From the Maxwell s equations to the String Theory: new possible mathematical connections

Transcription

1 o the Mwell equto to the Stg Theoy: ew oble thetcl coecto Mchele Ndell,, Roo Tuco Dteto d Sceze dell Te Uvetà degl Stud d Nol edeco, Lgo S. Mcello, 88 Nol, tly Dteto d Mtetc ed lczo R. Cccool Uvetà degl Stud d Nol edeco Polo delle Sceze e delle Tecologe Mote S. gelo, V Ct (uogott, 86 Nol, tly btct th e the Secto, we decbe the oble thetc coceg the ufcto betwee the Mwell equto d the gvttol equto. th Secto we hve decbed lo oe equto coceg the gvtogetc d gvtoelectc feld. the Secto, we hve decbed the thetc coceg the Mwell equto hghe deo (thece Kluz-Kle coctfcto d eltve coecto wth tg theoy d Plubo-Ndell odel. the Secto, we hve decbed oe equto coceg the ocouttvty Stg Theoy, clly the Dc-Bo-feld cto, ocouttve oe tg cto, Che-So coulg o the be, D-be cto d the coecto wth the Mwell electodyc, Mwell equto, B-feld d guge feld. the Secto, we hve decbed oe equto coceg the ocouttve qutu echc egdg the tcle cott feld d the ocouttve clcl dyc elted to qudtc Lgg (Hlto coected wth oe equto coceg the Secto. cocluo, the Secto, we hve decbed the oble thetcl coecto betwee vou equto coceg the guet bove etoed, oe l wth oe ect of Nube Theoy (Ru odul equto coected wth the hycl vbto of the uetg, vou eltoh d l coceg π, φ thece the ue to, the zet tg d the Plubo- Ndell odel tht l booc d feoc tg.

2 toducto Mwell Equto e ott coecto betwee thetc d hyc, tuet wthout whch we would ot hve eched the eet tte of owledge uch eltvty, qutu echc d tg theoy. e Cle Mwell Thee equto e of etody otce, eeclly fo hycl ot of vew, well thetc. The utho, th e, how how to tet the fo thetcl ot of vew wth techque of vecto clculu. the ecod t the utho ee, wth the ethod of the cocetul eeet d the "ubl clculu", "yetzto" of thee equto d we hve the queto of the coecte of the och d the etece of heoe tht hould be led the th fct. ll th doe wthout toducg the Mow ce d the equto of Ete-Mwell, but eeg fo lcty oly o Mwell equto. othe bc queto tht ccoe the whole e the oblty o ot to cete tgvty d oulo yte, th egd we hve etoed the EHT theoy, theoy to ufy eltvty d qutu echc, tht ult-deol (8 deo d tht c be elted wth M-theoy d uetg theoy.

3 . O the oble thetc coceg the ufcto betwee the Mwell equto d the gvttol equto. Mwell equto Mwell equto e: Lw Gu lw of electc feld Equto ρ E ( Gu lw of getc feld B ( dy lw o the electc feld B E ( t ee lw o the getc feld E B ( t whee: t the thetcl ybol "bl"(gdet t the thetcl ybol fo the dvegece t the thetcl ybol of the oto The: E B H c o the electc feld the getc ducto the getc feld the cuet dety the delectc cott of vcuu 8,888 - / the getc eeblty of vcuu π -7 H/ the eed of lght 8 / Soe ueful ule fo the follow: The dvegece of oto lwy zeo o ( C The oto of gdet lwy zeo o ( C C The oto of oto ted ( C ( C C The dvegece of gdet the Llc o ( C

4 Mwell equto vcuu We tt fo equto (. We dvde both de by d beg d tht: c whee c the eed of lght, we obt: E c B t Recll tht the Loetz foce gve by: ( E v B ( q ( The equto, cludg ( d (, e the equto locl fo d et the clculto of electogetc feld vcuu fo ow vlue of chge dety ρ d cuet dety. Mwell equto tel f we e de of the tel the the electogetc wve ut te ccout of othe hycl heoe due to the electc ducto D, the olzto P, the getc ducto M uch tht: D E P B H M ( P d M e the e vlue of electc d getc dole e ut volue. f we code the edu le d otoc tel, the equto e lfed : Thece, we obt: Lw Gu lw of electc feld D E B H (6 Equto D ρ ( Gu lw of getc feld B ( dy lw o the electc feld B E ( t ee lw o the getc feld D H ( t whee

5 the eltve getc eeblty The eq. ( c led to oethg oe teetg though (6 d the ecedet ule : B E t E t t E t D E ( ( H t t t thece, we obt: E E (7 t the vcuu ρ, d wth v tht the eed of lght, the we obt tht: v E E ( E ( E (8 v t whch the Dlebet wve equto vcuu (v c. o the vecto B we c ue l ocedue. Vecto otetl d cl otetl f ( the dvegece zeo ug the fct tht the dvegece of oto lwy gve ull eult, the thee et vecto otetl, uch tht: o ( thece, we hve tht: E t E t E t B B (9 Wth the lt eeo, ug the ule tht the oto of gdet lwy zeo, the, toducg cl otetl φ : E φ t thece, we hve tht: E φ ( t

6 6 Utlzg the eq. (9,( we c ewte the equto (,( follow: ρ φ φ t t E t equvlet to: ρ φ t ( o (, ted, : t t c φ ( t t c φ ( φ t t c c ( Guge tfoto Let ee wht he f o the vecto otetl d cl otetl φ e de chge le: t ψ ψ φ φ u u Codeg tht the dvegece of oto ull d tht the oto of gdet zeo, the the equto of vecto B d E do ot chge (ee (9 d (. So we e of guge vce. We c ue guge vce d chooe the vecto ote, fo ele: t c φ Now wth ( d ( we obt tht: t E φ ρ φ φ φ φ t c t t E o c t φ ρ φ ( f we ubttute ( we obt: t u u u (

7 Now ( d ( cottute yte of equto o fou-vecto tht decbe the wve dvcg ce-te wth eed c. fct ( the vecto d c be decooed to the cooet the decto, y, z. fct, you olve the Mwell equto by toducg vecto otetl d cl otetl, the guge vce eloted by educg eveythg to yte of dffeetl equto fou cl fucto follow: φ φ c t t y y t t z z Thee equto ut evdece tht the behvo of electogetc wve d of lght wee ect of the e heoeo. So f the clcl theoy. ρ z y Wth egd the oble of yetzto of the equto. ollowg He d Hue (ee [], we tt ou ltc flght of fcy, lttle Mwell, who dcoveed fo thetcl ot of vew tht g oethg to the ee lw d toduced the teol vto of electc ducto D. The to ee f we c obt the yetzto of equto, ddg the te ued d tht ctce e eglgble, d toduce foulto of lezed gvty, wthout ecely toducg the Mow ce d the equto of Ete-Mwell, but ttg ly by clcl equto of Mwell. f we loo t ( ( ( (, decbed below fo coveece, we ote tht, " ee", thee e two by two l, but betwee ( d ( thee obvou "yety beg", th t let fo fol thetcl ot of vew: D ρ ( B ( u u B E t ( u uu u D H ( t (, fo ele, thee t "dety of getc chge" ρ (, whle ( coed to ( thee t coeodg "dety of getc cuet". 7

8 utheoe betwee ( d ( e oote lo the g of the tl devtve. the elty hould be boe d tht getc chge o-etet, but fty of yetzto, eeclly fo cocetul eeet d "ubl clculu", led u to y tht the equto ( d (, whch e tue coed wth eeet, e uch becue the boudy codto volvg the eglgble o ull te "dety of getc chge" d "dety of getc cuet". We tt wtg ft fo of equto of yetzto: D ρ ( B ρ ( u u uu B E ( t u uu u D H ( t whee the ( ( we hve toduced the dety g. C we dd the gvty tegth to the Mwell equto? We beg othe flght of fcy: t oble to evge the toducto of gvty thee equto? O t let effect of feld tht tect wth t? We ow tht thee et logy betwee the Coulob lw d the Gvty lw; the Coulob lw eee the tecto betwee two electc chge q d q : π q q The tegth of gvty eee the tecto betwee two e d : whee G 6,67 - N Kg - G coeodece of the Coulob tegth we hve electc feld eeble by: E π q The gvttol feld c be eeed wth: X g G ( 8

9 C we cete electc feld equvlet to gvttol feld? f we equte the two feld we ee tht we ut toduce "chge equvlet to the ": o: q G qeq π G π q eq g g π G (6 g ³/² g We would lo clude chge dety equvlet to the ρ e though dety e ut volue ρ v : ρ ρ (7 e g v utheoe, f we wt to wte, wth egd the gvty, oethg of equvlet to ( we ut to code ete η uch tht: The (, the, could be wtte follow: g η π G (8 X g qeq πη (9 u u t th ot fo the gvty you eed to locte equvlet of D E (fo le d otoc u uuu tel, tht R η X g ; thece the equvlet of ( fo the gvty would be: whle: R ρ ρ ( g uu X v e ( The ( eult of the fct tht the gvttol feld hould be ottol o coevtve, but we wll ee tht we hve to wo g o th u d eh dcove ew ueected. Phycl coecte of the de. t coect fo the hycl ot of vew, wht we hve obted fo fol ot of vew? ctce, becue of the lty of the feld, t w d tht you could get electc feld equvlet to gvty (t let te of tegth. 9

10 Wht e the dffeece? the Coulob tegth we c hve both otve d egtve electc chge d tegth of ttcto f the chge e of dffeet g, eulve f the chge e of equl g, whle the gvttol tegth we hve oly otve d the gvttol tegth oly ttctve. Howeve, t the decto of the foce, whch educe o vldte the de, the decto ut be dffeet, but t cetly tue tht " flow of chge lo flow of e", o thee e electc cuet d cuet of e, the cuet of e e th ce both ouce of gvty feld d of electc d getc feld. So the oly vld hycl ee hee tht we e toducg feld equvlet to gvttol feld, o gvto-electc d gvto-getc feld. We ty to ufy the equto [b] Now uoe tht we could ut togethe ll the equto e thoe of Mwell, o to obt ufcto of electogetc d gvttol equto eug the yetzto d the g. How to oceed wth hyothetcl ufcto of equto? t volve: to toduce the te of coecto: the cuet dety of dlceet, eeeted by the tl devtve te of R; the cuet dety ; to toduce cott γ fo the deol cotecy of the hycl deo t yety of the equto d g (X hould hve fol ece l to E E d H ut hve, wth oote g, cotbuto of l becue the behvou the clcl Mwell equto We get ufed yte of equto: D ρ ( B ρ ( R g ρ v ρe (b u u uu uu B u D X g ( gγ ( (c u t u t u ( uu R uuuu B E γ t t ( u u uu u D uu R H t t γ π c π c (

11 The vefcto of lw [b] fte the fol-thetcl yetzto d ufcto we ut vefy tht the equto do ot volte the lw of coevto of chge d ottol feld. The evou equto, u R the bece of cuet of d te-vyg gvty feld, wll be uch tht d t ; the vto e due to oveet of. electc cuet ethe flow of chge but lo of. Coevto of totl chge Te the ( d clculte the dvegece of oto (tht ull. We obt tht: D R ( H D R t t t t o ( (b, we hve tht: ρ t ρ t e f we to the tegl of ufce, obt the lw of coevto of chge: Q t e u R t cle tht ce whee d we get the ctul lw of coevto of chge t Q tht we obt fo (, d tht eeet equto of cotuty. t the evet tht we wted to code ll the te of ( we obeve tht the flow of electo de u of electoc 9,9 - Kg, whle the cott π G,8 - Kg - - C N fo whch the flow of electo of the ode of -. f we ue the eece oly of the getottc feld, theefoe oly o te-vyg getc feld d the Qe bece of electc chge, the oly dffeet fo zeo d ; but lo hee t t eglgble, becue tg π G, of g e ecod cue cuet Q t ( et ottol electc feld? tcul, the ( would ee to volte the ottol electc feld, yg tht oe h electc feld eedcul to the oveet d ged ccle oud the. fct the ottol electc feld getottc feld vld, th becue tht the flow gfct eed to cheve eoou out le to. New heoeo?

12 The (c y tht the eece of o ttc electc d getc feld, the the gvttol feld ot ottol, tg to ccout tht thee o getc cuet dety (. fct fo wht h o f ugget to u the foul d clculto, the flow vey lttle fluece o electogetc feld, but t cetly oble othewe. de, theefoe, to elot the (c to cete the oote tegth to the gvtto, o to obt electogetc levtto (ot ut the tgvty! But we e vey cloe le effect.. O oe equto coceg the Gvtoelectc d Gvtogetc feld [] Codeg the Ete-Mwell foulto of lezed gvty, eble lty to the thetcl fo of the electogetc Mwell equto c be foud. logy to electoget thee et gvttol cl d vecto otetl, deoted by Φ g d g, eectvely. toducg the gvtoelectc d gvtogetc feld e Φ g d b g ( the gvttol Mwell equto c be wtte the followg fo: 6πG e πgρ, b, e, b ( c whee ρv the flu d G the gvttol cott. The feld e decbe the gvttol feld fo ttoy dtbuto, whee b decbe et gvttol feld oduced by ovg e. t ctcl teetue T c oe tel becoe uecoducto tht, the etce goe to. Suecoducto hve eegy g of oe E. T. Th eegy g ete uecoductg electo below fo ol electo bove the g. t teetue below T, electo (tht e feo e couled, clled Cooe, whch e boo. c We ote tht, th ce, thee the lcto of the eltoh of Plubo-Ndell odel,.e. ( G G f ( φ 6 R ρ νσ ν d g g g T ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, ( κ g geel eltoh tht l booc d feoc tg ctg ll tul yte. ottg uecoducto geete getc ducto feld, the o clled Lodo oet e B ω (6 e whee ω the gul velocty of the ottg g d e deote eleety chge. t hould be oted tht the getc feld T eeet [] oduced by the otto of the g, d ot by cuet of Cooe tht e ovg wth the g. T d h collegue ly otulte equvlece betwee the geeted B feld, eq. (6 wth gvttol feld by oog o clled gvtogetc Lodo effect. g c

13 Let R deote the du of the ottg g, the eq. (6 ut lt o the l llowble getc ducto, B, whch gve by B e BTC. (7 e R f the getc ducto eceed th vlue, the etc eegy of the Cooe eceed the u eegy g, d the Cooe e detoyed. The ottg g o loge B c T, uecoducto. Moeove, the getc ducto ut ot eceed the ctcl vlue ( whch the l getc ducto tht c be uted t teetue T, d deedet o the tel. th oble ceo, the getc ducto feld B equvlet to gvtohoto (gvttol feld b g. Theefoe, the followg elto hold, ovded tht B lle th B b g B B. (8 B oo B eceed B the gvtohoto feld vhe. C be deved the followg geel eltoh betwee getc d the eutl gvtohoto feld, b g : b g e B B ( ( e B (9 whee / d / 8. The deo of b g. etg eq. (6 to eq. (9, ug eq. (8, d dffeettg wth eect to te, eult b g t e e B B ( ( t B. ( tegtg ove bty e yeld b g t d g g d ( whee t w ued tht the gvtohoto feld, ce t gvttol feld, c be eted ccodg to eq. (, (. Cobg eq. ( d ( gve the followg eltoh g g d e e 8 B B B d. ( t o eq. (6 oe obt

14 B t e e ω&. ( Net, we ly eq. ( d ( clcultg the gvtohoto cceleto fo the -g cceleoete. t ued tht the cceleoete locted t dtce fo the og of the coodte yte. o eq. (6 t c be dectly ee tht the getc ducto h z - cooet oly. o eq. ( t obvou tht the gvtohoto cceleto the - le. Becue of yety eo the gvtohoto cceleto deedet o the zuthl gle, d thu oly h cooet the ccufeetl (tgetl decto, deoted by ê. Sce the gvtohoto cceleto cott log ccle wth du, tegto ove the e π eˆ z. etg eq. ( to eq. (, d cyg out the tegto the followg eeo fo the gvtohoto cceleto obted g g B ω& ( B whee the u g dcte cceleto oote to the ogl oe d t w ued tht the B feld hoogeeou ove the tegto e. The to of the getc feld w clculted fo the followg foul, obted by dvdg eq. (6 by the que oot of eq. (7 B B 7 e BT C ωr. ( etg etted vege vlue of ω 7d/, TC 9.K, d R 7., th to clculted e g, B.8 /K,. O the thetc coceg the Mwell equto hghe deo [] (-D, the feld tegth teo c be eeeted t-yetc t:. (6 We defe the cotovt cete oto vecto { } { ct,,, }, whee the lowece Gee lette eeet cete dce {, ν } {,,, }. We chooe the etc teo g dg {,,, } to hve otve tl cooet uch tht the g d loweg of dce oly chge the g of the teol cooet. The cooet of the feld tegth

15 ν teo { } e elted to the cooet of the cete otetl { } { V / c,,, } v ν ν ν. (7 The feld tegth teo tully dvded to teol d tl cooet. The thee deedet teol cooet e octed wth the vecto electc feld E / c (8 whle the thee tl cooet tully fo ecod t-yetc teo getc feld B. (9 Mwell hoogeeou equto, vcuu, e eeed cocely te of the feld tegth teo: ν ν ( whee the cete cuet dety h cooet { cρ,,, }. (-D, ρ the volue chge dety d ˆ ˆ ˆ eeet the dtbuto of the cuet ove co-ectol e. Mwell hoogeeou equto follow fo the eltoh betwee the feld d the cete otetl (eq te of the vecto electc feld E d teo getc feld B, Mwell equto dffeetl fo e: ρ E, E E cb, E B, B. ( c We do ot dtguh betwee covt d cotovt tl dce {,, } {,, } ce oly g o loweg of teol cooet volve g chge. Wth the le eleet d eeed t-yetc ecod- teo d the dffeetl e d eeed vecto v the followg equto C,! C, C C the tegl fo of Mwell equto e: S S E d B d V ec ρdv, Ed c B d,, C E! d. ( c B d C Sec S ec

16 Whe geelzg eq. ( to (N-D, the hghe-deol log of the dffeetl le eleet d wll hve N tl deo d the hghe-deol log of the dffeetl e d wll hve N tl deo. t lo ueful to wo wth tyetc ecod- teo G tht dul to : G! ν νρσ ρσ. ( Mwell hoogeeou equto e obted fo the dul teo G v G ν ν. ( ltetvely, Mwell hoogeeou equto y be etcted fo λνρ T, ( whee the t-yetc thd- teo T defed λν T λν. (6 λ ν The ot tghtfowd geelzto of Mwell equto to (-D beg by etedg eq. (8 to Φ Φ (7 whee the uece Gee lette {, } {,,,, } (-D ettvty d eeblty of fee ce deo th the (-D coutet d. Φ eeet (-D cete dce d the d, eectvely, hve dffeet The (-D feld tegth teo h deedet cooet, whch tully dvde to -cooet vecto electc feld E / c (8 d t-yetc ecod teo getc feld B (9 wth 6 deedet cooet. te of the vecto electc feld E d teo getc feld B, Mwell equto (-D dffeetl fo becoe E ρ, E E cb, E KL BK, B. ( c 6

17 Gu lw get, whch tte tht the et getc flu lwy zeo coequece tht getc ooole hve ot bee obeved, c ltetvely be eeed dffeetl fo : B B B. ( K K K ltetvely, the dffeetl volue y lo be eeed vecto dv v the followg equto C K.! KLCL, CL KLCK Wth the feld d dffeetl eleet eeed te of ote vecto d teo, (-D Mwell equto c be eeed tegl fo : V E dv KL V Wec B dvk ρdw, KL EdK c KL B dv,, S Vec E! dv. ( c B d S Vec Whle electc flu e cl, getc flu h fou cooet (-D: K Φ e E dv, Φ L KL V V B dv. ( K (-D, the cotuty equto ed c ρ. ( (-D, the dul teo G t-yetc thd- teo: G ΦΛ! ΦΛΣΩ ΣΩ. ( Mwell hoogeeou equto y be eeed te of the dul teo G v G ΦΛ Λ (6 o, equvletly, te of t-yetc thd- teo T v whee T defed (7 ΦΛΣΩ TΛΣΩ T. (8 ΦΛ Φ Λ 7

18 8 (-D, the dffeetl fo of Mwell hoogeeou equto e eeed cocely te of the 6 6 feld tegth teo Φ Φ (9 whee { } { },,,,,, Φ eeet (-D cete dce. Mwell hoogeeou equto c be eeed te of the dul teo G o, equvletly, t coutet T, Ξ Λ Φ Ξ G, Ω Σ Λ Ξ Ω Σ Λ Φ T, (6 whee! Ξ Ω Ξ Ω Σ Λ Φ Σ Λ Φ G, Λ Φ Λ Φ T. (6 te of the -cooet vecto electc feld E d the teo getc feld B, the dffeetl fo of Mwell equto (-D e: ρ E, B c E E, K M L K B, c E B, (6 whee { } { },,,,,. tegl fo, Mwell equto (-D e: W X ec dx dw E ρ, V W K M L K K M L K ec dw B c dv E, W K M L K dw B, V W ec dw c E dv B!, (6 whee dx the dffeetl fve-deol volue eleet. o olted ot chge (-D, the electc feld le dte outwd fou-deol ce. o Gu hyehee of du, the electc feld E llel to the dffeetl volue eleet V d eveywhee o the thee-deol volue boudg the (fou-deol hyehee defed by. (-D, the electc feld E due to ot chge thu foud to be V W ec q dw E dv E ρ π, q E ˆ π, (6 thece, we obt

19 EdV Eπ ρdw V Eπ. (6b ˆ W ec We hve utlzed the eult tht the (N -deol volue N-deol ce N N boudg old hee V hee N π N / N ( N / (6 whee ( z the g fucto. Geelzg to (N-D, the electc feld E due to ot chge E π ( N / q N / N N ˆ (66 whee the (N-D ettvty of fee ce N h dffeet deolty th t (-D coutet : [ ] [ ] [ L ]. (67 N N Thu, Coulob lw fo the foce eeted by oe ot chge q o othe ot chge q dlced by the eltve oto vecto R wy fo the ft / N foce lw: q, q π ( N / N / q q N N R Rˆ. (68 (N-D, the electc feld E( t feld ot locted t y ltetvely be deved though dect tegto ove the dffeetl ouce eleet dq : E ( π q ( N / N N Rdq / N R ˆ (69 whee the eltve oto vecto R eted fo the dffeetl ouce eleet dq locted t to the feld ot locted t. The (N-D electc feld elted to the (N-D cl otetl v E V. (7 t electottc, the (N-D cl otetl deved fo the followg tegl: 9

20 V ( π q ( N / ( N N R dq N / N. (7 The tdtol è loo ooted to two-deol è ufce (-D. utble choce fo tedy flety cuet ug log the - the twodeol ufce of old hee the thee-deol ubce. th ce, the oly o-vhg cooet of the getc feld teo B B. By yety, the getc feld cl, defed by the teo cotcto e { } B d { } B B B /!, cott eveywhee o the ufce d the cooet of the getc feld teo e elted by B B B B, whee, uch tht B B B B. Howeve, t le to wo hecl coodte, whee B d B B dω. Thu, lcto of èe lw yeld B π ec, B π S S d B dω B8. (7 (-D, the getc flu Φ cl d the getc feld le fo th fte tedy flety cuet e tdtolly dw cocetc ccle the Φ ccodg to eq. le. Howeve, (-D the getc flu fou-cooet vecto { } (. Th coeod to the fct tht ccle,, d lyg the,, d le, eectvely, e ll othogol to the cuet ug log the -. (N-D, the getc flu becoe t-yetc (N-- teo. Geelzg to (N-D, the getc feld due to tedy flety cuet B N π N ( ( / N. (7 (N-D, the foce e ut legth l tht oe tedy flety cuet eet o llel tedy flety cuet eted by the eltve oto vecto R : π N R R ( N / N, ˆ (-D, the teo getc feld ( l. (7 B t feld ot locted t y ltetvely be deved though dect tegto. o tedy flety cuet, π N R ( ( R d B N / N R d (7

21 whee the eltve oto vecto R eted fo the dffeetl ouce eleet d locted t to the feld ot locted t. f the cuet ted dtbuted ove oe-,two, B o thee-deol co ecto, the teo getc feld ( B B π N N N N R ( ( d d, B ( ( d d N / π ( ( ( R dv R dv N / R π N R N / N d, (76 whee,, d e the coeodg to the cuet dete. Wth gle o-coct et deo, two (-D electc chge q d q coucte v the echge of (-D hoto. Clclly, thee gle tght-le th wth whch the (-D hoto c ech q fo othe q. Oe eult of th gle ococt et deo tht Coulob lw becoe / foce lw. f ted the et deo coctfed o tou wth du R, thee et fte dcete et of th wth whch clcl (-D hoto could tvel fo q to q. The et foce eeted o q the ueoto of the (-D / Coulob foce (eq. 68 octed wth ech th:, π q q R Rˆ (77 whee R ( π R the th legth coeodg to clcl (-D hoto R wdg oud the tou te d ˆ ut vecto dected fo q to q R R Rˆ ˆ R (d ˆR ut vecto log the of the tou. te of t thee-deol d etdeol cooet, the et foce R Rˆ ˆ [ R ( πr ] q q π. (78 The et foce c lo be eeed te of the (-D cl otetl V ( R v q V. (79 The (-D cl otetl due to the ouce q t the locto of q ( R V q π R ( πr. (8 Note tht the (-D cl otetl eodc the et deo:

22 V ( R V ( R, πr, ±. (8 the lt tht the chge e vey cloe coed to the du of the et deo,.e. R << R d << R, the te dote d the et foce otely the (-D / Coulob foce:, q q π R q q Rˆ. (8 Th the lt whee the udelyg (-D theoy of electodyc.e. Mwell equto wth o-coct et deo gove the oto. the oote etee, whee the coct et deo vey ll coed to the eto of the chge,.e. R >> R tegl ovde good oto fo the u: V ( R q dy q 8π R R y y 8π RR. (8, Th the lt whee the effectve (-D theoy of electodyc.e. the uul (-D fo of Mwell equto gove the oto. ode fo the effectve (-D cl otetl eq. (8 to be cotet wth the uul (-D fo of Coulob lw, t ecey tht the (-D ettvty of fee ce be elted to the uul (-D ettvty v t follow tht. (8 πr π R (8 uch tht electogetc wve ogte t the uul eed of lght vcuu c. (86 N (N-D, Coulob lw / foce lw f the et deo e o-coct. f ted thee toodl coctfcto d the et deo e yetc..e. they ll hve the e du R - the, the lt tht R >> R, Coulob lw otely / foce lw the effectve (-D theoy. th ce, the (N-D ettvty of fee ce N elted to the uul (-D ettvty v (87 N N N ( R π

23 whee N >. Thece, the eq. (7, fo the eq. (87, c be wtte lo follow: V ( π whee ( z the g fucto. ( R π N N q ( N / / ( dq N N N R, (87b The devto the uul / fo of Coulob lw c be couted fo ecfed geoety of et deo though the lowet-ode coecto to the tegto eq. (8. The effect e l to the devto the uul / fo of Newto lw of uvel gvtto.. O oe equto coceg the ocouttvty Stg Theoy: the Dc-Bo- feld cto, coecto wth the Mwell electodyc d the Mwell equto, ocouttve oe tg cto d D-be cto [] [] [] [6] [7]. The Dc-Bo-feld cto: coecto wth the Mwell equto To obt the low-eegy effectve cto fo the guge feld, we eed to ed the woldheet theoy bout bcgoud feld X, d coute the (dveget oe-loo coutete: ( ( X Λ dτ, X (88 whee Λ ultvolet cutoff. Settg X, equvlet to woldheet cofol vce, o vhg of the β -fucto. Tht gve the cete equto of oto, fo whch the cto c be ecotucted. Pefog the bcgoud feld eo τ to zeo gve codto o the guge feld ( X X ξ (89 whee X bty clcl oluto of the woldheet equto of oto, we fd: S δs δ S ξ ξ... δx X X δx δx X X ( X S( X ξ (9 The le te ξ vhe X oluto of the equto of oto. The qudtc te ely evluted: δ S ξ ξ [ dσdτg ξ dτ ( X ξ ξ ξ ξ ] τ τ δx δx X X πα The coecto to the woldheet cto :. (9

24 πα ( τ, τ dτ τ X K τ τ, (9 whee we hve goed oble UV fte te. Recllg the foul fo the boudy ogto, we hve: l K τ τ ( τ; τ α G ( l Λ (fte (9 d hece the equto of oto : Whe we th of ( G ( g. (9 g g G the (vee oe-tg etc, the th loo ut le the fee Mwell equto. t tu out tht the deed oe-tg effectve cto : t oble to how tht: S NS NS [ ; g, B ] d det( g g. (9 δ δ ( det l ( g det( g G ( G ( l. (96 Sce the fcto det ( g ozeo d the t ( the bove eeo to zeo equvlet to: ( l lg (97 G vetble, t follow tht ettg whch the deed equto of oto. t lowet ode th equvlet to the Mwell equto: l g (98 but geel, we oted, t h ole coecto. The cto: S NS NS l [ ; g, B ] d det( g d det g πα ( B g ( (99 g clled the Dc-Bo-feld (DB cto. Edg th cto to qudtc ode, t ely ee tht t ootol to the uul cto of fee Mwell electodyc:

25 .e. we c wte the eq. ( lo follow: [ ; g, B ] S... ( NS NS d S [ ; g, B ] B dl E NS NS... C thece, we obt the followg teetg thetcl coecto: S NS NS ec ˆ d, (b dt S [ ; g, B ] d det( g d det g πα ( B g C g d B dl ec E ˆ d. (c dt S ( Th cotet wth the fct tht the lezed equto of oto e ut Mwell equto. We wll ow ect the DB cto dffeet fo. o th, let u ft defe We bbevte G G ( ( g g πα B g πα B πα B πα πα g πα B g πα B (. ( G by G. We lo defe the t G, bbevted G, to be the t vee of G. Thu we hve defed two ew cott teo teo g, B. tcul, G, te of the ogl cott G πα. ( g πα B t llutg to ewte the DB Lgg te of the ew teo. Th cheved by wtg: g det g ( g πα ( B det πα G πα ( det ( ( det G πα det G πα g G g G det det πα πα ( Defg.

26 ˆ, G G g det ( πα we ed u wth the elto g det ( g πα ( B det( det( G πα ˆ. (6 G The elto betwee d ˆ c be ely veted, ledg to: ˆ (7 ˆ fo whch t lo follow tht:. (8 ˆ Hece we hve: g det ( g ( B ( ˆ ( G πα ˆ πα det. (9 G det wht follow, we ut be ceful to eebe tht the bove equto wee obted the tct DB oto of cott. det ˆ the deoto, the ght hd-de loo le DB t fo the fcto ( Lgg wth ew tg coulg G, etc G d guge feld tegth ˆ, d o B- feld. Let u theefoe tettvely defe the cto: ( G ˆ S ˆ DB det G πα. ( Let u tt by edg the elto though whch we defed ˆ, to lowet ode : etg the defto of, we get: ( ˆ l Ο. ( l ( Ο( ˆ l. ( l l l l We c e ole edefto of to th ode, whch bob thee of the fou te le : 6

27 ( ˆ l l l Ο. ( We c fd tht ˆ ˆ ˆ l l. ( ˆ ˆ t cle tht thee o futhe edefto of  tht wll bob the lt te. Howeve, we ote tht th te : l ˆ ˆ ˆ, ˆ ( l { } whee {...,} the Poo bcet wth Poo tuctue. Thu, to le ode, we hve foud tht: { ˆ, ˆ } ˆ ˆ ˆ. (6 Th loo le o-bel guge feld tegth, ecet tht thee Poo bcet ted of coutto. We c y tht the feld tegth ˆ ocouttve feld tegth elted to t guge otetl  by: [ ˆ, ˆ ] ˆ ˆ ˆ. (7 The ˆ (8 ˆ l l l Ο( (9 ow the Sebeg-Wtte. We eebe tht the Sebeg-Wtte guge theoy et of clculto tht detee the low-eegy hyc ely the odul ce d the e of electclly d getclly chged ueyetc tcle fucto of the odul ce. Th oble d otvl guge theoy wth N eteded ueyety by cobg the fct tht vou ete of the Lgg e holoohc fucto ( coequece of ueyety d the ow behvo of the theoy the clcl lt. The odul ce the full qutu theoy h lghtly dffeet tuctue fo tht the clcl theoy The ocouttve decto etzed by the ocouttvty ete, the oe-tg etc G, the oe-tg coulg G, d decto ete Φ, te of whch the eltoh betwee cloed-tg d oe-tg ete gve by: 7

28 N g πα B πα G πα Φ, det ( g πα B det( G πα Φ g G. ( Now we wh to coe the u of the couttve DB cto S DB lu the devtve coecto to t SDB wth the ocouttve DB cto Ŝ DB, fte tg the Sebeg-Wtte lt o both de. The dlto coule to the ete Lgg dety, o we eed to code the full DB cto. We wll tt by etctg to te qudtc. To th ode, we hve: S DB ( g πα B ( πα ( πα ( tn det πα t( N t( NN.... ( g 8 the Sebeg-Wtte lt we hve N d theefoe: πα S DB SW ( g πα B det t( t( ( t... g 8. ( Let u ow covet the couttve feld tegth eg th eeo to ocouttve feld tegth ˆ, ug the Sebeg-Wtte. To the ode tht we eed t, th : ˆ l ˆ, ˆ ˆ, ˆ ( ( l b b b bl whee ˆ ˆ ˆ l ˆ, ˆ. ( b b b l b Hee we hve ued detty eltg the Moyl coutto d the oduct: [ f, g] f, g. ( etg the Sebeg-Wtte to eq. (, we fd S DB ( g πα B det ˆ b l ˆ, ˆ SW l b g b l ( ˆ ˆ ˆ ˆ l ˆ ˆ,, ( ˆ lb bl l 8. (6 Soe ulto of the lt few te et u to ewte th : S DB SW ( g πα B det ˆ l ˆ ˆ b l, ˆ, ˆ l lb g 8

29 l l ( ˆ, ˆ l ˆ ˆ l ˆ, ˆ l ( ˆ, ˆ l ˆ ˆ l l (7 8 8 whch the fo whch t wll be ueful. Let u ow tu to the ocouttve de. Hee, we oly eed to ee the te g fo eo of the Wlo le, ce ll othe te e ueed by owe of α the Sebeg- Wtte lt. The Wlo le gve u: Sˆ DB SW det ( G πα Φ G ˆ l l ˆ, ˆ. (8 fte oe egeet of te, th c be wtte: Sˆ DB SW det ( G πα Φ G ˆ l ˆ, ˆ b l ˆ ˆ l, ˆ, ˆ lb l 8. (9 l Now we c te the dffeece of eq. (9 d (7. The efcto fot of ech eeo the e, by vtue of eq. (. t fo th fcto d the tegl g, the eult : Sˆ DB SW S DB SW l ( ˆ, ˆ l ˆ ˆ l ( ˆ, ˆ l ˆ ˆ l l. ( To the ode whch we e wog, we y elce ˆ by eveywhee th eeo. Th, the, ou edcto fo the coecto SDB, to ode ( α d to qudtc ode the feld tegth, fte tg the Sebeg-Wtte lt. We ote tht th fetly hghedevtve coecto: t vhe fo cott, fo whch the oduct educe to the ody oduct. Edg the oduct to -devtve ode, we fd tht 8 S whch gve: DB SW 96 l l l l ( S DB SW ( πα 96 h h l h h l h h l h h l ( whee the t h defed : h g πα ( B. ( 9

30 Tg the Sebeg-Wtte lt, whch out to the elceet πα h (, d futhe etctg to te qudtc, we fd ect geeet wth eq. ( bove. Now we wll coe the coulg of the bul gvto to the eegy-oetu teo o the couttve d ocouttve de. O the couttve de, we tt g wth the eeo eq. (, but th te we ue the full fo of N defed eq. (: N g πα ( B πα M ( whee Thece, we obt: M G πα Φ. ( N g πα ( B πα G πα Φ. (b the le coulg to the gvto tt t ode ( α, we ow hve to go beyod the ledg te the Sebeg-Wtte lt. Hece we wll ee te u to ode M. Edg S DB oud th lt d eeg te to ode ( α, d ug the Sebeg-Wtte, we fd: S DB det πα ode ]. (6 ( g πα B πα { ( } ( πα,, l l M l l g ( tm ( t ( tm ( πα 8 tmm tm te ot volvg M πα Tug ow to the ocouttve cto, the gvto coulg obted by edg the DB cto oud the Sebeg-Wtte lt to ode ( α. We hve, oetu ce: G ( G ( ˆ Φ W (, C Sˆ. DB L det πα e G ( ( ˆ Φ ( ˆ Φ ( det GL πα tg G W, C e... (7. The ece of the bove eeo tht ode α h ledy bee couted ele fo the dlto coulg. t cotbute to the coulg of the tce of the gvto. The ew o tvl coulg gve by the ode ( α te. To coe wth the couttve de, t coveet to ed the bove cto dffeetly, te of M the th G. We get:

31 Sˆ DB G det ( G πα Φ L ( W (, C πα tm M, tm, t l l ( πα ( πα t M, M tm, tm.... (8 8 Now tg the dffeece of the ocouttve d couttve cto eq. (8 d (6, d edg the eult to -devtve ode, we get the edcto: S DB SW ( πα 96 πα l M 8 M M l l l M M l M l Note tht coty to ece, both of the bove te e of ode ( α oe et G lce of M the ft le, the eult vhe. l l. (9. Th becue f. The ocouttve oe tg cto, Che-So theoy d D-be cto Let u focu o tcul Che-So coulg, the oe volvg the Rod-Rod 6- ( 6 fo C. the couttve theoy, th ut ( C 6. ( The ocouttve veo of th coulg obted fo the couttve oe by g the elceet: ˆ ( ˆ whee ˆ the ocouttve guge feld tegth, ultlyg the cto by fcto det ˆ, d ug the Moyl -oduct defed the followg equto: ( f q q ( g( f ( e g(. ( To e coulg tht guge-vt eve fo ocott feld, th h to be cobed ( 6 wth oe Wlo le. The eultg eeo fo the coulg to C oe coveetly ~ eeed oetu ce, whee the oue tfo of ~ C ( C 6 ( ( 6 ( L det( ˆ ˆ ˆ W (, C ( C 6 ( :. e ˆ ˆ. (

32 W, oe Wlo le, d L the ecto of eg locl oeto log the Wlo le d th-odeg wth eect to the Moyl oduct. Evluto of the L ecto led to oduct. The bove eeo c ely be e-eeed oto ce, d t tu to: Hee ( C ( 6 ( C. ( Let u deote the u of ll devtve coecto to S CS SCS. Ou ttg ot the eeo S CS S CS πα dσddφ ( φ C e B ( whee C eeet the RR feld, d B the Rod-ecto boudy tte fo zeo feld R tegth. We e ug uece otto, fo ele φ X ψ d D the uecovt devtve. t oble wte the followg eeo: R S CS S CS C e d d πα σ (! ~ ~ ~ D ~ φ φ φ... φ ( dσ [ Ψ ψ ] X X ψ ψ.. (.. πα! e B... (6 whee ozeo ode hve tlde o the, whle the zeo ode e elctly dcted. We c do the ft eoetl fcto eq. (6 bove, well the ft feo ble ~ Ψ ψ the ecod eoetl. The, edg the eoetl to ecod ode, we get: ~ ~ ~ R π π l SCS SCS dσ dσ C ψ ψ ψ ψ πα ~ ~ ~ b ~ b X ( σ... X ( σ X ( σ... X ( σ... ( b... b l ( B. (7 R!! Now we eed to evlute the -ot fucto of the X ~. The elevt cotbuto hve ologthc fte t d coe fo ogto fo whch thee o elf-cotcto. Th eque tht. The we get cobtol fcto of! fo the ube of uch ~ ~ σ X σ. The eult : cotcto ( ( ( ( X S CS S CS π π b dσ dσ D! πα... l ( b... b l ( C B ψ ψ R ψ b ( σ σ... D ( σ σ ψ. (8 The feo zeo ode eectto vlue e evluted ug the ece:

33 ψ ( (9 ψ α whee the o the ght hd de dffeetl -fo. Thu we e led to: S CS b b SCS T... ; b... b ( whee T... ;... b b Thece, we obt:! πα! πα S CS π π b b ( α dσ dσ D ( σ σ... D ( σ σ SCS... b... Net we et the eeo fo the ogto: D b b π π ( α σ σ ( σ σ... d d D b D b ( σ σ b ( σ σ b ( σ σ ( σ α h e h e ( e σ (. (b. ( whee egulto, d h. ( g πα B we hve ee, th teo whe eded bout lge B h the fo: ( g h... πα πα ( whee the te the eo e ltetvely tyetc d yetc, d the two te ehbted bove e decto-deedet. Now we eglect ll but the ft te bove. t follow tht T... ; b... b! dσ π π ( α b h h b e σ e σ. ( fte evlutg the u ove, the eult, deedg o the egulto, T... ; b... b! σ π dσ [ ] ( ( α b e h l σ π ( e h ( b l e σ. (6

34 [ b] ( b Hee, h d h e, eectvely, the tyetc d yetc t of o Sebeg-Wtte lt cot of the elceet: b h. The lge- B [ b] b h, πα ( b h. (7 By vtue of the fct tht ll σ ( e σ ( e ( σ π (8 th lt led to eleety tegl. Evlutg t, oe flly obt the eult: S CS S CS C ( 6 ( ( 6 ( ( C (9 b b b! b whee the oduct defed by the followg equto: (, ( ( ( l q q q q l. (6 Th gee efectly wth eq. (, the edcto fo ocouttvty. Now we eted the clculto fo the ot of eq. (6. To go to ft ode beyod the Sebeg-Wtte lt, we e the elceet: [ b] b h, πα h ( b c db gcd (6 ( πα ( b... ; b... b d ee ll te tht e ft ode h. Deote by T ( the ft coecto to T (defed eq. ( wy fo the Sebeg-Wtte lt. The, we ee tht T... ; b... b ( ( (! ( π ( g b πα b b π dσ σ... ( σ π l e. (6 π The tegl the bove eeo vhe fo eve. o odd, we fd tht T ( ( (! ( π ( g b... ;... b b b b... πα (6 whee

35 dσ π σ! ( σ π l e ( (! ( ζ ( ( π whee ζ the Re zet fucto. Thece, we obt: dσ T... ; b... b ( ( (! ( π π b ( g b b πα... ( σ π l e ( (! ( ζ ( ( π σ π π!, (6. (6b t coveet to defe -fo W tht ecode the devtve coecto fo the coulg to ( 6 C : ( 6 ( 6 SCS SCS C C W. (6 The ledg-ode te W the lge- Blt : ( W. (66 The clculto ledg to eq. (6 out to coutg W to ft ode ( α oud the Sebeg-Wtte lt: ( b ( g b b π ( ( ( ( b b ζ π πα! W techgg the ode of the two uto, we fd tht the u ove c be efoed d led to the ece of the fl oduct. The eult, fte oe elbelg of dce, : (67 W ( π ( g cd ( ( b b ζ c d b b πα. (68 Ule the ledg te eq. (9, whch gle fte ee devtve ued by the oduct, hee we ee double fte ee. fte fog the oduct we tll hve ddtol ee whoe coeffcet e ζ -fucto(re zet fucto of odd guet. We wll ow ue th to etct the te coeodg to ou coutto eq. (68 d coe the two eeo. The coutto evluto of the ltude:, RR ; O O ; ( q V (, ; V (, y Α dy V (69

36 , whee V RR the vete oeto fo RR otetl of oetu q, the, ctue, d V O e vete oeto fo le guge feld of oetu d olzto,,. We defe: ; (7 π π t α α G d chge tegto vble v y cotπτ. The t follow tht the coeffcet of ovded by th coutto, to be coed wth the coeffcet of eq. (68, : t 6 ( t ( t ( t t dτ ( coπτ coπτ. (7 The ght hd de c be eded owe of t d, u to te of Ο ( t, oe h: ( t [ ( ψ ( ψ ( t Ο( t ] ( ( ( ( t t γ (7 d l. d The ft te c be ecoged the eel of the -oduct, ug the elto: whee γ the Eule cott d ψ ( the dg fucto ( ( ( π π. (7 Let u ow ee the ecod te oe cefully. We ue the fct tht: to wte: γ ψ ( γ ( ζ ( ψ (7 ( ψ ( ( Puttg eveythg togethe, we fd tht: W ( πα ( π ( ζ ( ζ ( O oue tfog, th detcl to eq. (68. Thece, we obt: ζ ( ( ( g. (7 ~ ~ ( ( (. (76

37 ( ( ζ t ( Ο( t t dτ ( coπτ t coπτ, (76b πα S CS ( π S ζ CS C C ( 6 ( 6 ( ( ( g ~ ~ ( ( (, (76c whee ζ the Re zet fucto. Now we code Che-So guge theoy wth the guge gou G cole Le gou uch SL (,C. Let Αbe coecto o G -budle E ove thee-fold M. Such coecto h cole-vlued Che-So vt W π ( Α T Α dα Α Α M, (77 whch we hve olzed to be guge-vt odulo π. ut the ce of coct gou, W ( Α guge-vt odulo π. The detecy W ( Α el, eve f the guge gou cole, becue cole Le gou cotctble to t l coct ubgou. The qutu theoy bed uo tegtg the eeo e (, d fo th fucto to be well-defed, ut be defed od π. Becue of the detecy Re W, t coeffcet ut be tege, whle the coeffcet of W y be bty cole ube. The cto theefoe h the geel fo W l ReW. C, l Z. (78 ltetvely, we c wte wth ~ ~ tw t W t t 8π M 8π T Α dα Α Α Α T Α dα Α Α Α ~ t l, t l. (8 M, (79 toduce cole vble z d cole-vlued olyol Now code the tegl g ( z z. (8 7

38 Z g ( g( z g( d z e z. (8 Th g coveget oclltoy tegl d cloe logou of Che-So theoy, ~ wth g( z d g( z coeodg to the te tw d t W the cto (79. Defe ew olyol d geelze the tegl Z g to ~ ~, (8 g ( z z ( g( z g( Z ~ g g, ~ d z e z. (8 The Z g, g~ cocde wth the ogl Z g f g ~ g. Deotg the deedet cole vble z d ~ z, the lytclly cotued tegl ( g( z g~ ( dzdz ~, ~ e z. (8 C Z ~ g g The tegl ove two-deol el tegto cycle C, whch the el lce ~ z z f g ~ g, d geel ut be defoed g d g ~ vy o tht the tegl e coveget. ut we ooted z to cole vble ~ z tht deedet of z, we wll hve to oote Α to ew G -vlued coecto Α ~ tht deedet of Α. Thu we code the clcl theoy wth deedet G -vlued coecto Α d Α ~ (we ecll tht G SL,C d cto cole Le gou uch ( ~ t t ~ ~ ~ ~ ~ ( Α, Α T Α dα Α Α Α T Α dα Α Α Α 8π M The we hve to fd ote tegto cycle C the th tegl C ~ 8π ~ ( ( Α, Α ~ D ΑDΑ e. (87 M. (86 The cycle C ut be equvlet to Α ~ Α f el, d geel ut be obted by defog tht oe ve o tht the tegl e coveget. Let u code ocouttve Euclde D-be wth eve ube of wold-volue decto. Gve collecto of locl oeto Ο ( o the be wold-volue whch tfo the dot ude guge tfoto, oe c obt tul guge-vt oeto of fed oetu by eg the locto of thee oeto log tght cotou gve by ξ ( τ τ wth τ, d ultlyg the oduct by Wlo le ( C W, log the e cotou, 8

39 W ( C ( τ ξ, e dτ ˆ ( ξ ( τ, (88 τ whee  deote the ocouttve guge feld, d ˆ the coeodg feld tegth. The eultg foul fo the guge-vt oeto : Q ( d ( dτ P W, π d L W (, C ( ( Ο e π. ( C Ο ( ξ ( τ e. (89 whee P deote th-odeg wth eect to the -oduct, whle L bbevto fo the cobed th-odeg d tegto ove τ. th foul the oeto Ο e eed ove the tght cotou of the Wlo le. Th ecto e by ttg wth the yeted-tce cto fo ftely y D-tto d edg t oud the cofguto decbg ocouttve D-be. Edg the Wlo le, we get Q ( ( π d Q ( e. (9 whee Q ( (...( Ο ( Ο( ˆ ( ˆ,...,,,..., (. (9! Thece, we obt the followg eeo:. Q ( d ( ( ( ( ( ( ( ˆ ˆ... Ο,..., Ο,,..., e π!. (9b Hee we hve toduced the otto f (, f ( f (,..., We ote hee the le foul fo : f the zeo-oetu coulg f fo the oduct of fucto. q q q (, g( f ( g( q. (9 ~ Σ Ο Σ, ( d ( X ( π Pf (9 9

40 whee the guge feld d X e the tvee cl, the the coulg t ozeo oetu gve by ~ Σ ( d L [ Ο (, X W ( C ] ( π Pf Σ, e.. (9 The cott fcto Pf det h bee wtte elctly, ted of bobg t to the defto of Ο Σ, fo coveece. ou ce the elevt cloed tg ode the RR guge otetl C ~ (, d, the ole of Ο lyed by the oeto PfQ Σ whee the be teo d l Q ˆ. (9 Hece we deduce the coulg of th be to the fo ~ ( ( d L ( det( ˆ W (, C π l ( C, oetu ce, to be: e.... C... O the othe hd, we ow tht the coulg of D-be to couttve decto by... ( ( ( ( ~ C δ. ( (96 ( C fo gve the equto (96 d (97 decbe the e yte two dffeet decto, they ut be equl. Thu we edct the detty: ( π ( ˆ W (, C ( (. δ d L det e. (98 Th out to yg tht the ght hd de ctully deedet of Â, the otvl fct. teetg to ee the eq. (98 the oeto fol. We ue the fct tht ( π to ewte the left hd de of eq. (98 follow: ( d t (99 Pf ( π ( π. ( Pfe. δ d e t. ( The ght hd de of eq. (98 c be coveted to yeted tce volvg X ˆ, d t becoe: (

41 ( π St. X ( PfQe ( whee St deote the yeted tce. lly, we ue [, ] d [ X X ] Q ( The eq. (98 te the elegt fo: t.. X ( Pf [ ] e St( Pf [ X, X ] e,. (,. th fo, t ey to ee tht eq. (98 hold fo cott ˆ, o equvletly fo cott Q. th ecl ce t c be oved by ullg PfQ out of the yeted tce o the ght hd de, d the ug eq. (99 wth elced by Q. Now let u tu to the coulg of ocouttve -be to the RR fo C. the couttve ce th fo e wedge oduct wth the -fo B. o the ocouttve be cott RR bcgoud, B ut be elced by the -fo Q wth Q gve by eq. (9. t follow tht the coulg the geel ocouttve ce (wth ( vyg C : ~ C π ( (... d L ( ( ( ( det ˆ Q W, C... o coo, the coulg of D-be to the fo gve by: e.. ( ( ( C te of couttve vble [ ]... ~ ~ ~ C Net, ewte (.... ( ( ( ( d B e C ( δ ( B ( ( Q π Q whee we hve ued the elto ( : ~ ( ( δ ( ( ˆ B ˆ ( ( ˆ ˆ ( B. Ug th elto d lo eq.(98, we c ewte eq. ( ˆ ˆ ( ˆ W (, C B d L det ( π... C... Equtg th to eq. (, we fd tht ~ ( ( π L ( ( ˆ ˆ ˆ W ( C d det, e.. (7 e (6 Th elte the couttve feld tegth to the o-couttve feld tegth ˆ, theefoe t out to cloed-fo eeo fo the Sebeg-Wtte....

42 The coulg of ocouttve D-be to the RR fo ( C fo the ce of cott RR feld : ( ( ( ( ( ( ˆ det ~ Q Q d C π (8 whee the -fo Q gve eq. (. o tlly vyg ( C we c theefoe wte the coulg : ( ( ( ( ( ( ( ] e C W Q Q L d C , ˆ det [ ~ π. (9 the DB oto of lowly vyg feld, the couttve coulg : ( ( ( ( ( e B B d C ~ π. ( etg eq. ( fo Q eq. (9, d cog wth eq. (, we fd tht the DB oto we ut hve: ( ( ( ( ( ( ( l l e C W L d d., ˆ ˆ ˆ ˆ ˆ det ~ π ( To ve t th eeo we hve de ue of the dette eq. (98 d (7. The oe Wlo le cludg tvee cl gve by: ( ( ( ( ( ( Φ ˆ ˆ e, τ ξ τ ξ τ τ ξ τ q d P C W. ( etg th defto lce of ( C W, eq. (9, d deotg the left hd de by ( Q, oe fd tht the coulg to geel tlly vyg uegvty ode e: ( Q ( (. e d Q π ( whee the Q e gve by ( ( ( ( ( ( ( ( ( ( ( q q Φ Φ Ο Ο ˆ,.., ˆ, ˆ,.., ˆ,,..,......! Q. ( Thece, we obt tht:

43 Q ( ( π ( ( ( q ( q ( ( ˆ ( ˆ ( ˆ Ο ( ˆ ( e,.., Ο,,..,, Φ,.., Φ.! (b t well-ow tht o-bps be uetg theoy lo coule to RR fo. couttve vble, thee coulg fo gle o-bps be e gve by: d ˆ ( B dt C e ( S CS T whee T the tchyo feld d T t vlue t the u of the tchyo otetl. Code the coulg of Euclde o-bps D-be wth eve ube of wold-volue decto, to the RR fo C T (, the couttve decto: T dt C... T (... ( d T ( C (... ~ ~ ( ( T ( C ( d.... (6 The ocouttve geelto of th coulg, fo cott RR feld, T ~ (... C ( d det ( ( ˆ Tˆ (... D (7 π whee [ ] ( Q X, Tˆ ( ˆ D T. (8 ( The, the e RR fo C coule to ocouttve o-bps D-be though the followg coulg fo ech oetu ode: T ~ ( ( d L ( det( ˆ D Tˆ ( W (, C π e.... C.... (9. O oe equto coceg the ocouttve qutu echc egdg the tcle cott feld d the ocouttve clcl dyc elted to qudtc Lgg [8] We code hee the ot le d uul ocouttve qutu echc (NCQM whch bed o the followg lgeb: [ ˆ, ˆ ] hδ, [ ˆ, ˆ ] h, [ ˆ ˆ, ] (

44 Θ the tyetc t wth cott eleet. To fd eleet (,t whee ( Ψ of the Hlbet ce ody qutu echc (OQM, t uully ued the Schodge equto h Ψ(, t Hˆ Ψ(, t, ( t whch elze the egevlue oble fo the coeodg Hlto oeto H ˆ H ( ˆ,, t, whee ˆ h. Thee othe och bed o the ey th tegl ethod Κ ( (, t;, t e S[ q] ( h Dq, ( whee Κ (, t;, t the eel of the uty evoluto oeto U ( t ctg o (,t D ( R Ψ L. Pth tegl t ot geel foulto cot tegto ove th the he ce R D {(, q } wth fed ed ot d, d o etcto o the tl d fl vlue of the oet,.e. Κ π h ( ( dt t t (, t;, t e [ q& H (, q, t ] DqD, ( The ey th tegl fo qudtc Lgg c be evluted lytclly d the ect eeo fo the obblty ltude : Κ (, t;, t ( h D S det π e S (, t;, t, ( h whee (, t;, t S the cto fo the clcl tectoy whch the oluto of the Eule- Lgge equto of oto. Thece, we obt tht: ( t (, t;, t e [ q& H (, q, t ] ( π h Κ ( h f we ow the followg Lgg t dtdqd S π det e S (, t;, t. (b D h (,, t α &, & β, & γ, δ, & η φ L &,, ( d lgeb (, we c obt the coeodg effectve Lgg

45 ( q, q, t α q&, q& βq, q& γ q, q δ, q& η q φ L &, (6 tht utble fo qutzto wth th tegl NCQM. Elotg the Eule-Lgge equto L d L q dt &,,,..., D q oe c obt clcl th q q ( t coectg gve ed ot q ( t d q( t th clcl tectoy oe c clculte cto S t (, t;, t L ( q&, q, t. t Pth tegl NCQM dect log of ( d t ect eeo the fo of qudtc S, t;, t cto ( dt. o Κ (, t;, t ( h D S det π e S h (, t;, t. (7 o tcle cott feld, the Lgg o couttve cofguto ce : The coeodg dt the t fo e: L( &, ( & & η η. (8 α, β, γ, η, η, φ, (9 δ, ( ητ whee ut t. We ow ote tht oe c ely fd α, β, γ, η η, ( η η δ τ, φ ( η η th ce, t ey to fd the clcl cto. The Lgg L ( q&, q, t L,. ( 8 ( q& ( η q& η q& η q η q ( η η q&. ( 8 The Lgg gve by ( le the Eule-Lgge equto q& η, & q η & &. ( The oluto e:

46 t q t η, ( ( t η tc C, q ( t td D whee C, C, D d D e cott whch hve to be deteed fo codto: (, q ( T, q ( q ( T q. (, fte fdg the coeodg cott, we hve q η t ( t t (, q ( t ( T η T Ug ( d (, we flly clculte the coeodg cto S T (, T;, L ( q, q, t dt ( ( & η t η T &,,. ( T [ ] T η( η( [ ] T [ η ( η ( ] T ( η η ( η η T 8 ccodg to (7 oe get Κ whee (, T;, h T π h (, T;, e S (, T;, Κ (, T;, e ( η ( T ( η η η, (7. (6 π h Κ elted to the Lgg (8 fo whch. Hece, th ce thee dffeece oly the he fcto. t ey to ee tht the followg coecto hold: Κ (, T;, Κ Tη, T;,, (8 whee Now tg. N oe c ewte the followg equto Κ N π (, t;, t l det( α e [ N R DN h D N ] h α q&, q& D γ q, q δ, q& η, q φ d q, (9 β q, q& 6

47 (: Κ t (, t;, t e L( q, q, t dt l det( α( t det( α π h t & N D N D N h h dq. (. Mthetcl coecto [9] Ru odul equto d Plubo-Ndell odel Now, we ote tht the ube 8, d thece the ube 6 8 d 8, e coected wth the ode tht coeod to the hycl vbto of uetg by the followg Ru fucto: 8 coπtw π w e d tlog cohπ πt w ( e φw tw 7 log t w. ( utheoe, wth egd the ube ( / d 8 they e elted to the hycl vbto of the booc tg by the followg Ru fucto: coπtw π w e d t log cohπ πt w ( e φw tw 7 log t w. ( Plubo ( h ooed le odel of the bth d of the evoluto of the Uvee. Plubo d Ndell ( hve coed th odel wth the theoy of the tg, d tlted t te of the ltte obtg: ( G G f ( φ 6 R ρ νσ ν d g g g T ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, ( κ g 7

48 geel eltoh tht l booc d feoc tg ctg ll tul yte. -dc, delc d zet-tg Le the ody tg theoy, the ttg ot of -dc tg cotucto of the coeodg ctteg ltude. Recll tht the ody cog yetc Veezo ltude c be eeted the followg fo: (, b g g R b DX e π whee h, / π d g d σ α X ( ( b ( b α X t u 8,.e. b c. The -dc geelzto of the bove eeo ( b ( c ( b c d σ e ( c ( ( c ( ( X g, ( T, d α (, b α ( t, c α ( u ( b g b, d, R ( ζ ( ( b ζ ( b ζ ζ ζ ( c ζ ( c wth the codto : ( b g b, d, ( Q whee... deote -dc bolute vlue. th ce oly tg wold-heet ete teted -dc vble, d ll othe qutte hve the uul (el vluto. Now, we eebe tht the Gu tegl tfy delc oduct foul wht follow fo R ( b d χ ( Q b χ, d P Q v ( b d λ ( Q, b Q, (6 b χ v v v χv v, v,,...,.... (7 Thee Gu tegl ly evluto of the ey th tegl K fo eel (, t;, t v χ Dvq, (8 t h v dt, t t (, t;, t L( q&, q, t, t K v of the evoluto oeto delc qutu echc fo qudtc Lgg. the ce of Lgg 8

49 9 (, q q q q L λ & &, fo the de Stte coologcl odel oe obt ( ( P T K T K, ;,, ;,, Q,λ,, Q T, (9 whee ( ( ( [ ] ( T T T T T T K v v v v 8 8, ;, λ λ χ λ. ( lo hee we hve the ube tht coeod to the Ru fucto tht h ode,.e., the hycl vbto of booc tg. Hece, we obt the followg thetcl coecto: ( ( ( [ ] ( T T T T T T K v v v v 8 8, ;, λ λ χ λ ( 7 log coh co log w t tw e d e tw t w w t w φ π π π π. (b The delc wve fucto fo the let goud tte h the fo ( ( ( ( Ω P Z Q Z \,, ψ ψ ψ, ( whee ( Ω f d ( Ω f >. Sce th wve fucto o-zeo oly tege ot t c be teeted dcetee of the ce due to -dc effect delc och. The Gel fd-gev-tte g d bet fucto e: ( ( ( ( R d ζ ζ χ, ( ( Q d χ, ( ( ( ( ( R b c b d b B,, ( ( ( ( ( c b d b B Q b,, ( whee C c b,, wth codto c b d ( ζ the Re zet fucto. Wth egulzto of the oduct of -dc g fucto oe h delc oduct:

50 ( u (, ( b B (, b u P whee b c B,, u,, u, b, c, ( P. We ote tht B (, b d ( b B, e the cog yetc tdd d -dc Veezo ltude fo ctteg of two oe tchyo tg. toducg el, -dc d delc zet fucto oe obt whee ( ζ e d π, (6 R ζ ( Ω( d, Re >, (7 Q ( ( π ( ζ ( ζ ( ζ ( ζ ( ζ, (8 P ( ζ ( ζ, (9 ζ c be clled delc zet fucto. We hve lo tht ζ ( ζ ( ζ ( ζ ( ζ ( ( P Let u ote tht e( π d ( e π d ( R Ω Q d. (9b Ω e logou fucto el d -dc ce. delc hoc ocllto h coecto wth the Re zet fucto. The let vcuu tte of the delc hoc ocllto the followg Schwtz-Buht fucto: whoe the oue tfo π ( e Ω( ψ, (6 P π ( χ ( ψ ( e Ω( ψ (6 P h the e fo ψ (. The Mell tfo of ( ψ Φ ( ψ ( d ψ ( d Ω( d π ζ ( R P Q (6 ψ. The ccodg to the Tte foul oe obt (9. The ect tee-level Lgg fo effectve cl feld ϕ whch decbe oe -dc tg tchyo d the e fo (

51 whee y e ube, etc wth gtue (... L ϕ ϕ ϕ, (6 g t the D-deol d lbet d we dot. Now, we wt to how odel whch coote the -dc tg Lgg etcted delc wy. Let u te the followg Lgg L L. (6 C L φ φ Recll tht the Re zet fucto defed g φ ζ (, σ τ, σ >. (6 Eloyg uul eo fo the logthc fucto d defto (6 we c ewte (6 the fo L φζ φ φ l( φ, (66 g whee φ <. ζ ct eudodffeetl oeto the followg wy: ~ ζ φ ( ( π ( whee φ ( e φ( d D e ζ ~ φ ( d, >, (67 the oue tfo of φ (. Dyc of th feld φ ecoded the (eudodffeetl fo of the Re zet fucto. Whe the d lbet guet of the Re zet fucto we hll cll uch tg zet tg. Coequetly, the bove φ oe cl zet tg. The equto of oto fo the zet tg φ ζ φ D e > ( π ζ ~ φ ( d φ φ (68 whch h evdet oluto φ. o the ce of te deedet tlly hoogeeou oluto, we hve the followg equto of oto t t ~ φ ( ( ( t ζ φ t e ζ φ d ( π >. (69 φ( t Wth egd the oe d cloed cl zet tg, the equto of oto e

52 ζ φ ( π D e ζ ~ φ ( d ( φ, (7 ζ ( π D e ζ ~ ( d ( ( ( ( φ, (7 d oe c ely ee tvl oluto φ. Now we te the eq. (87b of Secto. We ote tht e oble the followg thetcl coecto wth the Plubo-Ndell odel ( d the zet tg (68: ( N / ( N ( R π N N q V ( N N π R dq / R ν ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, (7 κ g 6 ρ νσ d g g g T( G G f ( φ ( N / ( N ( R π N N q V ( N N π R dq / ~ φ ( ( ζ > φ D e d π. (7 φ Now, we te the ght hd de of eq. (c of Secto. We hve the followg thetcl coecto wth the eq. (9, (7 d (8 d wth Plubo-Ndell odel: Sˆ DB SW det ( G πα Φ G ˆ l ˆ, ˆ b l ˆ ˆ l, ˆ, ˆ lb l 8 d B dl ec E ˆ d C dt S 6 R ρ νσ ν d g g g T Gν Gρσ f φ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, (7 κ g ( ( l

53 G ( G ( ˆ Φ W (, C S ˆ. DB L det e G πα. det GL... d B dl ec E ˆ d C dt S ( ( ˆ Φ ( ˆ πα tg G Φ W (, C e R ν ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, (7 κ g 6 ρ νσ d g g g T( G G f ( φ Sˆ DB G det ( G πα Φ L ( W (, C πα tm M, tm, t l l ( πα ( πα t M, M C tm, tm... 8 d ec E ˆ d dt S B dl R ν ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, (76 κ g 6 ρ νσ d g g g T( G G f ( φ thece, the thetcl coecto betwee ocouttve DB cto, Mwell equto d Plubo-Ndell odel. Now we te the eq. (6b e (76c of Secto. We ote tht e oble the followg thetcl coecto wth the eq. (68 coceg the zet tg: T dσ ( b b... ;... ( (! ( π b ( g b b πα... ( ( ( π σ σ π l e! ( ζ ( π (! ~ φ ( ( ζ > φ D e d π, (77 φ π

54 S CS S CS C C ( 6 ( 6 ( ( ( ζ g πα ( π ~ φ ( ( ζ > φ D e d π. (78 φ ~ ~ ( ( ( Now we te the eq. ( d (9 of Secto. lo thee equto c be coected wth the Mwell equto d wth Plubo-Ndell odel: ~ ( ( ( π ( ˆ ˆ ( ˆ ˆ ( ˆ W (, C d l d L det. C B dl ec d dt E ˆ d R ν ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν, (79 κ g 6 ρ νσ d g g g T( G G f ( φ S l e T ~ e (. ( ( ( ( (... C... W, C d L det ˆ D Tˆ π d B dl ec E ˆ d C dt S R ν ν ρσ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν. (8 κ g 6 ρ νσ d g g g T( G G f ( φ cocluo, we te the eq. (6. We ote tht e oble the followg thetcl coecto wth eq. (9, (8,.e. the DB ocouttve cto, eq. (79, (86,.e. the Che-So theoy, the Ru odul equto d the Plubo-Ndell odel:

55 S T (, T;, L ( q, q, t dt ( ( & [ ] T η( η( [ ] T [ η ( η ( ] T ( η η T ( η η det ( G πα Φ G ˆ l 8 ˆ, ˆ b l ˆ ˆ l, ˆ, ˆ lb l 8 t ~ t ~ ~ ~ ~ T Α dα Α Α Α T Α dα Α Α Α ~ 8π M M 8π coπtw π w e d tlog cohπ πt w ( t w e φw tw 7 log 6 R ρ νσ ν d g g g T( Gν Gρσ f ( φ g φνφ 6πG 8 Φ ( Φ Φ ( / ~ κ d G e R H T ν. (8 κ g l lo thee eeo vey evdet the l betwee π d φ,.e. the ue to, by the le foul cco φ, 879π. (8 cowledget The co-utho Ndell Mchele would le to th Pof. Bo Dgovch of ttute of Phyc of Belgde (Seb fo h vlblty d fedh.

56 Refeece [] Wlte Doche, oche Hue Scete hyc d dvced oulo cocet ec ttute of eoutc d toutc 6; [b] Scl Sbto "Setzzzoe delle equzo d Mwell co ltoduzoe del co gvtzole, ude bzz?", htt://t.volt.led.t/etee/e6/e6-w.ht []. W. McDvd d C. D. McMulle Geelzg Co Poduct d Mwell Equto to Uvel Et Deo [] Sul Muh Nocouttvty Stg Theoy S, Pceto, uly -. [] Sut R. D, Sul Muh d Ne V. Suyy Devtve Coecto fo Nocouttvty Xv:he-th/6v [] Sul Muh d Ne V. Suyy Oe-Stg cto d Nocouttvty Beyod the Lge-B Lt Xv:he-th/8v [6] Edwd Wtte lytc Cotuto Of Che-So Theoy Xv:.9v [he-th]... [7] Sul Muh d Ne V. Suyy Guge-vt Coulg of Nocouttve Be to Rod-Rod Bcgoud Xv:he-th/v 9... [8] Bo Dgovch, Zo Rc Pth tegl Nocouttve Qutu Mechc Xv:he-th/9v [9] Bo Dgovch: Zet Stg Xv:he-th/78v M 7. [] Bo Dgovch: Zet Nolocl Scl eld Xv:8.v [he-th] 8. Ste Eg. Roo Tuco htt://thbuldgbloc.blogot.co/ D Mchele Ndell (vou e o the tg theoy htt://ooe.vglo.t/tgtheoy/ htt://dell.oo.t/vglowzd/ CNR SOLR htt://.6../el/ue_et?ued6 ERTOSTENE gou htt:// 6