QED: Quantum Electrodynamics

Transcription

1 Chapter 2 QED: Quantum Electrodynamcs 2. Negatve-Energy States: Antpartcles 2.. Settng the Stage: Non-Relatvstc Quantum Mechancs In non-relatvstc Quantum Mechancs, t was seen that (n the standard poston representaton) essentally everythng can be derved by the substtuton E! t (2.) ~p! ~ (2.2) (remember that we have set h = ). Ths substtuton drectly converts the classcal Hamltonan nto the Schrödnger equaton H = ~p2 +V (~x) 2m! t y(~x)= ~ 2 +V (x) y(~x) 2m actng on the wave functon y(~x). Once we have found a wave functon y(~x) satsfyng the Schrödnger equaton, we can also take the complex conjugate expresson:! ~ 2 t y (~x)= +V (x) y (~x) 2m We then multply the orgnal Schrödnger equaton by y (~x), and ts conjugate by y(~x). Subtractng the two yelds y (~x) t y(~x)+y(~x) t y = y (~x) 2m ~ 2 y(~x) y(~x) ~ 2 y (~x).

2 It s easly seen that ths can be wrtten alternatvely as t y(~x) 2 = 2m ~ y (~x) ~ y(~x) Thus, ths leads us to the contnuty equaton wth t r(~x)+~ ~j(~x)=, r(~x) = y(~x) 2 and ~j(~x) = y (~x) 2m ~ y(~x) y(~x) ~ y (~x). y(~x) ~ y (~x) The quantty r(~x) occurrng n ths equaton s postve defnte, makng the nterpretaton of y(~x) 2 as the probablty densty of fndng a partcle at the poston~x a proper one Translaton to the Relatvstc Case The approach n the case of relatvstc Quantum Mechancs s exactly the same; however, ths tme t must be appled to the Hamltonan of specal relatvty. Restrctng ourselves to free partcles, V (x)=, the basc classcal equaton s then p µ p µ = m 2 or E 2 = ~p 2 + m 2. (2.3) When we agan make the substtutons of Eqn. 2.2, and make the resultng equaton act on a wave functon f(x) (ths notaton combnes the spatal and temporal dependence), the result s 2 ~ 2 t 2 + m 2 f(x)=, or, n explctly covarant form: ( µ µ + m 2 )f(x) ( + m 2 )f(x)=. (2.4) Ths s the Klen-Gordon equaton. Unsurprsngly, for our case of free partcles, ths equaton s easly solved to yeld plane waves just lke n the non-relatvstc case: f(x)=ne p x = Ne (Et ~p ~x), (2.5) wth N an a pror arbtrary normalzaton constant, and the four-momentum components E and ~p satsfyng our orgnal classcal Eqn But here we are n trouble! For the soluton to Eqn. 2.3 s E = ± p ~p 2 + m 2. 2

3 Whle the soluton wth the + sgn gves us a standard pcture, the soluton wth the sgn cannot be gnored. As a consequence, the system has no ground state (t s unbounded from below), and hence no meanngful physcal nterpretaton seems possble. To make thngs worse, also the contnuty equaton becomes problematc. As n the nonrelatvstc case, t s obtaned by takng also the complex conjugate of the Klen-Gordon equaton and multplyng t wth f(x), and combnng t wth the orgnal equaton multpled wth f (x). However, due to the fact that the Klen-Gordon equaton nvolves a second order rather than a frst order tme dervatve, ths tme we have to subtract the two. The result s (f (x) f(x) f(x) f (x) ) + t t t ~ (f (x) ~ f(x) f(x) ~ f (x)) =, whch can agan be consdered as a contnuty equaton, but wth r(x) = (f (x) f(x) f(x) f (x) ), t t ~j(x) = (f (x) ~ f(x) f(x) ~ f (x)). (2.6) Ths can agan be cast nto explctly Lorentz-covarant form: µ j µ (x)=, wth j µ (x)=(f (x) µ f(x) f(x) µ f (x)). (2.7) When we now substtute the free-partcle soluton of Eqn. 2.5 n Eqn. 2.6, we fnd that j µ (x)=2p µ N 2. (2.8) In partcular, we have r(x)=2e N 2. So n the case of a negatve-energy soluton, we also fnd that r(x) becomes negatve,.e., t can no longer be nterpreted as a probablty densty. Fnally, there s another problem wth negatve-energy solutons. Consder some localzed spatal wavefuncton at some tme t. It s then straghtforward to determne ts Fourer spectrum, and n general t wll be seen that ths wll contan both postve- and negatve-energy components, whch wll have opposte tme evolutons. Constructng the norm of the wavefuncton would then contan oscllatng terms; the correspondng ztterbewegung s not observed n realty. (Ths ssue s dscussed n more detal n Secton of Ref. [].) 2..3 Feld-theoretcal Interpretaton A proper nterpretaton can only be gven n the context of Quantum Feld Theory. In that context, f s a feld rather than a wave functon, and ts plane-wave expanson leads to partcle creaton operators for the postve energes combned wth antpartcle annhlaton operators for the negatve energes: f(x)= d 3 p (2p) 3 a(~p)e 2E ~p 3 p x + b (~p)e p x

4 The case of a real-valued classcal feld (we wll dscuss classcal felds n a bt more detal n Secton 2..4) then translates nto a hermtan quantum feld,.e., wth b(~p)=a(~p). So n terms of plane-wave solutons, the acton of the feld s ether to create a partcle wth four-momentum p µ or to annhlate one wth four-momentum p µ (where t can be shown, although dong so s outsde the scope of ths course, that the postve-energy soluton s assocated wth the annhlaton of a partcle, whle the negatve-energy soluton s assocated wth the creaton of a partcle). Note that the four-momentum here s n both cases the physcal (postve-energy) four-momentum. However, here we have to watch the other desred propertes of ths feld. Consder the case of a feld representng a charged partcle lke the electron. The acton of the feld must be to change the charge by the same one unt, rrespectve of whether creaton or annhlaton s concerned (motvatng ths requrement further would brng us too far n the realm of Quantum Feld Theory; a constructon can e.g. be found n the book by Peskn and Schroeder [2]). Ths means that n ths case, the equalty b(~p) =a(~p) cannot hold anymore (and hence also that f cannot be a hermtan feld). Ths can be acheved by makng b (p) represent the creaton of an ant-partcle wth four-momentum p µ ; we should therefore expect the exstence of a partcle much lke the electron, but wth opposte propertes such of ts charge. Ths partcle s called the postron. Of course, one would hope for expermental evdence of the exstence of the postron. It was frst observed n 932, n a cloud chamber exposed to cosmc rays (see Fg. 2.). Its dscovery earned Anderson [3] the 936 Nobel Prze. (The dscovery followed the predcton of the postron by Drac by only a year. Drac used a dfferent nterpretaton of negatve-energy states, though, whch s not approprate for the descrpton of bosons.) So what about the contnuty equaton, and the fact that there doesn t appear to be a conserved quantty (.e., one occurrng n a contnuty equaton) that can be assocated wth a probablty densty? The fact of the matter s that the (conserved) probablty densty s a concept that s useful n non-relatvstc quantum mechancs (non-conservaton would correspond to the creaton or dsappearance of partcles). However, n a relatvstc context, t s perfectly acceptable for (ant-)partcles to be created or annhlated (and the operator nature of quantum felds allows to descrbe such processes). So t doesn t make sense to ask for a conserved probablty densty Prncple of Least Acton and Euler-Lagrange Equatons; Noether Theorem Acceptng that we need a feld-theoretcal nterpretaton (per Sect. 2..3), we can now also use a dfferent startng pont for our computatons than the Klen-Gordon equaton. Gong back to a classcal sngle-partcle system of a sngle degree of freedom q(t), we can express the acton S as S = R t t dtl(q, q), where L represents the Lagrangan. Demandng that S be statonary under arbtrary but small changes of q(t) at each t results n the requrement ds = t t L L dt dq + q q d q =. 4

5 Fgure 2.: Photograph made of a postron bent n a magnetc feld and traversng (and losng energy n) a Pb plate. The postron hypothess follows from () the sgn of the curvature, ndcatng a postvely charged partcle; and (2) the track length after havng traversed the plate and before beng stopped, ndcatng a partcle much lghter than a proton. Interchangng the order of the tme dervatve and the d operaton and carryng out an ntegraton by parts then results n the condton t L d L dt dq =. q dt q t If ths equalty s to hold for arbtrary d q(t), then we mmedately arrve at the Euler-Lagrange equaton L d L =. q dt q In a feld-theoretcal settng, thngs work n much the same way. The essental dfference s that the Lagrangan L s obtaned as the spatal ntegral of the Lagrange densty L (f(x), µ f(x)), where µ f refers to the tme as well as spatal dervatves of f. The acton therefore ( f x µ ) 5

6 becomes a four-dmensonal ntegral convenent snce ths allows us to express t n a covarant form. The arbtrary changes are then n the feld f(x), and the prncple of least acton becomes L ds = d 4 x f df(x)+ L ( µ f) d µf(x) =. (2.9) The same manpulatons as for the above sngle degree of freedom then lead to the Euler- Lagrange equaton for the feld: L f µ ( L )=. (2.) µ f We wll make use of ths equaton, as well as of propertes of the Lagrange densty, later n ths and n other chapters. Note that n these lecture notes we wll follow the common partcle physcsts sloppness and smply call L the Lagrangan. For now, suffce t to say that the Klen-Gordon equaton can be recovered from the followng choce of Lagrangan: L = 2 µf µ f 2 m2 f 2. (2.) The Noether theorem s related to so-called nternal symmetres, whch we wll cover later n more detal, but whch for now we can llustrate usng the relatvstc wavefuncton f of Secton 2..2, whch we subsequently concluded should really be treated as a quantum feld. In the wavefuncton pcture, Quantum Mechancs dctates that the physcs should not depend on any complex phase of f. Now n the feld theoretcal context, t s qute well possble to post a real scalar feld; however as an alternatve we can post a complex scalar feld f, and stll make the assumpton that the physcs descrbed by the Langrangan ndeed does not depend on the phase of f. Ths s arguably the smplest example of an nternal symmetry. Under an nfntesmal phase change, whch we wll descrbe more generally as a group transformaton (see Appendx B for more detals), we can then wrte the transformaton of the feld f as f! f = f + at f, where a s the nfntesmal phase change, and T s the generator of the group transformaton. In the case of phase changes, we know the transformaton propertes: f! f = e a f = f + af, (2.2) so we smply have T =. We now requre agan that the acton be nvarant under ths transformaton, so we obtan the condton L ds = d 4 x f at f + L ( µ f) µ(at f) L = d 4 x µ at f + L ( µ f) ( µ f) µ(at f) L = d 4 x µ ( µ f) at f =. 6

7 Requrng that ths equalty hold for any a and ntegraton boundares, we fnd that µ j µ =, wth j µ = L ( µ f) T f Ths s the essence of the Noether theorem: every symmetry brngs wth t a conserved quantty Perturbaton Theory and Electromagnetc Interactons 2.2. Perturbaton Theory A theory descrbng only free partcles s not terrbly exctng... therefore, let us see how nteractons can be ncorporated. The am here s not to be entrely rgourous, but rather to provde a heurstc ntroducton to the computaton of scatterng ampltudes that can be understood as a reasonably straghtforward extenson of (tme-dependent) non-relatvstc perturbaton theory. Suppose that the Hamltonan of a system s descrbed by H = H +V (~x,t) and that the system correspondng to the unperturbed Hamltonan H can be solved exactly, H f n = E n f n wth d 3 xf n (~x)f m (~x)=d nm. (Here we are assumng that the system leads to a set of dscrete egenstates. That lmtaton does not affect the followng argument.) We now want to know the tme evoluton of a system that at a tme t s n the state y(~x). To ths end, we decompose y n terms of the egenfunctons of the unperturbed Hamltonan: y(~x, t)=âa n (t)f n (~x)e E nt. n Applyng the Schrödnger equaton then yelds ) Â n da n (t) f n (~x)e dt y(~x,t) t E nt = Âf n (~x)e n E nt E n a n (t)+ da n(t) dt = (H +V (~x,t))y = Â(H +V (~x,t))a n (t)f n (~x)e E nt n = Â(E n +V (~x,t))a n (t)f n (~x)e E nt n = ÂV (~x,t)a n (t)f n (~x)e E nt. (2.3) n Now assume that the nteracton V (~x,t) s swtched off for large tmes T!, so that the decomposton nto egenstates of the unperturbed system s the proper thng to do for such large 7

8 tmes. Multplyng Eqn. 2.3 by f f (~x)ee f t and ntegratng the result over all space then yelds da f (t) dt V fn (t) = = Â n a n (t)e (E n E f )t Vfn (t), wth d 3 xf f (~x)v (~x,t)f n(~x) Ths s just the well-known Dyson seres from non-relatvstc Quantum Mechancs. Also the soluton of ths ntegro-dfferental equaton proceeds n the same way as n nonrelatvstc Quantum Mechancs. In addton, suppose that before the nteracton s swtched on the system s n an egenstate of the unperturbed Hamltonan,.e., a n ( T )=d n. Order by order, we have a f (t) = d f + ( ) t + ( ) 2 Â t T +... dt V f (t )e (E E f )t T t n T dt V fn (t )e (E n E f )t dt V n (t )e (E E n )t At ths pont, we formulate the above equaton n a more covarant form by settng f n (x) f n (~x)e E nt. Retanng only the lowest-order (nontrval) transton, we then obtan t a f (t) = = T t T dt dt d 3 x f f (~x)e E f t V (~x,t ) f (~x)e E t d 3 xf f (x)v (x)f (x). Fnally, consderng ths quantty far after the nteracton, at t = T, and lettng T!, ths leads to the transton ampltude T f = d 4 xf f (x)v (x)f (x). (2.4) 8

9 2.2.2 Covarant Formulaton of Classcal Electrodynamcs Before proceedng to the mplementaton n Eqn. 2.4, t s useful to pay some attenton to the covarant formulaton of classcal electrodynamcs. The startng pont s the Maxwell equatons: ~ ~E = r (Gauss), (2.5) ~ ~B ~E = ~j t (Ampère), (2.6) ~ ~B = (Gauss), (2.7) ~ ~E + ~B t Eqn. 2.7 ndcates that ~B can be wrtten as = (Faraday). (2.8) ~B = ~ ~A, where ~A s called the vector potental. Combnng ths wth Eqn. 2.8, t follows that ~E can be wrtten as ~E = ~ F ~A t, wth F the scalar potental. Wth ths notaton, t then follows that Eqn. 2.6 can be wrtten as Fnally, we have ~ ~B ~E t = ~ 2~A + ~ ( ~ ~A) + ~ F t + 2 ~A t 2 = ~A + ~ ( ~ ~A + F t )= ~j. ~ ~E = ~ 2 F ~ ~A = r. t When we add and subtract here a term 2 F, ths last equaton can be rewrtten as t 2 F ~ ~A + F = r. t t The two rewrtten nhomogeneous equatons now have a very smlar form; defnng A µ =(F,~A) and j µ =(r,~j) allows us to fnally put the nhomogeneous equatons nto a manfestly covarant form: whch can also be wrtten as A µ µ ( n A n )= j µ, µ F µn = j n, wth F µn µ A n n A µ. (2.9) 9

10 The quantty F µn s called the electromagnetc feld tensor, and t turns out that ts elements are just ~E and ~B. (Of course, puttng these equatons nto a ncely covarant-lookng form does not guarantee the rght -known- behavour of ~E and ~B under Lorentz transformatons. But that can be verfed explctly and turns out to be n good order.) Even ths nce formula can be smplfed further. The feld tensor F µn encodes the physcal nformaton. Therefore, a change n A µ A µ! A µ = A µ + µ c, (2.2) wth c an arbtrary functon, does not affect the physcs. Ths s the gauge freedom of electromagnetsm. As a consequence, we can choose c such that n A n = : ths s called the Lorentz condton. So fnally A µ = j µ. (2.2) Ths choce for A µ s also called the Lorentz gauge. It s to be emphassed agan that the choce of gauge does not affect the physcs of the system (and other choces are ndeed used, such as the Coulomb gauge, n whch ~ ~A = ). A last ngredent that wll be extremely useful n the followng s the fact that the nteracton of partcles wth an electromagnetc feld can be descrbed smply by the mnmal substtuton : p µ! p µ + ea µ. (2.22) The usefulness of ths substtuton s that we can use t nstead of a proper feld theoretcal treatment of gauge symmetres: the so-called covarant dervatve correspondng to the U() symmetry group relevant for ths treatment of QED yelds precsely the same result The covarant dervatve, and mplcatons of U() symmetry As dscussed n the exercses, the use of the mnmal substtuton allows for a dervaton of the Lorentz force n classcal electrodynamcs. If we are to extend ths valdty to the realm of (non-relatvstc) quantum mechancs, ths results n a Schrödnger equaton 2m ( ~ q~a) 2 + qv y(~x,t)= y(~x,t) t (2.23) (where we have replaced e n the mnmal substtuton wth the more general charge q). However, the requrement that the gauge transformaton of Eqn. 2.2 should not affect the physcs (.e., should leave the form of eqn nvarant) now has a nontrval consequence. For t can be shown that ths nvarance s only acheved f smultaneously wth Eqn. 2.2, also the wavefuncton transforms: y(~x,t)! y (~x,t)=e qc(~x,t) y(~x,t). (2.24) The dervaton of ths property s lengthy and we wll not venture nto t here. More detals can be found e.g. n Jackson [4], Chapter 2. Also one of the exercses offers a partal justfcaton. 2

11 Although the above s done n the framework of non-relatvstc quantum mechancs, exactly the same concluson (Eqn. 2.24) holds n the relatvstc case. In concluson, we end up wth a space- and tme-dependent phase transformaton of the wavefuncton, whch does not affect any physcs. In group theoretcal terms, the U() symmetry group can be dentfed exactly wth all possble phase transformatons hence the statement that QED mplements a U() symmetry. But havng drawn ths concluson, matters can n fact be turned around: let us suppose that we requre that Eqn does not affect any physcs. Then t can be shown that the quantum mechancal analogue of Eqn. 2.22, µ! D µ µ qa µ, (2.25) precsely acheves ths. The quantty D µ s called the covarant dervatve. Note that ths phase change looks a lot lke the one encountered n Eqn An essental dfference s that rather than merely requrng nvarance under global phase changes, we now mpose ths requrement even for local (.e., space and tme dependent) phase changes. Another mportant dfference, although we wll not prove t here, s that we now requre not merely the acton to be nvarant under the transformaton, but also the Lagrangan tself! Of course, all of the above hnges on the known propertes of QED. However, t turns out that the gauge prncple (startng here wth the assumed phase transformaton property of the wavefuncton or feld and constructng the approprate covarant dervatve, whch then ultmately descrbes the nteracton of charged partcles wth the electromagnetc feld) s very powerful. The same prncple wll be used later to descrbe the strong and weak nteractons. Fnally, note that whle Eqn. 2.2 does not depend on the charge q of the fermon nvolved, the covarant dervatve and the phase transformaton so. Ths means that we can use the same prncple (and wth the same electromagnetc feld!) for partcles of dfferent charge. In group theoretcal terms, ths means that dfferent representatons of the underlyng phase symmetry are possble. Ths s a fact that wll be exploted later on Transton Ampltudes We now have all the requred ngredents n hand to proceed further. In the Klen-Gordon equaton, we make the mnmal substtuton of Eqn. 2.22; the resultng equaton can be recast as wth the potental V gven by ( + m 2 )y = V y, (2.26) V y = e( µ A µ + A µ µ )y e 2 A 2 y (note the operator character of the dervatve: t acts on y as well as on A). We wll neglect the last term n ths equaton, on account of the fact that e s small. Retanng only the frst two terms, we then have T f = d 4 xf f (x)v (x)f (x) = d 4 xf f (x)(e)(aµ µ + µ A µ )f (x). 2

12 The last term s amenable to ntegraton by parts, and neglectng the resultng surface ntegral the result becomes T f = d 4 xj µ f (x)a µ(x) wth j µ f (x)= e f f (x) µ f (x) ( µ f f (x))f (x). (2.27) Note that the quantty j µ f (x) looks almost exactly lke the quantty jµ (x) n Eqn There s however a dfference n that j µ f (x) nvolves two dfferent wavefunctons, those of both the ntal and fnal states. The proper nterpretaton of j µ f (x) s that of the current nvolved n the nteracton of a mcroscopc partcle. Ths s relevant n that the absorpton or emsson of a photon (we ll see later that ths pcture s approprate) may affect the partcle notceably. Eqn s approprate for the descrpton of the nteracton of a partcle wth a general electromagnetc feld. However, ths s not the stuaton typcally of nterest n partcle physcs. Rather, our nterest s n scatterng partcles off each other,.e., n electromagnetc felds caused by other partcles: the feld satsfes A µ = j µ(2) f (2.28) relatng t to the current of the other partcle (whch we wll also assume to be an electron). We wll also restrct the further dscusson to plane-wave ntal and fnal states (as approprate for our dscusson of scatterng experments where long before and after the scatterng process, the partcpatng partcles can be consdered as free partcles). In ths case, the current j µ(2) f takes on the smple form j µ(2) f (x)= e N 2 p (2) + p (2) µ (p (2) f e p (2) f ) x, and t s not hard to see that Eqn s satsfed by A µ (x)= jµ(2) f (x) q 2 wth q µ =(p (2) p (2) f ) µ. (2.29) Therefore, the fnal transton ampltude s gven by T f = = N 4 d 4 xj µ() f (x) g µn q 2 d 4 xe e(p () + p () f ) µ j n(2) f (x) (p() p () f + p (2) p (2) f ) x g µn q 2 e(p (2) + p (2) = N 4 (2p) 4 d 4 (p () + p (2) p () f p (2) f ) A few remarks are n order at ths pont: f ) n (2.3) e(p () + p () f ) µ g µn q 2 e(p (2) + p (2) f ) n. 22

13 . For clarty, a label () has been attached to the current representng partcle (the partcle that s scattered by the potental caused by partcle 2). However, Eqn. 2.3 s clearly symmetrc n the treatment of the two partcles under consderaton. Ths s n fact to be expected! For n our -now mcroscopc- setup, we are scatterng two electrons off each other, and there really sn t any physcs reason to treat them dfferently. 2. The factor (2p) 4 d 4 (...) arses from the ntegraton over all of spacetme of the plane-wave exponents. Its effect s to mpose conservaton of four-momentum, as desrable for these scatterngs. In fact, ths s not at all partcular to the process we are consderng here, but s nstead related to the assumpton of asymptotcally free states. 3. Implct n Eqn. 2.3 s the assumpton that the normalzaton N s ndependent of the momentum. Ths s n fact correct, but we wll not bother wth such normalzaton ssues. Therefore, n general we wll be smplfyng the dscusson of the transton ampltude to that of the so-called matrx element, genercally denoted by M. Ther relaton s defned by T f = (2p) 4 d 4 (p () + p (2) Â p j )NM, (2.3) j where the sum s over all partcles n the fnal state, and N takes care of the above normalzaton. In ths case, M s gven by Lmtaton M = e(p () + p () f ) µ g µn q 2 e(p (2) + p (2) f ) n. (2.32) The thng that makes the above dervaton heurstc s Eqn. 2.26, n whch a potental term s added to the equaton of moton for a free partcle (and not to the free partcle Hamltonan). Clearly ths s not a proper thng to do. Fortunately, t turns out that n a proper quantum feld-theoretcal context, we can use the actual Hamltonan for a complex scalar feld (whch we lack the formalsm to construct explctly), and t can be shown that the expresson for the transton ampltude s correct Feynman Dagrams and Feynman Rules The transton ampltude of Eqn. 2.3 s our way to Feynman dagrams. Apart from the delta functon and normalzaton factors, t contans three ngredents: two terms orgnatng from the currents nvolvng the two partcles (and whch are called the couplngs); and one term that represents the electromagnetc feld, as per Eqn

14 In addton, that same equaton shows us that the four-momentum q µ occurrng n the term correspondng to the electromagnetc feld corresponds precsely to the dfference between the ntaland fnal-state partcles, or n other words, ther momentum transfer. Ths leads us to a very smple pcture, especally gven that we are aware of the partcle nature of the photon: n ths process, a photon s exchanged between the two electrons, absorbng four-momentum from one electron and transferrng t to the other. The g µn /q 2 term s called the photon propagator. Ths pcture can n fact be translated easly to a graphcal equvalent, as shown n Fg. 2.2, called the Feynman dagram correspondng to ths ampltude. In t, the exchanged photon s clearly recognzable, as s ts couplng to the electrons. The correspondng Feynman rules (gven p!%# p!"# q p!%# $ p!"# $ Fgure 2.2: Graphcal representaton of the matrx element of Eqn wthout proof that s rather a topc for a course on Quantum Feld Theory) then tell us how to go back from the dagram to the matrx element:. Each Feynman dagram conssts of external and nternal lnes (n Fg. 2.2, the electron and photon lnes, respectvely) and of vertces, whch are assocated wth the couplngs of partcles to each other. 2. Each vertex nvolves a factor (2p) 4 d 4 (Âk ) e(p + p f ) µ where the delta functon expresses four-momentum conservaton at each vertex (all the k are taken to be ncomng; ths s a generc feature of all Feynman dagrams) and n the couplng e(p + p f ) µ the electron four-momenta follow the arrows, as n Fg Each nternal photon (.e., each photon propagator) s represented by a wavy lne and corresponds to a term d 4 q g µn (2p) 4 q 2, where q µ s the photon s four-momentum (meanng that each nternal four-momentum s ntegrated over). 4. The result contans a factor (2p) 4 tmes a delta functon expressng overall four-momentum conservaton. Ths factor s dscarded (but of course s to be kept n mnd when dong actual computatons); the result s equal to M. 24

15 5. The complete matrx element for a gven process (.e., for gven -completely specfedntal and fnal states) n general corresponds to multple Feynman dagrams, the ndvdual matrx elements of whch have to be summed. (In fact, to obtan the complete matrx element all possble Feynman dagrams need to be summed. Ths s a consequence of the Dyson seres: we have restrcted ourselves to the computaton of the frst term n perturbaton theory, and deally we would lke to compute hgher-order terms as well.) It may be noted that the photon s four-momentum q µ does not n general satsfy q 2 =. On the one hand ths s good (as otherwse the transton ampltude would dverge), but on the other hand the queston s how ths relates to the masslessness of the photon! The resoluton of ths ssue rests on the fact that the nteracton (.e., the exchange of the photon) takes place on very short tmescales. On such tmescales, the Hesenberg uncertanty prncple dctates that a photon of (squared) mass q 2 may exst for an amount of tme / p q 2. Such photons are called vrtual (snce they cannot propagate over macroscopc dstances) or offshell. In fact we wll also encounter many examples of other off-shell partcles beng exchanged n nteractons. On a more practcal note, whle the process under consderaton here s the elastc scatterng of two partcles, we could have equally well chosen to consder the scatterng of a partcle and an ant-partcle nstead (e.g., electron-postron scatterng). Now recall that n the Feynman- Stückelberg approach, ant-partcles are (loosely speakng) consdered as partcles movng backward n tme, and are assocated wth the negatve-energy solutons. In Feynman dagrams, ths dfference between partcles and ant-partcles s expressed by reversng the drecton of the arrows; so for ant-partcles the drecton of the arrows s always opposte the physcal propagaton n tme. As a corollary, the conservaton of (electrcal) current mples that the arrows n a sngle current lne (the external and nternal lnes featurng electrons and/or postrons) must always be n the same drecton along the lne. Returnng now to our computaton of electron-electron scatterng, t s not too hard to realze that the above Feynman rules gve rse to another dagram, even at the lowest order n perturbaton theory. Both of them are shown n Fg The second dagram arses because we are dealng wth ndstngushable partcles (ths s why t s not mmedately obvous that we dd not fnd t straght from our orgnal treatment of ths process, n whch we started out not makng any assumptons as to the nature of the other partcle). Ths process s called Møller scatterng. 2.3 The Drac Equaton 2.3. Drac s Attempt As mentoned n Sect. 2., n a quantum mechancal settng there are two problems wth the Klen-Gordon equaton (perceved problems, as they are addressed by a proper feld-theoretc treatment):. t nvolves a second order tme dervatve, gvng rse to negatve-energy states and a system that has no ground state; 25

16 p "&$ p "&$ % p "&$ p "&$ %! p "#$ p "#$ % p "#$ p "#$ % Fgure 2.3: Dagrams contrbutng (n lowest order) to the Møller scatterng process e + e! e + e. 2. and these same negatve-energy states lead to a contnuty equaton that s not amenable to a probablty nterpretaton. Even f n the context of feld theory there s no drect problem, Drac s attempt to address the above ssues by constructng an equvalent equaton that only nvolves a frst order tme dervatve has proven to be of great mportance, as t leads us to a proper descrpton of spn-/2 partcles (the dscusson above has not mentoned spn at all, but of course we know that electrons are spn-/2 partcles). The Drac equaton for free spn-/2 partcles (lke the Schrödnger equaton, n the poston representaton) s ( µ g µ m)y(x)=, (2.33) wth the quanttes g µ satsfyng the antcommutaton relaton {g µ,g n } g µ g n + g n g µ = 2g µn. (2.34) Clearly ths equaton cannot be satsfed by ordnary numbers, and therefore a (four-dmensonal) matrx representaton s used. Multple conventons are possble, but the one most often used (the Björken and Drell conventon) s g =, g s = s, (2.35) where the s represent the Paul matrces (so also the rght-hand sde of Eqn formally needs to be multpled by the 4 4 unt matrx ). Also y(x) cannot be a smple scalar-valued wavefuncton anymore; nstead t becomes a column vector of dmenson four, called a b-spnor. (That the Drac equaton s suffcent can be seen by multplyng t from the left by ( n g n + m). Ths smply yelds the Klen-Gordon equaton, so we have proven that t s a suffcent condton for the Drac equaton to be satsfed.) Clearly, gven that we are agan consderng free partcles here, t s to be expected that the solutons to the Drac equaton are plane waves. Now n partcular, let us consder those planewave solutons correspondng to a partcle at rest. Gven the 2 2 block form of the gamma 26

17 matrces, wrte y = ya y B. In ths case, the Drac equaton can be rewrtten as ( t ( t m)y A =, m)y B =. Clearly the soluton y A e mt corresponds to a normal postve-energy soluton; however, y B e +mt agan corresponds to a negatve-energy soluton. By now, however, aware of the antpartcle nterpretaton of E < states, we proceed undeterred. Lke n the case of the Klen-Gordon equaton, we take the hermtan conjugate of the Drac equaton. The result s µ y (x)g µ my (x)=. We manpulate ths by notng, from Eqn. 2.35, that g = g and g = g (snce the Paul matrces are hermtan). Usng Eqn. 2.34, ths can be wrtten concsely as g µ = g g µ g. So we have µ y (x)g g µ g my (x)=. Next, we multply the whole equaton by -g from the rght; the result s µ y(x)g µ + my(x)=, wth y(x) y (x)g. Wth ths conjugate equaton n hand, we proceed to construct agan a contnuty equaton. Ths s easly done by multplyng Eqn by y(x) from the left, the conjugate equaton by y(x) from the rght, and summng the result. Ths yelds µ (y(x)g µ y(x)) =. Consderng n partcular the tme component, we therefore fnd that we have y(x)g y(x)=y (x)y(x), So we have found a soluton where a probablty nterpretaton makes sense! However, agan because of the antpartcle nterpretaton we wll not make further attempts n ths drecton, but nstead consder ths as a conserved electrc current: j µ = ey(x)g µ y(x). (2.36) 27

18 2.3.2 Spn-/2 Partcles The vrtue of the Drac equaton s that t allows for a descrpton of spn-/2 partcles. Ths s perhaps to be expected already smply from the presence of the gamma matrces contanng Paul matrces (whch also n non-relatvstc Quantum Mechancs are assocated wth the spn operators for spn-/2 partcles). However, t can also be seen n more detal from consderng the general Drac equaton, and agan wrtng t n ts 2 2 block form, ua y(x)= e p x, u B.e., splttng off the plane-wave pece from the spnors u A and u B (at ths stage we haven t yet specfed whether the soluton nvolves postve or negatve energes). For nonzero momenta, we obtan coupled equatons for the spnors: (~s ~p)u B = (E m)u A, (~s ~p)u A = (E + m)u B. (2.37) Restrctng ourselves to the postve-energy soluton, we can now choose two ndependent solutons for u A : u () A = c() =, u (2) A = c(2) =. The second equaton n Eqn then yelds u (,2) (~s ~p) B = E + m u(,2) A. Smlarly, n the case of negatve-energy solutons, we choose two ndependent solutons for u B, u (,2) B = c (,2), and fnd (from the frst equaton n Eqn. 2.37): u (,2) (~s ~p) A = E + m u(,2) B. The mnus sgn has been carred over to the four-momentum components here. The reason s that n ths case, the physcal four-momentum contans an extra mnus sgn relatve to the fourmomentum occurrng n Eqn (Note that we mght as well have started wth the ndependent solutons for u B n the case of postve-energy solutons, and vce versa. The mportant pont s that n the non-relatvstc lmt, for postve-energy solutons, u A u B, whle for negatve-energy solutons, u B u A.) Summarzng, y represents four ndependent degrees of freedom, two for E > and two for E <. These two are of course nothng but the two solutons correspondng to dfferent spn states. When dong practcal calculatons, the four solutons are typcally wrtten as u (,2) (p)=n c (,2) (~s ~p) E+m c(,2)!, u (3,4) (p)=n (~s ~p) E+m c(,2) c (,2) In addton, the negatve-energy b-spnors are usually wrtten n terms of the physcal fourmomentum, leadng to v () (p) u (4) ( p) and v (2) (p) u (3) ( p). 28!.

19 2.3.3 Perturbaton Theory The step from free to nteractng spn-/2 partcles s made n exactly the same fashon as n the case of spn- partcles: by means of the mnmal substtuton (see Sect ). In that case, the Drac equaton s modfed to ( µ g µ m)y(x)=( t g + ~ ~g m)y(x)= ea µ g µ y(x). (2.38) The reason for separatng the tme and spatal components s that we can use ths equaton to construct explctly a Hamltonan suted for spn-/2 partcles. To do so, multply (from the left) by g ; we then have t y(x)=( ~ g ~g + g m)y(x) ea µ g g µ y(x), the rght-hand sde of whch ncely has the form H = H +V, so that we can dentfy the term ea µ g g µ wth a perturbng potental V. Insertng ths n Eqn. 2.4, we obtan T f = = d 4 xy f (x)( ea µg g µ )y (x) d 4 xj µ f (x)a µ(x), wth j µ f (x)= ey f (x)g µ y (x). Also here, we restrct ourselves to plane-wave states, and assume that the electromagnetc feld s generated by another partcle. Ths mples that we can agan nsert Eqn ths tme of course wth a current that s approprate for spn-/2 partcles. From here, t s not hard to see that also the rest of the computaton of the transton ampltude proceeds as for scalar partcles Feynman Rules for Spn-/2 Partcles Wthout further ado, we quote here the Feynman rules approprate for the computaton of matrx elements n QED:. The basc buldng blocks of Feynman dagrams are agan propagators and vertces. 2. Each photon propagator agan corresponds to a factor d 4 q (2p) 4 g µn q Each fermon propagator corresponds to a factor d 4 q (/q + m) (2p) 4 q 2 m 2. Note that we have ntroduced here the notaton /a a µ g µ for any a µ. 29

20 4. Each vertex corresponds to a factor (2p) 4 d 4 (Âk ) eg µ, where all four-momenta are agan taken to be towards the vertex. 5. External lnes now need to be dealt wth more precsely, as the fermons can be labeled by ther spns, and we also allow for external photon lnes correspondng to specfc spn states: ncomng fermon: u outgong fermon: ū ncomng antfermon: v outgong antfermon: v ncomng photon: e µ outgong photon: e µ 6. All approprate Feynman dagrams should agan be summed. A small refnement compared to the case of scalar QED, however, s that when combnng matrx elements that dffer only n the exhange of two dentcal fermons, a relatve mnus sgn must be added. (Ths s because wavefunctons must be fully antsymmetrc under exchange of any two dentcal fermons.) 7. The overall (2p) 4 d 4 (...) s agan dscarded, and the result s agan M. They are shown here mostly for completeness, as we wll not attempt to perform complete calculatons of Feynman dagrams; nevertheless, t s mportant to be aware of the dfferences wth the scalar QED case. Polarzaton states of spn- bosons The polarzaton vectors e µ mentoned n the above mert some further dscusson. Let us frst dscuss the case of massve spn- bosons. In ths case, one can transform to the partcle s rest frame, so that the polarzaton vectors from a non-relatvstc treatment are approprate: ~e A, ~e 2 A, ~e 3 for plane polarzaton states, and (takng the z axs as our quantsaton axs) ~e l= = 2 A, ~e l= = 2 A A, ~e l= for crcularly polarzed states. These polarzaton states are orthonormal: ~e l ~e l = d ll. 3 A

21 Next, we promote these polarzaton vectors to proper four-vectors and requre that they reman orthonormal: e(p;l) p =, e (p;l) e(p;l ) = d ll. For a boost e.g. along the z axs, the transverse polarzaton states (l = ±) do not change under ths transformaton. However, the l = ( longtudnal ) polarzaton state does change. From the orthonormalty condtons t s not hard to see that for a momentum p µ =(E,,, p), a vector e µ (p;l = )= m (p,,,e) s requred, where m s the partcle mass. (Note: t s far from obvous to see how the polarzaton vectors transform under general Lorentz transformatons! Suffce t to say that a proper covarant expresson can be found, n the form of the so-called Paul-Lubansk vector.) A fnal useful property of these polarzaton vectors s Âe µ (p;l)e n (p;l)= g µn + p µ p n /m 2. l (Ths can ether be verfed explctly, or by realsng that the result cannot depend anymore on any specfc polarzaton vector, and hence only terms proportonal to g µn and p µ p n reman. The orthonormalty condtons can then be used to determne the correspondng coeffcents.) Let us now consder the case of massless spn- bosons. As dscussed n Sect , the QED gauge freedom allows for transformatons A µ! A µ µ c, wth c an arbtrary functon. Specalsng to plane waves A µ µ e µ e q x, these gauge transformatons amount to changes of the polarzaton vectors e µ! e µ = e µ + aq µ. (Note that ths does not volate the orthogonalty condton e q = : after all, for on-shell massless partcles we have q 2 =.) Ths means that we can n fact choose c such that e =. Gven the Lorentz condton, ths mples~e ~q =. So only the transverse polarzaton states survve (but of course ths s well known from classcal electrodynamcs!). 2.4 The Electron s Magnetc Moment As a fnal applcaton of our manpulatons nvolvng spn-/2 partcles, consder the nteracton of an electron wth an external magnetc feld. The non-relatvstc quantum mechancal treatment of ths phenomenon s to post an nteracton term ~µ ~B n the total Hamltonan, leadng to the eeman splttng n the presence of a (weak) statc magnetc feld. In ths term, ~µ s the 3

22 electron s magnetc moment. It s typcally expressed n terms of the Bohr magneton µ B e/2m as ~µ = gµ B ~S, where ~S s the electron s spn vector. For electrons n a quantum mechancal treatment, we have ~S = ~s, ~s 2 denotng the Paul matrces as usual. In summary, we fnd a term n the Hamltonan equal to 2 gµ B~s ~B. (2.39) The ssue s that n a smple quantum mechancal context, the Landé factor g cannot be computed from frst prncples. The followng calculaton shows that QED does provde a predcton for g and a correct one at that! We start agan from the Drac equaton wth the nteracton wth an electromagnetc feld added through the mnmal substtuton, as n Eqn Wrtng n 2 2 block form, we have (cf. Eqn. 2.37) ~s (~p + e~a)u B = (E + ea m)u A, ~s (~p + e~a)u A = (E + ea + m)u B. Combnng these equatons yelds ~s (~p + e~a) 2 ua =((E + ea ) 2 m 2 )u A. Next, we smplfy the left-hand sde of ths equaton, but keepng n mnd the operator character of ~p! Ths yelds ~s (~p + e~a) 2 = s s j p p j + e 2 A A j + e(p A j + A p j ) = (d j + e jk s k ) p p j + e 2 A A j + e(p A j + A p j ) = p p + e 2 A A + e(p A + A p )+e(p A j + A p j )e jk s k = (~p + e~a) 2 + e( ~ ~A) ~s = (~p + e~a) 2 + e~s ~B. Here, repeated ndces are to be summed over (from to 3). Clearly, ths square almost trvally reduces to the frst term on the one-but-last lne. It s precsely the operator nature of ~p, p A j = A j p A j, whch leads to the nontrval addtonal term. Next, we consder the rght-hand sde of the equaton for u A, n the non-relatvstc lmt. Ths mples that the knetc energy and A are small compared to m, so ((E + ea ) 2 m 2 )=((m +(E + ea m)) 2 m 2 ) 2m(E + ea m). Wth that approxmaton and dvdng by 2m, we obtan! (E m)u A = (~p + e ~A) 2 2m ea + e ~s ~B 2m u A. (2.4) 32

23 The last term clearly corresponds to the nteracton of a magnetc moment wth an external magnetc feld, wth g = 2 (by comparson wth Eqn. 2.39). So s the equaton g = 2 exact? Not qute, n fact. The statc external magnetc feld s merely one form of an electromagnetc feld, and as such the nteracton that s of mportant at the dagrammatc level s that of an electron wth the photon,.e., a dagram consstng essentally only of the eeg vertex (ths s possble knematcally snce the external magnetc feld represents vrtual rather than real photons). But hgher-order perturbatve correctons, exemplfed n Fg. 2.4, need to be appled.! Fgure 2.4: Fundamental vertex and vertex correcton dagram descrbng the nteracton of electrons wth electromagnetc felds. In fact, the electron s anomalous magnetc moment a e (g e 2)/2 has been computed very accurately: a e = a a 2 a p p p It s one of the great achevements of QED that the measured and predcted values of a e agree wth each other, wthn exceedngly small uncertantes of several parts n 9. 33

24 34