Quantized 6-Flat Maxwell Fields

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Quantizd 6-lat Maxwll ilds E Guivnchy PhD Emitus Pofsso of Sobonn nivsity anc * bstact viw of th path intals is alizd in th quantizd ambit and th scala thoy to QED passin thouh of th canonical

1 Quantizd 6-lat Maxwll ilds E Guivnchy PhD Emitus Pofsso of Sobonn nivsity anc * bstact viw of th path intals is alizd in th quantizd ambit and th scala thoy to QED passin thouh of th canonical quantization Thn uls wll dfin to th six-flat Maxwll filds a obtaind to thi popaatos Kywods Canonical quantization path intals quantization scala QED S d x j whos au tansfomations a dtmind by: ' havin that [ ' S[ S whn j ( x (5 (6 I INTRODCTION NDMENTL ds of fdom a obtaind fom th -foms (x thn th stnth tnso to now E E E E B B B which can b wittn in th quivalnt diffntial foms E B E B B ( W dscib th omty of (x This omty is vy usful fo non-blian thois and lctomantism dfind in cuvd spacs Th fist omtical stac of th bul of th filds (x is to consid th complx lin bundl that can dfind in a Rimannian manifold that modl th spac-tim wh can xist chas as soucs whos fild a dscibd fo ths (x This complx lin bundl that w calld L wh a copy of C fo ach x M R ( M th spac-tim Locally L C (7 wh is a opn ion in M Globally nd not hav poduct stuctu d Thn th Maxwll quations stay as: ( d dd ( ( If w consid th local coodinats ( z x with x z C x th lmnts in (7 can b viwd as in th fiu and nd j ( iu Chats of L ov cuvd spac M with j an xtnal souc om th action [ sction of this lin bundl is a map Emitus Pofsso of Thotical Physics Studis Laboatoy in th Sobonn nivsity (-mail: uivnchy@sobonndu On analou can hav it with th Möbius stip which is isomophic to th L bundl ov S

2 : M L (8 wh x C ( x Physically dscibs chad paticl Th local dsciption of th sction stays ivn by th chats Thn th connction ivn by th -fom ( x z( x bins th followin advantas: D * D * i * ( i( * * In this point w will bin ou tatmnt (6 a povids divativs of sctions that is to say D z z( x i z( x b lats nihboin fibs C x (9 c this connction is simila o quivalnt to th Chistoffl connction in nal lativity Th analysis dvlopd thouh local coodinats stablishs that chan local coodinats by au tansfomations is ivn by z'( x i z( x ' i W want D z' D z thn by (9 w hav wh ( i ' i i ( i ( ( ' thn i i i ( ' ( If w consid th au oup as th st of points i (x that is to say ( as au oup of G wh psnts lctomantism plus oth focs implyin that non-blian natu ofg If z has cha z* has cha Thn th au tansfomation tas th fom to th conjuatd lmnt: II PTH INTEGRL QNTIZTION O Expssin th action (5 as quadic fom on S d d with xpct ( x( ( [ ( i [ ( wh a intstin poblm is to hav [ ( ( w hav ( (7 ( ' (8 (9 Thn is not th invs sinc D is divnt This poblm fom th au invaianc stablish that S whn Thfo is ncssay to fix au Indd w stictin to w hav G( to say G ( which is Lontz au that is Thn with this spcific au w can chaactiz th lctomantic fild (o classical chad scala fild + lctomantism thouh th addv-popov quantization [ z*' i z* D z * ( i ( Th action S dfind by (5 is invaiant if nt cha is ( D z*( D z is a classical chad scala fild + lctomantism to now; S D * D m * (5 5 Not j fom th (5 havs as souc th stats cha and quanta hav Q Evy au fild has an associatd host and wh th au fild acquis a mass via th His mchanism th associatd host fild acquis th sam mass (in th ynman-'t Hooft au only not tu fo oth aus In ynman diaams th hosts appa as closd loops wholly composd of -vtics attachd to th st of th diaam via a au paticl at ach -vtx Thi contibution to th S-matix is xactly canclld (in th ynman-'t Hooft au by a contibution fom a simila loop of au paticls with only -vtx couplins o au attachmnts to th st of th diaam ( loop of au paticls not wholly composd of -vtx couplins

3 W dfin Thn fomally G( D G ( G( dt ( Ralizin ctain wo (that lt to ou ads w hav: G( ( x which chaactiz ou au as ood au Thn fom ( w hav D c c' c' c D D D D D i i D D ( G( d x ( G( ( ( lina lobally ( G( ( c dt wh has bn usd S[ S[ and D D H wh ( is foc Thn what w can to affim to th intal i( SEM [ SI [ D f (? f and ( Sam aumnt is coct if ( a au invaiants To show this may b is nd involv foms and nd also DD * D D ( * which is coct if G is unitay Ralizin som invsions as th invsion of S considin th ptinnt unitay tansfomations w hav: [ fo invs [ C [ C D D C wh popaato is : D D S I D C ( (5 thn D ( / so th i i ( ( i ( If is physically indpndnt of that is to say (6 is c' c ( D ( D i ( S ( So in au th lctomantic action EM is cospondd to S d ( [ ( ( showin that is is [ EM D c' D [ ( is not canclld by hosts Th opposit sin of th contibution of th host and au loops is du to thm havin opposit fmionic/bosonic natus 6 Th S aus a a nalization of th Lonz au applicabl to thois xpssd in tms of an action pincipl with Laanian dnsity L Instad of fixin th au by constainin th au fild a pioi via an auxiliay quation on adds to th "physical" (au invaiant Laanian a au bain tm L Th choic of th paamt dtmins th choic of au Th Landau au obtaind as th limit is classically quivalnt to Lonz au but postponin tain th limit until aft th thoy is quantizd impovs th io of ctain xistnc and quivalnc poofs Most quantum fild thoy computations a simplst in th ynman 't Hooft au in which ; a fw a mo tactabl in oth S aus such as th Ynni au n quivalnt fomulation of S au uss an auxiliay fild a scala fild B with no indpndnt dynamics: L B B

4 had th Landau au If thn w hav th ynman m in au (pimay sinc in nal if li th Klin-Godon quation but as nativ nom stat which is unphysical [ casts to an au [ 5 6 Now w consid oth solution o appoach o xampl th ivn by th Lontz invaiant quantization (o mi/gupta/blul [6 7 5 To this w consid III METHODS L ( W consid ou functional divativ L ( and w want th bact ou ads is dmonstat! (7 n xcis to Th au invaianc implis that has two physical popaatos Som solutions a ivn to th spct o xampl th ivn by th adiation au which bas covaianc Without soucs w choos th au Thn lctin th fild: ix ix d ( a a E (8 W hav with th othoonal conditions ( and ( imply that ( ( as: [ a and considin and Thn by th Klin-Godon quation wittn ( a s ' s ( ' i j i j ij Thn th popaato (6 tas th fom: D i i j ij i ij ( (9 ( ( With soucs w lct th au j thn D includ th additional ffcts Th sult is th sam as Thn Imposin s Considin th xpansion of fild d s waly w hav ix ix ( a a E with commutation lations w obtain [ a ( a s ' s ( ' a a ( ' ' ( ( (5 (6 5 In QT th Gupta Blul fomalism is a way of quantizin th lctomantic fild W consid a sinl photon basis of th -photon vcto spac (which not confoms a Hilbt spac is ivn by th instats wh th -momntum is null ( and th componnt th ny is positiv and is th unit polaization vcto and th indx ans fom to So is uniquly dtmind by th spatial momntum sin th ba-t notation w quip this spac with a ssquilina fom dfind by a; b; ( ( a b wh th facto is to implmnt Lontz covaianc W a usin a th mtic sinatu h Howv this ssquilina fom ivs positiv noms fo spatial polaizations but nativ noms fo tim-li polaizations Nativ pobabilitis a unphysical not to mntion a physical photon only has two tansvs polaizations not fou a 7

5 wh a ' a host stats (th spac dfind fo thm not confom a Hilbt spac Thn is ivn th popaato ( au: D i i (7 Typ of Popaato Scala Popaato TBLE I COVRINT EYNMN RLES ynman Gaph p Popaato i m i Photon Popaato i ( ( i Thn th action ivn in (5 can b -wittn to ths solutions o appoachs as th six-flat-maxwll fild action: [ S *[ m ( i (( * * * (8 Thn w hav th followin covaiant ynman uls (s Tabl * vtx * vtx i p p ( i Th invaiant IV 6-LT MXWELL IELDS SO ( actions on Minowsian flat Maxwll filds bin a pictu of th Maxwll quations fom 6-dimnsional fam 6 Th shap of ths filds and thi othoonal conditions not vay To obtain 6-lat Maxwll filds fom th fild w apply th Gupta-Blul quantization schm mntiond in th pa foot with th bact latd with th functional divativ ivn in (7 Thn th au invaianc implis two physical stats on physical stats in th Minowsian fam and oth physical stat obtaind in th oc spac includin th stats obtaind in th scond quantization du Gupta-Blul Th solutions on th Minowsy spac can b obtaind as Extnal ds: on fo incomin/outoin scala incomin/outoin photon (th cas of puly ny stats of th fild quation havin that: bin Y lm (x c Y lm l lm * fo ( th hyp-sphical hamonics and th nomalization constant cl is chosn in od to iv (9 s consqunc w hav (5 But Minowsian mods can b xtndd via a Wyl tansfomation [8 7 thouh lm ml x M (9 ' lm lm l lm x M ( wh (x a th polaization vctos as hav bn dfind in th bfo sction vifyin th inn poduct (6 Th scala mods a solutions of th Minowsian SO( lm (x invaiant (o masslss scala fild quation 6 6 Th intsctin btwn a R null-con with 5-dimnsional sufac w obtain a confomally flat -dimnsional spac This null-con is SO( invaiant 8 which solv th Maxwll quation in th Eastwood-Sin au Th cipocal tansfomation can obys to th invsion of th Wyl tansfomation and usin th path intals appoximation to stuctu cofficints 7 In thotical physics th Wyl tansfomation is a local scalin of th mtic tnso: ( x which poducs anoth mtic in th sam confomal class thoy o an xpssion invaiant und this tansfomation is calld confomally invaiant o is said to possss Wyl symmty Th Wyl symmty is an impotant symmty in confomal fild thoy

6 Thn th popaatos to th six-flat Maxwll spac a th Wihtman functions [9 Oth pocdus could hav to obtain th Maxwll quations in th Minovsian spac fom oth n dimnsional spacs althouh could b ncssay us th hyp-complx analysis tools and chiality To it is ncssay to obsv oth constuctions of th m lat Maxwll filds thouh au filds m Scholium To obtain lat Maxwll filds that com fom th physical stats in n dimnsions with n m a ncssay solv th Maxwll quations in th fam-au cospondin to th scond quantization of ths n dimnsional physical stats with polaization vctos of th fild IV CKNOWLEDGMENTS I am vy atful fo th JPS invitation to submit a pap Rfncs [ Wylman H Path Intals fo Photons: Th amwo fo th Elctodynamics Capt of th Spac-Tim Jounal on Photonics and Spintonics Vol no pp-7 [ addv L D and Popov V N "ynman Diaams fo th Yan-Mills ild" Phys Ltt B5 (967 9 [ Jacson J D ( "om Lonz to Coulomb and oth xplicit au tansfomations" mican Jounal of Physics 7 (9: 97 [ ampton P (8 Gau ild Thois (d dición Wily-VCH [5 Goy S dins ynman uls of Coulomb-au QED and th lcton mantic momnt Phys Rv D6 99 (987 [6 Gupta S (95 Poc Phys Soc 6 (7: [7 Blul K (95 Hlv Phys cta (in Gman (5: [8 Wyl H Raum Zit Mati (Spac Tim Matt Lctus on Gnal Rlativity in Gman Blin Spin 9 [9 Hall D and Wihtman S: thom on invaiant analytic functions with applications to lativistic quantum fild thoy Matmatis-fysis Mddlls (957 [ Wihtman Quantum fild thoy in tms of vacuum xpctation valus In: Physical Rviw vol 956 p 86 9

E F. and H v. or A r and F r are dual of each other.

E F. and H v. or A r and F r are dual of each other. A Duality Thom: Consid th following quations as an xampl = A = F μ ε H A E A = jωa j ωμε A + β A = μ J μ A x y, z = J, y, z 4π E F ( A = jω F j ( F j β H F ωμε F + β F = ε M jβ ε F x, y, z = M, y, z 4π