On the Origin of Charge-Asymmetric Matter. I. Geometry of the Dirac Field

Transcription

1 Journl of Modern Physis Pulished Online pril 6 in SiRes On the Origin of Chrge-symmetri Mtter I Geometry of the Dir Field lexnder Mkhlin Rpid Reserh In Southfield MI US Reeived 5 Ferury 6; epted 5 pril 6; pulished 8 pril 6 Copyright 6 y uthor nd Sientifi Reserh Pulishing In This work is liensed under the Cretive Commons ttriution Interntionl Liense (CC Y) strt This work presents new round of the uthor s pursuit for onsistent desription of the finite sized ojets in lssil nd quntum field theory Current pper lys out n dequte mthemtil kground for this quest novel frmework of the mtter-indued physil ffine geometry is developed Within this frmework () n intrinsi nonlinerity of the Dir eqution eomes self-explntory; () the spheril symmetry of n isolted lolized ojet is of dynmi origin; () the uto-loliztion is trivil onsequene of nonlinerity nd wve nture of the Dir field; () lolized ojets re split into two mjor tegories tht re lerly ssoited with the positive nd negtive hrges; (5) of these only the former n e stle s isolted ojets whih explins the glol hrge symmetry of the mtter oserved in Nture In the seond pper the nonliner Dir eqution is written down expliitly It is solved in one-ody pproximtion (in sene of externl fields) Its two nlyti solutions unequivolly re positive (stle) nd negtive (unstle) isolted hrges From the uthor s urrent perspetive the so for otined results must e developed further nd pplied to vrious prtil nd fundmentl prolems in prtile nd nuler physis nd lso in osmology Keywords Dir Field ffine Geometry Loliztion Cosmologil Chrge symmetry Introdution This work ddresses the long-stnding puzzle of how the physil Dir field of rel mtter eomes finitesized prtile This puzzle suessfully withstood severl mjor ttks undertken sine erly 9s in oth lssil nd quntum ontexts for relisti fields of mtter nd for the d ho onstruted effetive field theories The importne of definitive onlusion goes fr eyond the purely demi re The present How to ite this pper: Mkhlin (6) On the Origin of Chrge-symmetri Mtter I Geometry of the Dir Field Journl of Modern Physis

2 Mkhlin unertinty of n nswer ffets numerous studies in theoretil nd experimentl physis (eg quntum ollisions of finite-size ultrreltivisti nulei []) nd rehes s fr s the origin of the oservle mtter in the Universe drmti differene etween the hrge symmetry of the visile Universe nd n pprent hrge symmetry oserved in trnsformtions of elementry prtiles hs never reeived rtionl explntion In mid- 96s D Skhrov [] mde n ttempt to onnet the osmologil hrge symmetry with the violtion of the CP-invrine nd nonequilirium proesses in the erly hot Universe ut this hypothesis nnot e verified y lortory experiment More speified (nd exoti) senrios were onsidered y Dolgov [] [] prtilly in onnetion with the prolem of ryogenesis For n extensive review with further referenes see Ref [5] The present study onludes tht for the Dir field C nd P do not exist seprtely nd tht oth re intimtely onneted to inevitle loliztion of the Dir field into finite-sized prtiles Furthermore it ppers tht only positive hrges re ple of stle uto-loliztion in rel world The time sle nd reltive weight of ll the underlying proesses nd/or mehnisms re not yet ler ut the Universe definitely hd enough time to ondut suh n experiment Moreover experimentl studies of the lst dede [6] reveled surprising exess of positrons (nd no exess of ntiprotons) in the osmi rys whih n e n indition tht retion of the hrge-symmetri mtter in the Universe is n ongoing proess The present work ws supposed to orret nd ugment the uthor s pper [7] whih ws foused minly on the trnsient proesses with lolized prtiles The ents hve hnged with the initil progress In this work nd then in pper [8] we pursue somewht nrrower gol to find n ext uto-lolized solution ( relisti Dir prtile) whih ould serve s n input for the study of trnsient proesses The prolem is posed nd solved in novel frmework of the mtter-indued ffine geometry whih dedues geometri reltions in the spe-time ontinuum from the dynmi properties of the Dir field Frmework is set in Setion y reviewing well-known lgeri identities etween the iliner Dir forms (the Fierz identities) t ny point in spetime ontinuum (the prinipl differentile mnifold ) there exist four fields of qudruples of these forms (the Dir urrents) whih re linerly independent nd Lorentzorthogonl nd n serve s lol lgeri sis for ny four-dimensionl vetor spe inluding the infini- tesiml displements in oordinte spe In Setion we use this sis of four Dir urrents s the Crtn s moving frme in spetime nd develop the tehnique of ovrint derivtives for the vetor nd spinor fields Relying on results of Setion nd Setion we metiulously derive in Setion vrious differentil identities from the Dir equtions of motion These identities re shown to e impertive for the geometry of the ojets ssoited with the Dir field to hve ovrint form nd e independent of oordinte kground We disover tht oordinte lines nd surfes nnot e hosen y fit the Dir field nnot e emedded (this oservtion hd triggered the present work strting from [7] where the key rgument regrding loliztion ws found) In Setion 5 the differentil identities for the divergenes nd urls of the Dir urrents re written down in terms of omponents nd properties of the ongruenes of the Dir urrents re nlyzed ll omponents of the onnetions re found s funtions of the Dir field These two steps finlize the forml design of the physil ffine geometry There re only few digressions regrding physil mening of some equtions the most importnt of whih is relted to the existene of the mtterdefined world time τ nd the lol time slowdown The ltter is the min physil mehnism ehind the utololiztion It ppers tht in order to e omptile with the Dir eqution its oordinte sis indeed nnot e holonomi The known onnetions mde it possile to exmine the properties of the dmissile oordinte systems mong four tetrd vetor fields we find in Setion 6 two integrle susets of three PDEs for the oordinte lines (two hypersurfes with the orresponding norml ongruenes) nd two two-dimensionl surfes In Setion 6 we study the internl geometry of these surfes s sumnifolds of It ppers tht the twodimensionl surfe of the onstnt world time nd rdius n e only spheril whih seems to e inevitle for n isolted stle ojet The generl properties of oordinte surfes in (like their spheril symmetry nd inherent stility) re disovered in the present pper without ny ssumptions on the nture of n mient spe or Dir field It ppers tht the min qulittive hrteristi of the sttionry Dir ojet is the diretion of the xil urrent whih n point only outwrd or inwrd It must e lerly understood tht the lolly defined notions of outwrd nd inwrd re prerequisites for ny resonle disussion of the loliztion phenomenon The frme- into oordinte sis ( x ) 588

3 Mkhlin work of the mtter-indued ffine geometry not only idelly fits this gol ut lso explins the uto-loliztion s it is seen in the rel world s n intrinsi property of the Dir field This pper is ontinued in Ref [8] where the pilities of the mtter-indued ffine geometry re employed to ddress speifi prolem of existene of the uto-lolized Dir wveforms We egin with writing down the nonliner Dir eqution nd putting it in prtilly solvle form The lolized onfigurtions of the Dir field re found nlytilly in the sene of externl eletromgneti field They require the Dir spinor to hve only up- or only down-omponents when the xil urrent is pointing outwrd or inwrd respetively The up-mode is stle hs ump of invrint density nd the negtive energy E = m while the down-mode is unstle hs dip nd the positive energy E = m t lrge sptil distnes the invrint density hs universl vuum unity vlue Therefore the two modes were (y fortunte oinidene!) properly interpreted s positive nd negtive hrges The dey of unstle mode is due to the hrged Dir urrents tht nturlly osillte s e imτ suh dey requiring only the presene of n externl eletromgneti field Possily these fts explin the vivid glol hrge (eventully ryoni one) symmetry in the Universe Lst setion of pper [8] summrizes ides methods urrent results nd perspetives Vetors t Point lger of the Dir Currents Mthemtil frmework We onsider s usully the mthemtil spetime s smooth fourdimensionl mnifold so tht every point P of hs n open neighorhood tht n e mpped one-to- x x x x From the viewpoint of the differentil topology one one onto n opened suset of points hs to strt with slr funtions ( P) f λ on the urves λ ) in order to uild t eh point P the liner spe T ( P) = ( ddλ) = ( h h = ) P( λ ) λ p P λ (determined y mp of tngent -vetors h () with the omponents h with respet to the linerly independent vetors P x of the oordinte sis in eing defined vi the mpping urve nd its tngent vetors re invrint ojets; only the omponents h of vetor expliitly depend on prtiulr hoie of oordintes in tion of opertor () on the funtions f ( x) = x yields the system of ODEs for the unknown x dx dλ = h ( x) It is sid tht u re omponents of vetor if they re trnsformed s omponents dx of displement dx ny four linerly independent vetors h = d d λ = h = (with the non-degenerte mtrix h det h ) n e used s the sis Then there lso exists the inverse mtrix h of the -forms h so tht h h = δ nd h h = δ Sine ny qudruple u of numers n e expnded over the sis h we hve u= uh = uh = u Therefore u hu = uh = nd dλ = h dx ut in generl dλ re not the totl differentils of ny independent vriles Physil frmework sis of Dir urrents In physil spetime of speil reltivity points P re ssoited with events The loks of the net tht register these events re synhronized y light signls; this results in Lorentz trnsformtions etween the oordintes of events mesured y the nets of different inertil oservers Speil reltivity is sed on independene of ll physil proesses from prtiulr hoie of n inertil frme nd thus from the oordinte sis tht is used to prmeterize the events s mtter of ft the oordinte sis is uilt into mteril referene frme nd thus is n invrint ojet ll mthemtil tretments of ffine or Riemnnin geometry strt with n ssumption of the independent tngent spe with n ritrrily oriented norml sis t every point of the ontinuum (differentile mnifold) While invrine with respet to the hoie of oordintes x is trivil there nnot e solute freedom of hoosing tetrd vetors t every point the omponents h ( x) of tetrd vetors must e ontinuous funtions of the oordintes Is there wy to endow the prinipl mnifold with sis of vetor fields tht would e invrint ojets without referene to urves nd/or derivtives t point? For the physil four-dimensionl spetime the nswer is ffirmtive euse there exists mtter field the Dir field ψ ( P) oordinte slr tht provides suh sis t eh point P of the mnifold nd ssigns the ltter the sttus of physil ojet The lgeri desendnts of the Dir field re the vetor-like ojets the so-lled Dir urrents J P = j P = ψ αψ J P = J P = ψ ραψ J P =Θ P J P =Φ P () [ ] [ ] [ ] [ ] 589

4 Mkhlin of whih the lst two re the rel nd imginry prts of the omplex mtrix element etween the two hrgeonjugted onfigurtions Λ = ( ψ ) αψ =Θ Φ i nd Λ = ( ψ) αψ =Θ Φ i where ψ is the hrge-onjugte spinor The omponents J of the urrents J ( P ) depend only on the Dir field nd on prtiulr hoie of the mtries α t the point P The numers J [ ] ( P) re the oordinte slrs ut re dued omponents of the vetor urrent nother four rel numers J [ ] ( P) re ssoited with the omponents of the xil urrent The ide to use Θ nd Φ s the tetrd vetors ws first spelled out in Ref [9] i i i In these definitions n expliit form of the Dir mtries α = ( α ) = ( ρσ ) ρ i nd σ ( = ; i = ) is not speified; it is only required tht they stisfy ommuttion reltions α βα α βα = βη α β βα = nd in generl they re not just numeri mtries One n resort to prtiulr set of numeril mtries α nd β only in onjuntion with the orresponding tetrd sis h Fierz identities Completeness of the sis It ppers tht the four qudruples J ( P) ( = ) long with the slr nd pseudoslr stisfy the following identities j j = = ΘΘ = ΦΦ = j j Θ Θ Φ Φ = η () j = j Θ = j Φ = Θ = Φ =Θ Φ = where dig ( ) j = η j The Dir urrents ( P) η = is the Minkowski tensor (whih ws not ontemplted to e here) nd J re lmost lwys linerly independent In wht follows unless stted otherwise we will onsider only regulr domins where > nd use insted of J the nor- mlized urrents V = J The mtrix VV V is not degenerte nd thus hs n inverse mtrix V = δ VV = δ () y virtue of Eqution () t every point P of the si mnifold the urrents V form omplete (in the sense of liner lger) system of orthogonl (with respet to the metri η ) unit vetors η VV = η η VV = η (5) The vetor V is timelike while the other three re spelike It is lso strightforwrd to hek the following identities 5 V = η η V V = η η V (6) Employing the Dir mtries we n define the four omponents of the vetor urrent j = ψ α ψ ψγ ψ the four omponents of the xil urrent J = ψ ρ α ψ ψγ γ ψ two hrged urrents Λ = ( ψ ) αψ =Θ Φ i nd Λ = ( ψ) αψ =Θ Φ i the slr = ψ ρψ ψψ nd pseudoslr = ψ ρ ψ 5 iψγ ψ Well-known re the six omponents of the skew-symmetri ten- sor Σ = ψ α ρα α ρα ψ (or its dul Σ = Σ d [] The hrge-onjugted spinor is defined s ψ = C ψ with rel-vlued mtrix C (eg d ) ll of them re interonneted y the so-lled Fierz reltions C = ρσ ) This is smll suset of the Fierz identities tht inludes 8 si reltions nd hundreds of derivle from them They were studied in detils in Ref [] s the sis for the mthemtil reonstrution theorem [] tht sttes tht Dir spinor field n e uniquely restored vi the Dir urrents (without ny ount for the dynmis) Within this pproh it is possile to reple tetrd vetors of ny oordinte system y n equivlent Dir field thus simplifying vrious lultions [] mong the ojets onneted vi the Fierz identities is [ ] present the skew-symmetri Σ The Σ ppers to e omintion of the skew-symmetri produts [ ] j nd Θ Φ nd slrs The uthor ws not wre of this ft nd wrongfully tried [7] to employ Σ to uild sustitute for the Θ nd Indeed the neessry nd suffiient ondition for the liner independene is tht the system of liner equtions Φ J = hs only trivil solution = ; the ltter is possile if nd only if mtrix tht hs these qudruples s its olumns hs nonzero determinnt det J The determinnt of the mtrix = i When ondition on ) > the four vetors ( P ) J equls = is equivlent to two rel equtions = = det J = where is the squred module of the omplex numer J re linerly independent nd n serve s sis of vetor spe over The whih determine singulr two-dimensionl surfe in (nd thus 59

5 Mkhlin nd lso tht the V = η V is the solution of the liner system V V = η Therefore ll indies re moved up nd down y the Minkowski η or η whih is nothing ut onsequene of the Fierz identities t every point P ny qudruple of slr fields U ( P) regrdless of its origin n e presented s P ψ P liner omintion of the si qudruples V determined y the Dir field U ( P) u ( P) V ( P ) u ( P) U ( P) V ( P ) = = (7) where u re the omponents of the U with respet to the sis V n intermedite tetrd sis The omponents V ( P) of qudruple V ( P) lerly nnot e ssoited with tngent vetor like () simply euse the former re defined only in terms of the invrint om- ponents stright in the prinipl mnifold (!) while definition of the ltter requires referene to n rith- meti nd its omponents re not invrint Despite eing omplete the system V nnot immeditely serve s sis for the tngent vetors () Its ompleteness is purely lgeri y nture while liner independene nd ompleteness of the system h = h is nlyti nd is lwys tred k to liner independene of the vetors of the sis ( x ) (the liner vetor spe over : x = x ( λ) λ ) n invrint representtion of vetor s is possile only together with system of the si vetors h ; then it n e repled y slrs the tetrd omponents of the vetor s s = h s s = hs Now one n use (7) to expnd the four slrs s over the system V s = hv s = e s s = V hs (8) nd interpret the quntities e = hv s the omponents of suh vetor e = in oordinte sis tht the slrs V = e h re the omponents of e in the sis h The system of ODEs for the unknown x dx ds = e defines the integrl lines of the vetor fields e It is lso ler tht the mtrix e = V h is the inverse of mtrix e viz ee = δ nd ee = δ Let s in Eqution (8) e one of the vetors of the sis h (or of the sis e ) Then h = hv h nd e = ehv whih results in Sine det V = the inverse mtrix hv e = δ V h = δ (9) V is uniquely defined; therefore h P = V P h P = V P () e must hve invrint vlues () These equtions together with normliztion onditions (5) nd unitrity det V = llow one to interpret V ( P) s the mtrix of lol Lorentz rottion etween the ses e ( P) nd h ( P) with prmeters tht re determined y the Dir field ψ ( P) So fr s long s we re onfined to point we must refrin from ssoiting this rottion with the physil Lorentz trnsformtions of speil reltivity Sine V ( ψ ) re immeditely defined s the fields over entire mnifold we expet tht if two systems e ( P) nd h ( P) do exist they re isomorphi not only in tngent T p ut even s fields over The question is whether the integrl lines of the vetor fields h ( P) nd/or e ( P) n form oordinte net 5 n uxiliry fundmentl tensor (not metri) It tkes simple lger to verify tht t the point P the ojets The omponents of the tetrd vetors h ( P) with respet to the sis ( P) g = η hh = η ee g = η hh = η ee () n e used to move the oordinte (Greek) indies up nd down Indeed g h = η hhh = η h = h Long go E Crtn [] pointed to diffiulty ie there re no representtions of the generl liner group of trnsformtions GL tht re similr to spinor representtions of the Lorentz group of rottions From the physil stndpoint this rgument is mrginl sine Lorentz trnsformtions re etween the referene frmes of inertil oservers nd not etween different differentile mppings Crtn stted the following theorem whih vetoed spinors in Riemnnin geometry: With the geometri sense given to the word spinor it is impossile to introdue spinors into lssil Riemnnin tehnique; ie hving hosen n ritrry system of o-ordintes x for spe it is impossile to represent spinor y ny finite numer of omponents ψ i suh tht j ψ i hve ovrint derivtives of the form ψ ψ ψ i; i i j = Γ where j Γ i re determinte funtions of x Of these two undersored reservtions of Crtn the first one ws investigted y Ne'emn et l [] who proposed to overome the veto y resorting to the infinite-dimensionl representtions of the Lorentz group The present study explores the window whih is left open y the seond reservtion 59

6 Mkhlin With g thus defined we lso hve the forml reltions g hh = η g ee = η () whih n e interpreted s orthonormlity reltions for the tetrd ses h nd e if we postulte tht this g determines metri in oordinte sis Indeed y virtue of the identities () the eqution ds g dx dx ηdλ dλ ηds d S = = = () determines n intervl whih is Euliden lolly nd invrint with respet to the hoie of the oordinte sis within domin where Most likely this is not the metri tht governs propgtion of signls t lrger sle It is remrkle tht Fierz identities determine system of unit vetors even efore notion of length is introdued Finlly when g is defined ording to () nd R = then ll four vetors e = Vh regrdless of the tetrd h ( x) whih oviously does not hve this property lso eome lightlike on two-dimensionl surfe = = in spetime Oviously in this se mtrix V hs no inverse Vetor nd Dir Fields in Spetime nlyti Preliminries From now on we look t the ψ σ ( P) s the physil Dir field over four-dimensionl mnifold The points P re mpped onto points ( x x x x ) The omponents ψ σ re thought of s smooth funtions of the ritrrily prmeterized points x = ( x x x x ) of the spetime So fr we hve verified tht the lgeri struture of iliner forms of the Dir field nturlly ontins n orthogonl qudruple of unit (with respet to Minkowski metri) vetors t generi point y the rgument of lgeri ompleteness this qudruple must e isomorphi to sis of ny four non-omplnr tngent vetors h ( P) in In oordinte spe the ltter re trnsformed s dx while the former re slrs In for given fixed λ we n onsider x λ = onst s the eqution of oordinte hypersurfe nd the lines long whih ll oordintes ut x λ re onstnt s oordinte lines Tngent vetors of these lines (whih re grdients of the liner funtion ϕ ( x) = x λ λ ( λ) λ ( λ) ) re h = x x = δ λ ( λ) Their ovrint ounterprts h = x x = δ re ( λ) ( λ the grdient vetors nd the system of equtions x h ) = ( x) is integrle ut there is no metri nd no wy to determine if its oordinte lines re orthogonl One my reple x y smooth funtions of other oordintes y x = f ( y) thus redefining oordinte lines nd surfes ut suh hnge does not lter ψ ( x( y) ) nd hs nothing to do with Lorentz rottions Thus we hve to ount for two different kinds of invrine One of them is the ovrine trivil mthemtil independene from the oordinte system The seond one is the invrine of the Dir field s the mtter nd it is dominnt on every ount euse ny oneivle mesurement requires the presene of the lolized physil ojets In this setion we onsider the Dir field s known funtion of oordintes nd do not employ its eqution of motion Dir Currents s Moving Frme in Spetime The Dir field ( P) y the Dir urrents ( P) ψ is oordinte slr ut it nturlly genertes n ffine entered vetor spe (spnned e ) t P whih is similr to the tngent spe T p of the four-dimensionl mnifold t P (spnned y the vetors or h ( P) P ) These urrents onstitute omplete sis they re of unit length nd orthogonl in the sense of Eqution (5) The ontinuous field of tetrd e ( P) is emedded into Therefore n infinitesiml hnge of the V (nd eventully of the e ) from point P to point P is predetermined s dv PP = ϖ PP V P () lso predetermined is the derivtive of the slrs V ( P) ( P) V ω V ( P) mening For given displement dx in the totl hnge d ( P) d omplete system V ( P) with the oeffiients ϖ ( PP ) ω ( P) d vtive = nd it hs very simple V = V x n e expnded over = More preise is the diretionl deri- x V hv P =ωh P P () 59

7 long n ritrry vetor h in y tking = D with the diretionl derivtive D = e D long e D s ojets in prinipl mnifold Mkhlin h e we immeditely reognize the onnetions ( P ) ω ω ω V P = P V P V P = P V P () D D D D d V V = Then ( ) ( ω D = V DV = V DV ) Sine ( ) viz the ωd V V = δ D we immeditely onlude tht C η ω = ω = ω () C D D D is skew-symmetri in the first two indies Covrint Derivtives t Point in In wht follows we ompute the ovrint derivtives of the vetor nd spinor omponents with respet to different ses nd estlish their interreltion The Dir tetrd Strting from Equtions (7) nd () nd following the Crtn s ide of moving u P = u P V P frme [5] we n ompute the ovrint derivtive of the omponents of ny vetor or in terms of omponents with respet to the sis u ω u u D P = V P D P D P P (5) e u = u ω u (6) D D D D where du = Du ds re the reltive hnges of the omponents nd u d S D D is their totl hnge We expliitly see tht the presene of the physil Dir field over the prinipl mnifold immeditely endows with n ffine onnetion It lso provides nturl definition of prllel trnsport s trnsformtion tht leves the omponents u of vetor unhnged with respet to lol sis even when the lol tetrd (or oordinte hedgehog) hnges its orienttion from point to point Eqution () is speil se of Eqution (6) when u = V Tking for u the omponents of the vetor urrent j = V j = ψαψ one n define the ovrint derivtive of the Dir field without leving the prinipl mnifold Indeed ssuming tht D ψ = ψ Γ ψ (7) nd ompring with Eqution (6) one redily otins the eqution tht determines the onnetion ( V ) D D D D Γ [6] Γ α α Γ = ω α = V α (8) where α = V α nd these mtries α depending on ψ must e onsidered s primry ojets in ritrry tetrds Knowing the ffine onnetion in the sis of vetors V we n find it in ny other sis h ( P) Indeed strting from Eqution (6) we rewrite ovrint derivtive in terms of the sis vetors h h γ D VV u = u u u P (9) d D d d d D where d = d nd γ d stnds for the expression γd = V dv ωdvvv d y virtue of Equtions D D (9) we hve γ d = V d DV V = h d Dh h Using Eqution () we otin (y definition DV = ; V is mtrix of Lorentz rottion) γ = ω VVV () D d D d These invrints re nothing ut the oeffiients of rottion of the si vetors e Conversely the eqution ω gives the oeffiients of rottion d d d D D d d D d D d h with respet to the sis = h h γ h h h h h = V V V () ω D of the si vetors Coordinte sis In the oordinte piture the sis vetors V with respet to the sis h h x re ssumed to e known in dvne 59

8 Mkhlin In this se one n derive the ovrint derivtive s where Γ σ stnds for ( d ) d σ hh u u σ u u = Γ () Γ = h h γ hhh h h h () d d σ σ d σ d σ nd (euse of the term with hσ ) it is trnsformed s onnetion under hnge of the oordintes lter D ntively we ould strt with u = e e( Du ) (or just sustitute γ d from Eqution ()) nd otin nother representtion of the sme onnetion Γ σ Γ = e e ω eee e e e () D D σ σ D σ D σ whih is now expressed vi quntities tht expliitly depend on the physil Dir field Finlly using Equtions () we n invert the lst two equtions to otin γ ω = h h h = e e e (5) d d D D whih is normlly tken s n d ho definition of the oeffiients of rottion of tetrd vetors when one prefers to sty in Notly Equtions (5) nd () determine the sme ω D lthough Eqution () pp- rently elongs to nd hs nothing to do with the This my e onsidered s n evidene tht the vetors h nd the onnetions γ d re the uxiliry quntities When h is vetor nd g ( x) is tensor (not neessrily determining metri) then the ovrint derivtive h with respet to g is lso tensor [7] Using Equtions () nd (5) it is strightforwrd to hek tht if g ( x) hs the form () then g = Indeed sine γd = γd we hve λ d λ g = η h λh h λ h = hhh λ γd γd = n ide of how to find this g prtilly will eome ler only in the next pper [8] where onrete solution ψ is found Strting from there one n tke the following pth ψ V ωc γ h e nd eventully expliitly determine the g ( ψ ( x) ) Connetions for the Dir field Strting from Eqution (9) for the vetor urrent j = ψ αψ or trnslting Eqution (8) into the sis mtrix Γ 5 : D j = j γ j j D j = j γ j j (6) h it is strightforwrd to otin the following eqution for the Γ α α Γ = γ α Γ α α Γ = γ α (7) d d d d d d D where Γ d = Vd Γ D nd nothing implies tht α must e numeril mtries 6 If we introdue α = α = e d D α nd Γ = h Γ = e Γ nd use (5) then Equtions (8) nd (7) n e rewritten entirely in d D Γ α α Γ = α (8) Equtions (7) nd (8) indite tht the Dir mtries α re ovrintly onstnt with respet to the onnetion Γ of the Dir field D α = D α = The sme is true for other representtions s well Either of Equtions (8) (7) nd (8) n e solved (lgerilly) for the orresponding Γ κ The most generl solution reds s Γ = ie igρ γ ρα ρα (9) d d d d where so fr e nd g re ritrry onstnts The term 5 Indeed multiplying oth sides y D = γ α d ie d in the onnetion (9) (or the field ) is V V d we will hve in the rhs VVV D h ( V ) = VVV D ( V h ) = V D ( V ) V d D h d h D d D h α α α = = 6 This is strightforwrd to show = Γ Γ Γ Γ where D α = α Γ α α Γ γ α = α Γ α α Γ ψ αψ γ ψ αψ ψ αψ D ψ αψ Dψ αψ ψ Dα ψ ψ α Dψ ψ α ψ ψ α α ψ ψ α α γ α ψ 59

9 Mkhlin unquestionly interpreted s the eletromgneti potentil The term igρ d (or field ) ould hve een interpreted s nother field tht interts with the xil urrent J 7 The onnetion (9) ommutes with the mtrix ρ so tht Eqution (7) remins the sme when α ρα So fr it neither ommutes nor ntiommutes with ρ nd ρ viz Γ ρ ρ Γ = gρ Γ ρ ρ Γ = gρ () Similr formule rise for the hrge-onjugted onnetion Sine Cρ C = ρ nd d d Γ α αγ = γαd ieα Γ α αγ = γαd ieα The ommuttion reltions for the Dir mtries α = h α nd α V α = re α ρα α ρα = ρg nd α ρα α ρα = ρη i i Cα C = α () in nd respetively We ssume tht the mtries α re ssoited with the sis h in the tn- gent T p while mtries α elong to the prinipl mnifold In wht follows we onsider Dir field s the primry mtter field; ovrint derivtives of its iliner funtions will e omputed only using Equtions (7)-(9) 5 Connetions in different ses Equtions () nd () re nothing ut the well known formule for trnsformtion of liner onnetion etween two non-oordinte (nholonomi) ses In these ses ll quntities re funtions of the point P in the prinipl mnifold nd thus independent of the oordinte sis in the For exmple we redily hve the oordinte-independent eqution of the prllel trnsport of vetor u long vetor v = ddλ viz u ( P) = v u ( P) = v u v ( P) = If we omit indies nd use the nottion for mtrix V (s well s for V Γ κ for γ κ nd Ω for ω ) then Equtions () nd () red s κ κ Γ = Ω Ω = Γ () κ κ κ κ κ κ whih re the universl expressions 8 for ll kinds of onnetions ssoited with lol trnsformtions Equtions (6) nd (9) ugmented y definition of the derivtives d h d = d nd D = VD d = e D re fixing the omponents of ny vetor with respet to the (moving) tetrds e nd h The existene of the field of unitry mtrix of the Lorentz trnsform = V (nd then of n ffine onnetion ω D) ppers to e n mzing onsequene of the Fierz identities for iliner forms of the Dir field Finlly it is strightforwrd to hek tht one h nd e re the omponents of vetors nd γ d nd ω D re slrs the onnetion Γ trnsforms under further hnge of the oordintes s σ x x x x x σ x x x x Γ σ = Γ σ σ σ whih gurntees tht the derivtive u trnsforms s tensor Trnsformtions () nd () re redued to this formul when the tetrds re formed y the grdient vetors κ y definition γ d = ( d h ) h κ were index κ n elong to ny of the ses Therefore Eqution (9) Γ = ie igρ ρα h κ ρα h in hs the required generl form () nd n e rewritten s d d d d κ 7 In the erly dys of the Dir theory it ws firmly estlished tht = ψ ρψ nd = ψ ρψ re Lorentz slrs whih however does not gurntee tht they re slrs with respet to the generl oordinte trnsformtions of the group GL V Fok [6] resorted to speifi hoie of the Dir mtries to demonstrte tht S ρ S = ρ nd S ρ S = ρ under speil Lorentz trnsformtions S For now we shll refer to the differentil identity () = m ; sine is vetor nd is oordinte slr so re nd then (due to the Fierz identity ()) This rgument is not geometri in its nture euse it relies on the eqution of motion Intriguing is tht nd re the oordinte slrs only due to equtions of motion t the moment we hve no onvining rgument tht would llow one to rejet the presene of d in the Γ d exept tht we hve no experimentl evidene tht exists s physil field Here suh n rgument is rehed lter (with the referene to the equtions of motion) from the physil (nd then mthemtil) requirement tht nothing in physil mnifold or in oordinte spe n depend on tetrd sis h For the ske of lrity some equtions will e ending with = { = } until we reh Equtions (6) nd () nd then prove tht = 8 In generl κ is not tetrd index 595

10 Mkhlin ρ ρα ρα in the oordinte 6 Symmetry of the onnetion Γ If we nively ssume tht the Minkowski signture σ η in Equtions () nd (5) determines the lol metri of n inertil referene frme t point P (with lol oordintes y ) nd tht g ( P) of Equtions () is otined y lol oordinte trnsformtion of the η then eing σ tensor the skew-symmetri prt Γ [ ] of the onnetion (the tensor of torsion) should e zero This rgument would require in its turn tht the ovrint tetrd vetors e the grdient vetors h ( P) y x = whih is P y no mens self-evident σ There is however nother reson for the symmetry of Γ whih is hinted y the Crtn s method of moving frme The field of tetrd e ( P) elongs to nd n e used s moving frme for ll vetors s inluding the vetors dx of infinitesiml displements Consider now losed pth through the point P nd tth the nturl tetrd e ( P) to its points Then every next point of the pth hs position with respet to the tetrd of the previous point Sine the tetrd e ( P) is hnging from point to point we hve no other hoie ut to speify the trnsport of vetor s the prllel Fermi trnsport (in the sense tht the omponents of vetor with respet to the lol tetrd do not hnge) long the hosen pth We will e le to get k to P (the imge of the pth in the moving frme will e losed) with the sme ψ ( P ) nd therefore with the sme tetrd e ( P) nd mtrix V ( P ) whih is impertive if nd only if the σ omponents Γ of the onnetion s they re defined in the oordinte sis x of the re symmetri in their susripts Then the torsion tensor vnishes nd only then will we e le to ontrt the entire pth to the point P Consequently the following formule tetrd sis nd s Γ = ie ig d d D D ψ = γ γ ψ ψ = ω ω ψ () d D n e onfidently used for ny oordinte slr ψ ( x) Differentil Identities for the Dir Currents s it ws pointed out ove Equtions (6) nd (9) with the predetermined oeffiients of rottion fix the omponents of vetor with respet to n priori ritrry tetrd sis One might expet tht these equtions n e trivilly used to fix the omponents of ny tensor field However the oeffiients of rottion of the geometri tetrd h nd those of the tetrd e of the normlized Dir urrents re interonneted y Eqution () Hene the dynmi n potentilly limit fesile hoie of the sis h The oordinte system (oordinte lines) n e not ritrry; not ll oordinte vriles n even hve the mening of oordintes Therefore it seems pproprite to postpone for s long s possile expliit use of ny oordinte sis nd tret the tetrd e ψ ( x) s n orthogonl moving frme [5] n ffine geometry will e onstrutive if nd only if ll the oeffiients ω C of rottion of the tetrd e n e determined from the equtions of motion In this setion we show tht this is indeed possile There ppers to e suffiient numer of identities for the Dir urrents to ompletely determine the oeffiients ω C nd the onnetions Γ in the ovrint derivtive Dψ = ( Γ ) ψ Therefore from now on we re deling with the physil mteril Dir field tht stisfies the Dir equtions of motion α D ψ = imρψ ψ D α = imψ ρ () with the derivtive D = Γ onnetion Γ defined y Eqution (9) nd the mss prmeter m The ltter is for now rel ritrry nd stnds for the rte of mixing etween the right nd left omponents of the Dir spinor The equtions of motion for the hrge-onjugted spinor re α D ψ = imρψ [ ψ ] D α = im ψ ρ () where Dψ = ( Γ) ψ y Equtions () Divergenes of the Dir Currents [ ] is the ovrint derivtives of the hrge-onjugte Dir field nd Γ is given From the equtions of motion () one immeditely derives two well-known identities Multiplying the Dir eqution y ψ from the left nd its onjugte y ψ from the right nd tking their sum we redily otin tht 596

11 Mkhlin D j = j = () This eqution lerly indites onservtion of the timelike vetor urrent (of proility) of the Dir field The seond identity is otined from the Dir Eqution () whih is multiplied y ρ from the left (nd its onjugte from the right nd noting tht ρρ = ρρ = iρ ) It indites tht the spelike xil urrent is not onserved D = = m () nd hs the pseudoslr density s soure Sine is lolized not less thn nd the vetor is spelike it defines the rdil diretion The existene of suh diretion is distint hrteristi of ny lolized ojet Similr identities n e derived for the vetors Θ nd Φ of Setion Using Equtions () nd () ( ± ) we immeditely rrive to ovrint derivtives of the mtrix elements Λ s ( ± ) ( ± ) d ( ± ) ( ± ) ( ± ) ( ± ) Λ = Λ γ Λd Λ Λ Λ (5) D ie ie Though these vetors re omplex nd expliitly depend on the phse of ψ this dependene is ompensted in the ovrint derivtive (5) y the guge trnsformtion of the vetor potentil The derivtives of Θ nd Φ eome D Θ = Θ e Φ D Φ = Φ e Θ (6) The fields of omplex urrents Θ ±Φ i look like eing hrged with hrge e From the equtions of motion () nd using Eqution (6) it is strightforwrd to get Λ = nd onsequently D ± Θ = e Φ Φ = e Θ (7) Similrly to the vetor of xil urrent these vetors re not onserved due to eletromgneti potentil Curls of the Dir Currents In order to ess the differentil identities for the urls of the Dir urrents one hs to ompute using the equtions of motion the derivtives of the ojets T P Θ Φ whih re tres of tensors (ojets) T = ψ α Dψ P = ψ ρα Dψ Θ = ( ψ ) α Dψ nd Φ = ( ψ ) D αψ respetively These tensors re neither rel nor symmetri nd we re not onerned here out their physil interprettion T tensor or not? One would expet the solute differentil of T eing omputed ording to the Leiniz rule e s follows d d DT = T γ T γ T T (8) d d σ σ nd this expression would fix similrly to Equtions (9) nd () the omponents of the tensor T = h ht with respet to the tetrd h If this expettion turns out justified then the usul ovrint derivtive will e immeditely reprodued s T σ Γ σ T Γ T σ = eee σ T = T σ (9) λ λ λ λ λ Contrry to the expettion of (8) the nswer reds D ψ α Dψ = ψ α Dψ ψ Γ α α Γ Dψ () d = ψ α Dψ γdψ α Dψ with the lst term of Eqution (8) missing nd no hope to reover the full geometri expression (9) of the ovrint derivtive of the tensor! Contrting here indies nd nd using equtions of motion we would rrive t [7] T γ T γ T = γ T iψ α ψ () 597

12 Mkhlin with the norml ovrint derivtive in the lhs The nd n norml term γ T in the rhs originte from the ommuttor of the ovrint derivtives D D Its rel prt is the Lorentz fore Re iψ α ψ = ej F [7] [6] 9 norml terms nd how they restore the GL() ovrine The norml term enters nother identity tht follows from the Dir eqution whih rises fter ontrting indies nd in Eqution () On the one hnd we formlly hve (Cf footnote 7 The S = ψ ρψ must e slr nd the lst term in the rhs must e sent) D ψ α Dψ = ψ α Dψ γψ α Dψ = im ψ ρψ γψ α Dψ () On the other hnd y virtue of the Dir eqution the first term on the rhs of () eomes imψ ρψ lterntively one n immeditely use the equtions of motion on the lhs nd only then differentite D ψ α Dψ = imd ψ ρψ = im ψ ρψ imψ Γ ρ ρ Γ ψ () Compring the lst two equtions nd using () we finlly find tht the norml term ( ) T h λ h h T σ λ σ γ = vnishes (or t lest n e expressed vi norml field ) { mg } γ T = = () thus restoring the ovrine of Eqution () Remrkly the usul ovrine in oordinte spe is restored due to equtions of motion Eqution () yields two nontrivil onditions on the struture of the Dir urrents s follows The Rii oeffiients re rel-vlued nd skew-symmetri in the first two indies The rhs of Eqution () is rel Therefore the imginry prt of Eqution () reds s ( T T ) D D ( j j ) γ Im = γ ψ αψ ψ αψ = γ = (5) In order to filitte further nlysis of the rel prt of Eqution () let us rewrite its lhs in terms of the stu s t u xil urrent Using the dul representtion of the xil urrent s = iψ α ρα ρα ψ ( stu ) nd employing the equtions of motion we otin stu s t t s D = iψ α Dψ iψ α Dψ iψ D αψ iψ D α ψ u t s s t where the rhs is four times the nti-symmetri Hermitin prt of the energy momentum tensor Therefore the rel prt of Eqution () reds s st γ = γ = { = mg } (6) s t More non-tensors nd norml terms Next onsider the stress tensor P = iψ ρα Dψ mostly following the sme protool nd strting from its ovrint derivtive We find tht d D ψ ρα Dψ = ψ α Dψ γdψ ρα Dψ (7) One gin the lst term of Eqution (8) is missing nd thus we hve no onfidene tht the ovrint derivtive is tensor For the immedite purpose of this work we only need the equtions tht emerge fter ontrting indies nd in Eqution (7) D ψ ρα Dψ = ψ ρα Dψ γψ ρα Dψ (8) y virtue of the Dir equtions the first term in the rhs eomes mψ ρψ lterntively one n immeditely use the equtions of motion in the lhs nd only then differentite (mtries ρ nd α ommute) 9 Three remrks re to e mde here: ) the Lorentz fore in the rhs llows one to ssoite the oservle j nd with vritions of the hrge density even without referene to the Mxwell equtions uniform distriution is not distint from vuum; ) if the sis were holonomi viz γ γ = then there would hve een no wy to hieve the desired ovrine In ft the norml term will d vnish ut only if the nontrivil onditions () re met; ) in generl = ( ) R ρα ρα ief d where R d nd F re the Riemnnin urvture nd the eletromgneti field tensors respetively h 598

13 Mkhlin D ψ ρα Dψ = md ψ ρψ = m ψ ρψ m g (9) Compring the lst two equtions we finlly get the eqution γ P = igm () whih is omplementry to Eqution () Sine γ is skew-symmetri in the first two indies the imginry prt in the lhs is due to P P = ( i) DJ Sine the xil urrent is vetor we n rewrite the imginry prt of the lst eqution s [Cf footnote 7 ] λ h γ = η ( h ) h ( λ λ ) = { = mg } whih is dul to Eqution (6) The skew-symmetri Hermitin prt ( P P ) ( P P ) () must vnish stu s t u sine the rhs of Eqution () is n imginry quntity Sine j = iψ ρα ρα ρα ψ stu this yields the eqution ut γ ψ ρα Dψ ψ D αρψ ψ ρα Dψ ψ D αρψ = γ Du jt = γ D j = () whih is similr to Eqution (6) nd dul to Eqution (5) full set of prerequisites for the ovrine Considered together Equtions (5) nd () onstitute liner system of eight equtions for the six unknowns j j In generl the rnk of its mtrix equls 6 Therefore it n only hve trivil solution Sine j re the invrints of true tensor j we hve the tensor eqution j j = () Equtions (6) nd () onstitute the system of 8 equtions for unknown quntities [ λ] nd These equtions lso expliitly depend on hoie of the uxiliry field of tetrd h ( x) whih is uneptle Insisting on independene s physil (nd then mthemtil) requirement nd relizing tht does not exist s physil field we must put = Then we hve the system of 8 homogeneous equtions for only 6 unknowns [ λ] with trivil solution = () λ λ whih is similr to Equtions () tht we hd for the vetor urrent More identities re redily otined long the sme guidelines s Eqution () Nmely dupliting ()- () we ompute D ( ψ ) α Dψ nd D ( ψ ) D αψ diretly nd using equtions of motion dding up the results we otin tht ( ) D γdd ψ αψ d = γd Λ d = ψ ρα Dψ Computing in the sme wy the dul quntities D up with ( ) d D γ d D ψ αψ d = γ Λ d = nd D ( ψ ) D whih one gin is system of 8 equtions for six unknowns with only trivil solution Sine symmetri in the first two indies nd is not zero we rrive t whih y virtue of (6) results in D ραψ (5) we end (6) γ d is skew- ( ± ) ( ± ) Λ DΛ = (7) Θ Θ = e Φ Φ Φ Φ = e Θ Θ (8) The differentil identities (5) () nd (8) for the Dir urrents re written down in the ovrint tensor form nd n e trnsformed further into tetrd representtion with respet to ny tetrd Therefore it is indeed possile to overome the Crtn s veto [Cf footnote ] relying on the seond reservtion in Crtn s sttement This omplishes the proof of the sttement outlined in the footnote7 599

14 Mkhlin 5 Dir Field nd Congruenes of Curves Eh of four liner prtil differentil equtions f = e f = determine ongruene of lines euse it is equivlent to the system of three ODEs for unknown x dx e ( x) = d S = The question is whether two or three of these PDEs n e solved together (if they form omplete system) The nswer is C enoded in the properties of the rottion oeffiients ω of the orthogonl net of the tetrd e These re not given priori ut it is possile to find them s dynmi quntities This is n immedite gol of this setion Tehnilly we will rely only on Eqution (5) e = e e e = e ω e e = ω = ω ω ω ω (5) D D 5 Vetor Current nd Timelike Congruene To nlyze the lines of the vetor urrent the two otined erlier equtions () nd () j j = j = (5) must e exmined together When the invrint density of the Dir (spinor) mtter is positive = j > the vetor field j ( x) is stritly timelike; its tngent unit vetor is e[ ] ( x) j = e[ ] Therefore Eqution () eomes [ ] [ ] [ ] [ ] e e e ln e ln = (5) Contrting this eqution with ee = nd using Eqution (5) we find tht ω ω = = (5) whih is neessry nd suffiient ondition for the ongruene e [ ] to e norml [7] [8] Nmely there exists suh funtion τ ( x) tht the vetor field [ e ] ( x) is orthogonl to the fmily of surfes τ ( x) = onst [ ] τ x = f x e x dτ = τdx = fe dx = fd S (55) [ ] where τ ( x) stisfies the omplete system of three equtions e ( x) ( x) oordinte slr Contrting Eqution (5) with [ ] [ ] e we get τ = = nd f x is ( ) ln = e [ ] ln ωe (56) where ln = e ln = ln S is the derivtive in the diretion of the r S Contrtion of Equ tion (5) with e[ ] e yields ln R = ω = (57) [ ] whih indites tht ongruenes of lines defined y the system of equtions dx ds = e[ ] must experiene permnent ending (elertion) whenever the invrint density ( x) of the Dir field is not uniformly distriuted The sptil grdient of ( x) nnot vnish for ny lolized stte dditionl informtion n e extrted from Eqution () ( e[ ] ) [ ] e[ ] = Then y definition e[ ] [ ] ln = ω = ω ω ω = (58) Hene we n rewrite (56) s [ ] ln = e η ω ωe (59) whih shows tht the rhs of Eqution (59) whih ontins only geometri ojets is omponent of grdient Together with ondition (5) this onstitutes neessry nd suffiient ondition tht the funtion τ ( x) defined y Eqution (55) is n hrmoni funtion [7] τ = g τ = (5) The prmeter τ x = τ = onst is the definition of the world time For the hrmoni funtion τ of 6

15 τ ( x) the onditions of integrility for system (55) of prtil differentil equtions reds s [7] Mkhlin [ ] ( ) ln f = e η ω ωe [ ] Compring it with (59) we find tht f ( x ) = so tht the world time τ nd the proper time S re relted y [ ] dτ = ds (5) Furthermore sine f ( x ) = nd system possesses the proper time we n rewrite Eqution (59) s j ( x) = τ ( x) whih ould hve een inferred diretly from Eqution (5) Then the hrmoni nture of τ ( x) immeditely follows from the urrent onservtion j = Sine dτ is the totl differentil nd the vetor urrent j elongs in ft to the prinipl mnifold so does the intervl of the world time τ x( τ ) [ ] τ τ = j ( x) d x ( d S ) x τ = (5) nd this intervl does not depend on the pth of integrtion (the time vrile τ is holonomi oordinte) [ ] Now we n drw the mjor onlusion: The proper time S flows more slowly thn the world time τ whenever Dir mtter hs mgnified density euse of the wve nture of the Dir field its loliztion is inevitle Sine the ongruene e [ ] ppered to e norml the hypersurfes τ ( x) = τ = onst represent spe t different times τ The sttes n e onsidered sttionry only with respet to τ ; one n hope to find them only fter repling i S y i τ in the opertor of energy! 5 xil Current nd Rdil Congruene Here we hve to del with the system of equtions = = m (5) λ λ whih is similr to Equtions (5) tht we hd for the vetor urrent The only differene is tht the xil urrent hs soure m Sine there is no flux of vetor urrent in this diretion (the mount of mtter inside losed surfe remins the sme) we ssoite the rdil diretion e[ ] ( x) with the xil urrent J = e[ ] Next oserve tht y virtue of the Fierz identity () = we n prmeterize = os = sin Then the seond Eqution (5) tkes form e[ ] e[ ] ln = m = msin (5) e On the one hnd y definition [ ] = η ω = ω ω ω On the other hnd ording to Equ- tion (57) we hve e [ ] ln = [ ] ln = ω Sustituting these expressions into Eqution (5) we otin n importnt reltion ω ω = msin (55) The first of Equtions (5) eing ontrted with ee yields ω ω = = (56) so tht the ongruene of lines [ ] onst tht e is norml nd there exists suh fmily of hypersurfes [ x n x e ( x) where ρ ( x) stisfies the omplete system of three equtions e ( x) ( x) ρ x = ρ = ρ = (57) = ρ = nd n x is oordinte slr In the sme wy s efore [f (56) (57)] ontrting the first of Equtions (5) with [ ] nd ee [ ] we will get ln [ ] [ ] ln = e ω e ln = ω = (58) nd this is omptile with the ondition for integrility ln n = e[ ] [ ] ln n e of the system (57) only when n( x) ( x ) = onst ω Next we my ompute the seond derivtive of ρ Using Eqution (57) nd Eqution (57) elow we rrive t e 6

16 Mkhlin g ( [ ] ) ne ρ = = n ω [ ] ln n = n [ ] ln [ ] ln n = n [ ] ln ( n ) From here we find tht if n( x) = ( x) then ( x) ρ is the solution of n inhomogeneous wve eqution [ ] ρ = g ρ = m ρ = e = e ρ = δ (59) for the potentil ρ with the soure density proportionl to the mss prmeter m of the Dir eqution nd pseudoslr density (in stti limit it eomes the Poisson eqution) Not surprisingly this soure is equl to the derivtive of the invrint density in the diretion of the xil urrent If the invrint density ws not hnging in rdil diretion the whole ide of lolized ojet would e vgue Similrly to (55) nd (5) we hve [ ] [ ] dρ = d x e dx ρ = = d S (5) From here we onlude tht the differentil form dx is integrle nd the rdil distne x( ρ ) [ ] ρ ρ = ( x) d x ( d S ) x ρ = (5) does not depend on the integrtion pth (the oordinte vrile ρ is holonomi) 5 Congruenes of the ngulr rs Here we must del with four equtions (6) nd (8) Tking e =Θ ( x) e =Φ ( x) [] [] (n lter- ntive hoie with e [ ] e [ ] will e disussed lter) strting from Eqution (6) nd dupliting the deri- vtion of Eqution (58) we rrive t the equtions ω ω ω ω = e[ ] = e[ ] Sine y the seond Eqution (58) we hve ω = [ ] ln nd ω = [ ] ln these equtions ompletely define ω nd ω ω = [ ] ln e [ ] ω = [ ] ln e[ ] (5) Putting further in Equtions (8) Θ e [ = ] nd Φ e [ = ] nd dupliting the sheme of Eqution (5)-(57) we otin ω ω = e η η ; (5) ω ln = e[ ] η = (5) ω ω = e η η ; (55) ω ln = e[ ] η = (56) Giving index in Equtions (5) nd (56) ll possile vlues we get the following onstrints ω = ω = [ ] ln ω[ ] = ω = [ ] ln = msin ; (57) ω = e[ ] [ ] ln ω[ ] = e[ ] [ ] ln (58) Equtions (58) nd (5) re mutully omptile only when [ ] ln = [ ] ln = nd ω = ω = [ ] ln = ω = ω = [ ] ln = (59) ie when the vetors of the geodesi urvture ω nd ω of the ongruenes [] nd [] of the vetor nd xil urrents hve no projetions on the lines of the ongruenes [] nd [] of the hrged urrents Hving no metri we ssume here geodesi of n ffine spe ie suh line x ( s) tht its tngent vetor d d trnsported (with respet to n ffine onnetion this mounts to = e ω e ω e p s = x s is prllel Γ λσ ()) long the line dp ds p In our prtiulr se of the tetrd vetor e 6

17 Mkhlin Together with the previously otined Equtions (58) (58) nd (5) they give ll ω in terms of derivtives of the invrint density nd eletromgneti potentils Nmely sine ω = [ ] ln we lso hve ω ω = whih together with the first Eqution (57) entils tht [ ] ln = ω = = (5) The seond of these equtions mens tht the ongruene [] is geodesi Quite remrkly this onlusion out stti hrter of the onfigurtion tht stisfies Dir equtions of motion is rehed only fter ll the differentil identities re onsidered together The dditionl onstrints tht follow from Equtions (5) nd (55) when indies nd re given ll possile vlues re s follows ω ω ω ω ω ω = = = e[ ] (5) ω ω ω ω = e[ ] = e[ ] (5) Comined with the previous results (Eqution (5) prtiulrly) they yield ω ω ω = e[ ] = = (5) ω = ω ; = ; (5) The lst of these equtions is the neessry nd suffiient ondition for the ongruenes of lines e [ ] e [ ] e eing nonil of the ongruene e [ ] [8] This property ppers to e yet nother onsequene of nd [ ] the Dir eqution of motion whih thus gurntees tht the orthogonl tetrd is Fermi-trnsported Finlly ompring Equtions (56) nd (5) we find tht ω = = ; (55) 5 Summry Coeffiients of Rottions Tht Completely Define the Mtter-Indued ffine Geometry y now we hve sueeded to find simple expressions for ll oeffiients ω C of rottion of the sis e of the normlized Dir urrents This is the lst step in the design of the mtter-indued ffine geometry From this point one n rely on the ommon tools of the differentil geometry We n divide the not vnishing omponents of ω C into two distint groups: ) Five geodesi urvtures ( the ω C with only two distint indies) ω = ω = ω = m = msin ω = e[ ] ω = e[ ] (56) ) Only two of the ω C with ll three different indies re nonzero These re ω = ω = e[ ] ω = ω = e[ ] (57) ) The oeffiients ω C whih depend on the potentil D re of the sme form ω D = ed (58) so tht presene of eletromgneti field uses rottion of the Dir tetrd in the () tngent plne This inter-tion mkes it impossile in generl to mth Dir eqution with the ll-orthogonl system of hypersurfes It is essentil tht the only diretionl derivtive tht survived ll onstrins is [ ] nd even it n e Keeping up with the promise given in Setion we ompute following Eqution () the oeffiients of rottion γ d of the sis h γ ω ω ω ω [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] = V D V V V V V V V V V V V d D d d d d Using Equtions (56)-(57) nd employing Eqution (5) s [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] η d d d d d V V V V V V = V V we otin [ ] [ ] [ ] [ ] [ ] [ ] = e d d ( V V V V ) m V V d d γ ψ sin η η (59) 6