Green's Functions at Zero Temperature

Transcription

1 MPP Cap Bo E Srliu MPP Cap Bo E Srliu Capr Gr' Fucio a Zro Tmpraur Wic i low iic mpraur vr acivd i a laboraory? Awr: 5m K wa acivd i 988 by a group a Ecol Normal Supériur: A Apc, E Arimodo, R Kair, N Vai, C CoTaoudji, Py Rv L 6, 86 (988) Sic r a b a icrad aciviy i lowmpraur pyic ad low mpraur i 997 i 6 K T Gr' fucio ciu ar ud w o ca o olv problm xacly L u aum a w ar ryig o dduc propri of a ym dcribd by Hamiloia H wic may o b olvd xacly T uual approac i o H H V, wr H i a Hamiloia wic ca b olvd xacly T rm V rpr all rmaiig par of H O ri o coo H o a ffc of V ar mall T baic procdur i o ar wi a ym complly dcribd by H T ffc of V ar iroducd, ad w ry o fid ow i cag ym w udrad Ti i baic procdur i maybody ory B Hibrg I i poibl o olv uaum mcaical problm aor way wic giv am awr bu y u mod a loo ui diffr T Hibrg rpraio a followig propri: T wav fucio ar idpd of im T opraor ar im dpd, ad i dpdc i giv by ih / ih / O () O( ) or, uivally, o i ryig o olv uaio wic i drivd from i: i O () [ O (), H] I pyic o i uually ryig o valua marix lm I Scrödigr rpraio, marix lm of opraor O() bw wo a i INTERACTION REPRESENTATION A Scrödigr Elmary uaum mcaic i aug i Scrödigr rpraio, wic i bad o formula i H j () j (), wic a opraor formal oluio / ih j() j( ) T u of i formula ruir om aumpio: T wav fucio ar im dpd Opraor ar a o b idpd of im ih / ih / j () O( ) j () j ( ) O( ) j ( ) I Hibrg rpraio o obai rul ih / ih / j ( ) O ( ) j ( ) j ( ) O( ) j ( ) T wo rpraio produc am rul C Iracio T iracio rpraio i aor way of doig ig Hr bo wav fucio ad opraor ar im dpd Ti i do by paraig Hamiloia io wo par, H H V,

2 MPP Cap Bo E Srliu wr H i uprurbd par ad V i prurbaio Ti paraio ca b do i diffr way Uually H i lcd a Hamiloia wic i xacly olvabl Opraor ad wav fucio i iracio rpraio will b dod by a car Tir im dpdc i giv by: Wav fucio av a im dpdc ih / ih / ih / jˆ( ) j() j( ) Opraor av a im dpdc ˆ( ) ih / ih / O O () I Hibrg rpraio im dpdc wa a away from wav fucio ad wa giv o opraor Hr, ju par of im dpdc comig from H i rafrrd L' cc if i produc am marix lm a bfor: ih / ih / ih / ih / ih / ih / jˆ ()ˆ( O)ˆ j () j ( ) O j ( ) ( ) ih / ih / j ( ) O j ( ) T im dpdc of opraor i govrd by uprurbd Hamiloia, H : i O ˆ( ) O ˆ( ), H [ ] T im dpdc of wav fucio i govrd by prurbaio, V: i i i ih H H ih j / / ˆ( ) ( ) j ( ) ih / ih / ih / ih / ih / ih / V j( ) V( ) j( ) 4444 ih / ih / ih / ih / 44V 44 j( ) V ˆ( ) jˆ( ) V ˆ( )ˆ( j ) MPP Cap Bo E Srliu 4 opraig wi im dvlopm opraor U(): ih / ih / U () Ti fucio a valu uiy a : U( ) Furrmor, i oby followig diffrial uaio: i U () VU ˆ( ) () W wi o olv i uaio O way of procdig i by igraig bo id of uaio wi rpc o im: i U () U( ) d Vˆ( ) U ( ) fl i U () d Vˆ( ) U ( ) If i uaio i rpadly irad, w g i i U () d Vˆ( ) Ê ˆ d d Vˆ( ) Vˆ( ) Ë L i Ê ˆ Â d d dvˆ( ) Vˆ( ) Vˆ( ) Ë L L Now w iroduc Tim ordrig opraor T I ac upo a group of imdpd opraor ad i ju a irucio o arrag opraor wi arli im o rig For xampl, [ ] > > TV ˆ( ) V ˆ( ) V ˆ( ) V ˆ( ) V ˆ( ) V ˆ( ) if I lp o iroduc followig p fucio: W from E () a wav fucio a im i obaid from o a by

3 MPP Cap Bo E Srliu 5 MPP Cap Bo E Srliu 6 ( x) if x > if x < if x Tu for wo opraor, xplici dfiiio of T ordrig giv [ ] TV ˆ( ) V ˆ( ) ( ) Vˆ( ) Vˆ( ) ( ) Vˆ( ) Vˆ( ) Now w rarrag igral by uig abov idiy:! [ ] d d T V ˆ( ) V ˆ( ) d d V ˆ( ) V ˆ( ) d d V ˆ( ) V ˆ( )!! T rm o rig ad id ar ual Tu w g! [ ] d d T V ˆ( ) V ˆ( ) d d V ˆ( ) V ˆ( ) Similarly w ca ow a L [ ˆ( ) ˆ( ) L ˆ( ) ] d d d T V V V! L d d d Vˆ( ) Vˆ( ) LVˆ( ) Now, av w rordrd opraor from rig o lf i acdig im ordr, or av w ju rordrd im argum? I i difficul o ll raig away ic opraor ar all am i i ca Prformig am drivaio a abov bu for diffr opraor w fid awr T awr i a w av ju rordrd im argum W opraor ar all am i i ju a uio of maic W ca ually wll rgard rordrig a a rarragm of opraor T im ordrig opraor i rally a opraor a rarrag ordr of opraor W will com bac o i lar If w ow rur o our xpaio of U(), w obai U () i! ( ) Â È i T xp Ê d Vˆ( ) ˆ Î Í Ë d d L d T Vˆ( ) Vˆ( ) LVˆ( ) [ ] (6) S MATRIX T im dvlopm opraor a wav fucio from zro im o im Now w iroduc a mor gral opraor a a wav fucio from ' o Sic i a wo argum i i calld a marix, S marix S(,'): jˆ( ) S(,')ˆ( j ') Now, i opraor i cloly rlad o opraor U: U ()ˆ( j ) jˆ( ) S (,')ˆ( j ') S(,') U(')ˆ( j ) fl U () S (,') U (') fl S (,') UU () (') W av i la p ud fac a U i a uiary opraor, i U U T S marix a followig propri: S(,) S (,') S(',) S(,')S(','') S(,'') 4 S(,') ca b xprd a a imordrd opraor, S i (,') VS ˆ( ) (,'), wic a oluio È i S (,') T xp Ê d Vˆ( ) ˆ () Î Í Ë ' A wav fucio (ad opraor) i r rpraio coicid A zro mpraur oly wav fucio of pcial ir i groud a wav fucio T ig w wa o calcula ar xprd a grouda xpcaiovalu For our Gr' fucio w all d o dfi j() a xac groud a wav fucio

4 MPP Cap Bo E Srliu 7 Sic oal Hamiloia i H, xac groud a mu av low igvalu of i Hamiloia T problm i a w do o ow ay igvalu or iga of i Hamiloia Ti i o of ig w wa o calcula Tu w av problm a all our formalim i bad o wav fucio wic w do o y ow T oly groud a wav fucio w ow i a of H, f Somow w av o drmi uow wav fucio j() i rm of ow fucio f Ti rlaio wa ow by GllMa ad Low o b: j( ) S(, ) f W aum a r i o prurbaio o ar wi T ym i i groud a of H A prurbaio i gradually urd o ad ym dvlop adiabaically io a j() W alo av o bor abou or xrm im limi O way i o aum a iracio i lowly urd off agai i fuur T ym will rur o oiracig groud a (a la if f i odgra) Ti a ca diffr from f by a mo a pa facor: MPP Cap Bo E Srliu 8 o craio ad o drucio opraor i ma a o paricl i rafrrd from o a o aor, o crad or aiilad W will ow fir dicu lcro Gr' fucio Ti dicuio i alo valid for or frmio wi propr modificaio i pi ummaio I i ligly diffr for boo wi ma Wr diffrc occur w will poi m ou Elcro: A zro mpraur lcro Gr' fucio i dfid a G( l, ') i Tcl ( ) cl ( ') T uaum umbr l ca b ayig dpdig o problm of ir, bu of w will a i o b uaum umbr of frlcro ga l (p,) Som am ar dd: T Gr' fucio i xprd i Hibrg picur; i ma a a ar im idpd ad im dpdc of opraor i giv by ad il f j ˆ( ) S(, ) j ( ) S(, ) f, il f S(, ) f ih / ih / cl() cl T a > i iracig groud a, i, iga of H wi low rgy W ould rmmbr a w ma baic aumpio a ym dvlop adiabaically from oiracig groud a io iracig groud a; w xclud ymmry braig, pa raiio ad caoic bavior GREEN'S FUNCTIONS I i cio w will dicu lcro ad poo Gr' fucio T poo ca i vry imilar o a for poo, bu mor complicad du o variou coic of gaug T poo i rad i cio of x boo T lcro par of Hamiloia w av dicud bfor ar valid o oly for lcro bu for all frmio, g H T oly diffrc i i ummaio ovr a diffr of pi uaum umbr I i furrmor valid for "ral" boo, g 4 H T poo ad poo ar mal boo, wic lad o poibiliy a umbr of paricl cag Hamiloia coai par wr paricl ar crad or aiilad T opraor for paricl wi ma alway appar a produc of T opraor c l ar dfid i rm of compl of a f l, wic ar iga o uprurbd Hamiloia H ; w ar uppod o ow a 4 T imordrig opraor T i ligly gralizd T opraor acig o vral opraor ordr m from rig o lf i acdig ordr ad add a facor () P, wr P i umbr of ircag of frmio opraor from origial giv ordr Ti dfiiio agr wi arlir o, bcau V' a w coidrd arlir i cocio wi T opraor coai a v umbr of frmio opraor Tu ac prmuaio of V' ma a v umbr of prmuaio of frmio opraor ad c o ig cagta rul cag ig for ac prmuaio of frmio opraor ma a rul for frmio ad boo ar diffr L u ry o g om flig for wa Gr' fucio ma For > ' w av G( l, > ') i cl ( ) cl ( ')

5 MPP Cap Bo E Srliu 9 A ' a paricl i addd o ym Afr i im ym dvlop i im Sic w a i o a iga of H addd paricl car io or igl paricl a Tr i om probabiliy a paricl i ill i a l a im T Gr' fucio i ju projcio of a o a c l (), i i i rlad o probabiliy a lcro a wa pu i a l a ' i ill i am a a For or im arragm ' > w av MPP Cap Bo E Srliu 4 il S(, ) S(, ) fl S(, ) S(, ) Similarly, w fid G( l, ' > ) i cl ( ') cl ( ), wr w av cagd ig ic wo frmio opraor av cagd poiio A a lcro i rmovd from ym ad pu bac a ' T Gr' fucio i a maur of probabiliy a a l i ill mpy a ' Aor way o loo upo i i a w cra a ol a im i a l ad Gr' fucio i a maur of probabiliy a ol i ill i a l a ' Now, w wi o xpr Gr' fucio i ow uaii; iad of iracig groud a w wa o u oiracig or uprurbd groud a >, i groud a of H ; u S(, ) S(, ) W u i o rplac lf brac i xprio for Gr' fucio ad fid G( l, ') i ( ') S(, ) S(, ) S(, ) S(, )ˆ cl () S(, ) S(, ) Nx w cag opraor o iracio rpraio: S (, ) S(,')ˆ cl (') S (', ) S(, ) S (,') S (', ) ih / ih / ih / ih / cl() cˆ l() U ()ˆ cl() U() S(, )ˆ cl () S(, ), wic lad o: i (' ) S(, ) S(, ) S(, ) S(, ')ˆ cl (') S(, ') G( l, ') i( ') S(, ) S(, )ˆ cl () S(, ) S(,')ˆ cl (') S(', ) S(, ) i (' ) S(, ) S(,')ˆ cl (') S(', ) S(, )ˆ cl () S(, ) S(, ) T im argum i i xprio go from o ; from o arli of ad '; from a im o ; from o la of ad '; from a im o ; from o W wa o rwri xprio i uc a way a im argum ar icraig W lard from prviou cio a S (', ) S(, c )ˆ l () S (, ) S(, ) S ( ', ) S (, ) Tu, Gr' fucio ca b rwri a: G( l, ') i ( ') S(, ) S(, )ˆ cl () S(,')ˆ cl (') S(', ) i (' ) S(, ) S(, ')ˆ cl (') S(', )ˆ cl () S(, ) Ti ca b xprd a

6 MPP Cap Bo E Srliu 4 MPP Cap Bo E Srliu 4 i Tcˆ l ()ˆ cl (') S(, ) G( l, ') S(, ) () T opraor S(, ) coai opraor wic ac i r im irval [, mi(,')], [mi(,'), max(,')], ad [max(,'), ] T T opraor auomaically or o a y ac i ir propr uc I do o mar wr w wri S(, ) i umraor, ic im ordrig opraor pu pic i rig plac A Gr' fucio ca alo b dfid for pcial ca wr iracio V ad c S marix i uiy Ti Gr' fucio, oiracig Gr' fucio, play a pcial rol i formalim, ad w diga i by G () : ( ) G ( l, ') i Tcˆ l ()ˆ cl (') I i alo ow udr am uprurbd Gr' fucio or fr propagaor Tr ar wo ui diffr yp of lcroic ym i wic w wa o mploy Gr' fucio aalyi T wo av ui diffr oiracig ad iracig groud a T wo ym ar followig A Empy Bad Hr w wi o udy propri of a lcro i a rgy bad i wic i i oly lcro A xampl i w w pu a lcro i coducio bad of a micoducor or a iulaor I i ca groud a i paricl vacuum, wic w do a > Ti a a propry a c a p, wr c p ad a ar drucio opraor for lcro ad poo, rpcivly Trfor bo H ad V giv zro w opraig upo vacuum I follow a S marix giv uiy w opraig upo vacuum: S (, ) Ti ma a bo of groud a, > ad >, ar vacuum T Gr' fucio ca xi oly for im ordrig G( l, ') i( ') cl ( ) cl ( ') T uprurbd Gr' fucio G () i paricularly ay o valua: ( ) i ( ')/ G ( l, ') i( ') l clcl i ( ')/ i( ') l T Fourir raform of G () (l,) wi rpc o i dfid a iw G( l, w) d G( l, ) To ma igral covrg, w d o add ifiiimal uaiy id o xpo 4 : ( ) i( w / id) G ( l, w) i d l G ( ) ( l, w) w l / id A Dgra Elcro Ga Our cod xampl i wr lcro ar i a Frmi a a zro mpraur T adard xampl i a impl mal I ca alo b a avily dopd micoducor T ym a a cmical poial m, ad all lcro a wi E < m ar occupid If uprurbd lcro (iga of H ) ar caracrizd by a rgy,, groud a > a all a, < m filld ad a, > m mpy T groud a > ad > ar o logr am T uprurbd groud a ca ill b coidrd a paricl vacuum if w coidr paricl abov m o b lcro ad paricl blow m o b ol Proc wr a lcro blow m i card o a a abov m i rgardd a a craio of a lcrool pair I i covi o maur lcro' rgy rlaiv o cmical poial, o dfi x,, m Somim w u m a a rfrc rgy ad omim w u boom of bad Wi m a a rfrc rgy w obai c c,, lim F x bx, b Æ, ( ) c c,, F( x, ) 4 Ti i o ju a mamaical ric T Gr' fucio w g w icludig ifiiimal id' i limi of iracig o w iracio go o zro W ca vr av xacly zro iracio or xprd i aor way: a lcro placd i a crai a will o ay r for vr Tu w av a propr limi

7 MPP Cap Bo E Srliu 4 T uprurbd Gr' fucio i ow ( ) ( F, ) F, G (, ') i Tc, ( ) c, ( ') ix i x x ( ')/ ( ') ( ) ( ' ) ( ), [ ] T fir par of i fucio giv coribuio for a ouid Frmi a ad i am a mpybad Gr' fucio T cod par giv coribuio oly if a i wii Frmi a T fir rm ca b rgardd a lcro par of Gr' fucio ad cod rm a ol par T Fourir raformd Gr' fucio i, i( w x, / id) È ( ) ( ) i( w x id G w i F x d / ) (, ) (, ) F( x, ) d ( ) F( x, ) F( x, ) G (, w) w x, / id w x, / id W av r addd a id i lcro par ad a id i ol par 5 Aor, mor compac way o wri G () i ( ) G (, w) ; d, dgx,, w x, / id, wr d, i a mall ifiiimal par wic cag ig a cmical poial I i i om iuaio prfrabl o u form i box If w wa o udy iglparicl xciaio of groud a, > i rplacd by a a wi a lcro ouid ad a ol iid Frmi a T xprio iid box i ow valid if occupaio umbr ar proprly modifid Poo: T Gr' fucio for poo i dfid a D(, l; ') i TA, l( ) A, l( ') ; A, l a, l a, l T ubcrip l rfr o polarizaio of poo Uually w ar ird i ju o 5 S fooo 4 Hr id' guara a a lcro or a ol placd i a crai a will o ay i am a for vr MPP Cap Bo E Srliu 44 id of poo wi Hamiloia wic do o mix polarizaio, o w all omi ubcrip irly I iracio rpraio o obai rul D(, ') i TAˆ ()ˆ A (') S(, ) S(, ) A zro mpraur r ar o poo Tu groud a > ad > ar agai paricl vacuum > No a i a lcropoo ym oaio > ma combiaio of groud a for lcro, poo, c Aloug poo ym a vacuum a i groud a, ir of wo lcro groud a ca b ud T uprurbd poo Gr' fucio i dfid a ( ) iw i i ' i ' i T a w a w a w a ( )( ) D (, ') i TAˆ ()ˆ A (') Now, { i ( ') i ( ' ) [ a a ] iw( ') iw( ') [ ]} w i ( ') w a a (' ) a a a a aa a a N bw Tu, aa a a N [( ) ] ( ) i ' i ' D (, ') i N w w N T Fourir raform 6 giv ( D ) È (, w) ( N ) Í w w id w w id N 6 To ow i i lf a omwor problm umbr 7 È Í w w id w w id

8 MPP Cap Bo E Srliu 45 W av r p poo occupaio umbr o b abl o ra iuaio wr a poo i pr i ym or w av a fii mpraur For zro mpraur ad i groud a w av N, wic ma a 4 WICK'S THEOREM ( D ) (, w) w w id w w id w w w id T Gr' fucio i valuad by xpadig S marix S(, ) i () i a ri uc a (): ig ( p, ') (i ) d d! L Tcˆ ()ˆ( V ) V ˆ(, ) Vˆ( ) c ˆ p L p, (') S(, ) Â (4) L u for mom igor facor < S(, ) > W all a car of i i Sc 6 Our immdia aim i o lar ow o valua im ordrd brac li Tc, V V V c, ˆ ()ˆ( ) ˆ( ) ˆ( p ) ˆ p (') (4) Suppo a V i lcrolcro iracio: ˆ( ) ˆ ( )ˆ V Âvc, c', ' ( )ˆ c', ' ( )ˆ c, ( ) v, ',, ' i ( c, c ', ' c ', ' c, x Âv x x x v, ',, ' ' ' )/ I i ca im ordrd brac (4) coai v craio ad v drucio opraor Ti ma a v lcro ar a away from groud a ad v ar pu bac bfor rulig a i projcd bac o groud a Exacly am MPP Cap Bo E Srliu 46 a av o b occupid afr opraio by all opraor a bfor opraio All ucciv a afr ac opraio ar iga o H o, ju a groud a i Tu a ar orogoal Ti ma a fial a a o b groud a Orwi brac will giv o coribuio W ca immdialy ruir a brac coai am umbr of craio a drucio opraor Ti i alway fulfilld for lcrolcro iracio ic ac V coai wo of ac yp of opraor I lcropoo iracio ca i i o o Oly brac wi a v umbr of V' urviv W alo a opraor av o b paird; y opra wo ad wo o am a, o craio ad o drucio opraor I i vry complicad o drmi valu of a brac ic r ar may poibl im ordrig ad may poibl pairig bw craio ad drucio opraor Howvr, oly a limid umbr of combiaio ar pyically irig Our aim i o or i a impl way, wic i acivd wi lp of om orm wic implify procdur T fir of i Wic' orm Ti orm i rally ju a obrvaio a im ordrig ca b a car of i a impl way I a a i maig all poibl pairig bw craio ad drucio opraor ac pairig ould b imordrd T im ordrig of ac pair giv propr im ordrig o ir rul For xampl, w g Tcˆ c a ()ˆ b ( )ˆ cg ( )ˆ cd (') Tcˆ a ()ˆ c ( ) Tcˆ b ( g )ˆ cd (') Tcˆ a ()ˆ c (') Tcˆ d g ( )ˆ cb ( ) dab dgd Tcˆ a ( c )ˆ a( ) Tcˆ g( )ˆ cg(') dad dgb Tcˆ a ()ˆ ca (') Tcˆ g ( )ˆ cg ( ) No a r i a imordrig opraor T i ac pairig brac I ca of craio ad drucio opraor r ar! poibl pairig Pairig rul: ) A ig cag occur ac im poiio of wo igborig Frmi opraor ar ircagd ) T cod rul cocr im ordrig of combiaio of opraor rprig diffr xciaio For xampl, coidr followig mixur of

9 MPP Cap Bo E Srliu 47 poo ad lcro opraor: Tcˆ ()ˆ c ( ) A ˆ ( )ˆ c ( )ˆ c ( ) A ˆ p p p p ( ) Bcau lcro opraor commu wi poo opraor, w do o car ow y ar ordrd wi rpc o ac or Tu w ca immdialy facor brac io para lcro ad poo par: Tcˆ ()ˆ c ( )ˆ c ( )ˆ c ( ) TAˆ ( ) A ˆ p p p p ( ) Ti paraio i alway poibl wi diffr id of opraor, i,wvr opraor commu Wic' orm alo appli o brac of poo opraor; for xampl, TAˆ ( ) A ˆ ( ) A ˆ ( ) A ˆ ( 4 4) TAˆ ( ) A ˆ ( ) TAˆ ( ) A ˆ ( 4 4) TAˆ ( ) A ˆ ( ) TAˆ ( ) A ˆ ( 4 4) TAˆ A TA A ( ) ˆ ( 4 4) ˆ ( ) ˆ ( ) d TAˆ A TA A 4 ( ) ˆ ( ) ˆ ( ) ˆ d ( 4) d TAˆ d 4 ( A TA A ) ˆ ( ) ˆ ( ) ˆ ( 4) d d TAˆ ( ) A ˆ ( ) TAˆ ( ) A ˆ ( ) 4 4 ) T ird rul w d i a mod of raig im ordrig of wo opraor wic occur a am im, uc a Tc c ˆ ( )ˆ ( ) I ca drucio opraor alway go o rig, d Tcˆ ( )ˆ c ( ) d F ( x ) ad rm i ju umbr opraor wic i idpd of im Ti covio i dpd o way w wro dow Hamiloia I corucig H w wr carful o pu drucio opraor o rig of craio opraor i all rm i Hamiloia W wo lcro opraor av diffr im argum i a pairig w covioally pu craio opraor o rig: Tc c d Tc c ˆ ( )ˆ ( ) ˆ ( )ˆ ( ) Ti rm ca b immdialy idifid a uprurbd Gr' fucio ig () (, ) Our prviou xampl ca alo b wri i rm of Gr' fucio: MPP Cap Bo E Srliu 48 Tcˆ a ()ˆ cb ( )ˆ cg ( )ˆ cd (') ( ) ( ) a, b g, d ( ) ( ) a, d g, b d d ig ( a, ) ig ( g, ') d d ig ( a, ') ig ( g, ) ; TAˆ ( ) A ˆ ( ) A ˆ ( ) A ˆ ( 4) 4 ( ) ( ) d d id (, ) id (, ) 4 ( ) ( ) 4 d d id (, ) id (, ) 4 d d 4 ( ) ( ) 4 4 id (, ) id (, ) I ummary, Wic' orm ll u a a imordrd brac may by valuad by xpadig i io all poibl pairig ad a ac of pairig will b a Gr' fucio or a umbr opraor F or B W all ow do a compriv xampl W all coidr rm of S marix xpaio of lcro Gr' fucio i (4) T iracio will b a a lcropoo iracio: V M a a M A c c v  r(, l, l )  v,, W av for impliciy ju icludd o poo polarizaio ad glcd ummaio ovr rciprocal laic vcor I xpaio rm i alway G () ad vai li r of rm wi a odd valu W obai ( ) ( ) i ig( p, ') ig ( p, ') d d! M M TAˆ  ( ) A ˆ ( ) v, Â,,,, p Tcˆ p ()ˆ c ( )ˆ c, ( )ˆ c, ( )ˆ c, ( )ˆ c, (') L T poo brac giv a iglpoo Gr' fucio: TAˆ ( ) A ˆ ( ) id (, ) d ( ) T lcro brac, uforualy, a! 6 poibl combiaio of pairig, ic i

10 MPP Cap Bo E Srliu 49 coai r opraor of ac yp W all giv ix rm ad u fac a Wic' orm giv rul p,,,,, p, Tcˆ ()ˆ c ( )ˆ c ( )ˆ c ( )ˆ c ( )ˆ c (') Tcˆ p, ()ˆ c, ( ) Tcˆ, ( )ˆ c, ( ) Tcˆ, ( )ˆ cp, ( ' ) ( ) ( ) ( ) idp G d ( p, ) id ( d G, ) idp d G (, ') p Tcˆ p, ()ˆ c, ( ) ˆ, ( Tc )ˆ c, ( ) Tcˆ, ( )ˆ c, (') p ( ) ( ) ( id d G ( p, ) id d G (, ) id d G ) ( p, ') p p Tcˆ p, ()ˆ c, ( ) Tcˆ, ( )ˆ cp, (') Tc , c, ( ) ( idp G d ( p, ) idp d G ) ( p, ') d F ( x ) ˆ ( )ˆ ( ) Tcˆ p, ()ˆ cp, (') Tcˆ , ( )ˆ c, ( ) Tcˆ, ( )ˆ c, ( ) ( ) d x d x ig ( p, ') F( ) F( ) Tcˆ p, ()ˆ c, ( ) Tcˆ, ( )ˆ c, ( ) Tcˆ , ( )ˆ c p, (') ( ) id G ( p, ) F ( ) p d d x ( ) idp d G ( p, ') Tcˆ p, ()ˆ cp, (') Tcˆ , ( )ˆ c, ( ) Tcˆ, ( )ˆ c, ( ) ( ) ( ) ( ) ig ( p, ') d d (, ) d d i G i G (, ) MPP Cap Bo E Srliu 5 ( ) ( ) i ig( p, ') ig ( p, ')! W av mad u of fac a N Â F ( x, ), d d Ï ÔÈ ( ) ÌÍ Â M id (, ) v ÓÔ ( ) ( ) ig ( p, ) ig ( p, ( ) ) ig ( p, ' ) ( ( ) ( ) ( ) ig ( p, ) ig ( p, ) ig ( p, ') ( ) ( ) ( ) ig ( p, ') ÂiG (, ) ig (, ) ), ( ) È (, v M id ( ) ( ) ) ( N( ig ( p, ) ig ( p, ') ( ) ( ) ( ) ig ( p, ) ig ( p, ') N ig ( p, ') ) )] } Tu fial rul i

11 MPP Cap Bo E Srliu 5 5 FEYNMAN DIAGRAMS A picuri mor a a ouad word W foud i prviou cio a v fir orivial rm i S marix xpaio of Gr' fucio i rar complicad ad o wri i dow a a lo of pac Ta coribuio producd ix rm Nx ovaiig coribuio produc 6 rm; o afr a 756 rm! I i obviou a a raigforward drivaio i impoibl if o i o aifid i pig ju fw fir coribuio Howvr, w will fid a may of rm cacl ou, may av idical valu ad om ould o av b icludd i fir plac T la am i idd for rm wic could av b limiad from Hamiloia from bgiig; am way a i wa do for our Hamiloia i cio 7 Fyma iroducd ida of rprig igral xprio by drawig T drawig, calld diagram, ar xrmly uful for providig a iig io pyical proc wic coribuio rpr T diagram a muc l pac Giv problm o ca immdialy wri dow diagram, ma maipulaio li ummaio of ubcla of diagram, xclud om cla of diagram, ad w fial diagram av b dcidd o o ca wri dow corrpodig igral xprio ad olv problm T diagram ca b draw bo for Gr' fucio dpdig o im a wll a for Fourir raformd vrio a dpd o w T diagram i im pac ar draw by rprig lcro Gr' fucio G () (p,') by a olid li wic go from ' o, a ow i Fig A arrow i of icludd o rpr dircio T arrow i moly for covic, ad i do o imply or ruir a > ' I a, owvr, omig mor o i If > ' ad arrow poi i dircio from ' o Gr' fucio rpr a lcro a If < ' Gr' fucio rpr a ol a T ol ca b viwd a a lcro goig bacward i im Of, a diagram coai om arrow poiig o rig ad om poiig o lf i figur Ti i ca i Fig (f) W > uppr Gr' fucio rpr a ol ad lowr a lcro W < uppr Gr' fucio rpr a lcro ad lowr a ol T poo Gr' fucio i rprd by a dad li I a o dircioal arrow ic if > ' i rpr bo a poo goig forward i im ad a poo goig bacward i im or uivally a poo ad a "poo ol" For < ' i i oppoi Howvr, Fyma diagram i wpac av poo wi a dircioal arrow, bu arrow ow dircio i wic momum ad rgy flow i diagram Nx w av o dcid ow o ra facor <c p ()c p ()> F (x p ) I i draw a a olid li wic loop ad rpr a MPP Cap Bo E Srliu 5 lcro li wic ar ad d a am poi i im Now w av all igrdi dd o coruc Fyma diagram for Gr' fucio rad a d of prviou cio To b abl o ra all problm ivolvig lcro ad poo w alo add Coulomb iracio li To diigui i from a poo Gr' fucio w draw Coulomb iracio a a wavy li T Coulomb iracio i iaaou wic ma a iracio li ar vrical i diagram () G (p,') () D (,') ' ' p <c p ()c p()> v p Fig I may ca w will av crd Coulomb iracio T iracio ar o logr iaaou ic crig d im o dvlop T cod ordr diagram i S marix xpaio of lcro Gr' fucio i prc of lcropoo iracio i giv i Fig of x boo T rm (c), (d) ad () ould o b icludd ic y all com from rm of Hamiloia for lcropoo iracio Ti rm ould av b xcludd i fir plac T rm (a) ad (b) giv coribuio Ty loo ali Ty diffr oly i lablig of variabl,, ± T ar dummy variabl i igral ad wo coribuio ar ual T la rm (f) i diffr from r of o w p i a i diagram coi of wo para par (i i alo ru for (d) ) I igral form i ca b wri a produc of wo facor; o of facor i a fucio of p ad(') ad or i a coa I blog o a pcial cla of diagram calld dicocd diagram Ti cla of diagram coai all diagram i wic o all par of ac diagram ar

12 MPP Cap Bo E Srliu 5 cocd W will com bac o diagram i x cio 6 VACUUM POLARIZATION GRAPHS W ow ur our aio o facor wic o i poi w av b igorig: ( ) i S( d, )  d TVˆ( ) Vˆ( )! L L Ti ould av b i domiaor of Gr' fucio Eac rm i i xpaio gra a ri of rm Eac of rm ar coa T grad Fyma diagram ar calld vacuum polarizaio grap If w wr o drmi i uaiy o cod ordr wi lcropoo iracio a V w would g a from zro ordr, o coribuio from fir ordr ad wo diagram from cod ordr coribuio T wo diagram ar xacly wo dicocd par of diagram i Fig (d) ad () If w wr o wri dow all diagram from xpaio of umraor of Gr' fucio, pic ou o paricular cocd diagram, ca roug all dicocd diagram ad pic ou all of a av am cocd par a o w av co w would fid followig: T dicocd par of all diagram w av picd ou gra xacly vacuum polarizaio grap Ti i ru for vry coic of cocd diagram Tu umraor of Gr' fucio ca b wri a a produc of wo facor; o coaiig all cocd diagram; o coaiig all vacuum polarizaio grap T la facor cacl xacly domiaor of Gr' fucio Tu i xpaio of Gr' fucio w ould procd a bfor ad xpad umraor ad from i xpaio dicard all dicocd diagramtu, (i ) ig ( p, ')  d d! L Tcˆ ()ˆ( V ) V ˆ(, ) Vˆ( ) c ˆ p L p, (') (cocd) MPP Cap Bo E Srliu 54 L u ow a Fourir raform of i xprio wi rpc o (') for ca w av udid Fir w rwri our obaid xprio afr all ocoribuig rm av b dicardd T rul i ig ( p, ') ig ( p, ') i d d ( ) ( ) È ( ) ( ) ( ) ( ) Í Â M id (, )( ig ( p, ) ig ( p, ) ig ( p, ') v Now w a ivr Fourir raform of poo Gr' fucio D (, ) dw' p ( ) iw'( ) ( ) D (, w'), ir i ad a Fourir raform of wol xprio T fial rul i S ( ) ( ) () G ( p, w) G ( p, w) G ( p, w) S ( p, w) ; [ ] dw' È ( p, w) ( ) i Í Â M D (, w') G ( p, w w'), p v () ( ) ( ) wr S () i lfrgy du o opoo proc W oic a Gr' fucio iid lfrgy a am pi idx a Gr' fucio w ar udyig Ti i ru i mo ca If o or ad iracio ca flip pi lf rgy will av o av wo pi idic Ti i alo ru for iracig Gr' fucio I a ca Gr' fucio ad lfrgi ar maric ad all produc of wo fucio ar rally marix produc Howvr, w will o coidr i ca r Nx w g rid of /! facor I ur ou o b ju! rm xacly ali i ac brac of rm i xpaio Tu if w coidr oly diffr rm, w obai rul  ig ( p, ') (i ) d L d (6) Tcˆ ()ˆ( V ) V ˆ( ) LVˆ( ) c ˆ (') ( diffr cocd ) p, p,

13 MPP Cap Bo E Srliu 55 MPP Cap Bo E Srliu 56 7 DYSON'S EQUATION If w um ri of rm o ifii ordr w fid a all igr ordr rm av a uprurbd Gr' fucio a bo d of diagram Ti ma a w ca wri ( ) ( ) ( ) G ( p, w) G ( p, w) G ( p, w) S ( p, w) G ( p, w), ( ) ( ) ( ) G ( p, w) G ( p, w) G ( p, w) S ( p, w) G ( p, w) ( ) G G ( p, w) ( ) G ( p, w) G ( p, w) * ( ) S ( p, w) G ( p, w) ( ) * ( ) ( p, w) S ( p, w) G ( p, w) * ( ) S ( p, w) G ( p, w) ; wr S i lfrgy Now, w d o iroduc om w cocp: A lfrgy irio i dfid a ay par of a diagram a i cocd o r of diagram by wo paricl li(o i ad o ou) A propr lfrgy irio i a lfrgy irio, wic cao b parad io wo pic by cuig a igl paricl li T propr lfrgy i um of all propr lfrgy irio I i dod by S * I follow from dfiiio a lfrgy coi of a um of all poibl rpiio of propr lfrgy, i * * ( ) * S ( p, w) S ( p, w) S ( p, w) G ( p, w) S ( p, w) Ti ca b wri a * ( ) * ( ) * S ( p, w) G ( p, w) S ( p, w) G ( p, w) S ( p, w) L * * ( ) S ( p, w) S ( p, w) S ( p, w) G ( p, w) S ( p, w), ad b olvd o giv S( p, w) S( p, w) * ( ) S ( p, w) G ( p, w) * Ti i Dyo' uaio for lfrgy Puig i io xprio for Gr' fucio giv T rlaio i box i Dyo' uaio for lcro Gr' fucio T lcro lfrgy i omim calld a ma opraor T Gr' fucio i xampl wi dgra lcro ga bcom F( x, ) F( x, ) G (, w) * * w x, / id S(, w) w x, / id S(, w) T lfrgy a ral ad imagiary par W all ow lar a imagiary par cag ig a w I i poiiv for gaiv w ad gaiv for poiiv w For poo w av ( ) D (, w) D(, w) * ( ) p (, w) D (, w) Ti i Dyo' uaio for poo Gr' fucio T poo lfrgy i omim calld a polarizaio opraor T poo Gr' fucio a zro mpraur ca b wri a w D(, w) w w id w p * (, w) T ral ad imagiary par of lfrgi ac av irpraio T imagiary par of lfrgi i irprd a cauig dampig of paricl moio Ty ar rlad o fii ma fr pa of xciaio or i rgy ad momum ucraiy T ral par ar acual rgy if of xciaio, wic may alo cag i dyamical moio T xciaio may alr i ffciv ma or group vlociy bcau of lfrgy coribuio I i acually S * ad P * a av dimio rgy ad ar acual lfrgi

14 MPP Cap Bo E Srliu 57 8 RULES FOR CONSTRUCTING DIAGRAMS Draw Fyma diagram for lfrgy rm, wi all poo, Coulomb, ad lcro li For ac lcro li, iroduc followig Gr' fucio: È ( ) F( xp, a ) F( xp, a ) Gab ( p, w) dab Í w xp, a / id w xp, a / id T d ab idica a lcro li mu av am pi a bo d of propagaor li Ti faur i impora i pi problm T Gr' fucio i valid for a dgra Frmi ym, bu ca alo b ud for mpy bad ca if cmical poial i aumd o b a boom of bad For ac poo li, iroduc followig poo propagaor: ( D ) w (, w) w w id Alo add a facor M / for ac poo Gr' fucio, wr M i marix lm for lcropoo iracio 4 Add a Coulomb poial v 4p / for ac Coulomb iracio No a w alway draw Coulomb li a a wiggly vrical li T Coulomb iracio i rgardd a appig iaaouly i im, ad im flow orizoally, from lf o rig, i our diagram O could, of cour, av im flow upward ad draw Coulomb iracio a orizoal wiggly li 5 Corv rgy ad momum a ac vrx Tu ac lcro li, poo li, ad Coulomb li av ir variabl labld o coform wi i rul 6 Sum ovr iral dgr of frdom: momum, frucy, ad pi T ummaio ovr momum ad frucy ad pi ould b prformd accordig o dw  v p, d dw ( p ) p  Ïfor box ormalizaio, Ì Ó ad dicr momum ummaio, for igraio ovr momum If o i calculaig a lfrgy rm S(p,w), all momum ad fruci xcp p ad w ar iral ad mu b ummd ovr MPP Cap Bo E Srliu 58 7 Fially, w muliply rul by facor (i/) m () F wr F i umbr of clod frmio loop T idx m i co a follow: a For lcro lfrgi, m i umbr of iral poo ad Coulomb li b For poo lfrgi, m i oalf umbr of vric 8 For ac poo li wic irac wi paricl roug ja iracio, ir a facor Ê ˆ Ë m  ( / ) D (, )( ' / m m w ) m wr D m (,w) i poo Gr' fucio ad ad ' ar wav vcor of paricl card a wo vric T or poibl iracio of a cargd paricl wi poo occur roug rm mc A ( r i)  r( ) Am ( ) Am ( ) m m,, W will lar a i iracio coribu a lfrgy rm of /m o lfrgy of poo, wr i diy of cargd paricl To obai mor gral rul for corucig ad irprig Fyma diagram w av o diribu (i/) m facor i diagram To do i w aac facor (i/) / a ac vrx(i ca b do ligly mor gral If w av a iuaio wr ym coi of paricl of diffr carg vrx facor ould b muliplid by valc Z for paricl card a vrx) T ac diagram coi of oly Gr' fucio (for paricl, poo or poo) ad vric T Coulomb iracio li ca b viwd a a poo Gr' fucio T Coulomb iracio bw wo paricl ca b rgardd a ircag of wo diffr yp of poo T Coulomb iracio a a Dyo' uaio v v( w) * vp (, w) v (, w), wr P* i propr lfrgy for Coulomb iracio or polarizaio opraor

15 MPP Cap Bo E Srliu 59 If oly low ordr coribuio of P* i p rul i c o, ucpibiliy iroducd a d of capr Wi i coic dilcric fucio i RPA dilcric fucio T Coulomb iracio i ow crd ad o logr iaaou L u udy followig par of a Fyma diagram: MPP Cap Bo E Srliu 6 wr p () (,w) i low ordr poo lfrgy irio Now, l u udy aor diagram (,w'w) v (p',w'') (i/) / v (,w) (p',w''w) (i/) / (,w') (i/) / (p,w') (i/) / (p,w'w) v T "bubbl" icludig wo vric i ju c o, ad diagram i irprd a v () i dw' ( ) ( ) c(, w) v  G (, w') G (, w' w), v p, wr miu ig com from fac a c o coai o clod frmio loop If w ow av a diagram wi poo li iad of Coulomb iracio li w g T diagram i irprd a D (M()/) (i) / (,w' w) (,w') () D (,w) () D (,w) (M()/)(i) / M ( ) (, w) p (, w) D (, w) c(, w), ( ) ( ) ( ) Ti dcrib carig of wo paricl, lcro ay, agai ac or via Coulomb iracio T lcropoo iracio giv ri o a vry imilar diagram, viz (p',w'') (p,w') () D (,w) (M()/) (i) / (M()/) (i) / (p',w''w) (p,w'w) Ti lad o a ffciv lcrolcro iracio coribuio Togr coribuio giv M ( ) v (, w) v D ( ) (, w)

16 MPP Cap Bo E Srliu 6 MPP Cap Bo E Srliu 6 L u wa w g for polar micoducor a w rad i of capr W fid v g ( ) ( ) v (, w) D (, w) v Ê p ˆ w LO LO Á w Ë ( w wlo id) v 4p w LO ( w wlo id) v ( w), d ordr ad p wr (w) i frucy dpd bacgroud dilcric fucio wic rul from combid crig by valc lcro (virual raiio acro bad gap) ad poo Ti dilcric fucio i ( ) w wlo ( w) w w LO i d 4 4 wto ( ) w wlo È Í ; wto Îw wto id w wto id ( ) ( ) 4 ordr p Ti dilcric fucio vai a logiudial poo frucy ad a a dla fucio imagiary par a ravr poo frucy T buildig bloc w av a our dipoal w drawig lcro Gr' fucio diagram, w iracio i lcropoo iracio, ar T corrpodig buildig bloc i ca of lcrolcro iracio ar

17 MPP Cap Bo E Srliu 6 ordr MPP Cap Bo E Srliu 64 9 TIMELOOP S MATRIX A Six Gr' Fucio W will dicu ix diffr Gr' fucio T yp w av iroducd o far i calld imordrd vrio T or fiv ar aiimordrd (dod by a bar ovr ), "G l" G <, "G largr" G >, rardd G r, ad advacd G adv For frmio ix Gr' fucio ar dfid a ad d ordr > G (, ) i c ( ) c ( ) < G (, ) i c ( ) c ( ) G (, ) ( ) G (, ) ( ) G (, ) < > G (, ) ( ) G (, ) [ ( ) ] G (, ) > < G (, ) ( ) G (, ) G (, ) i( ) c ( ), c ( ) r > < [ ] [ ] { } < > [ ] [ ] G (, ) ( ) G (, ) G (, ) adv W a imordrd Gr' fucio i ual o G > for poiiv ad G < for gaiv T rardd Gr' fucio i zro for gaiv For poiiv i i ual o G > G < T advacd fucio i zro for poiiv For gaiv i i ual o G < G > T uprurbd frmio Gr' fucio ar T rulig (d) ordr diagram i lcrolcro iracio ca loo xacly am a d (4) ordr diagram i lcropoo iracio ca xcp for a iracio li i ow wiggly iad of dad ( ) ix / G (, ) i ( ) ( ) ( ) ix / G (, ) i ( ) ( ) G (, ) i G < ix / > { [ ] [ ] } {[ ][ ] } (, ) i x G i (, ) ( ) ( ) i( ) ( ) i / ix / r ( ) adv ix / [ ] { [ ] } { [ ][ ] [ ] } G (, ) i ( ) ( ) [ ] ix / ix / i () Afr Fourir raformaio wi rpc o, wr a ifiiimal d a b addd o frucy xpo o guara covrgc i ± limi of im igral w obai

18 MPP Cap Bo E Srliu 65 G ( ) (, w) w x / id w x / id ( ) È G (, w) Í Îw x / id w x / id < G (, w) ip d( w x/ ) > G (, w) i[ ] pd( w x / ) ( ) Gr (, w) w x / id w x/ id w x/ id ( ) Gadv(, w) w x / id w x/ id w x/ id For boo fild uc a poo ad poo dfiiio ar imilar, viz > D (, ) i A ( ) A ( ) < D (, ) i A ( ) A ( ) D (, ) ( ) D (, ) ( ) D (, ) < > D (, ) ( ) D (, ) [ ( ) ] D (, ) > < D (, ) ( ) D (, ) D (, ) i( ) A ( ), A ( ) r > < [ ] D (, ) ( ) D (, ) D (, ) adv [ ] [ ] < > [ ] [ ] T mai diffrc i a D > ad D < av am ig, ic o ig cag i mad w ircagig poiio of boo opraor Alo diplacm opraor i Hrmiia, wic iroduc om rdudacy uc a D < (,) D > (,) T uprurbd boo fild Gr' fucio ar {[ ] } {[ ] } ( ) i ' i ' w D (, ) i N w N ( ) i ' iw ' > iw i D (, ) i N w N {[ ] } < iw D (, ) {[ iw ] i N N } ( ) r (, ) ( )i( w ) ( ) r (, ) [ ( )] i( w ) w D (, ) i N N D D MPP Cap Bo E Srliu 66 T Fourir raformd vrio rad È ( ) È D (, w) ( N ) Í N Í w w id w w id w w id w w id È ( ) È D (, w) ( N ) Í N Í w w id w w id w w id w w id < D (, w) ip N d( w w ) N d( w w ) > [( ) ] ( ) [ ] D (, w) ip N d( w w ) N d( w w ) È ( ) È Dr (, w) ( N ) Í N Í w w id w w id w w id w w id w w id w w id È ( Dadv ) (, w) ( N ) Í N w w id w w id w w id w w id È Í w w id w w id T frmio ad boo fild Gr' fucio ar o idpd I ca b fruiful o rmmbr followig rlaio: Gr G G Gadv G G < Dr D D > D D D adv < < Som word abou aalyical propri of all Gr' fucio ould b aid A zro mpraur rardd vrio av all pol i lowr alf of complx frucy pla Ti ma a y ar aalyic i uppr alfpla For advacd vrio i i oppoi T pol ar i uppr alfpla ad fucio ar aalyic i lowr alfpla T imordrd fucio av pol blow ral frucy ax for poiiv fruci ad abov for gaiv fruci I frmio ca i i ru oly if w av cmical poial a rfrc rgy for paricl rgy If o, pol ar blow ax for fruci largr a m/ ad abov for mallr fruci T oppoi i ru for aiimordrd fucio

19 MPP Cap Bo E Srliu 67 Rlaio o obrvabl W av r prd Gr' fucio wiou xplicily aig ow o u i All Fyma diagram coi of Gr' fucio ad vric; Coulomb iracio li ca alo b rgardd a poo Gr' fucio Apar from owig up a impora par of Fyma diagram iglparicl Gr' fucio ca b dircly ud o calcula impora propri of ym I giv: T xpcaio valu of ay iglparicl opraor i groud a of ym W av o far dicud Gr' fucio i momum rpraio I coordia pac i i dfid a G(,', rr ') i Tj(,) r j (',') r T xpcaio valu of a iglparicl opraor i O d r j (,) r O() r j(,) r dro() r j (,) r j(,) r lim lim r' Ær' Æ lim lim ± r' Ær' Æ dro() r j (',') r j(,) r dro() r Tj(,) r j (',') r lim lim ± i d ro() r G(,'; r r '), ' Æ Æ r r' wr plu ig i for boo ad miu ig for frmio T xpcaio valu of diy opraor bcom r() r lim lim ± i d r''( d r r'') G('','; r r ') r' Ær'' ' Æ lim lim ± ig(,'; rr ') r' Ær ' Æ ± i lim Â Æ v, dw iw G (, w), p MPP Cap Bo E Srliu 68 wr la p i valid for a omogou ym No a xpcaio valu i i ca i poiio idpd T xpcaio valu of iic rgy bcom T r lim lim ± i d r m G Æ (,'; rr ') r' r' Æ * dw i iw ± lim  m G Æ, * (, w ), p wr la p i valid for a omogou ym T grouda rgy of ym 7 Aloug Hamiloia of ym coai woparicl opraor i i poibl o xpr rgy of ym i rm of iglparicl Gr' fucio T rlaio i H ± i d r È i r lim lim Í G Æ (,'; rr ') r' r' Æ Î m* d i i È w w ± lim Â Í m G w (, w), Æ p, Î * wr la rlaio i valid for a omogou ym T xpcaio valu of iic rgy opraor i o wa w uually call iic rgy T iracio i ym cag i xpcaio valu Wi iic rgy w uually ma rgy i abc of iracio E T iracio rgy i diffrc bw rgy ad iic rgy Furrmor, a w will, i would b br o av a miu ig i fro of iic rgy opraor i abov xprio for E Wi a ric w will b abl o calcula iracio rgy ad xprio w g a i wad miu ig T Hamiloia i wri wi a variabl couplig coa H( l) H lv H() H H( ) H 7 I i cio paricl rgy i rlaiv bad boom iad of cmical poial Orwi w would av ad wm iad of w iid brac S Vr ad Walca p677

20 MPP Cap Bo E Srliu 69 L, H( l) Y ( l) E( l) Y ( l) Y ( l) Y ( l) fl E( l) Y ( l) H( l) Y ( l) T drivaiv wi rpc o couplig coa rduc o d d E dy ( l) dy ( l) ( l) H( l) Y( l) Y( l) H( l) l d l d l d E( l) Y( l) Y( l) Y( l) V Y( l) d l Y ( l) V Y ( l) Igra i uaio wi rpc o couplig coa from o d l E E Y V ( l) l Y( l) l dh( l) Y( l) Y( l) d l T if i grouda rgy i xprd olly i rm of marix lm of iracio lv d l E E Y H T ( l) ( l) Y( l) l d d i i È l w w ± Â Í m G l lim w (, w) Æ l p Î *, MPP Cap Bo E Srliu 7 Uig Dyo' uaio for Gr' fucio lad o [ ] dl dw E E i iw * l ± G l lim  S (, w) (, w) Æ l p, dl dw i iw * l ± G l lim  S (, w) (, w) Æ l p, T igraio ovr "" rm o fir li vai i propr limi W av r giv o way o driv iracio rgy of ym Ti formulaio i uful ic rul i giv i rm of propr lfrgy ad Gr' fucio uaii a w ow ow o calcula W will u a alraiv mod i cod par of i cour T xciaio pcrum of ym T fucio G(,w) a impl pol a xac xciaio rgi of iracig ym corrpodig o a momum Tu iglparicl Gr' fucio giv u mo of iformaio w d o udrad our iracig ym Howvr r i o impora ig miig W do' g ym' rpo o xral prurbaio A rcip for calculaig ig i giv by Kubo formula: Aum a a xral fild F(r,) giv ri o a prurbig rm i Hamiloia: Hx d ra(,) r F(,) r T ym rpod wi a iducd diy of om id; i may, g, b a carg diy, a curr diy or a pi diy For a omogou ym i liar rpo rgim iducd uaiy ca b rlad o xral fild by a ucpibiliy fucio a(r,) accordig o B (,) r d r' d' a( r r', ') F(',') r B id id (, w) a(, w) F(, w), wr i a( r r', ') ( ') [ B( r, ), A( r', ') ]

21 MPP Cap Bo E Srliu 7 Ti ucpibiliy or rpo fucio ca b o diffr form; rardd, advacd, imordrd,hr p fucio ma a rpo fucio i o rardd form, wic guara a rpo com afr prurbaio I a acual calculaio i i br o u imordrd vrio, ic w ca apply Wic' orm ad u rduc complxiy of problm dramaically A d i i ay o raform imordrd fucio io rardd o T rpo fucio or corrlaio fucio ar complx valud T ral ad imagiary par of rardd fucio ar rlad Somim i i difficul o calcula or maur o of par If or par i ow for all fruci fir par ca b obaid roug Kramr Kroig diprio rlaio T rlaio ar: L B(w) b aalyic i uppr alf of complx frucy pla ad B(w) Æ a w Æ T B( w) d ' B( ') P p w w w' w B( w) dw' B( w') P p w' w Rardd corrlaio fucio fulfill ruirm ad i i ca B( w) B( w) B ( w) B ( w), wic giv w' B ( w) d ' B P ( ') p w w w' w w B ( w) dw' B P ( w') p w' w Exampl of corrlaio fucio ar: (,w); /(,w); complx ivr rlaxaio im i GDA, /(w); rfraciv idx, (w); dyamical coduciviy, (w); pi ucpibiliy, c(,w)