SINGULAR STRUCTURE OF THE QED EFFECTIVE ACTION

Transcription

1 SINGULAR STRUCTURE OF THE QED EFFECTIVE ACTION arxv:84.233v [hep-th] 6 Apr 28 B.A.FAYZULLAEV Department of Theoretcal Physcs, Natonal Unversty of Uzbekstan, Tashkent 95, Uzbekstan September 2, 28 Abstract The equatons for the QED effectve acton derved n [3] are consdered usng sngular perturbaton theory. The effectve acton s dvded nto regular and sngular (n couplng constant) parts. It s shown that expresson for the regular part concdes wth usual Feynman perturbaton seres over couplng constant, whle the remander has essental sngularty at the vanshng couplng constant: e. Ths means that n the frame of quantum feld theory t s mpossble to swtch off electromagnetc nteracton n general and pass on to free electron. Key words: quantum electrodynamcs, effectve acton, sngular perturbaton expanson, vacuum expectatons.

2 Introducton Investgaton of analytc structure of quantum feld models usually s reduced to nvestgaton of S-matrx as analytc functon of ts varables such as angular momenta, energy, masses of partcles etc. Many results were reached n ths way - see, e.g. []. But there are no results concernng analytcty of quantum quanttes, e.g., vacuum expectatons or effectve acton. Especally, analytcty of vacuum expectatons of quantum felds n couplng constants. Expandng ampltude of any process n perturbaton seres over couplng constant we mplctly suppose analytcty of ths ampltude n ths couplng constant. But t s well known that perturbaton seres n quantum feld models are asymptotc ones. In ths artcle we want to nvestgate perturbaton seres n QED from the followng vewpont: s there any sngularty n ths seres at e =, where e s electrc charge (couplng constant). I.e., can we swtch off the electromagnetc nteracton n Quantum Electrodynamcs? In ths artcle we show that ths s mpossble. 2 Man equatons We wll use the followng notatons: the effectve acton W[η, η,j µ ] whch s generator of all weakly connected Green functons, s connected wth the generator of all Green functons Z[η, η,j µ ] (partton functon) as follows: Z[η, η,j µ ] = exp(w[η, η,j µ ]). Consequently, the vacuum expectatons (n the presence of external sources) for quantum felds are ψ = δw δ η, ψ = δw and A µ δη = δw. δjµ Applyng to the followng classcal acton for QED (here all the functons are classcal felds, not vacuum expectaton): S = 2 ψˆ ψ ( 2 µψγ µ ψ mψψ eψâψ+ 2 Aµ Dµν Aν, Dµν = g µν 2 ) µ ν, () the DeWtt operator [2] Λ = 2 δ 2 W δ 2 δjµδj ν j δa µ δaν j + δ2 W δ 2 δjµδη j δa µ δ ψ δ2 W δ 2 j δjµδ η j δa µ δψ + δ2 W δ 2 + (2) j δη δ η j δψ j δψ n accordance wth DeWtt equatons Λ δs δa µ = J µ, ΛδS δψ = η, Λ δs δ ψ = η t may be derved [3] the followng set of functonal equatons for the QED effectve acton

3 W (here all the felds are vacuum expectatons of quantum felds): e ψγ µ ψ +D µν A ν +e Tr γ µδ2 W = J µ ; δηδη [ ( ψ(x) )] δ 2 W +e Â+m +e δjµ δηγ µ = η ; ) ] [( eâ m δ 2 W ψ +e γ µ = η δjµδ η. Ths set of equatons concdes wth Schwnger equatons [4], of course. Takng nto account followng relatons (all dervatves over the grassmann varables are understand as left ones) (3) δ 2 j W = δ ψ δjµ δηj δjµ = δaµ δη j ; δ 2 W δψj = δjµ δ ηj δjµ = δaµ δ η j ; δ 2 W δη δη β = δψβ δη = δψ δη β (4) Eqs.(3) may be brnged to the more convenent for further consderaton form: δ ψ δψ e Tr δ η γ µ = e Tr δη γ µ = Jµ e ψγ µ ψ +DµνA ν ; e δ ψ γ δjµ µ = e δâ [ ( δη = η ψ(x) )] +e Â+m ; e γ µ δψ δj µ = e δâ [( ] δ η = η )ψ eâ m. (5) Usual way to solve these equatons s perturbatve expanson over small couplng constant e. I.e., at frst step we put couplng constant e =, thereby turnng these equatons nto equatons for free partcles whch can be solved easly. Further, nteractons between free partcles are taken nto account teratvely over small parameter e (really, e 2 /4π) gettng perturbatve power seres: W = W +ew +e 2 W 2 +. (6) Ths s usual way. But there s one crcumstance must be taken nto account. It s obvous that n front of each dervatve term there s couplng constant e (and ), ths means f we put e = then our (varatonal) dfferental equatons transform to algebrac ones. In the followng secton we show that n ths case a full soluton to equatons of such type (wth small parameter) should contans not only regular but sngular (n e) part too. In the Sect.5 we wll adopt ths method to QED vacuum expectatons. 3 The method of soluton The formulaton of problem under consderaton may be explaned n the followng smple example [5]: fnd soluton to equaton µ dx(t) dt = a(t)x(t)+b(t), x() = x, t < (7) 2

4 n the form of perturbatve expanson over small µ. It s very easy to fnd exact soluton to ths equaton obeyng gven boundary condton: x(t) = x exp µ a(s)ds + µ b(s) exp µ s a(z)dz ds. (8) t It s obvous that µ = s essental sngular pont for soluton to (7) and, consequently, regular perturbatve expanson for small µ can not exst. The reason for such stuaton can be seen from Eq.(7) tself: f we put µ = n ths equaton then t fals to be dfferental equaton and becomes to be algebrac one a(t) x(t)+b(t) =. (9) But soluton to ths (algebrac) equaton x(t) = b(t)/a(t) n general can not obey gven boundary condton: a()/b() x. It happens loss of boundary condton. Ths means that soluton to Eq.(9) can not be consdered even as frst approxmaton to exact soluton of Eq.(7). Ths consderaton underles the reason for sngularty at µ =. From above mentoned t follows that any expanson of a soluton to Eq.(7) around µ = may be sngular one only. Dervaton of a sngular perturbaton seres accordng to [5] consst of the followng steps. Frst, take the second term n (8) and ntegrate t by parts to get the followng seres: t µ b(s) exp µ s t a(z)dz ds = + [ b(t) a(t) +µ [ b() a() +µ As a result the followng seres s obtaned: [ b(t) x(t) = a(t) +µ b(t) a(t) a(t) + + ] + [ x + b() a() +µ ] b(t) a(t) a(t) + + ] b() a() a() + ] b() a() a() + exp µ exp µ Let s to make substtuton t = µτ, s = µζ n the ntegrand of the exponent. Then a(s)ds. a(s)ds. or, exp µ t τ a(s)ds = a(µζ)dζ = a()τ +µ a () µ 2 τ2 +µ 2a () 6 τ3 + [ ] a(s)ds = e a()τ +µ a () 2 τ2 +µ 2a () 6 τ3 +µ 2τ4 4 a 2 ()+ 3

5 So t has been derved the followng seres over µ : where x(t,µ) = x(t)+πx(τ), x(t) = x (t)+µ x (t)+ = b(t) a(t) µ s a regular part of the soluton and s a sngular one wth ( Π x(τ) = x + b() a() b(t) + () a(t) a(t) Πx(τ) = Π x(τ)+µπ x(τ)+µ 2 Π 2 x(τ)+ () ) [( e a()τ ; Π x(τ) = x + b() )a () τ2 a() 2 + etc. The terms Π k x(τ) are called boundary layer terms. It s easy to see that ] b() e a()τ a() a() (2) ( x (t)+π x(τ)) t= = x ; ( x k (t)+π k x(τ)) t= =, k. (3) Now we can present the algorthm of sngular perturbatve soluton n the followng form. Gven an equaton wth boundary condton (7). Then the soluton should be dvded nto two parts as follows: x(t) = x(t)+πx(τ) and Eq.(7) can be presented n a form: µ d x dt + dπx(τ) dτ = a(t) x(t)+a(µτ)πx(τ)+b(t), τ = t/µ. (4) Further one should to expand each term n both sdes of ths equaton n seres over µ. Equatng coeffcents n front of the same powers of µ, separately for terms dependng on t and terms dependng on τ, one obtans equatons for determnaton of terms of the expansons () and (). It s easy to check out that n ths way the seres () and (2) wll be obtaned. And t s not so hard to see, that ths algorthm s equvalent to consder Eq.(4) as dvded n to two parts - frst part ncludes terms dependng on t, and second part ncludes terms dependng on τ. Solutons of these equatons connect each other through boundary condtons (3). 4 The regular part of the effectve acton Let s to ntroduce new scaled varables ρ = η/e, ρ = η/e and j µ = J µ /e. And then present each feld n Eq.(5) n the form, dvded nto regular and sngular parts: ψ = ψ R (J,η, η;e)+πψ(ej,eρ, e ρ;e), ψ = ψ R (J,η, η;e)+πψ(ej,eρ, e ρ;e), A µ = A R µ(j,η, η;e)+πa µ (ej,eρ, e ρ;e). Further actng n accordance wth above mentoned (n the end of precedng secton) method one should to dvde Eqs.(5) nto part dependng on J,η, η and part, dependng on j,ρ, ρ. Let s for smplfcaton of equatons denote the sources as follows: J,η, η s and scaled sources as follows: j,ρ, ρ σ. These settngs allows one to wrte down equatons n more shorter form because now t s possble to denote: ψ R (J,η, η;e) = ψ R (s;e) for regular part 4

6 and Πψ(J,η, η;e) = Πψ(eσ;e) for sngular part of the feld ψ. The same notatons wll be used for other felds too. Then equatons for regular parts wll be (for sake of smplcty n the belove equatons for regular parts we wll omt the superscrpt R): e Tr δψ (s;e) δη γ µ δψ (s;e) = e Tr γ δη µ = Jµ eψ(s;e)γ µψ(s;e)+dµν Aν (s;e); e δψ (s;e) γ δjµ µ = e δâ(s;e) ( = η [ψ(s;e) δη )] +e Â(s;e)+m ; e γ µ δψ (s;e) δj µ = e δâ(s;e) δη [( ] = η )ψ(s;e) eâ(s;e) m. Solutons to these equatons wll be searched n the regular perturbaton form: (5) ψ(s,e) = ψ (s)+eψ (s)+e 2 ψ 2 (s)+ ; ψ(s,e) = ψ (s)+eψ (s)+e 2 ψ 2 (s)+ ; A µ (s,e) = A µ (s)+ea µ (s)+e 2 A 2µ (s)+. (6) It s very smple to fnd these seres by teratons. Equatons for zeroth order terms: Jµ +DµνA ν (s) = ; η ψ (s) ( ) +m = ; η m ψ (s) =. (7) Ther solutons: A µ (s) = D µν J ν ; Regular part of zeroth order effectve acton: Equatons for frst order terms: ψ (s) ( ) +m +ψ (s)â(s) = ; ψ (s) = η ; ψ (s) = η. (8) +m m W = η m η 2 Jµ D µν J ν. (9) m ψ (s) Â(s)ψ (s) = ; Solutons to them: Tr m γ µ +ψ γ µ ψ (s) = Dµν Aν (s). ψ (s) = m γ µ m ηdµν J ν ; ψ = η( +m) Â ( +m) ; A µ (s) = η( +m) γ µ m η + D µνtr m γν. (2) 5

7 Regular part of frst order effectve acton: W = η m γ µ m ηdµν J ν + J µ D µν Tr m γν. (2) Eq.(9) reproduces free propagators for electron-postron and photon felds. Eq.(2) reproduces frst order Feynman dagrams, ncludng one-loop tadpole dagram. Actng ths way one can to reproduce the Feynman dagrams of all order n couplng constant (electrc charge) e. Ths s why we have called ths seres as regular perturbaton ones. But the exstence of the couplng constant (electrc charge) e n front of terms wth dervatves n Eqs.(5) set one thnkng about possble sngularty at e =. 5 Sngular parts of vacuum expectatons n QED Equatons for sngular (boundary layer) parts are more complcated: δπψ(eσ;e) δπψ(eσ;e) Tr γ µ = Tr γ µ = eπψ(eσ;e)γ µ ψ(eσ;e)+ δρ δρ δπψ(eσ;e) γ µ = δπâ(eσ;e) δj µ δρ +eψ(eσ;e)γ µ Πψ(eσ;e)+eΠψ(eσ;e)γ µ Πψ(eσ;e) D µν ΠAν (eσ;e); (22) = Πψ(eσ;e) ( ) +m eπψ(eσ;e)â(eσ;e) eψ(eσ;e)πâ(eσ;e) eπψ(eσ;e)πâ(eσ;e); (23) γ µ δπψ(eσ;e) δj µ = δπâ(eσ;e) δρ = m Πψ(eσ;e)+eÂ(eσ;e)Πψ(eσ;e)+ +eπâ(eσ;e)ψ(eσ;e)+eπâ(eσ;e)πψ(eσ;e); (24) Recall that felds ψ(eσ;e), ψ(eσ;e) and A µ (eσ;e) are regular parts of correspondng felds, but they are functons not of s = (J µ,η, η) but of eσ = (ej µ,eρ,e ρ.) At frst step we should to extract zeroth order (n e) equatons from above mentoned ones: δπ ψ(σ) δπ ψ(σ) Tr γ µ = Tr γ µ = D δρ δρ µνπ A ν (σ); (25) δπ ψ(σ) δπâ(σ) γ µ = δj µ δρ γ µ δπ ψ(σ) δj µ δπâ(σ) = = δρ = Π ψ(σ) ( +m ) ; (26) m Π ψ(σ). (27) It s easy to fnd a general form of solutons to equatons for Π ψ(σ) and Π ψ(σ): [ ] Π ψ(σ) = exp 4 ĵ( m) f +cψ D ; [ Π ψ(σ) = f exp ( ) ] 4 +m ĵ +c ψ D, 6 (28)

8 where ψ D and ψ D are solutons to free Drac equatons, c - an arbtrary constant, f - some spnor feld. Now t s the tme to apply equaton for boundary condton (3). From Eq.(8) follows that regular parts of felds under consderaton vansh at J µ = η = η =. Further, from Lorentz nvarance t follows that vacuum expectaton for spnor feld n external source-free case vanshes: ψ = ψ =. Jµ=η= η= Jµ=η= η= These condtons gve us that f = cψ D. In general let s present the functon f as Jµ=η= η= follows: f = cψ D + c n ( ρρ) n+s ρ, (29) n= where s should be found from correspondng ndcal equaton (after defnng of correspondng dfferental equaton for f). After substtuton of Eq.(29) nto (28) we have followng expresson for sngular part of the spnor feld: [ [ Π ψ(σ) = c exp m) 4 ĵ( [ = exp m) 4 ĵ( ] n= ]] [ ] ψ D +exp m) 4 ĵ( n= [ c n ( ρρ) n+s ρ = exp m) 4e Ĵ( c n ( ρρ) n+s ρ = ] c n ( ρρ) n+s ρ (3) and conjugate expresson for Π ψ(σ). These expressons has essental sngularty at zero couplng lmt e. Concluson about essental sngularty at zero couplng lmt e can be referred to Π A ν too. So, anyvacuumexpectatonnqedhasessental sngulartyatzerocouplnglmte. n= 6 Concluson The sngularty at e = s very nterestng - ts exstence means that n general we can t swtch off electromagnetc nteracton and go to free electron. It s the tme to remember Dyson s proof [6] that perturbaton seres n QED s asymptotc one. Our consderaton shows that QED effectve acton can t be an analytc functon n the neghborhood of e =, consequently, any seres n ths regon can t be convergent one. In the lght of ths sngularty the noton of free electron should be revsed - because t s mpossble to swtch off the electromagnetc nteracton the exstence of free, nonnteractng electrcally charged partcle s questonable. But ths pont s very hard one and more accurate studes requred to be conclusvely establshed. References [] R.J. Eden, P.V.Landshoff, J.Polknghorne and D.I.Olve, The Analtc S-matrx, Cambrdge unversty Press, (966) [2] B.DeWtt, Dynamcal theory of groups and felds, NY, Gordon and Breach (965). 7

9 [3] B.A.Fayzullaev and M.M.Musakhanov, Two-loop effectve acton for theores wth fermons, Annals of Phys.(NY) 24 (995)394. [4] N.N.Bogolubov and D.V.Shrkov. Introducton to the theory of quantzed felds, ch. VI, Moskow, Nauka (984). [5] A.B.Vasleva and V.F.Butuzov, Asymptotc expansons of solutons to sngularly perturbed equatons, Moscow, Nauka (n russan) (973). [6] F.Dyson, Phys.Rev.85, 63(952). 8