Crisis in quantum field theory and its overcoming. (axiomatic approach versus heuristic)

Transcription

1 Crisis in quanum field heory and is overoming. (aiomai approah versus heurisi) Kyriakos A.G.* Many known sieniss have noed he presene of risis in fundamenal physis. Despie mahemaial suess, quanum heory no answers many quesions ha are asked by sieniss. Whih of our basi physial assumpions are wrong? Wha we need o hange? The proposed arile ries o answer hese quesions using a new approah... Crisis in Physis A onsiderable number of prominen sieniss says abou he risis in fundamenal physis, whih is refleed in he fa ha he las 4 years in his field of siene here are no new resuls (Smolin, 6; Woi, 7; Seh, 7; Shroer, 8; Shroer, 9; Horgan, 996; e).. So, well-known physiis Lee Smolin in his book (Smolin, 6) noes: The sory I will ell ould be read by some as a ragedy. To pu i blunly and o give away he punh line we have failed. We inheried a siene, physis ha had been progressing so fas for so ling ha i was ofen aken as he model for how oher kinds of siene should be done. For more han wo enuries, unil he presen period, our undersanding of he laws of naure epanded rapidly. Bu oday, despie our bes effor, wha we know for erain abou hese laws is no more han wha we knew bak in he 97s. How unusual is i for hree deades o pass wihou major progress in fundamenal physis? Even if we look bak more han wo hundred years, o a ime when siene was he onern mosly of wealhy amaeurs, i is unpreedened. Sine a leas he lae eigheenh enury, signifian progress has been made on ruial quesions every quarer enury Why is physis suddenly in rouble? And wha an we do abou i? These are he enral quesions of my book. The presene of he risis is also onfirmed by he philosophers: (Popper, 98) Today, physis is in a risis. Physial heory is unbelievably suessful; i onsanly produes new problems, and i solves he old ones as well as he new ones. And par of he presen risis he almos permanen revoluion of is fundamenal heories is, in my opinion, a normal sae of any maure siene. Bu here is also anoher aspe of he presen risis: i is also a risis of undersanding. This risis of our undersanding is roughly as old as die Copenhagen inerpreaion of quanum mehanis. I is hus a lile older han die original ediion of The Lope of Sienifi Disovery. In his par of die Possrip I have ried o make again a number of proposals inended o larify wha underlies his risis of undersanding. The quesion arises abou he auses of he risis of fundamenal siene... Whih of our basi assumpions are wrong? Alhough hey use differen erminology, physiiss and philosophers onverge o he same reason. Here is wha Popper says (Popper, 98): In my view, he risis is, essenially, due o wo hings: (a) he inrusion of subjeivism ino physis; and (b) he viory of he idea ha quanum heory has reahed omplee and final ruh. Subjeivism in physis an be raed o several grea misakes. One is he posiivism or idealism of Mah. I spread o he Briish Isles (where i had been originaed by Berkeley) hrough * a.g.kyriak@homail.om

2 Russell, and o Germany hrough he young Einsein (95). This view was rejeed by Einsein in his fories (96), and i was deeply regreed by he maure Einsein (95). Anoher is he subjeivis inerpre aion of he alulus of probabiliy, whih is far older and whih beame a enral dogma of he heory of probabiliy hrough he work of Laplae. Le us onsider wha he reasons are onsisen wih his in siene. Simplisially, we an say ha siene is a mehod of obaining he answer o a quesion in order o gain some benefi for people. Sine Naure is only one, only one answer o eah quesion mus eis as well as one piure of eah phenomenon. Suh an answer is usually alled rue or orre. Mehods ha are used in order o obain only one answers from Naure are named he mehodology of siene. In praie, mehodology of siene is a number of regulaions. The basis of mehodology of sienifi heory is nowadays a law (whih ondiionally an be named Franis Baon law of siene mehodology (SEPh, 3): «Sienifi ommuniy has aken ha any heory is rue, if i is in agreemen wih eperimenal resuls when hese eperimens are invarian wih respe o he spae, ime, eperimenaliss, ehnial means and some oher ondiions». In oher words, o announe a verdi abou he ruh of he heory, he eperimens should give idenial resuls in Mosow, Los Angeles, on he Moon or Aldebaran; a hundred years ago, oday, omorrow, afer a housand years; by eperimenaliss from USA, Argenines, Mars or Venus; by means of any devie, whih is fi for a given eperimen; and he resuls of he eperimen mus be mahemaially proessed and presened by known mehods. Assuming all of his, he Baon law an be summaried as follows: The oinidene of heoreial resuls wih eperimenal resuls is he ruh in siene. This law is regularly worked unil he early h enury. Bu, as he siene developmen shows, here is some inompleeness in he Baon law: his law says nohing abou he mehod of onsruion of heory and abou heory sruure. As i urned ou, he absene of his indiaion also leads o a risis in siene. In pariular, we assume ha one of he main auses of he urren risis is preisely his poin. Wha grounds are here for suh a saemen? Hisorially, here are wo aspes of mahemais. Proof-based mahemais is no he only form (Davis and Hersh, 98). "The mahemais of Egyp, of Babylon, and of he anien Orien was all of he algorihmi ype. Dialeial mahemais -- srily logial, deduive mahemais -- originaed wih he Greeks. Bu i did no displae he algorihmi.in Eulid, he role of dialei is o jusify a onsruion-i.e., an algorihm. I is only in modern imes ha we find mahemais wih lile or no algorihmi onen. [... ] Reen years seem o show a shif bak o a onsruive or algorihmi view poin. I urned ou ha his differene is also haraerisi for physis of XX-XXI enuries. Rihard Feynman augh he aenion of physiiss on his pariulariy. In a series of leures "The Charaer of Physial Law " (Feynman, 964), he analyed hese issues in deail. The following are ypial eerps from his book: "...here are wo kinds of ways of looking a mahemais, whih for he purpose of his leure I will all he Babylonian radiion and he Eulidean or Greek radiion. In Babylonian shools in mahemais he suden would learn by doing a large number of eamples unil esablishing he general rule... Tables of numerial quaniies were available so ha hey ould solve elaborae equaions. Under he Babylonian sysem, everyhing was prepared for alulaing hings ou. Bu Eulid (under he Greek mahemaial sysem) disovered ha here was a way in whih all of he heorems of geomery ould be ordered from a se of aioms ha were simple. The Babylonian

3 3 mahemais is ha you know all of he various heorems and many of he onneions in beween...". The ne sep is hen he guessing of physial equaions, whih, Feynman argues, failiaes he guessing of new physial laws in a way ha ommon-sense feeling, philosophial priniples, or models anno. Feynman (Feynman, 964) argued ha, "In physis, we need he Babylonian mehod, and no he Eulidian or Greek mehod." The Babylonian radiion and he Eulidean or Greek radiion in he framework of physis and mahemais an be named algorihmi approah and aiomai approah ; following Karl Popper (Popper, 98), hey an be alled "insrumenalism" and "realism"; realling he T. Kuhn analysis (Kuhn, 96), we an also name hese mehods Babylonian paradigm and Greek paradigm ; or neo-posiivisi approah and lassial approah (Mah, 897; Holon, 968)). In framework of Babylonian approah (see, for eample, he mahemaial uneiform ables of Mesopoamia, Egyp papyri, he Polemeus asronomy heory) he heory is formulaed in he form of regulaions, rules, reipes of alulaions found in any way, inluding hrough rial and error or he mehod of fiing. I is lear ha he number of hese regulaions, rules and presripions should be almos as grea as he number of quesions o be answered. Any mahemai apparaus an be invened here o obain he resul, wihou undersanding is onneion wih oher par of heory. In onras, aording o Greek approah for eah area of siene mus eis one of he equivalen sysems of aioms, and all mahemai resuls of he heory mus follow onseuively from his aiom sysem (for eamples see he Eulid geomery and lassial mehanis of Newon). Alhough boh approahes are no agains he Baon law, i is diffiul o disagree wih he fa ha a sienifi heory, whih enjoys a huge number of praial reipes and insruions, found by means of rial and error mehod, onradis o our inuiive undersanding of he uniy of he world piure (Plank, 9). "Is he physial piure of he world, only more or less an arbirary reaion of our mind, or, onversely, we have o admi ha i refles a real, oally independen from us, phenomena of naure?... If, on he basis of he above, I answer affirmaively his quesion, I am well aware ha he answer lies in a erain onradiion wih he direion of he philosophy of naure, whih is headed by Erns Mah and whih now enjoys grea sympahy among sieniss. Aording o his dorine, in naure here is no oher realiy oher han our own feelings, and every sudy of naure is, ulimaely, only he eonomial adapaion of our houghs o our feelings, o whih we ome under he influene of he sruggle for eisene. The differene beween he physial and menal is purely praial and onvenional; i.e. he unique elemens of world - his is our eperiene. Alhough I am firmly onvined ha in he Mah sysem, if i is onsisenly held, here is no self-onradiion, i seems o me no less signifian ha is value is, in essene, purely formal and does no onern he foundaions of siene. The reason for his is ha he Mah sysem is ompleely alien o he mos imporan aribue of any naural siene researh: he desire o find a permanen, independen of hange of imes and he people, world piure... The goal does no lie in he omplee adapaion of our ideas owards our sensaions, bu in he omplee liberaion of he physial piure of he world from he individualiy of he reaive mind. This is a more preise saemen of wha I desribed above as he eempion from anhropomorphi elemens. When he grea reaors of he ea siene - Copernius..., Kepler..., Newon..., Huygens, Faraday,... - inrodused heir ideas o siene, surely none of hese sieniss have relied on he eonomi poin of view in he figh agains he inheried beliefs and overwhelming auhoriy. The

4 4 suppor of all heir aiviies was he unshakable belief in he realiy of heir world view. In view of his undoubed fa, i is diffiul o ge rid of he fear ha he rain of houghs of leading minds would be violaed, he fligh of imaginaion weakened, and he developmen of siene would be faally delayed, if he priniple of eonomy of Mah really beame he foal poin of he heory of knowledge. Maybe i will aually be more "eonomial" if we give he priniple of eonomy a more modes plae?" Afer 4 years, in 95, E. Shrodinger even more learly epressed dissaisfaion wih algorihmi (Babylonian, neoposiivisi) developmen of modern physis (Shrödinger, 95): (Quoes from Par I) The innovaions of hough in he las o years, grea and momenous and unavoidable as hey were, are usually overraed ompared wih hose of he preeding enury; and he disproporionae foreshorening by ime-perspeive, of previous ahievemens on whih all our enlighenmen in modem imes depends, reahes a disonering degree aording as earlier and earlier enuries are onsidered A heoreial siene, where his is forgoen, and where he iniiaed oninue musing o eah oher in erms ha are, a bes, undersood by a small group of lose fellow ravellers, will neessarily be u off from he res of ulural mankind; in he long run i is bound o arophy and ossify, however virulenly esoeri ha may oninue wihin is joyfully isolaed groups of epers... The disregard for hisorial onneedness, nay he pride of embarking on new ways of hough, of produion and of aion, he keen endeavour of shaking off, as i were, he indebedness o our predeessors, are no doub a general rend of our ime There is, however, so I believe, no oher nearly so blaan eample of his happening as he heories of physial siene in our ime... There have been ingenious onsrus of he human mind ha gave an eeedingly aurae desripion of observed fas and have ye los all ineres eep o hisorians. I am hinking of he heory of epiyles. (Quoes from Par II) There is, of ourse, among physiiss a widely popular ene, informed by he philosophy of Erns Mah, o he effe ha he only ask of eperimenal siene is o give definie presripions for suessfully foreelling he resuls of any fuure observaions from he known resuls of previous observaions. If our ask is only o predi preisely and orrely by any means whasoever, why no by false mahemais? 3.. Algorihmi mahemais vs. aiomai 3.. Why is he modern heory of elemenary pariles alled he Sandard Model? Modern heoreial physis does no preend o eplain how somehing really happens in naure. Theoreial physis only laims ha i an offer mahemaial models ha desribe phenomena well, on he basis of whih i is possible o make prediions, and hen o es hem eperimenally. Therefore, nohing resris he mahemais ha is needed for heoreiians o build models. For eample, i is aepable o use omple numbers if i urns ou ha wih he help of omple numbers, i is possible o desribe somehing ha was no possible o desribe wih he help of real numbers; or, if i urns ou ha in order o desribe he elerial and magnei ineraions of bodies i is onvenien o inrodue he noion of an eleromagnei field ha is somehow "spilled" in spae, hen i is aepable o do so. If i urns ou ha i is more effeive as far as eplanaions and prediions are onerned, o use urved spae-ime o desribe graviy, his is also aepable. The ransiion from one mahemaial model o anoher does no neessarily have o be smooh, bu an be aomplished abruply. For eample, we have a se of eperimenal fas ha an no be desribed by he previous heory (say, lassial mehanis). In addiion i is no possible a a prinipled level; i.e., in lassial mehanis here is simply no plae for suh

5 5 phenomena. In his ase we have o inven anoher mahemaial formalism, in whih he main role will be played by oher objes. For eample, quanum mehanis is he kind of formalism ha does no ransiion smoohly from lassial mehanis, bu is based on anoher basis. If some srange varians appear in he ourse of he developmen of a heory, hey should be used if his mahemaial model wih is unusualness beer desribes he realiy han any oher models. The same is rue for he ransiion from quanum mehanis o quanum field heory. There, oo, he rules of he game hange: oher objes beome key-objes, and he formalism of working wih hem beomes more diffiul. Mos imporanly, his heory should suessfully desribe and predi phenomena ha ould no be desribed by quanum mehanis. In oher words, modern heoreial physis does no represen an aggregaion of knowledge in whih all resuls follow onsisenly from a limied se of saemens. I is raher a olleion of disparae reipes mahemaial desripion models, poorly onneed wih eah oher and aeped by agreemen by he majoriy of he sieniss of he world. Hene he name of he modern heory of maer: Sandard Model. Furher we will eamine he sruure of he onemporary heory of elemenary pariles - Sandard Model - and will noe is Babylonian diffiulies. 4.. Diffiulies of quanum field heory The quanum field heory (QFT), (in pariular, in he form of he Sandard Model (SM)), is he onemporary heory of elemenary pariles and heir ineraions. Is prediions agree wih eperimens. Bu i has very srange peuliariies. The mos peuliar feaures of quanum mehanis are quanum nonloaliy, indeerminism, inerferene of probabiliies, quaniaion, wave funion ollapse during measuremen. They and some ohers are basi priniples of quanum mehanis ha are generally aeped and alled The Copenhagen inerpreaion :.A sysem is ompleely desribed by a wave funion,.the desripion of naure is essenially probabilisi. The probabiliy of an even relaed o he square of he ampliude of he wave funion. 3. The wave funion represen he sae of he sysem, whih grows gradually wih ime bu, upon measuremen, ollapses suddenly o is original sie. 4. Heisenberg's unerainy priniple: i is no possible o know he value of all he properies of he sysem a he same ime; hose properies mus be desribed by probabiliies. 5. Wave-parile dualiy. An eperimen an show boh he parile-like and wave-like properies of maer; in some eperimens boh of hese omplemenary viewpoins mus be invoked o eplain he resuls, aording o he omplemenariy priniple of Niels Bohr. 6. Sine measuring devies are essenially lassial devies, i an measure only lassial properies. These peuliariies an no be eplained on basis of quanum heory. Copenhagen inerpreaion desribes he naure of he Universe as being muh differen hen he world we observe. The quesion arises, wha grounds eis for he adopion of hese oneps? I urns ou ha here are no bases, apar from he general agreemen of physiiss. As Niels Bohr (Bohr, 96) said: "Afer a shor period of ideologial disorder and he disagreemens, aused by shor erm of resriion of "presenaion", he onsensus abou replaemen of onree images wih absra mahemaial symbols, for eample as, has been reahed. In pariular, he onree image of roaion in hree-dimensional spae has been replaed by mahemaial haraerisis of represenaion of group of roaion".

6 6 Many physiiss have subsribed o he insrumenalis (or, aording o R. Feynman, Babylonian) inerpreaion of quanum mehanis, a posiion, whih is ofen equaed wih denial all inerpreaion. I is summaried by he senene "Shu up and alulae!". While epounding as he undispued leader of he Copenhagen shool, his peuliar miure of posiivism, realism, and eisenialism, Bohr unforunaely did no aniipae he long-range effes of his eahings on fuure generaions of physiiss who laked he philosophial raining or he sophisiaion required o disinguish beween suble philosophial nuanes and heir gross over-simplifiaions. Suh physiiss ondensed Bohr's enire philosophy ino simplified enuniaions of he priniples of omplemenariy, wave-parile dualiy and he purporedly "lassial naure" of he "apparaus," and simply ignored he res. Indeed, wha Karl Popper alls he "hird group of physiiss," who emerged righ afer World War II and soon beame he overwhelming majoriy, is desribed by him as follows(prugoveki, 99): "I onsiss of hose who have urned away from disussions [onerning he onfronaion beween posiivism and realism in quanum physis] hey regard hem, righly, as philosophial, and beause hey believe, wrongly, many younger physiiss who have grown up in a period of over-speialiaion, and in he newly developing ul of narrowness, and he onemp for he non-speialis older generaion: a radiion whih may easily lead o he end of siene and is replaemen by ehnology." (Popper, 98, p. ). 4.. Wha does he algorihmiy of modern heories lead o? Briefly and meaningfully abou his peuliariy of QFT spoke one of he reaors of SM, he Nobel laureae Murray Gell-Mann. (Gell-Mann, 98): Quanum mehanis, ha miserious, onfusing disipline, whih none of us really undersands bu whih we know how o use. I works perfely, as far as we an ell, in desribing physial realiy, bu i is a ouner-inuiive disipline, as soial sieniss would say. Quanum mehanis is no a heory, bu raher a framework, wihin whih we believe any orre heory mus fi. Aording o (Anhony, 985): The quanum mehanis says nohing abou he naure of he pariles, forming he Universe, and abou fores, whih operae beween hem. More likely, i is he se of rules, wih help of whih i is possible o find, wha will ake plae aording o he given dynami heory under erain ondiions Seven Weinberg in his book Dreams of he Final Theory (993) Chap. 4. Quanum Mehanis and Is Disonens wries: A year or so ago, while Philip Candelas (of he physis deparmen a Teas) and I were waiing for an elevaor, our onversaion urned o a young heoris who had been quie promising as a graduae suden and who had hen dropped ou of sigh. I asked Phil wha had inerfered wih he e-suden s researh. Phil shook his head sadly and said, He ried o undersand quanum mehanis. In his Leures on Quanum Mehanis (nd ed., 5), Ch. 3 : General Priniples of Quanum Mehanis he eplained his remark in more deail: My own onlusion is ha oday here is no inerpreaion of quanum mehanis ha does no have serious flaws. This view is no universally shared. Indeed, many physiiss are saisfied wih heir own inerpreaion of quanum mehanis. Bu differen physiiss are saisfied wih differen inerpreaions. In my view, we ough o ake seriously he possibiliy of finding some more saisfaory oher heory, o whih quanum mehanis is only a good approimaion I is neessary o reognie ha suh sruure of heory is ompleely aepable for he ehnial appliaions. Bu a he same ime, for his reason, SM does no answer many quesions

7 7 ha are eniled o be asked by any inquisiive mind (in framework of he QFT he answers are eiher separae posulaes, or laims ha our abiliy o know he miro-world is limied due o some of is feaures). Among hese, for eample, are: wha is he origin of he mass; why fundamenal pariles - eleron and quarks - don have sie (i.e., are poin); why he wave funion has no a physial sense. We do no know he physial meaning of quaniaion; unerainy priniple of Heisenberg; a wave-parile dualism; non-ommuaiviy of dynami variables; he operaor form of QM; saisial inerpreaion of wave funion; phase and gauge invariane; four-dimensional world; Pauli elusion priniple; The heory does no eplain elemenariness of he harge; he harge and fine sruure onsan values; he harges of weak and srong ineraions; universaliy of eleron harge; eising of plus and minus harge of he pariles; parile spin; heliiy; he eising of differen kinds of pariles: inermediae bosons, lepons, mesons, baryons; and why oher pariles don eis; onfinemen of he quarks; he sabiliy and insabiliy of he elemenary pariles; eisene of pariles and anipariles; sponaneous breaking of symmery; ierbewegung; e. We do no know he physial sense of he mahemaial haraerisis of Dira's eleron equaion: why he spinor equaion does onain wo equaions, and he bispinor - four equaions? Why ino he Dira equaions he maries are used, whih in he lassial heory desribe he roaion? Why do he Pauli and Dira maries form groups? Why he mahemaial heory of groups is he basis for he searh for invarians of physial heories? Why here are many equivalen forms of he Dira eleron equaion ha ransform ino eah oher hrough formal ransformaions of maries and he wave funion? E.. The undersanding of he fa ha quanum mehanis is no a heory, bu raher a framework, wihin whih we believe any orre heory mus fi, ause he desire o onsru wihin he framework of eising heory he ompleely aiomai heory of elemenary pariles. 4.. Is i possible o move o a differen paradigm? A quesion arises, of wheher he onemporary quanum field heory is already on ha sage, when i an be formulaed aiomaially (Smilga, ): In his well-known popular leures R. (Feynmann, 964) refles on he way physial heories are buil up and disinguishes wo suh ways or, raher, wo sages in he proess of heir onsruion: (i) he "Babylonian" sage and (ii) he "Greek" sage. I is no diffiul o guess ha he erm "Babylonian" refers o anien Babylon and he orresponding physial heory is jus, geomery. A Babylonian geomeer (he words "mahemaiian ' or "physiis were no ye oined) knew many fas abou irles, riangles, and oher figures, and his undersanding was no purely empirial beause he ould also relae differen suh fas wih eah oher In oher words, his heory desribed he observed eperimenal fas well and had dire praial appliaions. Our Babylonian olleague was laking, however, a, onsisen sruured sysem in whih a se of basi simple fas are hosen as aioms and all ohers are rigorously derived as heorems Feynman wries ha a modern physiis is a Babylonian raher han a Greek in his respe: he does no are oo muh abou Rigor, and his God and ulimae Judge is Eperimen. Srily speaking, his is no quie orre. Some branhes of lassial and also of quanum physis have now quie reahed he Greek sage. Regarding quanum field heory in general, we are living now in ineresing imes when we go over from he Babylonian o he Greek sage. Therefore, we an no elude an opporuniy of eisene of oher paradigm, whih are no breaking he mahemaial apparaus of quanum mehanis, bu give he essenially oher heory. "Is i possible o make differenly?" - he analysis of his quesion from known followers of de Broglie (Andrade and Loshak, 97) leads o following saemen of a quesion:

8 8 "From he poin of view of he sensible sienifi approah, here here is no alk abou wheher posulaes of he Copenhagen shool orre or false are. The disourse goes simply abou ha any philosophial posulaes have iself no evidenial fore, even if heir logi onneion wih quanum mehanial alulaions was perfe and he grea disoveries on is basis were made. Hene, we should se for ourselves a problem: o esablish, wheher i is possible, proeeding from oher posulaes, o onsru oher inerpreaion of quanum mehanis and, hus, o ome o he heory, whih are disin from hose, whih we know, and bringing new resuls. In oher words, wheher i is possible o make differenly or even beer? From he mos general poin of view his quesion seems quie perinen and i would be very muh desirable o answer i so ha, sine no way should remain wihou use, he similar enerprise will jusify he effors, spen for i ". As eamples of suessful physial aiomai heories serve, e.g., Newon s mehani and lassial elerodynamis. In hese heories on he basis of several posulaes (or, whih is he same, aioms) all formulas, neessary for alulaing of he physial values in hese area of siene, are derived. We propose as suh a heory o onsider he nonlinear quanum field heory. In framework of his heory, i an be shown ha all he peuliariies of modern quanum field heory arise due o he fa ha i is arifiially reaed as a linear heory. The mahemais of he nonlinear heory in he linear approimaion is idenial o he mahemais of eising QFT. A he same ime, all abovemenioned feaures of modern quanum field heory in he nonlinear heory have a naural physial eplanaion and do no require arifiial inerpreaions. Moreover, i appears ha all he iems of he Copenhagen inerpreaion are a mahemaial onsequene of he heory iself, hus,. jusifying Andrade and Loshak's hope. For simpliiy and ease of he omparison wih eising quanum field heory, we will onsider only he quasi-linear represenaion of he nonlinear heory. We will presen here very brief resuls of his heory, referring o he deails and proofs in he omplee heory (he laes, mos deailed version of he heory is published in he on-line journal «Prespaeime Journal" hp://prespaeime.om/ ) 5.. The aiomai nonlinear quanum field heory (shor review) In he presen heory (i.e., nonlinear quanum field heory NQFT) he quana of eleromagnei (EM) waves are inrodued as massless boson srings of Compon wavelengh sale. I is shown ha a urling up of hese srings wihin he srong eleromagnei field he losed srings, orresponding o he massive non-linear waves - solions, are formed. Noe ha his urling up is similar o he ransformaions of he gauge ype. The peuliar solions, whih are he onsiuens of his heory, are idenial wih he objes of Sandard Model heory. In pariular hey have masses, an be only in wo saes bosoni and fermioni, an have posiive and negaive harges, e. I is shown ha he equaions of his heory fully oinide wih quanum field heory equaions. The heory iniiaes he quesion, wheher beween he modern sring heory and Sandard Model heory some onneion eiss? 5.. Inroduion. The sring heory of Plank s lengh Sring heory was buil as a unified heory ha inorporaes quanum field heory and general relaiviy. Sring heory replaes he basi priniple of poin-like pariles of quanum field heory wih he idea ha he elemenary eiaions of our universe are no poin-like pariles bu srings of Plankian lengh -33 m. They are lile lines of energy, and when one ries o divide hem up, hey jus form new lile srehes of energy. This approah, from one side, makes i possible o avoid suh diffiuly of he quanum field heory as renormaliaion. On he oher side here are

9 9 no objeions o hinking of he elemenary onsiuens of naure as sring-like objes beause of he Plank lengh is so small ha anno be observed in eperimens. The main posulaes of he iniial sring heory of he Plank sale are he following. In naure here are some one-dimensional non-loal objes, whih are haraeried by he vibraion energy -srings. The simples srings are he open mass-free boson srings. The losed srings are formed by bending of he open srings as objes wih differen number of loops (fig. 5.): Fig. 5.. Moreover, losed srings an be divided ino oher losed srings, or wo of hese an be ombined in one losed sring (fig. 5.): Fig The low-energy EM sring of Compon wave lengh In aordane wih he Plank and Einsein heory of phoon (Frauenfelder and Henley, 974) a monohromai eleromagnei (EM) wave onsiss of N monoenergei phoons, eah of whih have ero mass, energy, momenum p, and wavelengh, whose values are muually unambiguously onneed among hemselves:, p k, p, (where k k is wave veor, is redued wavelengh). The number of phoons in he EM wave is suh, ha heir oal energy N N. Phoons are bosons, and oheren phoons are able o be full ondensed in EM wave (e.g., laser beam), whih has a erain frequeny. As i is known, in framework of QED (Akhieer and Bereseskii, 965) for onsruion of he heory of he phoons and heir ineraion wih oher pariles he Mawell equaions along wih he relaionship are suffiien. To obain he phoon wave funion he seond order wave equaions for EM field veors E and H are used. Faoriing he wave equaion o he equaions for rearded and advaned waves (Akhieer and Bereseskii, 965), we reeive wo equaions of firs degree regarding he funion f k, whih adequaes o a wave veor k and is some generaliaion of he EM field veors. The equaion for his funion is equivalen o he Mawell-Loren equaions. Bu he aemp o ener he phoon funion in he oordinae represenaion has srike on an insuperable diffiuly. Aording o analysis of Landau, L.D. and Peierls, R. (Landau and Peierls, 93 and laer of Cook, R.J. (Cook, 98a;98b) and Inagaki, T. (Inagaki, 994) he phoon wave funion is nonloal obje. Aually, he f ( r, ) funion is no defined by he value of he field E( r, ) in he same poin; i depends on he field disribuion in some area, whih sies are of he order of he phoon

10 wavelengh. This means, ha he loaliaion of a phoon in a smaller area is impossible and he value f ( r, ) will no have he sense of probabiliy densiy o find a phoon in he given poin of oordinae spae Eleromagnei sring hypohesis Being guided by above resuls le us inrodues he formal represenaion of phoon as he EM sring. Noe ha sine he phoon haraerisis are muually unambiguously onneed among hemselves, we an insis ha phoon has only one own independen haraerisi. Then, keeping in mind he wavelengh of phoon, i is possible o say ha phoon in framework of QFT is ondiionally one-dimensional formaion. The one-dimensional obje, whih, on he one hand, obeys he wave equaion, and on he oher hand is no poin, in physis is referred o as a sring (i is lear, of ourse, ha his supposiion an have no relaionship o he real sruure of a phoon). These allows us o inrodue he following posulae: The fundamenal parile of an EM field - phoon - is he open relaivisi EM sring wih one wavelengh sie, whih orresponds o is energy aording o Plank's formula. The main proof of validiy of his asserion is he opporuniy o onsru on is basis he heory, whih mahemaially oinides ompleely wih he resuls of quanum field heory (QFT). Sine a phoon is a boson, we an epe ha he phoon sring heory will be ognae o he iniial modern sring heory, in whih he boson srings are he soure maerial, from whih he losed srings, i.e. he elemenary pariles, are formed. Thus, we an suppose ha under erain eernal ondiions he EM sring an sar o move along he losed urvilinear rajeory, forming he losed srings (or in oher words, solions), whih an be onsidered as EM elemenary pariles. Noe, ha in he ase of he phoon sring he inroduion of suh posulae is no needed. Aually, he bending of a rajeory of an EM wave in he srong EM field follows already from he Mawell-Loren heory. I is obvious, ha due o he quanum naure of an EM sring, he formed losed srings should possess, a leas, a res mass and he angular momenum (spin). Moreover, he deail analysis shows ha suh EM elemenary pariles an have eleri harge, heliiy and all oher haraerisis and parameers of real elemenary pariles Comparison of EM srings heory wih he modern heory of srings Le s ompare he EM- sring heory wih modern heory of srings, as i is desribed by one of he founders of his heory (Shwar, 987): "Srings an have wo various opology, whih refer as opened and losed. The open srings are piees of lines wih free ends, while he losed srings represen loops wih opology of a irle and have no free ends Various quanum-mehanial eiaion (normal modes) of sring for eah soluion of he given heory of srings are inerpreed as a sperum of elemenary pariles... The heory of srings gives he uniform approah o he rih world of he elemenary pariles, onsidered as a various modes of eiaions of a unique fundamenal sring ". Elemenary pariles are simulaneously waves, and all equaions of he quanum field heory are wave equaions. This anno be abolished by any new heory, sine SM is very well heked eperimenally.

11 The heory of srings is represened as he generaliaion of he heory of elemenary pariles. Therefore i mus resul in he wave equaions. In oher words i mus have he Lagrangian and aion funion, whih orrespond o wave equaions. In he simples ase of real relaivisi wave he equaion of moion is wrien as: The Lagrangian, whih orresponds o i, is: L and also he funion of aion: S d V d The srings are some energy formaions, whih do no aahed o any onree physial objes. Bu in naure here is no energy wihou maer. In he elemenary parile s heory he objes, whih arry energy, are de Broglie s waves, desribed by - funion. Relaivisi Lagrangian of he moion of poin parile is used as iniial Lagrangian of he heory of srings (Larranaga, 3). On he basis of he las he aion funion of he Nambu- Goo is inrodued ino he heory: S d d ' ' To pass o quanum represenaion he square roo of he Nambu-Goo aion is reorded in he linear form. The equivalen aion of Polyakov, inrodued on his basis, is he iniial Lagrangian of he sring heory: S, T d ab d a Obviously, in order o pass o real elemenary pariles wihin he framework of SM we mus eamine he funions as he wave funion. Then as we see he above aion beomes similar o he aion of he wave equaion. The heory of srings has many ineresing and imporan possibiliies, bu i anno be verified beause of smallness of he Plank lengh sale (a leas unil now is onlusions have no been onfirmed on he hadron ollider).. A he same ime he modern elemenary parile heory Sandard Model is very well heked eperimenally. Below we will show ha here is he lowenergy sring heory of Compon s wavelengh sale, whih oinides wih he Sandard Model heory in is resuls. The proposed heory inlude he graviaional ineraion (no disussed here). b 5.5. Aiomai basis of he nonlinear quanum field heory (NQFT) The aiomai basis of he proposed heory is omposed by 6 posulaes, whih do no onradi o he resuls of onemporary physis. Noe ha posulae 5 is he basis for he formaion of massive elemenary pariles as nonlinear quanum eleromagnei fields.. Posulae of fundamenaliy of an eleromagnei field: he self-onsisen Mawell- Loren mirosopi equaions are he independen fundamenal field equaions.. The Plank s-einsein's posulae of quaniaion of eleromagnei waves: he eleromagnei waves are he superposiion of he elemenary wave fields named phoons, having he erain energy, momenum and ero res mass. 3. Posulae of dualism of phoons: phoons eis as real independen objes, whih have

12 he wave properies desribed by he wave equaion following from Mawell-Loren equaions, and quanum properies, desribed by quaniaion rules, p k. (where is phoon energy, p is momenum, k is wave veor, is irular frequeny, is he Plank onsan). 4. Posulae of EM sring: Wihin he framework of he presen heory he fundamenal parile of an EM field - he phoon - an be desribed as a relaivisi EM sring of one wavelengh sie, whih orresponds o is energy aording o Plank's formula. 5. Posulae of formaion of massive pariles: wihin he framework of he presen heory under he erain eernal ondiions he EM sring an sar o move along he losed urvilinear rajeory, forming he elemenary pariles. 6) The posulae of superposiion of EM srings: in he general ase elemenary pariles are he superposiion of elemenary losed EM srings. Le us use he abovemenioned posulaes o obain he equaions of eah ype of elemenary pariles. (see in deails he book (Kyriakos, 9) and speified ariles ha are freely available) 5.6 Equaion of phoon (see in deails (Kyriakos, )) Using he posulaes and 3, we an obain from Mawell's equaions he wave equaion of he phoon. An eleromagnei (EM) waves propagaing in any direion an be represened by wo independen waves wih plane polariaions or one wave wih irular polariaion. In boh ases hese waves onains only four field veors. For eample, in he ase of a direion along he y - ais, we obain he wave equaion ˆ ˆ ˆ ˆ o p, (6.) where ˆ i, pˆ i are he operaors of energy and momenum; ˆ ; ˆ ; ˆ ˆ 4 are Dira maries, while is erain mari; in his ase: i i, i i, (6.) where and are he veors of srengh of elerial and magnei fields) The harmoni funions are he soluion of his equaion: i( ky) * i( ky) oe oe, (6.3) i( ky) * i( ky) oe oe where and k are quanified aording o posulae 3: and k p. Faoriing of (6.), we will obain he sysem: ˆ ˆ ˆ o ˆ p ˆ ˆ ˆ, (6.4) pˆ o

13 3 These equaions, aking ino aoun he quaniaion of energy and momenum, are he known quanum equaions of phoon, equivalen o one equaion (6.). The physial sense of hese equaions is revealed wih he subsiuion of epressions (6.). As a resul we obain Mawell's equaions for he advaned and rearded waves: y y y y, (6.5 ) y y y y, (6.5 ) whih onfirms he EM naure of phoon. For waves of any oher direion he same resuls an be obained by yli ransposiion of indies, or by a anonial ransformaion of maries and wave funions. In he ase of plane polariaion here are wo separae phoons, ha move along he y -ais (in our ase wih he veors, and, ), whih are desribed by wo independen sysems of equaions: y y, (6.6 ) and y y, (6.6 ) Furher le us show, how he mass of elemenary pariles is generaed Equaion of inermediae boson ( massive phoon ) In he framework of nonlinear QFT pariles aquire mass hrough an inermediae massive boson. The las is generaed wih he roaion ransformaion of EM field of EM sring. We will use he posulaes 4-5 and produe he roaion ransformaion Rˆ of phoon fields : ˆ ' R, (7.) where ' is he new wave funion, whih appears afer he ransformaion of he roaion (see fig 5.3): Fig 5.3

14 4 ' ' ' i' i' ' ' ' ' 3 4, (7.) where ', ', ', ' are he veors of he eleromagnei field, whih appear afer he roaion ransformaion and are he wave funions of he new parile wihin he framework of quanum heory. Le us eamine he EM wave, whih moves along he irular pah, so ha veors E, H and Poyning's veor S move as shown in he figure : Displaemen urren in equaions (6.5) is deermined by he epression: j dis, 4 The eleri field veor of he epression (7.3), during he moion along he urvilinear rajeory, an be reorded in he form: n, (7.4) where, and n is he uni veor of he normal o he urve. Afer differeniaion he displaemen urren of he plane wave, whih moves along he ring, an be reorded in he form: j dis (7.3) n p, (7.5) 4 4 p m p where p, and mp p is a mass, whih orresponds o phoon energy p p ; j n n and j are he normal and angenial omponens of 4 4 displaemen urren of nonlinear EM waves, respeively. A more general epression an be obained, desribing roaion in he urvilinear geomery of Riemann. In his ase i ours ha he urrens are deermined by he onneions of field, i.e., by he symbols of Rii (or, in he mos general ase, by Chrisoffel symbols) The physial sense of he generaion of mass onsiss of he following. A he momen of roaion ransformaion, a self-ineraion of own fields ours in he phoon (mass-free boson). Due o his fa he phoon fields revolve in he small region of spae. In his ase is energy does no move from infiniy o infiniy wih he speed of ligh, bu i is loked in a small spae region. This onenraion of phoon energy is a massive parile, one of haraerisis of whih is he value m. p I is remarkable ha in nonlinear QFT he mass does no appear as primary haraerisi, bu as he raio of energy o he square of he speed of ligh. Is propery - o be oeffiien in he mehanial momenum, whih deermines he ineria of parile - an be shown by he Ehrenfes heorem. Beause of he roaion, his mass assigns an angular momenum of parile, i.e. spin (in his ase, equal o ). Simulaneously he angenial urren appears. Sine in his ase he urren is sinusoidal, elerial harge of massive phoon is equal o ero. As a resul of he ransformaion of roaion we will obain he equaion of inermediae boson ( massive phoon ):

15 5 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ o p o p ', (7.6) Or, aking ino aoun he value K (see above), we will obain his equaion in form: ˆ ˆ 4 p m ' (7.7), The Lagrangian equaion (7.7) an be reorded in he form: p L D ' D ' ' ' ' mp ', (7.8) where he erm, whih direly onains he mass of inermediae boson, an be represened as follows 4 ' ' ' m ' ˆ ' 4 ' ˆ ' p ', (7.9) 8 and desribes in he nonlinear heory he energy of self-ineraion. I is no diffiul o see ha he epression (7.9) has a similariy wih Higgs's poenial. In he following seion we will eamine he quesion of he generaion of massive harge lepons: eleron and posiron 5.8. Equaions of eleron and posiron (see in deails (Kyriakos, 4)) In he ase of he plane-polaried iniial phoon he equaion (7.7) gives he possibiliy o obain wo opposiely harged pariles wih half-inegral spin of he ype of eleron and posiron. For his, we will make, ondiionally speaking, he breaking of he inermediae boson symmery. Muliplying equaion (7.7) o he lef on ' and making faoriing, we will obain he equaions of wo pariles, whih are loaed in he field of eah oher: ˆ ˆ ˆ ˆ ˆ p m, (8. ) o p ˆ ˆ ˆ pˆ ˆ m o p 4, (8. ) E E Here is lepon wave funion, whih orresponds o eleromagnei field ih 3 ih 4 afer he breakdown of inermediae boson (his -funion is no he veor, bu a so-alled - by L.D. Landau - semi-veor, i.e. spinor). In he simples ase of he produion eleron-posiron pair m m, and from (8.) we have: ˆ ˆ ˆ ˆ ˆ o p me ˆ ˆ ˆ pˆ ˆ m o e p, (8. ), (8. ) I is obvious ha in order o beome free, he eleron and posiron mus spend energy. I is no diffiul o alulae, ha during heir removing from eah oher an amoun of energy mus be spen, equal o he amoun, whih is neessary for he formaion of parile hemselves. The eernal field of pariles arises due o his proess. Using a linear wriing of he energymomenum onservaion law, we will obain for he eernal field of he parile: ˆ m ˆ p e e ˆ A, (8.3) e e e e where e indiaes eernal ; hen, subsiuing (8.3) in (8.), we obain Dira's equaion wih he eernal field: e e

16 6 ˆ ˆ ˆ ˆ ˆ e p pe me, (8.4) A a suffiienly grea disane beween he pariles, when hese fields are no imporan, we obain Dira's equaions for he free pariles - eleron and posiron: ˆ ˆ ˆ ˆ ˆ p m, (8.5 ) o e ˆ ˆ ˆ pˆ ˆ m o e 5.9. Physial sense of wave funion and is normaliing, (8.5 ) Normaliaion of he wavefunion in quanum heory In quanum mehanis and quanum field heory, he sae of he sysem is represened by a omple wave funion r,. This wave funion evolves in ime aording o equaion of eah elemenary parile ( for eleron his is Shrödinger's of Dira equaions, e.) Aording o Born's rule he probabiliy densiy of finding a parile in a erain plae P r, is proporional o square of is absolue value: pr, r,. The Born rule ells us ha he inegral r, P d, (9.) (where d is elemen of spae volume), orresponds o a probabiliy of finding a parile in a erain spae volume r, d over all possible saes mus be :. And so i follows ha he inegral r, d P, (9.) by a Tha's alled normaliaion of he wave funion. Normaliaion is jus saling r, onsan o make sure his inegral is indeed uniy Wave funion normaliaion in NQFT In he NTEC, he wave funion has a physial meaning: i is he srengh of a nonlinear eleromagnei field. Mahemaially, i is a omple quaniy. Obviously, in his form i is unnormalied. I is no diffiul o show ha, wih appropriae normaliaion, his wave funion an haraerie he posiion of parile in aordane wih he Born rule We will use he indies 'n' and 'un' as normalied and unnormalied wave funions, respeively. The value un r, un r, E H 8 r, r,. The energy densiy is equal o energy, is he energy densiy of he eleron field enlosed in some volume, divided by his volume: If we divide he non-normalied funion by he energy or mass of he parile m in aordane wih he Einsein formula m, we ge a normalied funion: un r, un r, n r,, (9.3) 8 8m Then he probabiliy densiy of finding an eleron a a given poin of spae-ime will be epressed as follows:

17 7 p unn unn unn r, nor nor unn, (9.4) 8 8m In fa, he normaliaion epresses he affimen of he parile o self-energy or mass of his parile. Indeed, he inegral un und, (9.5) deermines he energy of parile. The main normaliaion requiremen is obained by going over o a normalied funion. n n d, (9.6) Thus, he absene of physial meaning of he wave funion in quanum heory based on he fa ha in he quanum heory he energy of an elemenary parile is aken as a dimensionless uni. Clearly, his imposes no requiremen o he parile o be poin. 5.. Equaion of he massive neurino (see in deails (Kyriakos, 5)) I an be easy shown ha from he irularly polaried phoon field, massive neurino is formed wih all is known properies. I is noieable ha in his ase he heliiies of neurino and anineurino are muually opposie and no ransformaion an hange his propery. In oher words, he neurino has always he lef spiraliy, and anineurino righ spiraliy (noe ha in SM his propery is no eplained and is aeped as a posulae). 5.. Equaion of he hadrons (see in deails (Kyriakos, )) The formaion of differen hadrons is also onneed wih he desribed haraerisis of lepons. Aording o he posulae 6, wave fields an form superposiions. I is possible o show ha wih he superposiion of elemenary fields, whih are equivalen o lepons, differen hadrons an be formed, whih are desribed by Yang Mills equaion. Moreover, by he superposiion of wo lepon-like fields mesons an be formed, by he superposiion of hree lepon-like fields - baryons The non-linear quanum field heory wihou formulas Phoon as par of EM wave (i.e. as EM-sring) has he following graphi represenaion (whih, noe again, doesn have any onneion wih he unknown o us real phoon sruure) (see fig.6.). Fig.6. Le's onsider under aeped posulaes he produion of elemenary pariles from EMsring. We shall begin wih he mos simple parile - eleron. Le s onsider he reaion of eleron-posiron pair produion in nulear field (see fig. 6.; ompare wih fig. 5.):

18 8 Fig.6. We know nohing abou wha happened in he birhplae of a pair. We only see he beginning and he end of he proess. Wha ransformaion ould ake plae wih massless phoon in a field of an aom nuleus, whih has led o he ourrene of wo, ondiionally moionless pariles, boh wih mass and spin, equal o half of energy and spin of a phoon, and also wih muually opposie eleri harges? Aording o a posulae 5 for parile formaion a phoon should o be wirled ino a ring, and aording o fig. 6. i should hen be divided in wo halfs, i.e. ino oher wo rings, whih an move now wih a speed oher han he speed of ligh. Obviously, a wirled phoon ges he mass ha is equal o energy of a phoon, divided on a square of ligh speed and as i is easy o show, has a spin, equal o one. Apparenly, afer he phoon dividing we reeive wo pariles wih res mass equal o half of mass of a wirled phoon and wih spin equal o half he spin of a phoon. Le s ry o find he heorei desripion of his proess. Reurn again o a fig. 6. of he eleron-posiron pair produion proess. Condiionally speaking, we see, how from he lef he Mawell equaion of eleromagnei wave "flies ino" a very srong eleromagnei field of a nuleus. On he righ we see hen wo Dira equaions (one for eleron, anoher for posiron) fly ou (fig. 6.3). Fig. 6.3 Thus, aording o our sheme i follows ha he Dira equaions are he field equaions eah of wo pars of he wirled EM wave. Does his assumpion onradi o he eising field heory? Aually, as i is known, he Dira equaions have ohers ransformaion properies, han Mawell-Loren equaions: he wave funion of Mawell-Loren equaions is veor, whereas he funion of Dira equaion is named spinor (from o spin ). Bu remember in his onneion, ha he Dira equaion in he fifieh years is named he semi-veor equaion, and heir wave funion semi-veor beause he las are onneed wih he veor field by erain relaions (see, for eample, (Goenner, 4)). In addiion, as is known, he Mawell-Loren ime depending equaions onain si veors and si equaions (he soure equaions are possible o onsider as he iniial ondiions). A he same ime he spinor Dira eleron equaion onains wo wave funions and wo equaions, and he bispinor - aordingly, four. I is easy o see, ha here does no eis any onradiion. In EMsring heory here is a quesion abou he eleromagnei waves, no abou EM field generally. The las do no onain he longiudinal field omponens and his propery is Loren-invarian. In our ase one plane polaried EM wave onains wo field veors and generaes one spinor. A he same ime, one irle polaried EM wave onains four field veors and generaes wo spinors, i.e. bispinor.

19 9 Obviously o adjus hese requiremens, i is neessary he division of he wirled phoon o be a speial proess. Bu how an he wirled phoon be divided so ha wo anisymmerial pariles wih spin half appear? Unique opporuniy of suh proess is he division of he wirled phoon ino wo wirled half-periods of phoon aording o following sheme (fig. 4): (Fig. 6.4) Thus, ondiionally speaking, from one veor parile we reeive wo semi-veor pariles, (wo spinors) whih aording o fig. 6.4 are fully anisymmerial. In he presen heory i is shown onsisenly from mahemai poin of view, how an eleromagnei equaion of he wirled wave (no he lassial Mawell-Loren equaions, bu some nonlinear equaion of EM field!) is derived from he EM wave equaion. Then from he las he equaions of he wirled half-period waves are dedued, whih in he mari form are he Dira equaions. Furher i is also shown, ha all quanum-mehanial values and haraerisis (inluding saisial inerpreaion of wave funion, bilinear forms, e., e.) in elerodynamis of urvilinear EM waves have simple physial sense. Thus, NQFT inludes quanum mehanis as he formal linear mahemaial sruure, and, erainly, does no anel any of is resuls, bu only eplains hem and yields addiional resuls. In he researh i is shown ha he urren (harge) of eleron (posiron) is an addiional par of he Mawell displaemen urren, whih appears due o he ranspor of elerial wave veor along he urvilinear rajeory (fig. 6.5): Fig. 6.5 Here hree veors elerial, magneial and Poyning veor omprise he rihedron, orresponding o rihedron of uni veors normal, binormal and angenial whih are known in he differenial geomery as Frene-Serre rihedron. I also appears, ha his addiional erm orresponds o onneion oeffiiens of Rii (in ase of lepons) or of Crisoffel (in ase of hadrons), whih haraerie he urns of field veors a heir moion in urvilinear spae. Sine eleron and posiron orrespond o wo wirled half-period waves of one phoon, i follows from his fa ha in Universe he numbers of posiive and negaive harges mus be always fify-fify (his leads o he harge onservaion law and he neuraliy of Universe).

20 In he framework of NQFT he ineraion among pariles in he eleron equaion appears auomaially in he momen of break of he neural wirled phoon ino wo harged pariles. I orresponds o he epression of he minimal ineraion, whih in eising quanum elerodynamis is enered by "hand" or by means of gauge ransformaion (he las one, as is known, represens, aording o formal erminology of QFT, he desripion of roaions "in inernal spae of symmery" of pariles). Are here sill he bases o aep his approah? Yes, here are, and very serious ones.. In his ase he opis-mehanial analogy of Hamilon, from whih all quanum heory began, finds is subsaniaion (wave-orpusular dualism of de Broglie). Aually, NQFT is he opis of urvilinear waves, whih simulaneously an desribe he moion of he maer objes.. The ourrene of Pauli's maries, whih desribe he roaion in lassial mehanis in D spae in he Dira eleron and posiron equaions, reeives an eplanaion, as well as he ourrene of Gell-Mann maries in he Yang-Mills equaions, whih desribe he roaion in 3D spae. 3. The neessiy of a nuleus eleromagnei field reeives an eplanaion: i serves as he medium wih he big refraion inde, in whih he ligh sring bens (obviously his requiremen is idenial o he requiremen of onservaion of sysem momenum). 4. The formed EM pariles are simulaneously boh waves and pariles (i.e. he wave - parile dualism is inheren o hem). 5. Sine he wirled phoon (as boson) has ineger spin, he wirled semi-phoons have spin half (i.e., hey are fermions), we auomaially reeive an eplanaion of division of all elemenary pariles ino bosons and fermions. 6. I is easy o see, ha he fig. 6.4 refles he proess of sponaneous symmery breakdown of an iniial phoon and ourrene of mass of elemenary pariles, whih have plae in presene of a nuleus field, as some aalys of he reaion. 7. In he heory of sai spherial eleron of Loren lassial heory here are no he eleromagnei fores, apable o onsrain he repulsion of eleron pars from eah oher and herefore i is neessary o ener Poinare's fores of non eleromagnei origin. In our ase i is easy o see, ha here, owing o presene of a urren, here is he magnei par of full Loren fore direed agains elerosai fores of repulsion and ounerbalaning hem. Thus, suh eleron does no demand he inroduion of eraneous fores of an unknown origin and is sable. Abou some oher onsequenes, whih follow from he suggesion abou phoon wirling, we will briefly alk below. In he researh i is also shown ha a plane wirling and division of he irularly polaried iniial phoon are produed he neural massive lepons of he same ype as neurino and anineurino, whih are desribed by Dira bispinor equaion. Figure 6.6 shows he disribuion of he eleri field onneed wih he irularly polaried wave of he posiive (righ) and negaive (lef) heliiy (fig. 6.6): Fig. 6.6 The wirled half-periods of suh phoons give he EM pariles wih inner heliiy (fig. 6.7).

21 Fig In his ase neurino as wirled helioids represens Moebius's srip: is field veor a end of one oil has he opposie direion in relaion o he iniial veor, and only a wo oils, omes bak o he saring posiion (fig. 6.8; see also animaion of Moebius srip in hp://mahworld.wolfram.om/moebiussrip.hml). This propery of he EM-lepon veor orresponds o he same propery of wave funion of Dira lepon heory. Fig. 6.8 The mass of a parile is defined by inegral from densiy of energy, whih is proporional o he seond degree of field srengh. In his ase he inegral is always disin from ero if he field srengh is disin from ero. A he same ime he parile harge is defined by inegral from densiy of a urren, whih is proporional o he firs degree of field srengh. Obviously, here is a hane, when he subinegral epression is no equal o ero, bu he inegral is equal o ero. I is easy o hek, ha we will reeive suh resul in ase of EM neurino, sine he sub-inegral funion hanges under he harmonious law. I is ineresing ha aording o R. Feynman (Feynman, 987) he parile, whih has he Moebius srip opology, mus obey he Pauli elusion priniple. Thus in he framework of NQFT he EM elemenary pariles mus be behave as fermions of quanum field heory. Furher in researh i is desribed he ourrene of spaial pariles, as he superposiion of he wirled half-phoons. The equaions of suh pariles oinide wih Young-Mills equaions for hadrons (mesons and baryons). In his ase he spaial superposiion of wo wirled semi-phoons generaes he mesons, and spaial superposiion of hree wirled semi-phoons leads o ourrene of baryons, e.g. proon (fig. 6.9): Fig. 6.9