg of nraon and nrg of lron nraon aordngl: U ρϕ ε ϕ (.) and also h dns of onu and h onu of lron nraon aordngl: g ρa Thn fro (.9) w shall rv: ε A A ε ε (.) ( ) Sn ( ) ( ) o (.) ak la hs rssons an b also wrn down as: ( ) ε ( ε ) ( ) A (.3) Usng h abov-sad rsuls for non-lnar ar of h M Lagrangan ( ) LN f A A M ± f ( ± A A ) ( ± ϕ A ) (.) w an a h rsson: Usng of Dra ars s as o rv h followng dooson: A ( ε ± ) # (.5) ha gvs for non-lnar ar of Lagrangan h rsson: Takng no aoun ha L N M β ( ε ± ) # (.6) ( ε ) (.7)

33 33 w s ha w an nr n h Lagrangan h ass r of h Dra quaon. Thus s ossbl o assr ha M Lagrangan an b ransford o hav h for of h Lagrangan of h non-lnar fld hor orrsondng o h hor of h lron of CWD and QD also. Th us of hs rssons lads o h Dra quaons of lron and osron and gvs h bass o h Wl s a o nrr h asr of boh sors of lr no n onnon wh ass bu n onnon wh dsnon arl - anarl. Thus h assuon of M ha nrnal rors of lron ar dsrbd b an lroagn fld orrsonds o h rsuls of CWD. Auall n h har w rsul ha h nrg ass and harg of arl ar dfnd b h nnr onals of hs arl. If o a ha onals nsd a arl orrsond o an nrg-onu of h arl fld aks h onals h hsall ran valus whh howvr ar no asurabl ousd of a arl. In ohr words h onals ar hr h hddn arars of lnar arls. Do hs rsuls onrad o h rnal rsuls of odrn hss? As s known n lassal lrodnas h onals la h rol of h ahaal aular valus and hav no hsal sns. Bu as aars n frawork of quanu hans h onals hav hsal sns ha s rovd b Aharonov-Boh rn (Aharonov and Boh 959; Fnan Lghon and Sands 989). As an al of alulaon of lron arars n frawork of lassal hor w wll onsdr h rsuls of h Born - Infld hor (Born and Infld 93) and show ha hs rsuls an b onsdrd as so aroaon of CWD soluon. 3.. Born-Infld nonlnar hor M. Born and L. Infld rvvd Ms hor and roosd a sf odl. Th Born-Infld hor (Born and Infld 93) rss on h sls ossbl Lagrangan: h squar roo of h drnan of a sond rank ovaran nsor. Suh a sruur auoaall guarans h nvaran of h hor undr arbrar oordna ransforaons akng h full rlavs and gaug nvaran non-lnar lrodnas. M. Born and L. Infld rodd fro h da of a ld valu of h lroagn fld srngh of h lron (whh s dnal o da of a ld s of h lron as s shown blow). Ths rasons ld h o h followng Lagrangan of h non-lnar lrodnas n h vauu: ( ) L BI π (3.) whr s h au fld of lron. W wll onsdr h os oran as of h lrosa fld of h on (shral sr) lron. Pung gradϕ ρ( ξ ) δ ( r ) δ ( s) w wll fnd aordng o (3.): L n r ϕδ ( r ) π Thn b h hl of h varaon rnl w oban: D π r r L ϕ whr D s h lral nduon vor (D-fld): L r Dr π r whh orrsonds o h quaon: soluon of whh s: dvd r πδ ( r) r

34 3 r D r (3.) 3 r As w s fro on of vw of h D-fld h lron should b onsdrd as on arl. For h lr fld (-fld) w oban: Dr r r (3.3) D r r r r whr r harar h lron s. In hs as.. fro on of vw of h lr fld (-fld) h lron s no a on arl. Ths s vr oran sf of h non-lnar hor n oarson wh h lnar hor whh an lan wh rns on sarng of lron an b nrrd so ha h lron looks as a on arl (whl h rnoralaon rodur allows o lna h nfns). Fro abov h lron harg dns dsrbuon an b found: ρ dv r 3/ π π r( r r ) (3.) Thus n rs o h lr fld h lron harg an b onsdrd as dsrbud anl n volu of radus r sn b r >> r h dns wll qukl asr o ro. Thrfor h s r an b onsdrd as ffv radus of lron. Usng known valus for ass and harg of lron and sd of lgh s ossbl o rv 3 r 8 см whh s raall qual o lassal radus of lron. Also s as o fnd valu for h aal fld of lron bng a fld n h nr of h lron (a r ): 5 V 98 CGS 75. r As s known h wo s of flds and h wo dfnons of h harg dns orrsondng o h ar also dsrbd b h hor of h dlrs. Th valu: r D r r ε (3.5) whh s hr a funon of h oson an b onsdrd as a dlr rabl of lron. On r larg dsans fro a harg whn ε aqurs a valu qual o un as n usual r lrodnas. I s ossbl o ll ha nsad of h rsson of nrg Born and Infld ak r and hn h rduon of r s onsad b nras of ε so h full nrg rans as ε r fnal. (I s ossbl o assu ha h rsn of hsal vauu should ak h andn o valu of dlr rabl and a h sa o valus of onal of lron s s and ohr hararss). Thus rodng fro so foral hohal non-lnar gnralaon of lrodnas aard ossbl (Ivannko and Sokolov 99):. o rov h hor of sabl.. o rov ha n h non-lnar hor h lron s sabl whou nroduon of fors of non-lroagn orgn;. o rv h fnal nrg (ass) of lron; 3. o rv h fnal s of s lr harg;. o rv h fnal s of s lroagn fld.

35 Ohr Lagrangans of nonlnar hors Also ohrs Lagrangans hav bn offrd for ron of h non-lnar hor. So. Shrodngr usd h followng arbrar obnaon for Lagrangan: L Sh ln (3.6) 8 π I was nod (Ivannko and Sokolov 99) ha varous varans of foral non-lnar lrodnas lad o los valus of offns f o ak no aoun ha h lron radus s qual o lassal radus of lron. I was also nod ha h bas df of hs hors as wll as of M hor was h arbrar ho of Lagrangan whh had no onnon wh h quanu hor n arular wh Dra hor and dd no ak no aoun rors of lron rvald b h las. W wll show ha hs hors an b onsdrd as aroah of h CWD and ha h ar ahaall onnd o h Dra lron hor... Th Born-Infld hor as an aroaon of CWD Sn n gnral as h CWD (s har ) s h non-lnar M hor hrfor s Lagrangan an onan all ossbl rs wh h nvarans of lroagn hor. Takng no aoun h gaug nvaran h CWD Lagrangan an b wrn as so funon of h followng fld nvarans: L f whr I ( ) I ( ) L ( I I ) (.) ar h nvarans of lroagn fld hor. Aarnl for ah robl h funon f L has s sal for whh s unknown bfor. W an suos ha n all ass hr s an anson of h funon f L n h Talor MaLaurn owr srs wh so unknown anson offns. I s also obvousl ha for h os s of h funons f L I I ) h anson onans aroal h sa s of h rs whh ar ( dsngushd onl b h onsan offns an of whh an b qual o ro (as an al of suh anson s ossbl o on ou h anson of h quanu lrodnas Lagrangan for arl a h rsn of hsal vauu (Akhr and Brssk 965; Wsskof 936; Shwngr 95). Gnrall hs anson looks lk: whr L L M 8π ( B ) L (.) ( B ) β ( B) γ ( B )( B) 3 ξ ( B ) ζ ( B )( B)... (.3) s a ar whh s rsonsbl for h non-lnar nraon (hr β γ ξ ζ... ar onsans). Th Lagrangan of h Born-Infld non-lnar lrodnas an b also andd no h sall arars a << and a B << whr a so ha a ( ) [( ) ( ) ] L BI B B B O( ) (.) 8π 3π whr O( ) s h srs rs wh h rs onanng vors n owrs whh ar hghr han four. Obvousl undr ondons a << and a B << on larg dsan fro h nr of a arl (whr hr s a aal fld) h rs of hs srs rall qukl onvrg bu on sall dsan fro h nr s aarnl norr and hr nds o ak no aoun h rs of hghr dgrs. In h har w hav shown ha a h frs aroaon Lagrangan of CWD n lroagn for an b rrsnd as followng: [ ( )] L N 8π ( B ) b ( B ) ( B) (.5)

36 36 whr b s so onsan. Takng no aoun (.) w an wr: LN L BI (.6) and rv n h frawork of CWD for M lron h aroah soluon lk h soluon of Born - Infld hor sad brfl abov. For hs rasons an slarl show ha h CWD Lagrangan aroal onds wh Lagrangan of Shrodngr and ohrs offrd Lagrangans of non-lnar hors allowng us o alula h orrsondng hararss of lron. Thus s no dfful o answr wh varous fro h hsal on of vw varans of foral non-lnar lrodnas lad o los valus of offns: as anson of non-lnar Lagrangan (.3) shows all of h ar aroal qual aong hslvs and onsqunl ld los rsuls. A h sa sn Lagrangan and quaons of CWD oll ond wh Lagrangan and h quaons of quanu lrodnas h M hor and s varan h Born - Infld hor s losl onnd wh h Dra hor. Char 5. Th assv nurno hor.. Inroduon. Nurno of Sandard Modl hor Th rsn saus of h robl of nurno hor s suard n (Blnk Gun and K ; Glashow 96; Wnbrg 967; Sala 969). Nal h hor of lrowak nraons nludng nurnos obnd wh h Quanu Chroo-Dnas (QCD) s now alld h Sandard Modl (SM)... Nurnos faurs In h Sandard Modl nurnos ar srl asslss ; all nurnos ar lf-handd wh hl - and all annurnos ar rgh-handd wh hl ; lon fal nubr s srl onsrvd. Bu odrn rnal vdn ndas ha all of hs sans ar n fa doubful (Blnk Gun and K ).... l and Chral In h nurno hor h onons of hl and hral la h oran rol. In h Sandard Modl hor nurno and annurno hav oos hl. I s ahaall ossbl ha hs s n fa h onl dffrn bwn nurnos and annurnos.. a rgh-handd nurno would b an annurno. Parls of hs sor ar alld Maorana arls. As long as h nurno s asslss s hl s oll dfnd and a Maorana nurno would b a dffrn arl fro s annurno. Bu f nurnos hav ass and hrfor do no ravl a al h sd of lgh s ossbl o dfn a rfrn fra n whh h hl would b fld. Ths ans ha hr s ffvl a ng bwn h nurno and h annurno (volang lon nubr onsrvaon). l rfrs o h rlaon bwn a arls sn and dron of oon. To a arl n oon s assoad h as dfnd b s onu and s hl s dfnd b h roon of h arls sn s on hs as: h hl s h oonn of angular onu along h onu: s h s. Thus h hl oraor ros ou wo hsal sas wh h sn along or oos h dron of oon - whhr h arl s assv or no. If h sn s rod aralll on h dron of oon h arl s of rgh hl f h roon s anaralll o h dron of oon h arl has lf hl. Sohng s hral whn anno b surosd on s rror ag lk for al our hands. Lk our hands h hral obs ar also lassfd no lf-hral and rgh-hral obs. In h Sandard Modl hor h asslss nurno s dsrbd b h Dra lon quaon whou h ass r: whh s also sasfd b 3 5 : (.)

37 37 ( 5 ) (.) 5 whr h obnaon of h ars 3 { } 5 has h rors. Ths allows us o dfn h hral oraors whh ro ou lf-handd and rghhandd sas: L ( ) and ( 5 ) 5 R (.3) whr L and R sasf h quaons 5 L L and 5 R R so h hral flds ar gnflds of 5 rgardlss of hr ass. W an rss an fron as L R so ha a assv arl alwas has a L-handd as wll as a R-handd oonn. In h asslss as howvr dsngras no sara hl sas: h Dra quaon sls no wo ndndn ars rforulad as h Wl quaons whr σ σ σ σ ( ± ) ± ( ± ) 5 5 (.) s h hl oraor rssd n rs of h Paul ars σ. Th Wl frons.. h asslss hral sas ( 5 ) 5 and ± ar hsal sn h orrsond o gnsas of h hl oraor. A asslss arl whh s n rual oon hus has an unhangabl hl. Th rason s ha s onu anno b alrd and s sn of ours rans unhangd.... lroagn hararss of nurno I s nrsng ha n s of nural nurno osssss lroagn hararss. Th analss of hs hararss (Trnov ) hls us o undrsand h naur of nurno ass. lroagn rors of Dra s and Maorano s nurno aar o b ssnall varous. Dra s assv nurno as a rsul of h aoun of nraon wh vauu rvs h agn on. And h nurno agn on s drd lnghwas a sn and h agn on annurno - agans a sn. Thus h arl and h anarl dffr b h dron of h agn on. For assv Maorano nurno dnal o s anarl aars ha anno hav nhr agn nor lr on. I also aard ha h ass and h agn on of nurno ar ol nonlnar funons of fld srngh and nrg of a arl. Movng n an rnal fld alongsd wh h agn on h Dra nurno gs as wll h dol lr on d ν. Calulaons show ha h lr on of assv Dra s nurno ovng n a onsan rnal gnral vw fld s rooronal o a sudo-salar ( ) whh hangs sgn a h rflon of. Thus h lr on s ndud b an rnal fld f for hs fld h sudo-salar ( ) and s sn dos no onrad o Т-nvaran of Sandard Modl. In ohr words h dol lr on of Dra s nurno as wll as h agn has dna naur. No also ha hr s on lroagn harars of Dra s nurno whh aks la also for Maorano s nurno: h anaol (or orodal dol) on. Blow w wll show ha n h frawork of h CWD h assv nurno has h onsrvd nnr (olodal) hl owng o whh h abov faurs our and s full dsrbd b h Dra lon quaon... Nurno of CWD In h rvous hars - n frawork of CWD w hav onsdrd h hor of lron and oband h Dra lron quaon. W hav shown ha h lron s h wrld lan-olard s-hoon M arl.

38 38 Blow w wll show ha h soluon of h Dra quaon n bsnor for dsrbs also h oon of h wrld rular-olard s-hoon whh s a nural arl wh half sn bu wh non-ro ass. W wll also show ha arl and anarl of CWD hav oos nnr srals. Ths faurs and an ohrs ol us o dnf hs arl wh nurno. 3.. Plan and rularl olard lroagn wavs lroagn wavs d b hargd arls ar n gnral rularl (or llall) olard (Ivannko and Sokolov 99; Grawford 97). lroagn wavs ar also ransvrs n h sns ha assoad lr and agn fld vors ar boh rndular o h dron of wav roagaon. Crularl olard wavs arr nrg ε and onu as wll as angular onu J U onu dns g SP and 8 π whh ar dfnd b nrg dns ( ) angular onu flu dns whh s gvn b S P π s r g π r (3.) whr s h Ponng vor ndas no onl h agnud of h nrg flu dns bu also h dron of nrg flow. For sl lroagn wavs h Ponng vor s n h sa dron as h wav vor. Th angular onu flu dns an b hkd b h rularl oon of h lron n h rularl olard wav fld (Grawford 97). Th fgur blow shows roagaon of lr fld assoad wh a rularl olard wav wh osv (rgh) and ngav (lf) hl. Fg. Posv hl s h as whn h lr fld vor s road suh a rgh srw would ov n h dron of wav roagaon (no ha n h os s alld lf hand rular olaraon). Ngav hl (rgh hand olaraon n os)) rfrs o roaon n h oos dron. Th dron of h nd of h hl ndas h had of h lr fld vor whh s roang around h as (as on fg.). Sn s ossbl b an ransforaon for h saal rflon o ransfr h rgh (lf) sral o h lf (rgh) sral h rular olaraon of hoons s hr ngral harars k a all ransforaons of h rror ransforaon. Sn h hoon hl s onnd o h fld roaon n lassal lrodnas h also alk abou roaon of a hoon and h nr h hoon roaon harars h angular onu or sn of a hoon. In quanu hans h sn arbung of a hoon has so ondonal harar. As s known as h sn s nad h nrnal angular onu of a arl n hos sss n whh h onsdrd arl s n rs. Thrfor n as of a hoon whos sd an no b ohr han h lgh sd s orr o alk or abou h hoon hl han abou h sn (Gofrd and Wsskof 98). In hs as s ossbl o dfn as hl h vor (Grawford 97). whr h ε h h s h ± (3.) s h un Ponng vor ε h and ar h hoon nrg and rular frqun h. orrsondngl. Aarnl h angular onu valu of hs vor s qual o h! Thn aordng o our hohss h hl vor of nurno as of a wrld s-hoon should hav angnal dron o h urvlnar raor of wrld wav oon and s angular onu should b qual o half of h abov valu h!. ν

39 39.. Quanu for of h rularl olard lroagn wav quaons L us onsdr h lan lroagn wav ovng for al along - as. In gnral as suh wav has wo olaraons and onans h followng four fld vors: { } Φ ) ( B analog wh h hod usd n har fro wav quaon w an oban wo quaons n quanu for: ( ) Φ o ε (. ) ( ) Φ o ε (. ) Choosng h wav funon Φ as (and onl as) Φ ( ) Φ (.) and ung (.) n (.) w rg h followng Mawll quaon of advand and rardd wavs : (.3) (.) Th lroagn wav quaon (and h quaons (.) and (.) also) has h followng haron soluon vw (n rgonor and onnal for orrsondngl): ( ) δ Φ r k A sn (.5) ( ) δ ε Φ r A! (.6) whr 3 A ar h aluds and δ s h onsan has. Pung hr A A δ w oban h followng rgonor for of h quaon soluons: ( ) ( ) ( ) ( ) k A k A k A k A sn sn os os (.7 ) ( ) ( ) ( ) ( ) k A k A k A k A sn sn os os (.7) L us show ha h vors and roa n h XOZ lan. Auall ung w oban: ( ) k A k os sn (.8) ( ) k A k sn os (.8) and

40 whr ( os k sn ) ( sn k os ) k A (.9) k A (.9) k ar h un vors of h OX and OZ as. I s no dfful o show b known algbra analss (Jakson 999) ha w hav oband h l olarsd wav. Bu o k n vdn w wll anals hs rlaons fro goral on of vw. Takng no aoun ha h Ponng vor dfns h dron of h wav oon: S (.) P ( ) π π whr s h un vors of h OY as and alulang h abov (.) w hav for (.3 ) and (.3 ): S P A π S P A π (.) (.) orrsondngl. Thus h hoons of h rgh and lf sss (.8) and (.9) ov n h onrar drons. Fng h vor osons n wo sussv nsans ( and ) w an dfn h roaon dron. Th rsuls ar agd on h fgurs and 3 orrsondngl: Fg. Fg. 3 As w an s h quaon ss (.3 and (.) dsrb h wavs wh rgh and lf rular olaraon orrsondngl. Obvousl b h rng wrlng of h rular olard hoon s hl dos no dsaar bu nsd h orus bo olodal hl (or -hl ). A h sa h ovn of flds of a hoon along a rular raor fors ohr hararss of an lnar arl - nal h angular onu of a arl or sn. Aarnl h sn of a assv arl and s olodal angular onu (-hl) ar dffrn hararss. Sn hs hararss ar h own nrnal hararss of a hoon n h non-lnar lroagn hor h sn and h olodal hl of a arl ar ndndnl onsrvd valus. W wll show now ha h Dra quaon wh ass r an b onsdrd as h quaon of h wrld rularl olard wavs. 5.. Th quaon of assv nurno of CWD Aarnl h assv nurno us b dsrbd b h h Dra lon quaon wh h ass r. Consdr h ass r aarng n hs as. L h rular-olard wav wrld wh so radus Κ whh hav h fld oonns { } r n h lan ( O Y ) b X of a fd o-ordna ss

41 ( X Y Z O ) so ha rndular o (s fg. ) ar aralll o h lan ( O Y ) ar X and Fg. whr h rular arrows shows h rgh ( R ) and h lf ( ) L roaon of hoon flds. Ls rla hr h un vors of h Carsan oordna ss as of aragrah 3 { } { n b} onnd wh a wav b h vors of h Frn-Srr rhdron τ aordngl. Thn for h lr and agn vors w hav nsad of (3.): ( ) ( ) n b n b ( n b ) ( n b ) (5.) (5.) r as wll as n h as of h lnar olard M srngs h un vor of a noral n urns around of OZ as and bvor b rans aralll o B analog o h rodur sad n har s as o rv h quaons of h wrld s-hoons.. h Dra quaon wh a ass r. Unlk o a as of wrlng of h lan olard hoons onsdrd n har w do no hav bass n advan o arov ha hr agn urrns ar qual o ro. Rall n frs as h agn vor was aralll o h roaon as and k a onsan dron n sa and lr onnuousl urnd around of hangng dron n sa. In h gvn as h agn vor roas around of a raor of oon and s ransord along a raor us as an lr vor. Usng h rodur sad n har w shall show ha n hs as hr ar boh lr and agn urrns. Ls n h quaons of h nal M srngs (.) onsdr h rssons π and. (Rnd ha afr wrlng of M srng a fld vors of an nal π wav ar ransford n a fld vors of h wrld wav dsgnad b us n h lroagn for as b and n h quanu for as ). Takng no aoun ha w shall rv fro (5.) and (5.): n n b (5.3) n n b (5.) n whr сκτ τ and r C!. Thus w rv h lr and agn r C angnal urrns a arular of whh s ha h ar alrnang:

42 τ π τ π τ os (5.5) τ π τ π τ os Κ Κ (5.6) Now s no dfful o s ha h quanu for of h quaon of rular olard shoons wh oos sral ar h Dra quaons wh ass rs. Takng no aoun h rvous son rsuls w oban h wrld s-hoon quaons n quanu for whh ar quvaln o h Dra quaons: β r C (5.7 ) β r C (5.7 ) and lroagn for of hs quaons: (5.8) (5.8 ) whh ar h ol Mawll quaons wh agnar angnal alrnang urrns - lr and agn. W an shaall rrsn h flds oon of arls dsrbd b hs quaons n h followng wa (fg. 5): Fg. 5 Aordng o fgs. and 3 h s-hoons (fg. 5) hav h onrar -hls. In h frs as h hl vor and h Ponng vor hav h sa drons; n h sond as h ar onrar. Thrfor n h non-lnar hor w an dfn h nnr or -hl as h roon of h olodal roaon onu on h onu of h rng fld oon. I s no dfful o show (Davdov 965) ha auall h hl s dsrbd n CWD b ar 5. Mullng h Dra quaon on β 5 and akng n aoun ha σ β 5 whr σ σ σ ar h sn ar and 5 5 β β β w oban: ( ) 5 5 ρ σ ε β (5.9) ( ) 5 5 ρ σ ε β (5.9) fro whh w an rg for h hl ar h followng rssons:

43 ha onns h σ 3 ( βε ) σ ( βε ) (5.) (5.) ar wh hl bu s no as n h as. Aordng o our hor nsd h arl h oraor $ 5 dsrbs h olodal roaon of h flds (fg. 5). Rbrng ha aordng o h lroagn nrraon (s h har 3) h valu 5 s h sudosalar of lroagn hor 5 w an affr ha n h CWD h -hl s h Lorn-nvaran valu for h assv arls and auall s h orgn of h ar non-onsrvaon of h assv arls. A quson arss abou how h wrld s-hoon an has a ass and sulanousl dosn hav a harg. I s as o undrsand h orgn of hs dffrn. 6.. Th harg and ass of M nurno 6.. Donsraon of h nurno harg absn Th harg s dfnd b ngral on so volu fro a urrn dns whh s rooronal o h frs owr of fld srngh. Obvousl s ossbl o hav a as whn h subngral rsson s no qual o ro bu h ngral slf s qual o ro. I s as o hk ha w wll rv suh a rsul n as whn subngral funon hangs aordng o h haronous law. I s no dfful o alula h harg dns of h wrld s-hoon arl: ρ π π r τ (6.) Th full harg of h wrld s-hoon an b dfnd b ngrang τ q τ ρ dτ whr s h volu of arl flds. % Usng h fg. and akng () l oban: (6.) whr l s h lngh along h rular raor w q o os k l dτ (6.3) π S - h ara of orus ross-son ds s k - h wav-vor). (hr o s h alud of h wrld hoon wav fld h ln of h ross-son surfa dl - h ln of h lngh I s as o undrsand hs rsuls: baus h rng urrn s alrna h full harg s qual o ro. 6.. Donsraon of h nurno ass rsn Th arl ass s dfnd b ngral fro nrg dns whh s rooronal o h sond owr of h fld srngh. In hs as h ngral s alwas dsn fro ro f h fld s dsn fro ro. To alula h ass w us alula frs h nrg dns of h lroagn fld: 8 π ( ) ρ ε (6.) In lnar aroaon (n Gauss s ss) w hav ρ ε π. Thn (6.) an b wrn so: (6.5)

44 Usng (6.5) and a wll-known rlavs rlaonsh bwn a ass and nrg dnss: w oban: ρ ρ ε ρ o os π π (6.6) s (6.7) k l Usng (6.7) w an wr for h s-hoon ass: o ρ τ os d π k l dτ s s (6.8) Obvousl hs rsson nvr an b qual o ro. Thus auall n h frawork of CWD hr ar h ass whn h arl ass do no qual o ro bu h arl harg s qual o ro. 7.. Toologal ulars of nurno-lk arl sruur Aordng o our analss h lon ar h wrld half-rods of hoon. In hs as nurno as wrld hlod rrsns h Mobuss sr: s fld vors (lr and agn) a h nd of on ol asss o a sa wh oos dron n oarson wh h wrld hoon vors and onl b wo ols h vor os o h sarng oson (s fg. 6) Fg. 6 (fg. 6 s fro fro (Mobus Sr MahWord): h://ahworld.wolfra.o/mobussr.hl whr h anaon shows a srs of gars arrangd along a Möbus sr as h lr and agn fld vors oon) Sr vrfaon of h abov onluson abou nurno flds sruur follows fro h analss of ransforaon rors of h wrld s-hoon wav funon. (Rbr also ha a lan lroagn wav an b onsdrd as vor obnaon of wo rularl olard wavs roang n oos drons). As s shown n quanu fld hor (Gofrd and Wsskof 98; Rdr 985) h roaon ar osssss a rarkabl ror (s also son 6 of h har ) wh s llusrad arularl vsual b analss of h nurno sruur. If h roaon ours on h angl θ π around an as (hrfor ours h rurnng o h nal ss of rfrn) w fnd ha U nsad of U as was ossbl o. Dffrnl h sa vor of ss wh sn half n usual hr-dnsonal sa has abgu and asss o slf onl afr urn o h angl π (whh aords hr o h on wav lngh of lroagn wav). Fro abov follows ha s-hoon an aar onl n CWD and n lassal lnar lrodnas do no s. 8.. Paul luson rnl As w alrad non aordng o R. Fnan arl whh has h Mobus sr oolog us ob h Paul luson rnl. L show hs. Th Paul luson rnl an b wrn n followng for: arls of half-ngr sn hav ansr wavfunons and arls of ngr sn hav sr wavfunons. Th answr o h quson (Fnan Lghon and Sands 963) wh arls wh halfngral sn ar Fr arls whos aluds add wh h nus sgn undrls h Fr sass and hrfor h Paul luson rnl.

45 5 Thr s (Gofrd and Wsskof 986; Gould 995) a rarkabl ror of lon n hr dnsonal sa: whn a lon s road 36 dgrs (wha ans ha h wav funon has shfs on 36 dgrs) rurns o a sa ha looks h sa gorall bu ha s oologall dsn wh rs o s surroundngs: a ws has bn nrodud. A sond full roaon (a oal of 7 dgrs) brngs h ob bak o s orgnal sa. In hs las lur R. Fnan (Fnan 987) skhd an lnar argun for abov quson (s fg 7 whh was akn fro R. Fnan ar)): Fg. 7 To s hs frs gras h wo nds of a bl on nd n ah hand; hn nrhang h oson of our hands. So w hav nrodud a ws whh s oologall quvaln o havng road on nd of h bl b 36 dgrs. Thus whn frons ar nrhangd on us k rak of hs ld roaon and h has shf sgn hang and dsruon nrfrn o whh gvs rs. For al f A()B() dsrbs lron n sa A and lron n sa B hn h sa wh lrons nrhangd us b -A()B() and hr suroson s A()B() - A()B() Sn n h frawork of CWD h lons hav h oolog of Mobus sr h us bhav as frons of quanu fld hor. No also ha n CWD wo nurno wh lf and rgh olodal hl for on wrld rular olard hoon. Ths orrsonds o h hor of Lus d Brogl abou h nurno naur of lgh (Brogl 93a; 93b; 93) f w an h wrld hoon no h lnar. 9.. Th nurno of SM and h nurno of CWD L s oar h faurs of nurno of CWD wh nurno of SM.. s a lon.. s dsrbd b h Dra quaon;. dosn hav lr harg; 3. arls and anarls ar dsngushd onl b hl;. has all nssar nvaran rors aordng o h hor of h wak nraon. On h ohr hand h CWD nurno has ass and h SM nurno s srl asslss and as suh s dsrbd b h Wl quaon. Bu odrn rns nda ha hs sans s doubful. Thus a hang of SM s nssar whh dosn vola ohr advanag of SM. I was shown ha h nrnal oon of s-hoon flds of nurno on a rular raor s dsrbd b h Dra lon quaon wh ass qual o ro.. b h Wl quaon. In ohr words n frawork of CWD h Wl quaon s h quaon of nrnal oon of a nurno flds as h asslss arl (hoon flds). As follows fro soluon of h Wl quaon and as w hav shown abov h nrnal olodal roaon (-hls) allows h assv nurno o hav rors of h asslss nurno. On h ohr hand fro h ousd a sod wrld lroagn wav look as h assv nurno. If w dnf h CWD nurno wh h nurno of h Sandard Modl w lna h dffuls of h hor of Sandard Modl wh nu alraon of h hor. Auall all ha s nssar o ovro h dffuls s o rogn ha h nurno has an nrnal oon dsrbd b h Wl quaon. Char 6. On hadrons hor As s known h hadron hor s basd on h Yang-Mlls quaon... Inroduon. Dra and Yang-Mlls quaons of SM As follows fro h Sandard Modl hor (Ph ; Pak and Varvll ; Okun 98). h quark fal s analogu o h lon fal and h Yang-Mls quaon s h gnralsaon of h Dra lron quaon. Th Dra quaon for h lron n h rnal fld an b wrn n h for (Shff 955):

46 6 whr 3 ( ) β { ε } whr!! ε A (.) ε ar h oraors of nrg whr ε ϕ A ar h and onu rsvl; { } lron nrg and onu n h rnal lroagn fld; rsvl; ( A ) ϕ s - onal of h rnal fld; s h lgh vlo ar h lral harg and ass of h lron orrsondngl. In Quanu Chroodnas whh s dsrbd b Yang-Mlls quaon w hav quarks nsad of lrons and gluons nsad of hoons bwn whh hr ar h srong nraons nsad of h lroagn nraons. Th Yang-Mlls quaon for on quark a b wrn (Ph ; Pak and Varvll ; Okun 98) slarl o (.): q ( ) β q q q (.) q 8 a a whr q ar h quark flds gg wh G G λa s h onal of h gluon fld λ ar h Gll-Mann ars srong harg and quark ass rsvl. a g q.. On quark hor of hadrons Forall w an sa (Pak and Varvll ; Okun 98) ha hadron s dsrbd b wo or hr Dra lron quaons of (.). Thus ondonall w an na h Dra lron quaon as h on quark quaon. Bu hr w nd o ak no aoun ha h Dra quaon (.) s no h fr lron quaon. On h ohr hand h quaon (.) s ndd h quaon of h fr quark. Th rnal fld rs ar usd n h QD for h dsron of h nraon bwn h lron and ohr arls. Th slar rs n h Yang-Mlls quaon ar h nrnal fld whh dsrbs h quark-quark nraon of h sa hadron. 3.. Th drvaon of Yang-Mlls quaon n frawork of CWD In h rsn har w wll show ha h CWD rrsnaon allows us o nrr h Yang- Mlls quaon as h urvlnar lroagn wavs suroson. Obvousl o oban h Yang-Mlls quaon w us su hr on quark quaons whou ass and urn on h wrlng of h flds. 3.. lroagn fors of hr quark quaons As h Paul ars ar (Rdr 985) h gnraors of h D roaon for h hr quark lroagn rrsnaon w us us h gnraors of h 3D roaon whh ar h known hoon sn 33-ars Ŝ of h O(3) grou [3]: 3 S S S (3.) As h hr quark quaons for h arl and anarl w wll us h Dra quaons (.) n h followng for: [( 6 6 ) 6 oε β ] 6 6 [( 6 ) oε β ] (3.) whr h lf ur nd 6 ans ha hs ars ar h 66-ars of h followng : 6 S S S 6 6 S 6 S β S (3.3)

47 7 r S and wav funon 6 s h 6 ar. As s no dfful o s h abov ars gv h rgh lroagn rssons of h blnar for of h hor: h nrg: U π h Ponng vor (or onu): π P S and h s salar of h M fld: ( ) ν ν π F F Thr-quarks quaon whou ass-nraon rs Fro h abov follows ha h roon quaon an b rrsnd b hr on quark quaons.. hr lron quaons or hr ars of h salar Mawll quaons (on ar for ah oordna). Obvousl hr s a ossbl of wo drons of roaons of ah quark (h lf and h rgh quarks). Thrfor h 66 salar quaons for roon dsron us s as wll as h 66 quaons for h anroon dsron. L us fnd a frs hs quaons whou ass-nraon ung h ass-nraon rs qual o ro. Usng (3.3) fro h quaons (3.) w oban h Mawll quaon whou urrn: b b a a (3.) b b a a (3.5) As s no dfful o s ha ah ar of h quaons ab dsrbs a sara rng; h flds vors of quaons (3.) ar rolld u n h lans XOZ ZOY YOX and slarl h flds vors of h quaons (3.5) ar rolld n h lans XOY YOZ ZOX Th onsaon or gaug flds n h odrn hor Th odrn arl hor s also known as h gaug fld hor baus h nraons bwn h arls ar nrodud n h fld quaon va h gaug ransforaons. I s known (Kaffr 965; Rdr 985) ha hs rodur s ahaall quvaln o h fld vor ransforaons n h urvlnar sa whh lad o h ovaran drvav aaran. I s no dfful o show (Rdr 985) ha h lroagn fld aars naurall as a onsqun of h rqurn of h Lagrangan nvaran rlavl o h gaug ransforaons of h loal roaons n h nrnal sa of h ol fld whn h Lagrangan has h sr O() or U(). Mahaall hs s rssd hrough h rlan of h sl drvavs wh h ovaran drvavs. Th gnralaon of hs rsul on a as of 3D-sa s h Yang - Mlls fld. Th lnar gnralaon of hs sr s h non-ablan grou SU ();.. h quson s abou h hor of h non-ablan gaug flds. L s onsdr (s dals n (Rdr 985)) h roaon of so fld vor F & n hrdnsonal sa around so as on an nfnsal angl. r h valu ϕ s a roaon angl and h vor ϕ ϕ / ss h dron of h as of roaon. Th ranson fro h nal oson of a vor o h fnal oson wll b dfnd b h ransforaon:

48 8 F F F ϕ F (3.6) Th robl s o ra ndndn roaons n varous ons of sa. In ordr o onsru orrl h ovaran fld drvav w should ak aralll ransor of h vors no h sa nsad of on a fla urv as n h abov as of snoral hor. Th orrsondng analss (Rdr 985) allows us o rv an rsson slar o h rsson whh aars b h snor ransor on a fla urv. I an b shown also (Kaffr 965; Rdr 985) ha hs rsson dfns a ovaran drvav of h fld whh s ransford aordng o so rrsnaon of a grou: D D ( gm a A a ) (3.7) d whr h ars M a ar h gnraors of h roaon. I s no dfful o ak sur ha hs rsson gvs h sa ovaran drvav as found arlr n h as of lron hor and an gv h ass-nraon rs. In h followng son w wll onsdr h lroagn dsron of h ass-nraon r aaran Th lroagn dsron of h ass-nraon r aaran Th snoral hor shows ha h aaran of h nrnal ass-nraon rs s boundd wh h hr vors SP ovng along h urvlnar raor. Ths vors rrsn h ovng rhdral of h Frn-Srr (Gra 997). In h gnral as whn h lroagn wav fld vors of hr-quark arls ov along h sa urvlnar raors no onl h addonal r dfnd b h urvaur aars bu also h rs ha ar dfnd b h orson of h raor. Auall n hs as w hav: n n (3.8) b b whr b s h bnoral vor. Aordng o h Frn-Srr forulas w hav: n υ Κτ υ Τb (3.9) b υ Τn whr Τ r s h orson of h raor and Τ Τ urrns an b wrn n h for: n Κ τ Τ b π π π b Τ n π π whr w na h orson angular vlo. Τ υ r Τ Τ r s h orson radus.thus h dslan (3.) Thus w an oban h followng lroagn rrsnaon of h hr-quarks quaons:

49 9 3 3 b b a a (3.) 3 3 b b a a (3.) whr k ( 3 k ) ar h urrns of ah quark: b k k k π τ π Τ Κ and n k k π Τ (3.3) As w nod a h analss of lroagn rrsnaon of h lron quaon h harg ass and nraon bwn arls ars sulanousl durng wrlng and dvson of a hoon. In ohr words h aaran of urrns a wrlng a lnar hoon sulanousl dsrbs h aaran of a harg asss and lron nraons. Sn n hs as w hav ondonall sakng hr lroagn lron quaons s nssar o onlud ha M asss of quarks hr hargs and nraons bwn h ar dsrbd b nn urrns of h quaons (3.) or (3.). I s ossbl o assu ha hr fro h whh ar angnal lr urrns dfn hargs of quarks and arall h asss. Whhr h ohrs urrns (hr lr bnoral and hr agn noral) nsr so andns no hs arars s dfful o ll Th dsron of of h ass-nraon r aaran n h frawork of Ran gor Th aaran of addonal r follows fro h gnral hor of h vor oon along h urvlnar raor. Ths h was sudd n h vor anals n h dffrnal gor and n h hrol nubr hor hundrd ars ago (Madlung 957; Korn and Korn 96) and s wll known. Blow w onsdr so onlusons of hs hors. An vor ) ( r F an hav h followng fors (Korn and Korn 96): ) ( ) ( F F F F F F F F F r F (3.7) whr 3 3 F F F F F F F F ar h nvaran and o-varan vor odulus and and ar h bass vors whh n gnral as ar hangd fro on o on. Whn vor ovs along h urvlnar raor h aral drvavs g h vw: F F F F F (3.) whr h followng noaons ar usd: k k k k Γ Γ (3.5) (hr 3 k ) Th offns k Γ ar nad Chrsoffl sbols or bound offns. Thus for h - dron hoon

50 w oban: k Γk 3 Γk k Γk 3 k Γk (3.6) k (3.7) Th sa w an oban for h ohr drons of h hoons. Thus n h gnral as whn h lroagn fld vors of hr-kno arls ov along h urvlnar raors h addonal rs of h sa whh w oband n h as of Yang-Mlls quaon aar. No: n h frawork of CWD h Chrsoffl sbols ar no h absra ahaal valus. On on hand h ar h hsal valus; nal h ar h urrns whh aard hanks o h wrlng and orson of h lroagn vors. On h ohr hand h hav goral sns: h ar rooronal o h urvaur of h raor Κ and o h orson of h raor Τ... Th nroduon of h rs of nraons of quarks L us an h foraon of hadrons (for al roon) fro h on of vw of h raon of hooroduon γ (.) N N γ ar a gaa-quanu (hoon) roon and anroon rsvl and N s h whr nular fld n whh s aolshd h sr brakng of hoon and as onsqun h aaran of assv arls. I should onlud fro (.) ha quarks hslvs ar rodud sulanousl wh nraon bwn h. Rbr ha w had h sa n h as of h hooroduon of lron-osron ar wh h onl dffrn ha n h las as h nraon was rnal. As h onsqun of hs w oband h doubld valu of ass br. Consqunl nsad of (3.) w wll hav 6 6 [( 6 ) oε β l ] 6 6 [( 6 ) oε β l ] ( l... 9 (.) whr hr hrough l ) w ondonall dsgna h arora ass and urrns whh dsrb boh h quarks and h gluons. Aordng o (.) w hav 9 quark urrns and 9 gluon urrns so ha h suar nrgs ah of hs urrns W l 9 l us b l h sa or n ohr words suar nrg of roon s dvdd n half bwn h quarks and h gluons. I s no also dfful o lan wh h nnr vrual hoons alld gluons nsd h hadron aqur h urrns: n h srong nrns fld of quarks h us b bn aqurng so rors of assv arls. Ths onlusons n ssn do no onrad rnal and horal daa oband whn h frawork of sandard odl. 5.. M hadron odls Aordng o SM hr ar wo sors of hadron: barons whh onan hr quarks and sons whh onans wo quarks.

51 5 5.. Thr quarks odl W an suos ha n lroagn rrsnaon a baron s oologall h suroson of hr knos and has h sh of h rfol kno (fg. ): Fg. Th fgur s akn fro wbs (Möbus sr rfol kno MahWorld): h://ahworld.wolfra.o/trfolkno.hl whr h anaon shows a srs of gars oon along a Möbus sr rfol kno as h lr and agn fld vors oon. A kno s dfnd as a losd non-slf-nrsng urv bddd n hr dnsons. Kno hor was gvn s frs us whn Lord Klvn roosd a hor ha aos of Dorus s vor loos (Klvn867). Th rfol and s rror ag ar no quvaln. In ohr words h rfol kno s hral ob. I s howvr nvrbl. Th quaon of on loo (.. rng) s h Dra quaon ha has a haron soluon. Thrfor an b suosd ha h M hadrons ar h 3D suroson of wo or hr haron osllaons. On ohr words h M hadrons ar slar o h sa wav aks. Aordng o Shrudngr (Shrudngr 96) (s also (Jar 967) son 6.) h wav aks bul fro haron wavs (osllaons) don hav a dsrson.. h ar sabl. Thus w an as a fs aroaon buld h hadrons odl as h sa ak of h 3D suroson of hr harons osllaons. r us b nod ha h surosons of haron osllaons (.. Lssaous fgurs) ar no h oologal fgurs as knos baus h ar h slf-nrsng urvs. Bu w an o suos ha durng h hadron foraon as 3D Lssaous fgurs h loos wll no nrs on aoun of h rulson of urrns. Th odls wr onsrud b us of MahCAD-rogra. Probabl h blow odls dffr a lo fro h ral CWD arls and an no b usd for alulaon of h arl faurs. Bu h gv so rrsnaon abou h. Thus w suos ha h hr-loos odl (baron) s bul fro hr harons osllaon. L s hoos h followng osllaon arars: π π φ φ 3 r r r 3. φ Th argun has h vw: π : whr N : :.. N k :.. N vk : k N Th haron osllaons ar dsrbd b funons (for ah o-ordna as): X r sn( ) Y k : φ k : r sn( φ k : r3 sn( 3 φ3 Z ) As rsul w oban h followng hr-loos fgur (fg.): )

52 5 hav: Fg. To show h fld lan wrlng and wsng w hang h arar o :.Thn w. Fg Two quarks odl To buld h wo-loos odl (son) n h abov roon odl quaons w hoos: and φ and u : Z. Thn w oban h fgur : k Fg. I s nssar o no ha dndng on olarsaon of h urvlnar hoon (lan rular ll) h abov odls an hav nurous dffrn faurs. W ho ha h furhr nvsgaons wll allow us o buld or rals odls whh wll gv us h oorun o alula h CWD arl faurs. Dsusson Th abov lroagn rrsnaon of h Yang-Mlls quaons allows us o dsuss so arulars of h QCD fro on of vw of CWD odls:. Th fraonal harg of h quarks: aordng o h abov rsuls h lr fld raor of h quarks no onl has a urvaur bu also a orson; hn h angnal urrn gnrad b h lral fld vor ransor alrnas along h sa raor. Consqunl h lr harg of on kno as an ngral fro hs urrn wll b lss han h lron harg. Bu h oal harg fro all knos an b qual o h harg of h lron.. Quarks onfnn: f quarks ar wo or hr onnd knos quarks anno s n a fr sa. 3. Th hargs and asss of h quarks: n h CWD odl quarks ar dfnd b h roaon frquns of ah kno. Fro hr-kno odl follows ha fgur has a sad sruur onl a h ran rular frquns rao 3 : 3 :. So for baron odl wo of quark hargs and asss us b qual aong h and no qual o ass of hrd quarks. Analogall wo-kno fgur hav frquns rao o 3 and hrfor h son odl has wo dffrn quarks.. Non-lnar of h Yang-Mlls quaon: obvousl h Yang-Mlls quaon as h suroson of non-lnar lroagn wavs s h non-lnar quaon.