Green Functions, the Generating Functional and Propagators in the Canonical Quantization Approach

Transcription

1 Grn Functons, th Gnratng Functonal and Propagators n th Canoncal Quantzaton Approach by Robrt D. Klaubr 15, 16 Mnor Rv: Spt, 16 Sgnfcant Rv: Fb 3, 16 Orgnal: Fbruary, 15 Th followng s not a complt documnt n that t dos not covr som matral and dos not provd vry stp for th matral t dos covr. It s not a stand alon prsntaton of Grn functons and gnratng functonals. Rathr, t s ntndd only as an ad for thos usng othr books, partcularly Mandl and Shaw, for crtan matral I dd not covr n Studnt Frndly Quantum Fld Thory (SFQFT. If I wrt a scond book on QFT, ths wll all b xpandd and mprovd pdagogcally for ncluson thrn. Thr s a wholnss chart summary of th matral bgnnng hrn on pag wth th ttl Summary of Grn Functon Mthodology of QFT, th lnk for whch may b found nar th bottom of th SFQFT hom pag. S Canoncal Approach column n that chart. Th path ntgral approach column s covrd n that chart, but s not covrd n ths documnt. Plas not that I hav yt to gt rvw/fdback on ths documnt from othrs, so t probably has som rrors n t. Hopfully, thy ar only typos. Usual Grn Functon Dfnton/Drvaton from Mathmatcs W want to solv an quaton whr u(x s th soluton, L s an oprator, and f(x s a known functon of form Lu ( x = f ( x. (1 Th Grn functon for ths quaton s G(x,s, whr (L = L * for L Hrmtan Th soluton u(x s (, δ LG x s = x s. ( (, u x Proof: Puttng (3 nto (1, thn usng (, w hav = G x s f s ds. (3 = (, = (, = δ ( = ( Lu x L G x s f s ds LG x s f s ds x s f s ds f x Usual Math Dfnton of Grn Functon for QFT Fld Equatons Th Kln-Gordon (Scalar Equaton Grn Functon ( m + = (5 φ Th Grn functon for (5 s found from (whr a mnus sgn on RHS dosn t chang ssnc of th ssu and s convntonal n QFT and w us y n plac of th s of ( ( m G( x, y δ ( x y + =. (6 Soluton to (6: Convrt (6 to momntum spac va a Fourr transform 1 ovr x. ˆ (, ky k + m G k y =. (7 1 kx Our convnton s th on typcally usd n QFT. ˆf ( k = d x f ( x whr kx = k x, and Not also that from Fourr transform tabls (or takng th tm to prov t yourslf f ( x = k ˆf ( k 1 kx f x = d k f k ( π.

2 Th soluton to (7 s = Gˆ k, y k ky m. (8 Convrtng (8 back to poston spac va a rvrs Fourr transform, w gt k( x y 1 G( x y = d k = F x y, (9 k m ± ε ( π whch s just th scalar propagator. (Not that w add th nfntsmal quantty ε to th dnomnator of th ntgrand n (9 for rasons dlnatd n Klaubr, Chap. 3, pg. 76. Essntally, th ntgrand would othrws blow up at k = m. Aftr dong th work on nds to do wth (9 (ffctvly ntgratng t, th ε can b takn to zro (and w actually gt a sutabl rsult. Th Maxwll Eq (Photon Grn Functon In analogy to th Kln-Gordon (scalar cas, nstad of (5, w hav (whr m = for photons ( m ( π + A =. (1 In smlar fashon to (7 through (9, w gt ν k( x y ν g ν G ( x y = d k D = F x y. (11 k m ± ε Gnral Concluson for Grn Functons of Usual Mathmatcs and QFT Fld Equatons Th Grn functon (dfnd as ngatv of usual math dfnton for ach partcl typ fld quaton quals th Fynman propagator. For scalars, ths s (tms a factor of For photons, Smlar for frmons. { } ( = ( = φ φ. (1 G x y x y T x y F { } ( = ( =. (13 ν ν ν F G x y D x y T A x A y So w can s why propagators ar somtms calld Grn functons. Smplfd Ovrvw of Grn Functon Mthodology of QFT A Somwhat Dffrnt Approach to Grn Functons Whn studnts frst s th us of Grn functons n QFT, thr s typcally confuson, as th mthodology thrn sms qut dffrnt from th usual math Grn functons, as dscrbd n th forgong. So, w wll consdr two dffrnt approachs to Grn functons, th usual math on and, what w wll trm th QFT Grn functon mthodology. Aftr ntroducng th scond of ths, w wll compar th two and s how, n som crcumstancs, thy gv th sam rsult,.., th sam Grn functons. (Th QFT Grn functon mthodology rsults ar mor xtnsv, ncludng not only th usual math Grn functon rsults, but mor. Brf Ovrvw of QFT Mthodology Grn Functons: In Words Th QFT mthodology Grn functon approach, found n dtal n txts such as Mandl and Shaw, provds a fnal rsult, th Grn functon for a partcular ntracton, that can b radly transformd nto th Fynman

3 3 ampltud for that ntracton. Th transformaton nvolvs straghtforward substtuton, as xpland n words n th nxt paragraph and n pcturs n th scton aftr ths on. As t turns out, th QFT mthodology Grn functon ylds a fnal rsult that has only propagators (for xampl, S F (p and no xtrnal ln rlatons (for xampl, u 1 (p, unlk Fynman ampltuds. On obtans th approprat Fynman ampltud smply by substtutng an xtrnal ln xprsson for ach ncomng/outgong partcl (whch s a propagator n a Grn functon n th partcular ntracton of ntrst (for xampl, substtut u 1 (p for S F (p. Thr ar othr subtlts, such as gttng th sgns rght on th -momnta, but that s th tchnqu n ts ssnc. Thrby, on turns a Grn functon nto a Fynman ampltud. Brf Ovrvw of QFT Mthodology Grn Functons: In Pcturs Th Grn functon s a mathmatcal ntty, just as th transton ampltud s a mathmatcal ntty. And just as th transton ampltud can b rprsntd pctorally by Fynman dagrams, so can a Grn functon b rprsntd by dagrams. Fg. 1 dsplays th rlatonshp btwn a typcal Grn functon dagram and a Fynman dagram. Fgur 1. Changng a Grn Functon Dagram to a QFT Fynman Dagram for Bhabha Scattrng Not from th LHS of Fg. 1 that, as mntond n th prvous scton, th Grn functon, and pctorally, ts assocatd dagrams, ar mad up wholly of propagators. In th LHS, th ncomng lctron and postron of Bhabha scattrng (only lowst ordr and only on of two ways t can occur ar shown ar rprsntd by propagators. So ar th outgong lctron and postron. To convrt th dagram (and mathmatcally, th Grn functon to a Fynman dagram (and mathmatcally, th transton ampltud, w mrly substtut an xtrnal partcl for ach ncomng/outgong propagator. Gnrally, w can conclud that both th Grn functon and th transton ampltud carry th sam nformaton. Ethr can b radly obtand from th othr. Fg. shows us what, gvn th forgong paragraphs, should not b a bg surprs. That s, a Grn functon s comprsd of many (an nfnt numbr of hghr ordr trms (dagrams n pctoral form, just as w onc larnd a transton ampltud s composd of many (an nfnt numbr of hghr ordr trms (Fynman dagrams n pctoral form. On may ask what valu ths has, as w alrady know how to construct ampltuds for any gvn ntracton. On nds to know a lot mor thory to answr ths quston. Full dvlopmnt of Grn functons and thr rlatonshp to somthng calld th gnratng functonal can hlp sgnfcantly n advancd QFT. Addtonally, t tachs us that th physcal world can b rprsntd n dffrnt, yt consonant, mathmatcal ways.

4 Fgur. Th Grn Functon Corrspondng to Bhabha Scattrng Includs All Ordrs for th Sam Incomng and Outgong Propagators Th Math Bhnd th Grn Functon Mthodology of QFT Mathmatcal Form of th Grn Functon In ths sub-scton, w frst smply prsnt, n (1, th dfnng xprsson for Grn functons n th QFT mthodology, wrttn G ν.. (x 1, x,,..y 1,..z 1. W thn xplor that rlaton and do a coupl of xampls, wth dffrnt ncomng/outgong partcls (calld lgs, n ordr to show how th Grn functon as dscussd n pror sctons arss from (1. ν { ( 1 ( ψ ( ψ ( 3 } T SA x A x... y... z... ν... Grn functon G ( x1, x,... y1,.. z1,... = M & S ( 1.8 pg 5 (1 S Whr S s th famlar (at last by now t should b S oprator from th canoncal quantzaton approach to QFT; A, ψ, and ψ ar th usual QED quantum flds from that sam thory; and T ndcats tm ordrng. Th notaton M & S rfrs to Mandl and Shaw, nd d. (1, Wly. For now, w ar not gong to worry about th dnomnator n (1, but focus on th numrator. In Appndx A w provd hopfully hlpful, ntutv nsght nto how th dnomnator arss mathmatcally and nds up bng lttl mor than a phas factor. Th prmary acton s n th numrator. Don t look at Appndx A untl aftr you gan som famlarty wth what s gong on n th numrator. W can r-xprss (1 by nsrtng nto t th full xprsson (15 for th S oprator (whr w undrln dummy ntgraton varabls to dstngush thm from th x 1, x, n th Grn functon (1, and th last ln s spcfcally for QED, n { } I ( x d x S T L = = T ( x 1.. ( x d x 1..d x n! L L I I n n n= 1 1 { 1 } 1 1 I! I I = I + L x d x T L x L x d x d x +... (15 ( ψ γ ψ ( ψ γ ψ ( ψ γ ψ = I + A d x1 T A A d x! 1d x... x + 1 x1 x Ths gvs us

5 ν... (,,...,..,... G x x y z ν { ( 1 ( ψ ( ψ ( 3 } T A x A x... y... z... = + S ν ( ψ γ ψ 1 ( 1 ( ψ ( ψ ( 3 T A d x A x A x... y... z... x1 S ν ( ψ γ ψ ( ψ γ ψ 1 ( 1 ( ψ ( ψ ( 3! T A A d x d x A x A x... y... z... x1 x S (16 How Only Propagators Rman n Any Grn Functon Rcall from Wck s thorm how w can convrt th tm ordrng of (16 nto normal ordrng, and along th way w gt trms wth propagators n thm. S Klaubr, Studnt Frndly Quantum Fld Thory, Chap. 7, pg. for Wck s thorm,.., (whr A,B,C tc. ar flds such as A, ψ, and ψ T {( AB......( AB... } = N ( AB......( AB... { } x1 xn x1 xn x 1 x x 1 x xn 1 xn { } { } { } + N AB.. AB N AB.. AB N... A...Z A...Z + N ( ABC... x1 ( ABC... x... + N ( ABC... x1 ( ABC... x _ _ + (all normal ordrd trms wth thr non-qual tms contractons + tc. Usng (17 n (16 w wll nd up wth, thanks to normal ordrng, a lot of trms havng dstructon oprators on th RHS. Any of thos opratng on th vacuum kt n th numrator of (16 wll rsult n zro. So, th only trms survvng ar thos wth only propagators (.., contractons n thm and no oprators, snc propagators ar only numbrs and not oprators. Ths trms ach hav a numbr sandwchd btwn a vacuum bra and a vacuum kt, so w can tak th numbr outsd th brackt, and th brackt = 1. Concluson: W can forgt about all trms arsng n (16 xcpt for thos havng only propagators and no oprators n thm. Thy ar th only trms survvng n (16. Thus w s how th clam mad arlr that Grn functons comprs only oprators s tru for th rlaton dfnd by (1. It rmans to justfy that ths qual our famlar transton ampltuds whn w substtut xtrnal partcl rlatons for th ncomng/outgong propagators (.., th lgs. An Exampl: Bhabha Scattrng In Bhabha scattrng w hav an ncomng lctron, an ncomng postron, an outgong lctron, and an outgong postron. So, n (1, w tak our lgs to b th four flds corrspondng to thos partcls. S Fg. 1. Not that w hav no photon lgs for ths cas, so w hav no A (x factors and no suprscrpts on G. { ψ ( 1 ψ ( ψ ( 1 ψ ( } T S y y z z G ( y1, y, z1, z = For Bhabha scattrng. (18 S Not w want to crat th lctron vrtual partcl at z 1, so w us th quantum fld ψ ( z 1 for that. To crat a postron at y 1, w us ψ..(y 1 ; to dstroy an lctron at y, ψ..(y ; and to dstroy a postron at z, ψ ( z. (17

6 6 From Appndx A w know that th dnomnator n (18 rprsnts a phas factor, so w wll concntrat on th numrator. (In fndng probablts w work wth th squar of th absolut valu of transton ampltuds and phass of th ampltuds drop out. Usng th last row of (15 n (18, w hav numrator of G y, y, z, z 1 1 ( ψ γ ψ = I A d x + x 1 1 T ψ ( y1 ψ ( y ψ ( z1 ψ ( z T! ( ψ γ A ψ ( ψ γ A ψ d x1d x +... x1 x ψ ψ ψ ψ ψ γ ψ ψ ψ ψ ψ = T! x1 x x1 ( y ( y ( z ( z + ( A d x ( y ( y ( z ( z T ( ψ γ A ψ ( ψ γ A ψ d x 1 d x ψ y 1 ψ y ψ z 1 ψ z From Wck s thorm (17 and our knowldg that, du to normal ordrng, only trms wth propagators and no oprators survv, w gt (19 as (whr I apologz for th crud rndtons of contractons [propagators] du to th lmtatons of my softwar ψ ( y1 ψ ( y ψ ( z1 ψ ( z + ( ψ γ A ψ ( A y y z z! x ψ γ ψ 1 x ψ ψ ψ ψ d x1d x. ( Not that th scond trm n th last row of (19 droppd out bcaus thr s no way w can mak a photon propagator out of a sngl A fld. Not also that all quantts nsd th bra and kt ar numbrs, not oprators, so w can mov thos numbrs outsd th brackt wth = 1, and just dal wth th numbrs nsd th larg parnthss of (. Th frst trm n ( s smply two unconnctd propagators that don t ntract. That trm rprsnts a postron at y 1 travlng to z and an lctron at z 1 travlng to y. For th S oprator approach, such a trm corrspondd to no ntracton btwn th lctron and postron, whch s on of th ways th partcls could bhav. Th ncomng lctron and postron rman unchangd and ar n th sam stat outgong as thy wr ncomng. Such a happnng has a probablty, just as th ntracton of th two has a probablty. Hr that trm dos not rprsnt xtrnal partcls, howvr, but propagators. But rcall that w obtan th Fynman ampltuds by substtutng xtrnal partcl rlatons for th lgs n a Grn functon. That s what w would do hr, so w would nd up wth th bhavor dscrbd n th forgong paragraph. Lookng now at th scond trm n (, w s t rprsnts th LHS of Fg. 1. That s, th contractons of (, ar th vrtual partcls (propagators of th LHS of Fg. 1. Th factor n front of that scond trm n ( s th sam factor w hav whn w valuat Fynman ampltuds. W can surms that that trm, whn w Howvr, whn w ntgrat ovr all spac and tm n th standard approach to QFT Fynman ampltud calculatons, th partcls thn hav to ntract, as thy must vntually ovr nfnt tm. Hnc ths trm dosn t com nto play n that approach. S Klaubr, Chap 17, whch xplans th ratonal for ntgraton ovr all tm and spac. (19

7 7 rplac th ncomng and outgong propagators wth xtrnal partcl rlatons,.., tak th approprat S F to v p. th approprat u r (p 1, v r (p, r u p, or r As xampls, lookng at Fg. 1 and th contractons/propagators of (, w would mak th followng convrsons to gt th Fynman ampltud. momntum momntum ( ( p ( ( p S z x S p u S y x S p v F 1 spac F 1 r1 1 F 1 spac F r momntum momntum ( ( ( p ( ( ( p S x y S p u S x z S p v F 1 spac F 1 r1 1 F 1 spac F r B awar that ( also contans hghr ordr trms whch would b assocatd wth th othr dagrams of Fg.. It also would nclud th othr way Bhabha scattrng can occur, as n th last dagram of Fg. plus non-dsplayd hghr ordr dagrams for that othr way. Anothr xampl: Th photon propagator Consdr th photon propagator, wth Fynman dagrams shown n Fg. 3 to th frst two ordrs (1 Fgur 3. Th Photon Propagator to Frst Two Ordrs Now consdr th Grn functon (16 wth only two photon lgs, whr agan w focus on th numrator. G ν ( x, x 1 ν { ( 1 ( } T A x A x = + S ν ( ψ γ ψ 1 ( 1 ( T A d x A x A x x1 S ν ( ψ γ ψ ( ψ γ ψ 1 ( 1 (! T A A d x d x A x A x x1 x S +... In convrtng th tm ordrng to normal ordrng va Wck s thorm, th scond trm wll drop out, as t has an odd numbr of photon flds, and only wth an vn numbr can a non-zro trm rsult. Thus, w nd up wth ν ( x x numrator of G, 1 = ν α β ν A ( x1 A ( x! ( ψ γ Aα ψ x ( ψ γ A A ( x1 A ( x d x 1 βψ x 1d x +... Agan ths quals smply th quantts nsd th larg parnthss, snc thy ar numbrs and = 1. Not w ar not convrtng th lg propagators to xtrnal partcls, snc w ar dalng hr smply wth a propagator (photon propagator and not an xtrnal partcl ntracton. + ( (3

8 8 W should b abl to s that th trms shown n (3 ar smply th lowst ordr propagator (frst trm and th scond ordr corrcton (scond trm, shown dagrammatcally n Fg. 3. Ths ar just th rlaton drvd n th standard QFT canoncal approach for th photon propagator to scond ordr. Thus, w s that th Grn functon for two photon lgs s smply th photon propagator. (W showd ths to scond ordr and can surms that f w ncludd yt hghr ordr trms n (, w would gt th hghr ordr trms for th photon propagator as found n standard QFT wthout Grn functons calculatons. Comparng th Standard Math Grn Functon wth QFT Grn Functon Mthodologs Grn Functon Fr Photon Propagator whr From th rsult of (3, w concludd that = ( + ( hghr ordr ν ν G x, y DF x y, ( no loops D ν F ( k = Grn Functon by Two Dffrnt Approachs = Propagator frmon-antfrmon loops ν g. (5 k + ε So now w s th conncton btwn th usual math Grn functon approach and th QFT Grn functon mthodology. Usng th formr wth th QFT fld quatons, w gt th no loops trm n (,.., (11 or (13, contanng th famlar photon Fynman propagator, and for that rason alon, w can s why t s common to call th propagator a Grn functon. It s th usual math Grn functon for Maxwll s quaton. But thr s anothr rason. Th QFT Grn functon mthodology also gvs rs to th propagator as w hav known t (apart from a factor of, D ν F, n (. Th full propagator (ncludng hghr ordr trms of ( s a Grn functon, as found va th QFT Grn functon mthodology. Gratr Extnt of QFT Grn Functon Mthodology As shown abov, w can us th QFT Grn functon mthodology to obtan th math form of th propagators. Howvr, th QFT Grn functon mthodology has a wdr rang of applcaton. At ts cor s (1, and that rlaton can b usd to fnd a Grn functon for any ntracton, n addton to just thos of th fr propagators (as w dd for Bhabha scattrng n th frst xampl abov. Summary of Gnratng Functonal Anothr, usful way of dtrmnng Grn functons mploys us of somthng calld th gnratng functonal dfnd blow n (6. W wll show how ths s don aftr xplanng what w man by th symbols n (6. Dfnton of gnratng functonal whr S Z [ Jk, σ, σ ] = M & S ( 1.83 pg 65, (6 S

9 { I ( x d x S T L = } = n= n n! { LI ( 1 LI ( LI ( n } 1 n... T x x... x d x d x... d x M & S 1.8 pg65 9 κ σ ψ ψ σ LI = LI + LS = LI + Jκ x A x + α x α x + α x α x M & S 1.77 &1.78a pg 6 L Not th nwly ntroducd fld sourcs Jκ ( x... α ( x... α ( x functons, not oprators, whras th flds A ( x... ( x... ( x S σ σ n L S abov ar classcal flds (just ψα ψ α ar quantum (oprator flds. Not furthr that J κ (x s NOT th frmon -currnt of QFT,.., t s not ψγ κψ. It s a nw (fcttous ntty that wll b usd as a hlpful tool. In what follows, w don t solv for, or us, algbrac forms of Jκ ( x... σα ( x... σ α ( x. W just mploy thos fld sourcs as symbols that ad us n dvlopng usful rlatons. Dfnton and Exampl of Functonal Dffrntaton of a Functon W frst not th followng. δ Jσ ( x ( = δ σ δ ( x x δ J x M & S ( 1.55 pg 59 (8 (8 may look strang n th sns that t mans th (functonal varaton of a functon wth rspct to tslf (.., whn σ = and x = x s nfnt, whn w mght xpct t to b on. Howvr, t works bcaus all functonal opratons ar nvolvd wth ntgraton. Usng (8, as w wll s n at last on xampl, works. It lads to a consstnt thory. As such an xampl whr w us (8 wth a functonal, consdr functonal δ δ Jν y Jν y ( δ Jσ y ν ν f y Jσ y dy = f y dy = ( f y δσδ y y dy = δσ f ( y δ Not ths parallls mor lmntary calculus,.., Ky Functonal Drvatv ( g y g y y f ( y ' g ( y = f ( y ' = f ( y' = f y ' y = f y g y g y g y y From (6 wth (7 and (8, (sorry for mssng stps 1 δ S 1 δ Z J δ J ( x T { S A ( x } = = δ J x S S [ k, σ, σ ] wth smlar rlatons for σ ( x and σ ( x Rlaton Btwn Grn Functon and Gnratng Functonal y δ M & S 1.9 pg 66 Th rlatonshp (3 blow s provn by substtutng (6 nto (3, usng (31 (ncludng th rlatons for σ and σ, and comparng wth (1. (agan, sorry for mssng stps n n δ Z [ Jk, σ, σ ] n 1 G ( x1,... y1,.. z1,... = ( 1 M & S ( 1.91 pg 66 δ J x δσ y δσ z n = total drvatvs n = drvatvs wth rspct to σ flds. (7, (9 (3 (31 (3

10 1 Gnratng Functonal Spcal Cas: Fr Fld Photon Propagator (S Mandl & Shaw, Sct (pg Th fr photon propagator cas conssts of a (vrtual sngl photon propagatng, but not ntractng wth anythng. W dfn th symbol Z as th fr fld gnratng functonal whr th subscrpt sgnfs fr fld, no ntractons. Th Mandl and Shaw tratmnt of ths sms to m lk t would b hard to follow. Th tratmnt blow s mor straghtforward and probably asr to undrstand, at last concptually. Fr photon propagator Z from gnral Z dfnton For th fr fld cas, S S = 1, th gnral dfnton for gnratng functonal (6 bcoms S S Z J,, = = = S M & S 1.9 pg 67 (33 [ σ σ ] k S 1 In (7, th ntracton Lagrangan L I =, so L = L S. Thus, { I ( x d x S T L = } n= n { LS ( 1 LS ( LS ( n } 1 n... T x x... x d x d x... d x M & S 1.93 pg67. n! = (3 Substtutng th valu for L S n (7 nto (3, w hav ( J ( x1 A ( x1 + σα ( x1 ψα ( x1 + ψα ( x1 σα ( x1 ( σα ψα ψα σα ( J ( xn A ( xn σα ( xn ψα ( xn ψα ( xn σα ( xn n S =... T J x A x x x x x... d x1d x... d xn n= n! + +.( Bcaus w sk a rlatonshp for th fr photon fld, w now gnor th frmon flds,.., tak σ = σ =. Latr, w wll gnralz our rsult to nclud both frmons and photons. W also smplfy by only lookng at th frst fw trms n (35. Dong that, w gt ν κ { ν κ } n = n n= n! S T J x A x J x A x J x A x d x d x d x 1 ν { } ν { } = 1 + T J x A x dx T J x A x J x A x d x d x +... Whn w us Wck s thorm to convrt th tm ordrng to normal ordrng, th trms wth just propagators arsng from th quantum flds n (36 (th photon flds wll b just numbrs. Wth (36 n (33, ths wll survv (not b qual to zro. Any trms wth oprators (th quantum photon flds lft n thm, upon normal ordrng wll hav dstructon oprators on th RHS of (36. In (33, ths wll act on th vacuum kt and rsult n zro. Hnc w can gnor all trms n (36 that hav oprators lft aftr usng Wck s thorm. That s, w only hav to b concrnd wth th trms arsng that consst solly of on or mor contractons (propagators. Rstrctng our ffctv S to ths for us n (33, w hav (not that w nd an vn numbr of photon flds n th scond ln of (36 to gv rs to a contracton (36

11 1 ffctv trms n S = 1 J x A x A x J x d x d x ν 1 1 ν 1 1 ν = 1 J ( x1 DF ( x x1 Jν ( x d x1d x +... Thus, from (33 (whr Z s now only a functon of J wthout frmon sourcs, ν Z [ Jk, σ, σ ] = Z [ Jk ] = S = S = S = 1 J ( x1 DF ( x x1 Jν ( x d x1d x +... (38 Compar th RHS of ths to our rsult (5 from Appndx B, wth th dfnton (51 ν ( J x D x y J y d xd y F ν ν = 1 J x DF x y Jν y d xd y +... (37 (5 J ( x DF ( x x J ( x d x d x symbol σ ρ Jσ DF J = ρ, (51 and w s that, at last to lowst ordr, for only photons and no frmons, (38 s ( F ( ( Z J Z J = = [, σ, σ ] [ ] k k Jσ x D x x Jρ x d x d x J σ D F J ρ. (39 Hopfully, w can accpt that f w carrd out th abov stps and thos of Appndx B for th hghr ordr trms, th rlaton (39 stll holds. Showng Z Ylds Grn Functon for Fr Photon Propagator From th dfnton (3, whr th Grn functon w sk has no frmons ( n = and two photons (n =, so t s δ Z [ J k ] δ ν 1 G ( x y = M & S ( 1.18 pg 69 δ J x J y ν Insrtng (39 wth (6 nto (, w fnd, to lowst ordr, J D J 1 δ δ J x δ J y δ J x δ J y [ ] F Z J σ ρ k δ = ν ν ν J DF J ν δ ( 1 δ = + Jσ ( x DF ( x y J ρ ( y d x d y... δ J ( x δ Jν ( y δ J ( x δ Jν ( y +... = Trm X From th rsults of th frst and last lns of (61 n Appndx B, w s trm X n (1 s ( (1

12 δ δ Trm X = Jσ ( x DF ( x y J ρ ( y d x d y δ Jν ( y δ J ( x ( x δ δ Jσ δ J ρ = DF ( x y J ρ ( y d x d y + Jσ ( x DF ( x y d x d y δ Jν ( y δ J x δ J x δ = + δ Jν ( y 1 ( y ( x D x y x y d x d y δσ δ ( x x DF ( x y J ρ ( y d x d y Jσ ( F ( δ ρ δ ( σ va sym n D ndcs & F δ ρ σ = ( DF x y J ρ y d y + Jσ x DF x x d x swtch dummy ndcs δ J y ν = ρ σ, gt nxt row swtchng dummy δ σ σ = DF ( x y Jσ ( y d y + DF x δ Jν ( y ( ( x Jσ ( x d x ntgraton varabl x y = gvs nxt row σ ( F σ y δ = D x y J y d y δ Jν So, Trm X s. J = σ δ Jσ ( y σ ν ( ( δσδ δ J y J = (3 J = J = Trm X = DF x y d y = DF x y y y d y ν ν F = D x y. Thus, from ( and (1, Z [ Jk ] δ ν 1 δ ν F δ J x Jν y G x y = = D x y + ( hghr ordr trms Rsult ( shows that th usual Grn functon of mathmatcs for Maxwll s quaton (11 quals th lowst ordr Grn functon from th QFT Grn functon mthodology. Gnralzng Fr Fld Photon Propagator to Includ Frmons Th abov procdur for photons can b carrd out n paralll for frmons. (W nd to us classcal Grassman sourc flds σ and σ nstad of th classcal photon sourc fld J., as w dd abov. Grassman varabls do not commut lk ordnary varabls [such as th mor usual classcal flds] do, and thr s a far amount of study nvolvd n larnng about thr algbrac and calculus rlatd bhavor. W do not do ths hr. Carryng out th abov procdur for Grassman sourc flds, and ncorporatng th rsult wth what w obtand n (39 for photon sourc flds, w fnd [ ] [ σ σ ] [ ] [ σ σ ] k k k J Dαρ F J α ρ σ S Fσ J = ( Z J Z J,, = Z J Z, =. (5 Usng (5 n (3 wth only Grassman fld drvatvs (and all J κ =, would gv us a Grn functon quvalnt to th fr frmon propagator. J = (

13 13 Appndx A. Som Background for (1 To vsualz what s happnng n (1 (rpatd blow, { ψ ψ } T SA x... y... z poston spac G ( x,... y,.. z,... = M & S ( 1.8 pg 5, (1 S whr th bras and kts ar all vacuum stats at t =, consdr th followng xprsson, whch s th sam as th numrator of (1 xcpt th bra s at t = +, { ψ ψ }, t = + T SA x... y... z..., t =. (6 W know, snc th ntr S ( = U(+, oprator taks a gvn stat to ts fnal stat, and th vacuum at t = can only vary from th vacuum at t = +, by a phas factor, kts, t = + =, t = not rlvant hr, but notd, t = + S, t = = 1 bras, t = + =, t = φ φ 1, t =, t = S, t = =, t =, t = + =, t =, t = = = = φ, t = + us ths φ φ, t =, t = + =, t = S, t = wth ths (7 Usng th last ln of (7 n (6 ylds (1. So (6 s th sam as (1,..,... (,...,..,... G x y z = { ψ ψ }, t = T SA x... y... z..., t =, t = + S, t = { ψ ψ } =, t = T SA x... y... z..., t =. To vsualz what (1, th frst ln of (8, mans, t s asr to us th scond ln of (8,whch acts mor lk th ampltuds w ar famlar wth. That s, t taks an ntal stat at an arlr tm (t =, hr to a latr tm (t = +, hr. In arlr work, to fnd an ampltud, w gnrally startd wth a stat of crtan partcls, whch was actd on by th oprators n S to produc a fnal stat of partcls. Th tm ordrd S oprator workd fn for that. Hr ( nd row of (8, on th othr hand, w start wth a vacuum stat at t = and nd wth a vacuum stat at t = +. So to gt anythng manngful, w hav to crat som partcls. Th flds A x... ψ y... ψ z... can b usd for that. Whch ons w us dpnds on th problm at hand. If w want to xamn a problm wth an ntal lctron and postron, w would us ψ (z to crat an lctron at z and ψ (y to crat a postron at y. Thn th S oprator would tak thos flds forward n tm and caus thm to ntract (.g., to annhlat cratng a vrtual photon whch thn transforms nto an outgong lctron and postron, as n Bhabha scattrng. But thn w hav to gt back to th vacuum agan (at t = +. So w us addtonal flds from A x... ψ y... ψ z... to dstroy th rsultng lctron and postron and lav th vacuum. In our xampl, w would hav a fld thrn of form ψ (z to dstroy th postron at z and ψ (y to dstroy th lctron at y. (8

14 But not th S oprator can b xprssd as 1 (, (, (, (, (, (, S = U + = U + y U y z U z y U y z U z. (9 Snc S and th othr oprators n th nd row of (8 ar tm ordrd, ach of th U oprators n S of (8 oprats n th approprat tm squnc. In our pror xampl, U(z, taks th ntal vacuum to th tm at z, whr ψ (z crats an lctron. U(y, z thn taks that stat to y, whr ψ (y crats a postron (y and z could b th sam D pont, tc. untl w arrv at th fnal vacuum stat. Th frst row of (8 s just a dffrnt mathmatcal way to xprss th ampltud assocatd wth ths unfoldng of vnts. But n t, th fnal bra s dffrnt n that t rprsnts th dntcal vacuum stat of th kt, both bng at t = --. Dfnton not: Th flds n A ( x ( y ( z... ψ... ψ... ar calld lgs. For rasons that ar obvous whn on ss thy show up as,.., corrspond to, xtrnal partcls n Grn functon dagrams. Addtonal not: Unlss th fnal and ntal stats ar qual (8 wll qual zro. W can only gt th sam fnal and ntal vacuum stats n th frst row of (8 f w hav th sam numbr of flds ψ flds as ψ flds. Snc n QED, L I = ψ A ψ, S wll always contan th sam numbr of ach, so w must hav th sam numbr of ach n A ( x ( y ( z... ψ... ψ...,.., n th lgs. Snc thr s only on photon fld n L I, on could consdr that any numbr of photon flds could xst as lgs,.., n A ( x ( y ( z... ψ... ψ.... Howvr, as Mandl and Shaw show on pg. 7 (last paragraph, for th spcal cas wth no frmon lgs, an odd numbr of photon lgs, (8 wll vansh. So for th cas of no frmon lgs, thr must b an vn numbr of photon fld lgs.

15 15 Appndx B: Exponntal Expanson of a Functonal Not frst th stramlnd notaton [AKB] AKB = A x K x, y B y d x d y (5 [ ] For A=B= th fld sourc J σ (wth dffrnt dummy subscrpt n th sourc Lagrangan L S of (7, and K = th no-loop photon propagator D F, (5 bcoms th spcal cas = J σ DF J ρ J σ x DF x y J ρ y d xd y. (51 Th ntr purpos of ths appndx s to show or ν ( J Dν F J ν ν = 1 J DF J ν J x DF x y J y d xd y ν ν Not how (5 parallls th lmntary calculus rlaton +... = 1 J ( x DF ( x y Jν ( y d xd y +... (5 x = 1+ x +... (53 Rlaton (5 s all you nd to rmmbr from ths appndx. Th proof follows Proof of (5. Rcall (8, δ J σ δ J ( x ( x Also, rcall that for a smpl functon g(x, σ ( ( = δ δ x x M & S 1.55 pg 59, (8 d g ( x dg x 1 1 g ( x = g ( + x + x +... = g ( + g ( x + g ( x +... dx dx x= x= (5 f ( x f ( f ( ( ( For ( 1 f f g x = = + f x + f ( + f ( f ( x +... On mght xpct an analogous rlaton to hold for functonals, as follows, whr th prms ndcat functonal drvatvs wth rspct to J, [ ] [ ] [ ] [ ] [ ] F J F 1 F F F For = + F J + F + F F J +... mght xpct. (55 Howvr, snc F[J] s an ntgral ovr spactm and s a numbr, th LHS of (55 s a numbr, not a functon of spac. But ach J n (55, would hav a form such as J (x or J ν (y,.., t would b functon of spactm poston. Thus th RHS of (55, as wrttn, would b a functon of spactm poston (x, y,.. for xampl. But t can t b, snc th LHS s not a functon, but a pur numbr. From ths, w can glan that th trms n (55 must b ncludd nsd th ntgrals ovr x, y, tc. Thus, th corrct form for th xpanson of [ ] F J s

16 F[ J ] F[ ] F[ J ] δ = + F ( J J ( x d x δ J ( x = Trm A 16 1 F[ J ] δ δ + F J J x J y d xd y ν δ J ( x δ Jν ( y = Trm B actual form ( of xpanson 1 F[ J ] δ δ + F ( J F ( J J δ J ( x δ Jν ( y ( x Jν ( y d x d x = +... Trm C As an xampl of th applcaton of (56, w xamn a form for F w wll fnd vry usful. That s th functonal w lookd at n (51 abov (wth dffrnt dummy ntgraton varabls that wll hlp n th proof, F [ J ] = J DF J σ ρ Jσ ( x DF ( x x Jρ ( x d x d x =. (57 Wth (57 n (56, notng that all F[J] whr J =, hav F[J] =, and thus J D J J x D x x J x d x d x F[ J ] σ F ρ σ F ( ρ = = F J δ = 1 + d x d x Jσ ( x DF ( x x J ρ ( x J ( x d x δ J ( x = Trm A F J 1 δ + d x d x Jσ ( x DF ( x x J ρ ( x J ( x Jν ( y d xd y δ J ( x δ Jν ( y = Trm B F J 1 δ d x d x Jσ ( x DF ( x x J ρ ( x + δ J ( x = (56 F[ J ] F[ ] = = = 1, w fnd F J δ d x d x J ( x D ( x x J ( x J ( x J ( y d xd y δ Jν ( y ( , J = σ F ρ ν = Trm C Carryng out th functonal drvatvs for ach trm abov usng (8, w gt th followng.

17 17 Trm A F J δ Trm A J x D x x J x d x d x J x d x = σ F ρ δ J ( x = ( ( δσ δ ( x x DF ( x x J ρ ( x d x d x J ( x d x J = ( ( + Jσ x DF x x δρ δ ( x x d x d x = J x d x J = = + J = chang dummy ndcs ρ σ ( DF ( x x J ρ ( x d x Jσ ( x DF ( x x d x = J ( x d x ρ ν, σ ν ; dummy + J = varabl x x for nxt ln ν ν = ( Jν ( x DF ( x x d x Jν ( x DF ( x x d x J ( x d x + J = ρ σ ρ ( DF ( x x J ρ ( x d x Jσ ( x DF ( x x d x J ( x d x ( DF sym n ρ and for nxt ln ν ( ν ( F ( J = = J x D x x d x J x d x. Trm C ν ( ν ( F ( J =, snc J ( x. Trm A = J x D x x d xd x J x (6 = = ν From (59 and (6, w should b abl to s rght away (I hop that both factors n Trm C of (58 qual zro. So th only thng w hav to worry about (at last to lowst ordr s Trm B. Trm B (59 F J 1 δ Trm B = d x d x J ( x DF ( x x J ( x J ( x J ( y d xd y σ ρ ν δ J ( x δ J ( y ν = δ A = D x x J x d x d x J x J y d xd y ( from trm ρ F ρ ν δ J ( y ν = ρ ν ( D ( F x x δ ρδ ( y x ν = ntgrat ovr x, ν ( F ν = D x y J x J y d xd y. J = Rturnng to (58, w s or as w statd n (5. ν ( d x J x J y d xd y J = F ν ν = 1 F ν J x D x y J y d xd y J Dν F J ν ν = 1 J DF J ν J x D x y J y d xd y +... Back to hom pag (6 (61