Guidelines to Quantum Field Interactions in Vacuum

Transcription

1 Journal of Modern Physcs, 7, 8, 8-44 htt:// ISSN Onlne: 5-X ISSN Prnt: 5-96 Gudelnes to Quantum Feld Interactons n Vacuum Frédérc Schuller, Mchael Neumann-Sallart, Renaud Savalle Laboratore de Physque des Lasers, Vlletaneuse, France Groue d Etude de la Matère Condensée, CNRS/Unversté de Versalles/Unversté de Pars-Saclay, Pars, France CNRS/Observatore de Pars-Meudon, Pars, France How to cte ths aer: Schuller, F., Neumann-Sallart, M. and Savalle, R. (7 Gudelnes to Quantum Feld Interactons n Vacuum. Journal of Modern Physcs, 8, htts://do.org/.46/jm.7.86 Receved: January 7, 7 Acceted: February 5, 7 Publshed: February 8, 7 Coyrght 7 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.. htt://creatvecommons.org/lcenses/by/4./ Oen Access Abstract In ths treatse we stress the analogy between strongly nteractng many-body systems and elementary artcle hyscs n the context of Quantum Feld Theory (QFT. The common denomnator between these two branches of theoretcal hyscs s the Green s functon or roagator, whch s the key for solvng secfc roblems. Here we are concentratng on the vacuum, ts exctatons and ts nteracton wth electron and hoton felds. Keywords Quantum Vacuum, Quantum Electrodynamcs, Quantum Feld Theory, Relatvstc Quantum Mechancs, Feynman Dagrams. Introducton It s the am of ths treatse to ay trbute to Feynman s roagator method and ts vsualzaton n Feynman dagrams. Ths method has alcatons as wde as e.g. many electron theores, condensed matter hyscs and quantum feld theory. It conssts on one hand of showng for ntrcate mathematcal exressons of the underlyng hyscs, and on the other hand, of alyng re-establshed rules to these grahs, to set u these exressons. Here we are not gvng a lecture on these rocedures; we are merely alyng them to vacuum exctatons nteractng wth electron and hoton felds. Startng from routnely used technques as e.g. develoed n the book by M. E. Peskn and D. V. Schroeder [], we ntroduce some noveltes n the dervaton of fnal results. In artcular, a dscusson of the electron self-energy result n terms of a Ztterbewegung s resented. In a frst ntroductory art, we recall the basc facts of the second quantzaton DOI:.46/jm.7.86 February 8, 7

2 of the Klen-Gordon and the Drac feld and dscuss the resultng consequences. Then we defne roagators for the Drac and hoton felds and use them to treat nteractons of these felds wth the vacuum. More secfcally we study the electron and hoton self-energes. We do not concern ourselves n general wth collsons between elementary artcles, although ths s one of the man subjects met n Quantum Feld Theory. As an exceton we consder however electron-electron scatterng because of ts connecton wth vacuum olarzaton. The resultng hyscal facts are dscussed extensvely.. Partcles and Felds It s the am of ths secton to recall how, n relatvstc quantum hyscs, negatve energy states are avoded by adotng the feld vewont. For ths urose we chose as the smlest ossble case that of an uncharged artcle obeyng the Klen-Gordon equaton. The essental arguments develoed here then aly equally to the case of more general systems. Negatve energy states, causalty. In quantum mechancs we assocate a artcle wth a wave functon Ψ ( xt, deendng on tme and sace coordnates t and x resectvely. The wave functons are solutons of a dfferental equaton known as the Schrödnger equaton. In a heurstc way ths equaton can be derved by relacng the energy and momentum of the artcle by oerators, accordng to the relatons E = ħ and = ħ t For a artcle we then have n the non relatvstc case t ħ m Ψ= E = yeldng m (. 4 In the relatvstc case we start from the relaton E = mc + c and obtan, after nsertng the relevant dfferental oerators mc + Ψ Ψ = (. c t ħ Ths relatvstc verson of the Schrödnger equaton s called the Klen-Gordon equaton. It s mortant to note that n contrast to the non relatvstc Equaton (. the Klen-Gordon equaton contans the second tme dervatve meanng that t allows for negatve energy solutons. Usng from now on natural unts ħ =, c =, we wrte exlctly + Ψ m Ψ = (. t Settng Equaton (. reduces to ( xt ψ ( x Ψ, = e Et 8

3 E ψ m ψ = (.4 where we have used =. For lane wave solutons wth = const, we then have the energy relatons E = + m (.5a E = ± + m (.5b Hence there are negatve energy solutons. The queston arses whether these solutons cannot be dscarded as non hyscal. But n that case we would not have a comlete set of basc functons snce these solutons are art of t. In actual calculatons ths could yeld erroneous results. Furthermore, n a less obvous way, omttng these solutons leads to a volaton of the rncle of causalty as we shall demonstrate now. Consder the amltude At = xe Ht x for the evoluton of a free artcle from an ntal to a fnal oston durng the tme nterval t. Dscardng negatve energy states ths amltude would be t + m t + m At = xe x = d xe x (.6 Insertng the wave functons x x x = e ; x = e (.7 π π we have ( x x t m At = + (.8 ( π Usng olar coordnates as follows: d e e x x = x x cosϑ ; d = π snϑdϑd we arrve after ntegraton over ϑ at the exresson t + m At = d sn ( x x e π x x (.9 For smlcty we set X = x x. Wth a convergence factor Λ + m e, Λ >, nserted the value of ths ntegral s known []. Settng b = t + Λ ts value s roortonal to the Bessel functon K m X + b u to a ratonal functon of X and t. For large values of ts argument the Bessel functon reduces essentally to the exonental for Λ = to the result e m X ( b + [], leadng e m X t (. Gven ths factor n the exresson of At we have a non-zero amltude outsde the lght cone, thus volatng the rncle accordng to whch sace lke searated events cannot be causally connected. Consequently volaton of the causalty rncle occurs f only ostve energy functons are taken nto ac- 84

4 count. There are however other shortcomngs contaned n the relatvstc artcle theory. One could argue that any ostve energy state must be unstable snce after some tme the artcle would fall nto a lower energy state, n the same way as an atomc electron n an excted state falls nto the ground state after some short lfetme. In the case of fermons ths can be revented by assumng, followng Drac, that all negatve energy states are occued already. Ths stuaton s due to the fact that, accordng to the Paul rncle, each state can only receve one electron. The comletely flled negatve states consttute the Drac sea. Moreover, ths cture has led Drac to the redcton of the ostron,.e. a ostvely charged electron, aearng as a hole n the Drac sea when by some rocess an electron s removed from t. It s however ossble to gve a less artfcal descrton of relatvstc quantum artcles by adotng the feld vewont whch wll be resented now. Lagrangan feld method We consder a feld functon φ deendng on the tme-sace vector x= ( tx, wth comonents x α, α =,,,. Dstngushng between contra- and covarant α α αβ comonents, x, xβ resectvely, we further have x = g xβ and a smlar αβ relaton wth g = gαβ = the metrc tensor As usual, Greek ndces belong to the Mnkowsk four-sace, Latn ones to ordnary sace, wth x = t. In analogy wth classcal mechancs, we ntroduce a Lagrange functon, havng here the character of a densty, gven by the exresson ( φφ,, α, where we have set φ, φ = φ =. x α α α Note also the comlementary relaton φ, α α φ = φ =. We now defne an x acton ntegral S over a regon Ω bordered by a closed surface ( Ω, as follows: 4 = d ( φφ,, S Ω x α (. Varyng ths ntegral n the usual way accordng to the relaton Ω α 4 δ S( Ω = d x δφ + δφ, Ω α φ φ, α (. and usng the denttes we arrve at δφ = δφ + δφ; δφ = δφ, φ, α φ, α φ, α α α α α α 85

5 4 δ S( Ω = d x α δφ + α δφ Ω φ φ, α φ, α (. The last term n the arenthess can be seen as the four-dvergence of a four-vector roortonal to δφ. Therefore wth Gauss s theorem t can be transformed nto a surface ntegral over the border ( Ω. Snce the Lagrange method ostulates δφ = at the surface, ths term dsaears. On the other hand, f the acton ntegral S has to be an extremum, δ S must vansh for any value of δφ. Ths leads to the famlar Euler-Lagrange equatons or more exlctly α = φ φ, α (.4 =. (.5 α φ x φ α x These equatons aly to classcal felds, e.g. one comonent of the electromagnetc vector otental, as well as to wave functons n artcle quantum mechancs. As an examle let us therefore consder the Klen-Gordon wave functon. Settng α = {( φ, α φ, m φ } (.6 α αβ we wrte φ, φ, = g φ, and hence α yeldng wth β αβ α = αg φ, β = { φ, φ, } = φ φ, α t = φ m φ the Klen-Gordon equaton t m φ + = (.7 The Hamltonan. In order to establsh a lnk wth classcal mechancs, we frst conceve the sace coordnates x as a countable set, each element occuyng an nfntesmal sace segment δ x. Consderng the classcal exresson of the Hamltonan H = q L (.8 wth the canoncal varable obeyng the relaton = L q (.9 we have the corresondence q φ; δx = πδ x φ defnng the canoncal varable 86

6 π = φ. (. Wth these defntons we obtan for the classcal relaton (.8 the followng equvalent exresson: H = ( πφ δx (. Swtchng now to the lmt of contnuous sace coordnates, ths result takes the form H = d x π x φ x φφ,, = d x x (. { ( α } where reresents the Hamltonan densty x = π x φ x (. wth π ( x the canoncal momentum gven by π ( x =. (.4 φ Let us consder as an examle the Klen-Gordon case. Accordng to Equaton (.6 the Lagrange densty can be wrtten as We then have π ( x ( φ φ m φ =. (.5 = φ and hence ( x = φ φ + ( φ + m φ = φ + ( φ + m φ. (.6 Second quantzaton. Smly seakng, a gven wave functon s quantzed f t s relaced by an oerator. Ths s famlar n quantum electrodynamcs where e.g. one comonent of the vector otental s relaced by hoton creaton and annhlaton oerators. A smlar rocedure can be aled to quantum mechancal wave functons and n ths latter case one then talks of second quantzaton, snce the wave functons are already obtaned by a frst quantzaton rocedure. Note however that the term second quantzaton s not unversally acceted. Here we consder agan as an examle the Klen-Gordon case, whch consttutes the smlest one, as t concerns snless artcles lke K or π mesons. Let us frst swtch from x sace to sace by ntroducng the followng transformatons: φ d x, = e φ,. (.7a ( xt ( π ( π ( t d x φ x, t = e φ, t (.7b ( π d x φ( xt, = π ( xt, = e π ( t, (.7c The Hamltonan densty then takes the form 87

7 ( π ( π { π π ( ( φ φ( } d d ( + x ( x = e + + m (.8 Snce we want to quantze the system by relacng wave functons wth oerators n the Schrödnger cture, we dsregard t n ths exresson. Integratng over the sace coordnates, we thus arrve at the followng exresson for the Hamltonan n terms of functons n sace: wth d H = d x x = + (.9 ( π { π π ωφ φ} ω = + m (. To obtan Equaton (.9 we have made use of the relaton ( + ( d xe x = δ + The arenthess nsde the ntegral of Equaton (.9 remnds one of the Hamltonan of a harmonc oscllator. q ( + ω In the latter case quantzaton s acheved by ntroducng creaton and destructon oerators a, a, accordng to the relaton q = a+ a = ω a a ω ( ; ( wth the commutator aa, =. We therefore try n Equaton (.9 the substtutons π ω = (.a ( a a φ = + (.b ω ( a a The arenthess nsde the ntegral n Equaton (.9 s then found to be gven by the exresson { π π( + ωφ φ( } = ω( aa + a a Snce comlete summaton over takes lace, we can dsregard the mnus sgns of the ndces and wrte { } π π + ωφ φ = ω aa + aa = ω aa + a a, We thus obtan for the Hamltonan the followng result d ω, H= aa + a a (. ( π Accordng to general rules of quantum hyscs, the commutaton relaton for 88

8 canoncal varables takes the followng form n the resent case: ( x, ( x = ( x x φ π δ. (. Insertng nto the commutator the transformaton relatons gven By equaton s (.7a, (.7c we wrte d d x x φ( x, π ( x = e e φ, π ( (.4 ( π ( π Substtutng for φ, π ( the exressons gven by Equaton s (.a, (.b we obtan after a lengthy but straghtforward calculaton Adotng the tral rule Equaton (.5 reduces to, ( = { a, a + a, a } φ π δ (.5 a, a = π (.6, ( = ( π ( + φ π δ (.7 Substtutng ths result nto Equaton (.4 we recover the commutaton relaton of Equaton (.. Ths confrms the valdty of the tral rule of Equaton (.6. In the feld equatons develoed above the number of artcles concerned s not secfed. Let us now be more secfc by ntroducng sngle artcle states assumed to consttute an orthonormal set n a gven nertal frame. Actng wth the Hamltonan of Equaton (. on one of these states, e.g., and usng Equaton (.6 for the commutator, we obtan the formal exresson d H = ω + ω δ (.8 ( π The second term on the r.h.s. of ths equaton contans the nfnte quantty δ and moreover t nvolves an nfnte sum over energes ω. Mostly ths term can be consdered as some sort of ground state energy E whch cannot be detected exermentally and thus can be gnored. In order to establsh the tme deendence of the oerators φ and π one has to relace them by Hesenberg oerators accordng to the relaton Ht Ht φ( xt, = e φe and smlarly for π ( xt,. Startng from the exressons (.a, (.b we evaluate the corresondng Hesenberg oerators of a and a as follows: Actng on an egenstate of H, accordng to Equaton (.8, the nfnte zero-ont energy term cancels n the oerator roduct snce t s a c number. We are thus left wth the exresson ω t Ht Ht e a e = e a usng Ht e a = = a. 89

9 Smlarly we have Ht Ht ω t e ae = e a Hence the requested oerator equatons are e ω t a t = a (.9a ω e t a t = a (.9b Wth Equaton (.b the quantzed form of Equaton (.7a becomes ( π ωt x ωt x ( d (.4 ω φ xt, = ae e + ae e where Equaton s (.9a, (.9b have been used. Introducng the Lorentz nvarant scalar roduct x = x x n four sace, wth = ω and x = t, we obtan for the quantzed feld the exresson d x x (. (.4 ω φ xt, = ae + ae ( π Causalty agan. As mentoned earler, two onts x, y wth sace lke searaton ( x y corresonds to the regon outsde the lght cone, the commutator φ( x, φ( y < are not causally connected. Ths means that n ths case, whch must vansh. Startng from Equaton (.4 the commutator s gven by the exresson ( x φ( y d ( x y ( x y ω φ, = e e (.4 ( π where the oerator commutaton rule of Equaton (.6 has been used. In order to obtan zero for ths quantty, the nverson transformaton x y ( x y has to be aled to the second ntegral. However, ths s only legtmate f ths transformaton leaves the value of the ntegral nvarant. Ths we shall dscuss now. Frst set x y = = (,, wth = t, = x. Then we have x y = = ω (.4 Now we defne a sace lke surface [] ; = > (.44 K ; K Wthout loss of generalty we can restrct ourselves to the lane (, where the surface of Equaton (.44 aears as the curve = K see Fgure (.45 Now take a artcular ont (, frame n both terms of Equaton (.4 from (, to (, One then has the relatons on ths curve and rotate the coordnate. = cos ϕ; cosϕ = + (.46 9

10 Fgure. Transformaton dagram for sace lke coordnates n connecton wth the causalty roof dscusson. The quanttes and reresentng the vertcal and the horzontal axs are defned by Equaton (.4. Hence the transformed quanttes are, =, = = cosϕ yeldng the followng result n terms of rotated quanttes: ( x y ( x y e e e e Now the cumbersome factor e has dsaeared and the transformaton leaves the value of the second ntegral unchanged, snce n ths ntegral one can change the sgn of the ntegraton varable wthout affectng ts value. The fact that for any ont on a gven curve the corresondng coordnate rotaton can be made, and that ths s true for any curve, roves the statement that the commutator vanshes at any ont outsde the lght cone. Insde the lght cone,.e. for tme lke searatons, the commutator does not vansh so that n ths regon onts can be causally connected. It s however nterestng to note that the corresondng commutator s nvarant wth resect to roer Lorentz transformatons as shown e.g. n ref. []. Note fnally, that n many calculatons the nfnte energy of the vacuum state s elmnated by erformng normal orderng of oerators. It conssts n reshufflng oerator roducts n such a way that destructon oerators always stand on the rght of creaton oerators. Generalzatons [4] [5] [6]. Partcles obeyng the Klen-Gordon equaton do not bear any electrc charges. In order to treat charged artcles, comlex wave functons have to be ntroduced nto the theory. Even more rofound modfcatons are necessary n the case of electrons accordng to the Drac theory. Here, due to the resence of sn, wave functons are reresented by snors consstng of four functons as comonents of a vector. An even more strkng dfference occurs f second quantzaton s erformed. In ths case, the fermon character of the artcle s 9

11 taken nto account n ostulatng ant-commutaton rules for the feld oerators nstead of the commutaton rules ertanng to bosons. However, the general dea of avodng negatve energy states by means of second quantzaton, already aled to the Klen-Gordon case, remans essentally the same n ths and other stuatons.. Symmetry Transformaton Relatons An essental feature of relatvstc artcles and felds s ther behavour wth resect to transformatons of the Lorentz grou. Transformaton oerators We recall that the elements of ths grou are three rotatons n the xy, xz, and yz lanes around the z, y and x axs resectvely, comleted by three seudo-rotatons belongng to the xt, yt and zt lanes resectvely. These transformatons can be vewed as an nfnte successon of nfntesmally small rotatons whch generate a reresentaton of the grou. Desgnatng the rotaton oerator ν wth resect to the lane x, x as J ν, and the corresondng rotaton arameter as ω ν, then an nfntesmal transformaton s generated by the oerator Λ ω J ν ν (. yeldng for the fnte Lorentz transformaton oerator the exresson ex ω J ν Λ= ν. (. Recallng that the famlar exresson for rotatons n ordnary sace can be generalzed to Mnkowsk sace as ν ν ν ν ν J = ( x x, J = J, (. we can generate a four dmensonal reresentaton of the roer Lorentz grou x, x. Usng the relatons by actng wth ths oerator on the vector J x = x, j j J x = x (.4 j j we consder the examle ω = θ = ω, all other ω ν equal zero. Equaton (. then yelds the matrx θ M Λ (.5 θ Ths matrx thus corresonds to a rotaton by an nfntesmal angle θ n the xy lane as can be shown by multlyng the matrx by the vector ( x, x. As a second examle we consder the Lorentz boost n the x drecton by settng ω = β = ω wth all others equal zero. Then the relaton J x = x, J x = x x = x x = x (.6 wth = substtuted nto Equaton (. leads to the result 9

12 M Λ β β (.7 Note that the factor n Equaton (. dsaears because n both examles two equal terms are accounted for. Note also that by multlyng the matrces x of Equaton s (.5 and (.7 by the column vector one recovers the usual x relatons for the corresondng nfntesmal rotatons and Lorentz boosts. Alyng a Lorentz transformaton as exressed by the oerator Λ of Equaton (. to wave functons Ψ ( x, one obtans the followng change: ( x ( x Ψ Ψ Λ (.8 The crteron for the corresondng wave equatons to be vald s ther Lorentz nvarance. Ths roerty can be establshed by rovng that the Lagrange densty, from whch a gven wave equaton s derved, s a Lorentz scalar. We shall now demonstrate ths ont n the artcular case of the Klen-Gordon equaton. We cast the Lagrange densty of Equaton (.6 n the form αβ { ( βφ φ } = g m (.9 wth only one tye of dfferental oerator. Wth the transformaton of Equaton (.8,.e. the scalar roerty of and wrte φ * ( x φ( Λ x (. φ s obvous. We therefore focus on the quantty ( βν βν g ( ( x ( ( x g ( Λ ( Λ σ φ β φ β φ φ φ ν = β ρ ν ρ βφ (. where we have omtted on the r.h.s. the argument Λ x of the φ functons. Note also that the horzontal shft of the lower ndces on matrx elements allows us to dstngush between lne and column ndces. Snce matrx elements are c-numbers, ther roduct can be treated searately. It s suffcent to do ths n the lmt of nfntesmal rotatons. The more abstract general treatment can be found n the lterature e.g. n ref. []. Accordng to Equaton s (. and (. we wrte Wth the defnng relaton g ( ( Λ Λ = g ( δ β + λ β ( δ ν + λ ν βν βν ρ ρ σ σ β β (. λ = ω J ϕψ ϕψ (. Treatng only the change ntroduced by the transformaton and gven the fact that λ s an nfntesmal quantty, we consder the exresson g δ λ + g δ λ βν σ βν σ ρ β ν ν β (.4 9

13 In the frst term the ndces β and ν are elmnated yeldng wth β = ρ, ν = β = g ρρ λ no summaton whereas for the second term we fnd wth ν = σ, β = ν = σ, Hence the fnal result g σ ρ ρ σσ λ g no summaton σ λ + g λ no summaton (.5 ρρ σ σσ ρ ρ σ Suose now that ρ and σ belong both to ordnary sace.e. ρ =, σ = j then the g elements are both equal to, but as shown by Equ- j aton (.5, we have λ = λ j and the sum n Equaton (.5 s zero. In the ooste case of Lorentz boosts wth e.g. ρ =, σ = we have g = and g = whereas, accordng to Equaton (.7 λ λ = and the sum s agan zero. Ths roves the statement that Lorentz transformatons do not affect the Lagrangan densty functon, excet for the argument of the wave functons, and hence t s a Lorentz scalar. The resultng Euler-Lagrange equaton,.e. the wave functon, has therefore a Lorentz nvarant form. The roof gven here for nfntesmal varatons s generally vald, snce fnte transformatons nvolve an nfnte successon of nfntesmal ones. As already mentoned, more formal roofs are found n the lterature, but we thought t nstructve to aroach the roblem by exlct calculatons as well. Snors. Havng treated as an examle the case of a structure less artcle obeyng the Klen-Gordon equaton, we are now movng to the case of the electron, where n addton to sace coordnates sn varables have to be consdered, together wth the exstence of an electrc charge. + + Introducng sn functons uu, vv, wth the + sgns ndcatng sn varables +, n a gven frame, the wave functon n four sace can be wrtten n the form Consderng comonents ψ u + can also wrte Ψ= ψ u + ψ u + ψ v + ψ v ( etc. as elements of a vector n sn sace, we Ψ Ψ ΨA Ψ Ψ Ψ = =, Ψ A = Ψ B = Ψ ΨB Ψ Ψ4 Ψ4 (.7 where the functons Ψ etc deend on both the sace and the sn varable. The column vector of Equaton (.7 s known as a snor. Its Lorentz transformaton can be exressed as follows: Ψ Λ Ψ Λ (.8 x where t s understood that the oerator Λ acts only on sn states. 94

14 We now defne oerator matrx elements S ρσ by ntroducng for Λ the lmtng exresson ω S ρσ ρσ (.9 wth ω ρσ beng the usual rotaton and boost arameters. We now recall that sn functons transform under rotatons n ordnary sace accordng to the Paul sn matrces σ wth σ = σ = σ = Then clearly, ordnary sace rotatons occur accordng to the relaton (. S j k k ΨA σ ΨA σ Ψ A = k = k ΨB σ ΨB σ Ψ B (. j k n normal order. Remark: normal order means that j k are all dfferent and that startng wth an odd number of ermutatons ntroduces a mnus sgn. One may ensure ths roerty automatcally by multlyng wth a quantty known as the εjk -tensor. The queston now arses, what haens n the case of Lorentz boosts? Wthout enterng nto detals, we only state the answer gven by Drac s theory accordng to the relaton S Ψ Ψ σ Ψ = = A σ A B ΨB σ ΨB σ Ψ A (. Hence the matrces of Equaton s (. and (. consttute a four-dmensonal reresentaton of the Lorentz grou known as the Drac-Paul reresentaton. The Weyl reresentaton. The Drac-Paul reresentaton s reducble snce ts matrces can be brought nto dagonal form by a untary transformaton nvolvng the matrces, M M = = Wth these matrces we have σ σ M M = σ σ and hence (. S = σ σ (.4 whereas the j matrces reman unaffected. Desgnatng as left and rght handed snors Ψ L and Ψ R the snors whch now relace Ψ A and Ψ B, we have nstead of Equaton s (., (. the relatons 95

15 S j k Ψ L σ Ψ L = k Ψ R σ Ψ R and S Ψ L σ Ψ = Ψ R σ Ψ L R (.5 Takng as an examle the values ωj = θ, ω = β wth artcular fgures for j and all other ω s equal, Equaton s (.9 and (.5 then yeld the equaton showng that the functons Ψ R, called Weyl snors, transform ndeendently from each other. k Ψ L σ Ψ L σ Ψ L θ β 4 k + 4, (.6 Ψ R σ ΨR σ Ψ R Ψ L and Clearly, these relatons can be generalzed for arbtrary rotaton and boost arameters descrbed by vectors θ and β resectvely. Ths leads to the transformaton relatons σ σ ΨL θ β Ψ σ σ ΨR θ + β Ψ L R (.7a (.7b Hence the Weyl snors Ψ L, Ψ R consttute the bass for two-dmensonal reresentatons of the Lorentz grou, nstead of the reducble four-dmensonal reresentaton of the Drac-Paul bass. In order to exlan the desgnatons of Ψ, Ψ as left and rght handed snors, we consder the fact that they are egenstates of the helcty oerator h σ = ˆ σ snors. L R wth egenvalues for left and + for rght handed As an examle the snors ntroduced n Secton 4 are rght handed for those of Equaton s (4.a, (4.5a and left handed for those of Equaton s (4.b, (4.5b. Ths can be shown by alyng the helcty oerator wth = to these snors. Connecton wth wave equatons. The wave equaton for snors Ψ s Drac s equaton, whch can be derved from the Lagrange densty ( γ m, γ =Ψ Ψ Ψ=Ψ (.8 as the corresondng Euler-Lagrange equaton aled to Ψ, wth the result Note that for notaton. ( m ( x γ Ψ =. (.9 γ and smlar roducts Feynman has ntroduced the slash The γ matrces enterng the Lagrange densty are of vtal mortance, snce n choosng them n an arorate way, one meets the condton that has to be a Lorentz scalar, necessary for the corresondng wave functon to be vald. As a consequence, there s clearly a connecton between these matrces and the Lorentz transformaton roertes of the snors. The corresondng relatons 96

16 are derved n many textbooks and wll be gven here only n ther fnal form. Accordng to Drac, the followng equatons hold: = γ, γ 4 (. S ν ν γ, γ = g ν ν + (. where the + ndex ndcates an antcommutator. Note that later n ths text the antcommutator wll be desgnated by the symbol { }. Gven the fact that the matrces S are dfferent n the Drac and the Weyl reresentaton, one would exect a smlar dfference n the γ matrces. Substtutng n Equaton s (., (. the secal values =, ν =, one obtans γ s equal n both the Drac and the Weyl reresenta- Makng the guess that ton,.e. for S one obtans the result By settng γ γ = S and γ γ S σ = σ = (. Drac and S σ γ = σ γ = γ = γ = γ = one then obtans the followng relatons: = σ σ Ψ R σ Ψ R γ Ψ= γ Ψ= Ψ L σ ΨL Weyl (. Drac (.4 Weyl (.5 The Drac equaton, gven n ts general form by Equaton (.9, then takes n the case of the Weyl reresentaton the form of the followng two couled equatons: wrtten n matrx form as ( σ ( σ R m L + Ψ Ψ = (.6a L m R Ψ Ψ = (.6b m ( + σ ΨL ( σ m Ψ (.7 R As can be seen from these equatons, the mxng of the two Lorentz grou reresentatons Ψ L and Ψ R occurs because of the mass term n the Drac equaton. 97

17 Noether currents. Let us now consder some contnuous symmetry transformatons on the wave functons, whch leave the Lagrangan densty nvarant. In the nfntesmal lmt we then wrte ( x ( x φ φ + δφ (.8 The corresondng change n the Lagrange densty ( φ, φ reresented by the exresson ( φ s then δ = δφ + δ ( φ (.9 φ Wth the obvous relaton δ φ = δφ (.4 we then have δ = δφ + δφ (.4 φ ( φ Usng the dentty δφ = δφ + δφ ( φ ( φ ( φ the second term on the r.h.s. of Equaton (.4 can be rewrtten wth the result δ = δφ + δφ (.4 ( φ φ ( φ Now the second term of ths equaton, set equal to zero, reresents the Euler-Lagrange equaton as gven by Equaton (.4. For =, accordng to the nvarance condton of the Lagrange densty, we then wrte δφ = ( φ Introducng Noether currents by the defnng relaton α j (.4 = δφ, (.44 ( φ Equaton (.4 nvolves the four-dvergence of ths quantty for whch we thus have = (.45 j Integratng ths exresson over the entre ordnary sace, and alyng Gauss theorem to the corresondng three-dvergence, wth vanshng contrbuton at the nfnte surface, we are left wth the exresson Hence the sace ntegral d. (.46 j x = all sace j d x s a conserved quantty. all sace 98

18 In order to nterret ths quantty, let us consder the Drac equaton. The corresondng Lagrange densty functon s gven by Equaton (.8. Ths equaton s nvarant under the hase transformaton Ψ e α Ψ, or n nfntesmal form δψ αψ (.47 and For Noether s current we then have, accordng to Equaton (.44 α j = Ψγ α Ψ (.48 j =ΨΨ (.49 =. As can be seen, j reresents the robablty densty, whch multled by the electron charge, consttutes the charge densty. Hence Equaton (.46 exresses the fact that the electrc charge of the electron s a conserved quantty. where we have used the fact that n any reresentaton ( γ 4. The Drac Feld As an entrance door to the Drac feld let us consder free artcle solutons of the Drac Equaton (.9. These solutons can be vewed as suerostons of lane waves of the form. ( x u e x Ψ = wth = m (4. Pluggng ths exresson nto Equaton (.9, yelds the equaton ( mu γ = (4. Ths equaton s most easly solved n the rest frame, where only the comonent = m s dfferent from zero, so that we have = = ( mγ m u m u( where for γ the Weyl exresson (.5 has been used. Introducng two-comonent snors ξ, the soluton s u( m ξ = ξ (4. (4.4 where the factor m has been chosen for future convenence. Let us now look for a more general soluton wth two comonents = E and and = m becomng E = m (4.5 Ths soluton can be obtaned by erformng a Lorentz boost on the revous one, whch n nfntesmal form can be wrtten as E m η = + (4.6 Ths relaton can be deduced by analogy from the matrx of Equaton (.7 99

19 notcng that all satal drectons are equvalent whereas the nfntesmal arameter η, called radty, relaces the revous β. For fnte values of η we therefore have E m m mcoshη ex η coshη snhη = = + = (4.7 msnhη The second exresson on the r.h.s. s obtaned by exandng the exonental and notcng that even owers of the matrx yeld the unt matrx, whereas odd ones leave ths matrx unchanged. Now we aly the same boost to the amltude u( of Equaton (4. and wrte u m ξ = Λ ξ (4.8 From the nfntesmal oerator as gven by Equaton (.4 wth I =, we deduce the relevant Lorentz transformaton oerator Consderng the matrx σ Λ = ex η σ σ =, wth (4.9 σ the unt matrx, an even ower of the matrx n the exonent of Equaton (4.9 yelds the unt matrx, whereas an odd one yelds ths same matrx. The seres exanson of the exonental oerator of Equaton (4.9 therefore leads to the followng matrx exresson: Exlctatng σ cosh η Λ = snh η σ (4. σ and addng all matrces, a lengthy but straghtforward calculaton yelds the followng dagonal matrx where the relaton has been used. E E+ Λ = m E+ E cosh η± snh η= e = E ± η / ± We now go back to Equaton (4.8 and calculate the amltude secal snors ostve and negatve m (4. (4. u for two ξ = and ξ =, corresondng to sns orented n the x drecton resectvely. The matrx of Equaton (4. 4

20 then mmedately yelds the results u u = = E E + E+ E (4.a (4.b So far we have ut the mnus sgn on the exonent of the defnng relaton gven by Equaton (4.. Consder now the case of a lus sgn wth e x v x Ψ = (4.4 We choose however to mantan > and hence E >. Deste ths choce ths case corresonds to the negatve energy solutons whch consttute the famous Drac sea. Ths s only aarent f e.g. the Hamlton densty s calculated. At ths stage we take t only as a known fact. We are not reeatng a calculaton smlar to the revous one, but ndcate only the relatons relacng Equaton s (4.a, (4.b. For these secal stuatons one fnds E v = E + E v = E + (4.5a (4.5b Defnng as usual u = u γ and v = v γ, t s nstructve to calculate the roducts uu, vv and uu, vv. Consderng the secal case of Equaton (4.a we have ( (,, (, ( u = E E+ u = E+ E Wth u gven by Equaton (4.a we thus obtan uu E = m, u u E = = (4.6 A smlar calculaton for the case of Equaton (4.5b yelds the result vv mv v E = = (4.7 For the case of an arbtrary sn orentaton axs we ntroduce the notatons 4

21 r r u, v wth r =, desgnatng the two ooste sn drectons. Then the relatons (4.6, (4.7 have to be comleted as follows: Furthermore we have the relatons The Hamltonan. uu = mδ u u = Eδ (4.8 r s rs r s rs vv r s = mδ rs v r v s = Eδ rs. (4.9 s r s ( =, v u r u v = (4. Startng from the exresson (.8 of the Lagrangan densty ( γ m γ = Ψ ΨΨ = Ψ and from the exresson of the conjugate varable densty s gven, accordng to Equaton (. by the exresson = ΠΨ More exlctly we then have wth γ = γ m, Ψ=Ψ and ( γ Π=, the Hamltonan Ψ = Ψ Ψ+Ψ Ψ Π = Ψ (4. In the exresson of the term ΠΨ thus cancels the frst term n Equaton (4. and we are left wth the result Involvng the sngle artcle Hamltonan The amltudes u( and D ( γ γ mγ = Ψ + Ψ (4. h = γ γ + mγ = γ γ + mγ (4. v of Equaton s (4. and (4.4 are egen- functons of ths Hamltonan wth egenvalues E and E resectvely. To see ths, multly the Drac Equaton (.9 by γ =. rememberng that ( γ γ mγ γ and wrte + Ψ= (4.4 Ths equaton can be exressed n the form h D Ψ= (4.5 Relacng Ψ by the free-artcle exressons of Equaton s (4. and (4.4 we then have Ψ= u (4.6a e x Ψ= v (4.6b e x Introducng these exressons nto Equaton (4.5 yelds the egenvalue relatons stated above wth E. h u D = u, h v v = > Hence the amltudes D =, (4.7 v corresond to negatve energy solutons whch consttute the famous Drac sea. As n the Klen-Gordon case, ths nconvenence s crcumvented by means of a fully quantzed treatment. 4

22 Second quantzaton. In relacng the wave functon Ψ ( x by an oerator, we frst consder the tme-ndeendent Schroednger oerator ψ ( x whch, n analogy wth Equaton (.4, we wrte n the form (summaton rule wth s =, d ( π s s x s s x ( e e ψ x = au + bv (4.8 E or equvalently d x s s s s e ( (4.9 E ψ x = au + bv ( π a s Defnng an emty state t s understood that we must have s = b =. Introducng the total Hamltonan H = d x we obtan usng Equaton (. n the Schoednger cture H d xψ hd = ψ (4. After substtutng the exresson (4.9 and ts adjont we wrte H = d d q ( q x d x e 6 ( π E Eq r r r r s s s s ( aq u ( q b qv ( q Eau Eb v + (4. Invertng the order of ntegraton, we take advantage of the relaton δ ( q x d xe = π q and notce that, accordng to Equaton (4., the cross terms n the roduct of the ntegrand n Equaton (4. dsaear. We are thus left wth the exresson r s r s r s r s H= ( Ea au ( u Eb bv ( v (4. d E ( π where, gven the ntegraton over all values of, the relacement has been made. Elmnatng the amltudes by means of the relatons (4.8, (4.9, we thus arrve at the fnal exresson d H= Ea a Eb b (4. ( r r r r π At ths stage t has to be remnded that n the resent case of fermons the oerators obey ant-commutaton relatons, whch n contrast to the boson relatons (.7, are of the form { a }, a aa a a π δ ( + = (4.4 Ths relaton allows us to deal wth the embarrassng negatve energy term n the ntegrand of Equaton (4.. Wrtng by means of the rule stated by Equaton (4.4 4

23 d H= Ea a + Eb b E ( r r r r π d (4.5 we have cast the negatve energy nto an nfnte constant term whch can be gnored f the orgn of the energy scale s shfted adequately. r r A next ste conssts n nterchangng the order of b and b. Ths s a trck justfed n detal n ref s [] [4] [6]. Here we ndcate only that t has to do wth the fact that n the one-artcle case, accordng to the Paul rncle, we have β = so that by nterchangng β, we recover the fundamental relaton =. Normal orderng β A rocedure of elmnatng negatve energy terms n the Hamltonan conssts n what s called normal orderng. It means that all oerator roducts are reshuffled n such a way that annhlaton oerators stand always on the rght of creaton oerators. These oeratons are symbolcally exressed by the letter N n front of the roducts. Alyng ths conventon to the exresson (4., suosed second quantzed, we thus wrte { ( γ } = Ψ + Ψ (4.6 H d xn m where Ψ, Ψ are tme-deendent Hesenberg oerators gven by the exressons, smlar to Equaton (4.8 and ts conjugate d ( π s s. x s s. x ( e e Ψ ( x = au + b v E ( π (4.7a d s s x s s x ( e e (4.7b E Ψ x = bv + a u Here the tme deendence of the oerators has been absorbed nto the exonental factors. Moreover, the nterchange taken nto account. b b dscussed above, has been r r A calculaton smlar to that develoed above, wth only the cross terms contrbutng, then leads to the exresson d d H = E N a a + b b = E a a + b b s s s s s s s s (4.8 ( π ( π Ths s exactly the result obtaned revously f n Equaton (4.5 the nfnte negatve energy term s gnored and f the oerator and state changes dscussed there, are accomlshed. Thus clearly normal orderng merely ntegrates these facts. 5. Proagators The retarded Green s functon. Let us frst consder roagaton amltudes gven by the exressons d s a b s a b E ( π s ( x y Ψ x Ψ y = u u e (5.a 44

24 d s b a = s a b E ( π s ( x y Ψ y Ψ x Ψ v v e (5.b These exressons are obtaned by usng the fact that n the roduct of the wave functons of Equaton s (4.7a, (4.7b only cross terms contrbute. Ths s because n the other terms annhlaton oerators are on the rght and therefore elmnate these terms n the mean values of Equaton s (5.a, (5.b. Furthermore the oerator relaton (4.4 has been accounted for. We now evaluate the sn sums aearng n Equaton s (5.a, (5.b. Usng Equaton s (4.a, (4.b and (4.5a, (4.5b we obtan the followng tensor roducts: E m E uu = ( E+ E = E+ E+ m where the relaton A smlar calculaton yelds E = m has been used. uu m E+ = E m For the sn sum we therefore arrve at the result m E s s m E+ uu, a b = uu a b + uu a b = (5. E + m E m It s now an easy matter to show that ths matrx s dentcal wth the exresson γ E γ + m or by extenson γ + m. Consequently we obtan wth after a smlar calculaton s s uu, a b = γ + m (5.a ( γ s s vv, a b= + m (5.b Makng these relacements n the exressons (5.a, (5.b and addng them afterwards, we obtan an antcommutator of the form { a b} ( π ( x y ( x y (( γ ( γ d (5.4 E Ψ x, Ψ y = + m e + m e wth = E and hence x y = x y x y = E x y x y (5.5 We now want to lnk the above commutator to an ntegral n four sace. For ths urose we ntroduce a quantty defned by the relaton 45

25 4 ab d SR ( x y = + m E = ( π d ( π 4 ( x y ( γ e d e m e ( ( γ + x y π E ( x y (5.6 where has now become an ntegraton varable. The ntegral can be evaluated by consderng a closed crcut n the comlex lane wth two sngulartes at = ± E as shown n Fgure (a. The corresondng resdua are for E ( x y e E γ γ = Ea = E + + m π for = Ea = E + + me ( γ γ π E ( E x y for the lower clockwse crcut, corresondng to for the ntegral the value x ( γ E + γ + m > y, we therefore obtan ( E x y e = π res = E (5.7 E ( x y ( γ E + γ + m e whereas for the uer crcut, corresondng to x < y, the ntegral s zero. Insertng the value gven by Equaton (5.7 nto the comlete ntegral gven by Equaton (5.6 we can, wthout loss of generalty, relace n the second term by and n ths way we obtan for and for ab R ( π x > y ( x y ( x y (( γ e ( γ e S x y = + m + m x < y d (5.8a E ab R Comarng wth Equaton (5.4 we thus fnd where ( x y S x y = (5.8b ( x y { a b( y } Θ Ψ, Ψ (5.9 Θ s the Heavsde ste functon. (a (b Fgure. (a Comlex ntegraton ath for evaluatng the ntegral of Equaton (5.6 n the Green s functon case. (b Smlarly n the Feynman s case. 46

26 Gong back to Equaton (5.6 we notce that the denomnator E can be wrtten as m = m. One can also rove that = ( γ so that n the end we have Wrtten n the form S ab R = d ( π ( γ + m ( γ m 4 ( x y e ( ab d SR = S e ( π ths quantty can be regarded as the Fourer transform of or n Feynman slash notaton ab R ab ( x y 4 R (5. ( γ + m γ m S = (5. ( + m ab S R = m (5. Ths exresson s known as the Drac roagator. Its Fourer transform reresented by Equaton (5. s a Green s functon of the Drac oerator defned n Equaton (.9. To see ths, we frst notce that for lane wave states ths oerator can be wrtten as γ m. Actng wth t on the exresson gven by Equaton (5. the denomnator cancels and we obtan ( γ d (5. 4 ab ( x y 4 m SR = e = δ x y 4 ( π thus rovng the Green s functon relaton stated above. Note however that the ntegral n Equaton (5. can be evaluated along dfferent aths. The way chosen so far yelds the artcular exresson (5.9, called the retarded Green s functon. Ths s because t s only non zero durng the tme erod x > y. The Feynman roagator. A dfferent ath for evaluatng the ntegral of Equaton (5.6 s that shown on Fgure (b. Desgnatng by the lower crcut,.e. x > y and by the uer one,.e. x < y, the theorem of resdua then yelds, nstead of Equaton (5.7 the followng two contrbutons: πa ( E e E x y m E γ γ = = + + (5.4a ( E x y = πa = ( γ E + γ + m e (5.4b E Insertng these exressons nto Equaton (5.6 we obtan the Feynman Green s functon F d ( x y ( γ e (5.5a E S = + m x > y ( π 47

27 F d ( x y ( γ e (5.5b E S = + m x < y ( π where agan n the second lne. Comarng these exressons wth Equaton (5.4 we see that we have Ths can also be wrtten as S F b Ψa x Ψ y for x > y = Ψ b y Ψ a x for x < y (5.6 S = TΨ x Ψ y (5.7 F a b where T s the tme orderng oerator whch ensures that the earler tme always stands on the rght, wth the addtonal condton of a mnus sgn f the oerators are nterchanged. In the Feynman case the ntegraton aths can be slghtly modfed wth resect to those of Fgure (b f we relace Equaton (5. by the exresson S F = d ( π ( γ + m ( γ + 4 ( x y e (5.8 4 m ε Wth the denomnator equal to E + ε, the sngulartes are now shfted ε away from the real axs to = ± E so that ths axs s now entrely art E of the ntegraton aths. Interretng Equaton (5.8 as a Fourer ntegral we thus obtan for the Feynman roagator the exresson or n slash notaton ( γ + m γ + S F = (5.9 m ε F ( + m S = (5.9 m + ε These exressons are basc elements n Many-Body tye calculatons. The hoton roagator. In analogy wth Equaton s (5.6 and (5.7 reresentng the Feynman roagator n the Drac case, we defne a hoton roagator by the relatons A x Aν y x > y Aν y A x x < y Corresondng to the tme-ordered roduct Here A ( x, A ( y (5. D = TA x A y (5. ν ν ν are oerators of the quantzed vector otental accordng to the exresson d q * = r + = Eq ( π r r qx r r qx ( qε e q ε e A x a q a q (5. 48

28 ε and r n a chosen bass. Let us frst consder the roduct r The quanttes ( q d q d q r* ε are olarzaton vectors labeled by the ndex r r r r * qx q y aa e e q q ε r r qε q = ν (5. = ( π ( π Eq Eq Postulatng the rule ( π δ ( r r q q δrr aa = q q (5.4 wth Ths exresson reduces to d q f E ν q ( π ( q q( x y e (5.5 r r* ε ε f q = q q (5.6 ν r = ν The value of the quantty f ( q ν deends on the choce of a artcular gauge. In the case of the Lorentz-Feynman gauge ths value reduces to the metrc tensor g ν, as shown n standard textbooks. As n the Drac case we now lnk the exresson (5.5 to an ntegral n 4 dmensonal sace of the followng form: 4 d q gν q( x y d q q( x y dq q ( x y e g e e 4 ν ( π q E = q ( π π q E (5.7 q Performng the ntegraton over q along the aths ndcated n Fgure (b we obtan the two exressons d q gν q( x y e x > y (5.8a π E ( π q d q g < ν q( x y e x y (5.8b Eq Comarng ths wth Equaton (5. we see that the two ntegrals corresond to the exressons defnng the roagator D ν. Fnally, as n the Drac case, the exressons (5.8a, (5.8b can be obtaned by relacng the ntegral (5.7 by the modfed exresson 4 d q gν q( x y e (5.9 4 π q E + ε Settng q Eq = q q = q we therefore obtan the exresson for the roagator n momentum sace g D ν ν = q + ε (5. as the fnal result for the hoton roagator n the Lorentz-Feynman gauge. 6. Interactng Felds: The Radatve Electron Mass Shft Introducton. q 49

29 Consder an electron n the form of a ont charge-e, then the surroundng statc electrc feld ossesses the energy wth e e α dr = dr = 4πr r (6. = = the Sommerfeld fne structure constant. Recall that through- 4π 7 out ths treatse we use natural unts settng ħ = c =. In order to make the ntegral n Equaton (6. fnte, a lower cut-off radus r = Λ has to be ntroduced yeldng the value for the energy α = Λ (6. In ths way the energy tends lnearly towards nfnty wth the cut-off arame- ter Λ. Alyng n a naïve manner Ensten s relaton = mc or n our unts = m, we see that the electromagnetc mass of the electron aears as a lnearly dvergng quantty. Attemts have been made to mrove thngs by alyng the formalsm of quantum feld theory to ths roblem. In ths treatse we resent a slghtly renewed verson of these calculatons. As a result the lnear dvergence of the sem-classcal theory s brought to the form of a logarthmc one however wth no quanttatve soluton at the end. The roagators. Prelmnary remark: as s customary n quantum feld theory we desgnate vectors and ndces n 4 dmensonal Mnkowsk sace by ordnary letters and l.c. greek letters (e.g. =,,, resectvely and the corresondng objects n dmensonal Eucldean sace by bold letters and l.c. Latn letters (e.g. =,,, resectvely. Moreover, summaton over reeated ndces s assumed and furthermore γ are Drac s gamma matrces. We now consder an electron movng freely through vacuum and defne a correlaton functon by the exresson where ψ ( x and ( y ( y ψ Ω Tψ x y Ω (6. = are oerators relacng n second quan- ψ ψ γ tzed theory the usual wave functons. In exresson (6. the Dyson oerator T stands for the tme ordered roduct. The resence of ground states Ω nstead of zero electron states ndcates that we are consderng nteracton of the movng electron wth the surroundng electromagnetc vacuum feld. The easest way for evaluatng the correlaton functon (6. conssts n alyng Feynman rules accordng to the Feynman dagram of the fgure (Fgure whch shows that the electron-vacuum nteracton can be conceved as the emsson and reabsorton of a vrtual hoton vsualzed by the wavy lne. The elements of ths dagram corresond to Feynman roagators n momentum sace gven by the exressons 4

30 Fgure. Feynman dagram for evaluatng the electron correlaton functon of Equaton (6.. ( + ( k + m m, m + k m + (6.4 for the electron of mass m n momentum state and k resectvely and the roagator exresson g ν k + ε (6.5 for the hoton. In ths way durng the rocess the total momentum of the system s conserved at every ste. In addton the exressons ν eγ, eγ (6.6 descrbng the electron-hoton nteracton have to be nserted at the vertces. In these exressons the Feynman slash notaton abbrevates the sums γ and γ k, whereas g ν s the metrc tensor reresented by a 4 dmensonal dagonal matrx wth g =, g etc. =. Assemblng these relatons, known as the Feynman rules, we see that the above dagram corresonds to the roduct ( ( k ( + m + m + m e γ γ m k m + ε k + ε m (6.7 ν where the relaton gν γ = γ has been used. Notcng that n both the Weyl and the Drac reresentaton the sum γ γ s equal to 4 tmes the unt matrx, we condense the exresson (6.7 nto the form ( + ( ( + Σ m m m m where the central art s gven by the exresson ( π ( k m 4 4( e 4 Σ = (6.8 d k + (6.9 k m + ε k + ε after addng an ntegraton over all ossble ntermedate 4 momenta. The ndex on Σ ndcates that the alcaton of the above dagram reresents n fact a lmtaton to second order of a erturbaton exanson. An extenson to all orders under secal condtons wll be dscussed below. The ntegraton rocedure. Before startng the ntegraton n the exresson for Σ we use Feynman s 4

31 trck based on the dentty = dx ab ax b x { + ( } Comarng wth Equaton (6.9 we thus wrte the k ntegral n the form 4 d k k + m dx (6. 4 ( π ( k + ε x+ ( k m + ε( x { } Followng a common rocedure we now change varables accordng to the relaton q = k x (6. Then the arenthess n the denomnator of the ntegrand takes the form ( q x x m + + (6. An essental smlfcaton arses f we restrct ourselves to the zero th order contrbuton n m wth The ntegral n (6. then reduces to k = q + mx (6. 4 d q q + m( x+ dx (6.4 4 ( π q m ( x + ε { } Note that the same letter m matrces as well as scalars recognzable from the context. Searatng the q art from the sace art q and extendng the ntegral over q comonents from to +, the q term n the numerator does not contrbute and we are left wth the exresson d q dq m dx x (6.5 + ( + ( π π ( { q q m x + ε} where now all matrces are relaced by scalars. The evaluaton of the second ntegral s resented n Aendx leadng to the result Settng d 4π d d q m x x q m x 8 π d ( + + (6.6 q = q q and ntroducng an uer lmt cut-off we relace the exresson (6.6 by the followng ntegral 8 π Λ d ( + d + ( (6.7 m x x q q q m x Introducng the dmensonless varable 6π wth the lmtng exresson λ = q m we also have Λ m d ( + λ dλ λ + ( (6.8 m x x x 4

32 6π dλ Λ mlog (6.8 λ 6π m Λ m m = where we have assumed that the cut-off value Λ m s large as comared to unty. Pluggng ths result nto Equaton (6.9 we thus arrve at the fnal exresson Λ Σ = e mlog (6.9 8π m Renormalzaton. Let us suose that the change n the correlaton functon reresented by the resultng exresson (6.9 can be reroduced by renormalzng the mass n the free electron roagator,.e. by addng a correcton δ m to the ntal mass. Assumng ths correcton suffcently small we then consder the exanson ( δ + m + m mδ m + + m m m m (6. Equatng the correcton term wth the exresson (6.8 wth the exresson (6.9 for Σ nserted, we have ( + δ ( + Λ ( + m m m m m = e m log m m 8π m m + + by ts domnant art m + Equaton (6. yelds the result [7] [8] Aroxmatng on the r.h.s. ( m( m ( m (6. m e log log 8π m Λ δ α m π m Λ = = (6. m Ths s the result derved n the lterature by varous methods, showng that the fully quantzed theory reduces the lnear convergence of the classcal exresson (6. to a logarthmc one. Dscusson. There seems to be no ndcaton how to estmate the cut-off arameter Λ. Clearly the logarthmc dvergence makes the mass shft less senstve to the value of ths arameter. Moreover t can be argued that a large mass shft should show u n exerments. Nevertheless, n order to get a number out of the calculatons, one could for nstance consder the fact that the roton mass consttutes a natural uer lmt on the mass scale of conventonal artcles. Identfyng t wth Λ would lead to the result: Λ δm α Λ = 86 = log =.6 m m π m a number that seems realstc. Naturally ths estmaton has to be taken merely as an examle among others that one could magne. However, deste the fact that the true numercal value of the electromagnetc electron mass shft s as yet unknown, ts correct qualtatve evaluaton, as revewed n ths secton undoubtedly consttutes an mortant fact. Ztterbewegung The fact that quantum feld calculatons lead to a logarthmc dvergence of 4

33 the electron self-energy nstead of the lnear classcal result of Equaton (6., can be understood f one takes nto account the sread of the electron oston due to quantum fluctuatons [7]. Ths s equvalent wth attrbutng the electron a fnte dmenson of the order of the Comton wavelength m, whereas classcally the electron s ont lke. More generally, quantum fluctuatons of a artcle oston of the order of m are known as Ztterbewegung. It s ths effect that we are studyng now n the statonary case. In order to determne the oston x occued n the average by the electron wth resect to some central oston =, let us consder the roagator roduct Interretng the central art ( y ( y ( x Ψ Ψ Ψ Ψ (6. ( y ( y ρ = Ψ Ψ (6.4 as a densty oerator we obtan the desred average by takng a trace reresented fomally by the exresson ( ρ P x = Tr Ψ Ψ x (6.4a Wrtng out exlctly the roduct of (6. we obtan from the defnng relaton (5.4 the result under the condton d d q + + y q ( x y ( m e ( m e ( π E (6.5 ( π Eq y > x (6.5a Furthermore, the varable x can be secalzed as x (, x = so that the condton (6.5a becomes y >. Wth these smlfcatons the roduct of (6.5 reduces to d m d q m q x + q y e e ( π E (6.6 ( π Eq The takng of the trace n Equaton (6.4a amounts to ntegratng over the varable y and afterwards relacng the matrx m by the scalar 4m. The re- δ + q then leads to the result sultng delta functon d m x P( x = e (6.7 E ( π The ntegral of Equaton (6.7 s elementary, yeldng wth the result d π sn E = + m m r P x = m = r cosθ e snθdθ d ( π + m ( π r (6.8 + m Wth the last ntegral on the r.h.s beng equal to π e mr, we obtan for the robablty the fnal result 44