arXiv:physics/0202025v1 [physics.atom-ph] 8 Feb 2002 [email protected] Paris 75252 UMR7 Min´eralogie-Cristallographie, CNRS de Laboratoire field external Brouder an Christian in QED of Renormalization aeo sn nymaual uniis rtemr mea- more the or ones. quantities, surable measurable only using advan- of physical tage the and distributions, standard nipulating derivatives. solutions functional working as involving given of equations are Ref.[6] which of idea functions Schwinger’s Green use with directly to prefer There- we measures. fore, ill-defined uses approach integral path and Schr¨odinger ap- the [5] and distributions Schr¨odinger rep- operator-valued field manipulate and quantum proaches integrals The [4]. path are resentations textbooks field, in quantum used formalisms the solid-state three for main convenient The most physics. is which version the choose go deal- to when framework especially excitations. safe methods, with a LDA ing present is the rela- theory Moreover, beyond field [3]. quantum theory tivistic functional spin-density prob- obvi- nonuniqueness of func- the look meet lem density not no does relativistic [1],[2] do theory instance, tional that For problems relativistic. for ously even proved have powerful, methods the very relativistic From view, effects. tiny of huge measure point made to theoretical able has now the is spectroscopy be- From and view, adequate advances reasons. of seems theoretical it point and beautiful, experimental experimental be of to approach cause true an such too Although look sound. may is calculation ba- the the of that sis garanties theory, successful and accurate most photon quantum of on calculation based the matter towards of electrodynamics. step spectroscopies electron a and is paper This Introduction 1 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ hsapoc a h ahmtclavnaeo ma- of advantage mathematical the has approach This to have first We QED. of presentations many are There a is which electrodynamics, quantum from Starting Abstract. 2018 7, November PACS. pr are e diagrams. equations in Feynman three for propagator the electron and of an relations) each give as for used are rules is equations renormalization that these function of Green solutions The perturbative fie weak The called ics. are which derivatives, functional involving cedex 22.mQatmeetoyais–1.0G Renormalizat 11.10.Gh – electrodynamics Quantum 12.20.-m 5 France 05, h cwne qain fQDaerwitni he differe three in rewritten are QED of equations Schwinger The 9,UM/D/PP ae15 lc Jussieu place 4 115, Case UPMC/UDD/IPGP, 590, rnpoaao.Teeeutosaesle trtvl us- elec- iteratively solved the are and equations These potential Schwinger propagator. electromagnetic tron the derive the we for functions Then, Green equations ex- Lagrangian. the to QED for the used and with notation and mag- start sample the we to of the introduction, definition nuclei, this the to the After applied to sources. light external fields due ternal This potential electric the field. or of netic external made an is with field QED formulate to densi- current be and can charge function electronic Green ties). the many the (e.g. but of measurable, measured elements directly matrix not Green also diagonal one-particle is the This with field function. computed). elec- function electron be the can Green measurable density describe photon We energy the the photon the be and from which fields) can (from gauge magnetic but given and measurable tric a directly in not calculated is (which potential this to possible as view. much of as point stick experimental to spec- try UV/visible We etc. absorption, troscopies, x-ray RHEED, photoemission, LEED, scattering, framework: BIS, inelastic spectroscopies scattering, this of photon Many within scattering, beam electron measured. described another is be measured. sample, can photons being the or system with electrons the interaction the by that its mean influenced we After not classical com- photons By is or source. source electrons classical of a beam from spectroscopist a ing experiments The sample the principle. scattering are on different measurements shines a the physics, on state of based and solid usually states useful, In defined. most out well is are and matrix in S the where where physics, particle d togfil,adSFqatmelectrodynam- quantum SCF and field, strong ld, o plctost h pcrsoyo atr eneed we matter, of spectroscopy the to applications For electromagnetic the with field photon the describe We from clearly comes QED to approach standard The vddbt nannprubtv a (Dyson way perturbative non a in both ovided ntrso prpit ena diagrams. Feynman appropriate of terms in n c aei icse ndti.Tegeneral The detail. in discussed is case ach ion twy sitga equations integral as ways nt 2 Christian Brouder: Renormalization of QED in an external field ing three different methods (weak field, strong field, self- where consistent field). The boundary conditions, such as the 1 number of electrons in the system, are determined by the D0 (q)= g (1 1/ξ)q q /q2 . µν −ǫ (q2 + iǫ) µν − − µ ν unperturbed Green function which is discussed in detail. 0 The proper definition of the current is established through the use of a bilocal operator. Then we describe the renor- It is a solution of the equation malization of QED in an external field. The non perturba- µ ǫ ✷gµν (1 ξ)∂µ∂ν )D0 (x, y)= δ δ(x y). tive Dyson relations are given to express the relation be- 0 − − νλ λ − tween bare and renormalized Green functions. The renor- In the last equation the D’Alembertian ✷ and the deriva- malization rules for Feynman diagrams are discussed. tives ∂µ∂ν act on variable x. The following assumptions are made in the present pa- The free electron propagator is per: the external field is weak enough not to create charges (this is true for all stable atomic nuclei [7]) and the ex- dq S0(x, y)= e−iq.(x−y)S0(q), ternal field is zero at infinity (we exclude constant electric Z (2π)4 and magnetic field, for which specialized monographs are available [8],[9]). Moreover, we do not consider IR diver- where gences. −1 S0(q) = (~cγ q mc2 + iǫ) . · − 2 Notation It is a solution of the equation (i~cγ ∂ mc2)S0(x, y)= δ(x y). In this section, we specify the notation that is used in · − − the paper. The charge of the electron is e = e . The −| | Analytic expressions for these (and other) propagators pseudo-metric tensor gλµ is are given in [11] and [12] section 2.3. Notice that these Green functions are neither advanced nor retarded. Their 10 0 0 physical meaning is discussed in [13], [14] and [15] p. 57. 0 10 0 g = . 0− 0 1 0 00− 0 1 − 2.2 The QED Lagrangian in SI units We choose the standard gamma matrices ([10] p.693) To compare our results with those of nonrelativistic many- body theory, it will be useful to write the QED Lagrangian I 0 0 σj γ0 = γj = . in SI units [16]. 0 I σj 0 0 α − − We have x = ct, we define the 4-current J = (ρ, J/c), and the 4-potential Aα = (V,cA). The Pauli matrices σj are defined by The field-strength tensor is ([10] p.8)
x y z 1 01 2 0 i 3 1 0 0 E E E σ = σ = − σ = . x − − z − y 10 i 0 0 1 αβ α β β α E 0 cB cB − F = ∂ A ∂ A = y z − x . − E cB 0 cB The charge conjugation and mass reversal matrices are, Ez cBy cBx − 0 respectively − The Maxwell equations are ([10] p.9) 0 iσ2 0 I C = 2 − , and γ5 = . iσ 0 I 0 αβ 1 β − ∂αF = J , ǫ0 They satisfy the following identities αβγδ ∂αǫ Fγδ =0. C = CT = C† = C−1, Cγ C−1 = γT , − − − µ − µ The photon Lagrangian is ([10] p.12) −1 5 µ µ γ5 = γ5 = γ , γ5γ γ5 = γ . ǫ ǫ − = 0 F F αβ = 0 ( E 2 c2 B 2). Lγ − 4 αβ 2 | | − | | 2.1 Free propagators The electron Lagrangian is
0 ~ ¯ 2 ¯ The free photon Green function D (x, y) is defined by its e = i cψγ ∂ψ mc ψψ, µν L · − Fourier transform µ where γ ∂ = γ ∂µ. The interaction Lagrangian is dq · D0 (x, y)= exp[ iq.(x y)]D0 (q), µν 4 µν = J A = eψγ¯ Aψ. Z (2π) − − LI − · − · Christian Brouder: Renormalization of QED in an external field 3
−3 −1 2 2 α 4−d/2 −2 −1 ǫ0 L M T C A L MT C From the definition of σ, A (x) can be written as a func- − − − µ J α L1 dC L L3 dMT 2 tional derivative with respect to jµ(x): − − ψ L(1 d)/2 e Ld/2 2C 3−d/2 −2 −2 −1 −d−2 2 D L MT C S M L T ¯ δσ iσ/~c −3−3d/2 −1 2 2 2−d −2 Z Aµ(x) = (A, ψ, ψ) e Π L M T C Σ ML T h i − Z D δjµ(x) −2−3d/2 δΣ/δa L C ξ 1 δZ ~ 3 −2 (5−d)/2 −2 ~ c ML T η ML T = i c µ . (2) − δj (x) I ML3T 2 Similarly, we shall use
The current is 2 ¯ ~ 2 δ Z Z T ψα(x)ψβ (y) = ( c) β α , Jµ(x)= eψ¯(x)γµψ(x). h i δη (y)δη¯ (x)
The total Lagrangian in SI units, including the gauge term and is 2 2 δ Z ǫ Z T Aλ(x)Aµ(y) = (~c) . = 0 F F αβ + ψ¯(i~cγ ∂ eγ A mc2)ψ h i − δjµ(y)δjλ(x) L − 4 αβ · − · − ǫ ξ 0 (∂ A)2. Finally, the following property will be essential − 2 · ψ(x) = ψ¯(x) =0. h i h i 2.3 Dimensions These equations are derived from the fact that the ground state of the system is an eigenstate of the charge. If Q is The following table gives the SI units of the quantities used the charge operator, then [Q, ψ(x)] = ψ(x) ([10] p.147), in the paper, in a space-time with dimensions d (d = 4). thus −
ψ(x) = G ψ(x) G = G Qψ(x) G + G ψ(x)Q G . 3 Derivation of the functional equations h i h | | i −h | | i h | | i The ground state is an eigenstate of Q with eigenvalue N, The Schwinger equations were presented in Ref.[6], and thus various derivations of them are available [17], [18] (p. 416- G ψ(x) G = N G ψ(x) G + N G ψ(x) G =0. 32) and [10] (p. 475-81). Our derivation follows Ref.[10]. h | | i − h | | i h | | i We define a generating function Z = Z(j, η, η¯), where More physically, no anticommuting operator is measur- the photon source jµ(x) and the anticommuting electron sources η(x) andη ¯(x) enter the total action as [10] able, only products of an even number of anticommuting operators can be observed. It may be stressed that the ¯ µ mean values of ψ(x) and ψ(x) are zero only when the (un- σ = I dx jµ(x)A (x)+ dx η¯(x)ψ(x)+ ψ¯(x)η(x), − Z Z physical) electron sources η(x) andη ¯(x) are set to zero. where I = dx (x). The minus sign before j (x) in the L µ definition ofRσ was chosen so that jµ(x) is a standard elec- 3.1 The photon equation tromagnetic current. It is the opposite of the convention used in Ref.[10]. To derive the Schwinger equations we use the fact that The generating function can be written, up to a nor- the derivative of an integral is zero (assuming that the malization factor, as a path integral ([10] p.476) integrand is zero at infinity):
~ ¯ iσ/~c δeiσ/ c Z(j, η,¯ η)= (A, ψ, ψ)e , (1) (A, ψ, ψ¯) =0. Z D Z D δAµ(x) or as the mean value of an operator ([10] p.210, 261) Thus Z(j, η,¯ η)= T exp i(σ I)/~c . h − i ¯ δI iσ/~c (A, ψ, ψ) µ jµ(x) e =0. (3) It is also possible to write Z(j, η,¯ η) in terms of a Z D δA (x) − Z (j, η,¯ η) without interaction [10] p. 445 and [19] p.246. 0 A direct calculation leads to Anyway, we do not calculate Z, we only need it to calcu- late mean values of Heisenberg operators. For instance δI = ǫ ✷g (1 ξ)∂ ∂ Aν (x) δAµ(x) 0 µν − − µ ν ¯ iσ/~c Z Aµ(x) = (A, ψ, ψ)Aµ(x)e . eψ¯(x)γ ψ(x). (4) h i Z D − µ 4 Christian Brouder: Renormalization of QED in an external field
According to the method of generating functions, a When the external electron sources η andη ¯ are set to factor Aν (x) in the integral can be replaced by a func- zero, we showed that ψ(x)= ψ¯(x) = 0, and we obtain our tional derivative of Z with respect to jν (x) (up to a factor first basic equation i~c). This is what we did for Eq.(2). In the case of an- ✷ ν ticommuting variables such as ψ(x) we must be a little ǫ0 gµν (1 ξ)∂µ∂ν )A (x) − − more careful. Functional derivative with respect to an an- +ie~ctr[γµS(x, x)] = jµ(x). (10) ticommuting source is very similar to that with respect to a function. The difference can be summarized in the It is a bit clumsy to use an electron source η(x) just identity to conclude that no such source exists which leads to the cancelation of ψ(x) and ψ¯(x). The way out of this difficulty δ δA δB AB)= B + ( 1)|A|A , is to use Rochev’s bilocal source η(x, y) [20]. δη δη − δη µ ¯ where A is a product of A anticommuting variables. For σ = I dxjµ(x)A (x)+ dxdyψ(x)η(x, y)ψ(y), | | µ − Z Z instance, Aµ = 0, ψ = 1, η¯ = 1, ψγ¯ ψ = 2, etc. | | | | | | ¯ | | Thus, each factor ψ(x) (resp. ψ(x)) in the path integral where η(x, y) is now a physically reasonable source of is replaced by a functional derivative of Z with respect to electron-positron pairs. Such a source would lead imme- η¯(x) (resp. η(x)). Paying attention to the signs and the diately to Eq.(10). We used the more standard electron factors ~c/i we can rewrite Eq.(3) as sources to follow the textbook derivations [10]. Equation (10) means physically that an induced cur- δI ~cδ ~cδ ~cδ − , , − Z = Zj (x). (5) rent ie~ctr[γµS(x, x)] must be added to the external cur- δAµ(x) iδj iδη¯ iδη µ − rent jµ(x) as a source of electromagnetic potential. Therefore, Eqs.(5) and (4) in Eq.(3) yield ~cδZ 3.2 The electron equation ǫ ✷g (1 ξ)∂ ∂ ) − 0 µν − − µ ν iδj (x) ν The second equation is obtained by varying ψ¯(x). Follow- ~ ′ ~ cδ ss cδZ e − γ ′ = Zj (x). ing [10] p.478 we obtain − iδηs(x) µ iδη¯s (x) µ
The value of the vector potential and the electron wave- δI ~cδ ~cδ ~cδ functions are obtained by [10] ¯ − , , − Z = Zη(x). (11) δψ(x) iδj iδη¯ iδη − ~c 1 δZ A (x)= , (6) The functional derivative of the action yields µ − i Z δjµ(x) ~c 1 δZ δI 2 ψ (x)= , (7) = (i~cγ ∂ mc )ψ(x) eγ A(x)ψ(x). (12) s i Z δη¯s(x) δψ¯(x) · − − · ~c 1 δZ ψ¯ (x)= . (8) Introducing this into Eq.(11), we obtain s − i Z δηs(x) ~ ~ ~ 2 cδZ cδ cδZ To simplify the notation, we have written A (x), ψ (x) Zη(x) = (i~cγ ∂ mc ) eγµ − . µ s − · − iδη¯(x) − iδj (x) iδη¯(x) and ψ¯ (x) for A (x) , ψ (x) and ψ¯ (x) . µ s h µ i h s i h s i Thus, we obtain From Eq.(7) we can write Zǫ ✷g (1 ξ)∂ ∂ )Aν (x) 0 µν − − µ ν ~c δZ ′ = Zψs(x). δ ss s ei~c γ Zψ ′ (x)= Zj (x). i δη¯ (x) − δηs(x) µ s µ Hence We define now the electron Green function by ~ 2 ~ δψ(x) δ2 log Z η(x) = (i cγ ∂ mc )ψ(x) ie cγµ ′ ~ − · − − δjµ(x) Sss (x, y)= i c s′ s δη (y)δη¯ (x) eγ A(x)ψ(x). − · δψs(x) δψ¯s′ (y) = ′ = . (9) −δηs (y) − δη¯s(x) Finally, we take the functional derivative of this equa- tion with respect to η(y), we use Eq.(9), the fact that Thus, the equation becomes ψ(x) = 0 and we obtain our second basic equation
δ(x y) = (i~cγ ∂ mc2 eγ A(x))S(x, y) ν ǫ0 ✷gµν (1 ξ)∂µ∂ν )A (x) − · − − · − − ~ δS(x, y) eψ¯(x )γ ψ(x)+ ie~ctr[γ S(x, x)] = j (x). ie cγµ . (13) − µ µ µ − δjµ(x) Christian Brouder: Renormalization of QED in an external field 5
Again, the same result can be obtained without elec- 4 Three integral equations tron sources η,η ¯ by using the electron-positron source η(x, y) [20]. In this section, we derive various integral equations which The equation for the electromagnetic vector potential correspond to the differential equations of the previous is section. 0 ν Aµ(x)= aµ(x) ie~c dsD (x, s)tr[γ S(s,s)], (14) − Z µν 4.1 Weak external potential where aµ(x) is the external potential created by the ex- ternal current jν (x): When the external potential is weak, we can multiply 0 ν Eq.(15) by the free electron Green function S0(x, y). This aµ(x)= dyDµν (x, y)j (y). Z gives us the following coupled equations
In terms of the external potential aµ(x), the functional equation (13) becomes S(x, y)= S0(x, y)+ e dzS0(x, z)γ A(z)S(z,y) Z · (i~cγ ∂ mc2 eγ A(x))S(x, y) = δ(x y) · − − · − ~ ′ 0 δS(z,y) 0 ′ +ie c dzdz S (x, z)γµ Dνµ(z ,z), ~ µ δS(x, y) 0 Z δaν (z′) +ie cγ ds Dλµ(s, x). (15) Z δaλ(s) ~ 0 ν Aµ(x)= aµ(x) ie c dyDµν (x, y)tr[γ S(y,y)]. On the other hand, we can take the functional deriva- − Z tive of the action with respect to ψ(x) The weak field approach to atomic physics was re- δI = i~c∂µψ¯(x)γ + ψ¯(x)(eγ Aµ(x)+ mc2).(16) viewed recently by Eides and coll. [21]. δψ(x) µ ·
We denote the external current jµ(x) by a star , so We can now repeat the calculation that was done start- the external potential a (x) is denoted by the Feynman ing from Eq.(12). This gives us another equation for the µ
electron Green function diagram . δ(x y)= i~c∂µS(y, x)γ S(y, x)(eγ A(x)+ mc2) − − x µ − · ~ δS(y, x) ie c γµ. (17) 4.1.1 Feynman diagrams − δjµ(x)
The coupled equations generate the following series for the 3.3 Photon Green function electron Green function. The propagator is oriented from right to left and the electron loops are oriented anticlock- For spectroscopic applications, it is useful to know the wise. photon Green function δA (x) D (x, y)= µ . µν δjν (y) 0 0 e = S → In physical terms, the photon Green function gives the lin- 1
ear response of the electromagnetic potential to the vari- e ation δjν (y) of the external source by →
ν 2
δAµ(x)= dyDµν (x, y)δj (y). e +
Z →
To obtain an equation for Dµν (x, y), we solve Eq.(10) for Aµ(x).
e3 + → 0 λ ~ λ Aµ(x)= dzDµλ(x, z) j (z) ie ctr[γ S(z,z)] . Z −
A functional derivative with respect to jν (y) gives us the + + equation for Dµν (x, y)
0 0 λ δS(z,z) Dµν (x, y)= D (x, y) ie~c dzD (x, z)tr[γ ]. µν − Z µλ δjν (y) + (18) 6 Christian Brouder: Renormalization of QED in an external field
4.2 Strong external potential
When the external potential is strong, we multiply Eq.(15) 1 by the Green function in the presence of aµ(x): e
→ −1 Sa = i~cγ ∂ mc2 eγ a(x) . · − − ·
e3 + Now the coupled equations become →
a a ′ S(x, y)= S (x, y)+ e dzS (x, z)γ A (z)S(z,y) If s (resp. a ) denotes the number of Feynman dia- Z · n n grams for S(x, y) (resp Aµ(x)) at order en, the coupled ′ a δS(z,y) 0 ′ +ie~c dzdz S (x, z)γµ D (z ,z), equations yield Z δaν (z′) νµ s =1, a =0, ′ ~ 0 ν 0 0 Aµ(x)= ie c dsDµν (x, y)tr[γ S(y,y)]. − Z an = sn−1, n−1 ′ In this case, the total potential is Aµ(x)= aµ(x)+ Aµ(x). s = s a + (n 1)s . n i n−i−1 − n−2 Strong field QED was reviewed recently in Ref. [22]. Xi=0 Strong field QED can easily accomodate bound states, but the nuclear potential is not screened. Thus, s2n = 1, 2, 10, 74, 706, 8162, 110410, ... and s2n+1 = 0. If s(x) is the generating function for the sequence sn, then
4.2.1 Feynman diagrams ∞ f(x) s(x)= s xn = , n 1+ x2f(x) For the electron Green functions, the propagator is ori- nX=0 ented from right to left and the electron loops are ori- ented anticlockwise, except for the tadpoles, where a loop where diagram is half the sum of a clockwise loop and an anti- ∞ clockwise loop. f(x)= (2n + 1)!! x2n. nX=0
0 a For similar results, see [10] p.467.
e = S →
e2 + 4.3 SCF external potential
→ Æ Finally, the most accurate results are obtained when the external field is taken as the complete vector potential
e4 + → Aµ(x), which is determined by a self-consistent field pro- cedure. Therefore, the potential that will be used in the initial Green function is Aµ(x) instead of aµ(x). This cal-
+ + culation can be found in Refs.[18] and [10] p.479. To eliminate the external current jν (y) from Eq.(13) we need the functional relation
+ + δS(x, z) δS(x, z) δAµ(s) ν = ds ν δj (y) Z δAµ(s) δj (y) δS(x, z) = ds Dµν (s,y).
+ + Z δAµ(s)
With this relation, the functional equation for the pho- ton Green function Eq.(18) becomes
+ + 0 0 Dµν (x, y)= D (x, y) ie~c dzdsD (x, z) µν − Z µλ µ For the potential A (x) we have the following Feynman λ δS(z,z) tr[γ ]Dλ′ν (s,y), (19) diagrams × δAλ′ (s) Christian Brouder: Renormalization of QED in an external field 7 and the functional equation for the electron Green func- For the potential, we have the following diagrams tion Eq.(13) becomes
(i~cγ ∂ mc2 eγ A(x))S(x, y) = δ(x y) 1 · − − · − e
→ µ δS(x, y) +ie~cγ ds Dλµ(s, x). (20) Z δAλ(s) e3
Then, we multiply Eq.(20) by the Green function in → the presence of Aµ(x):
−1 Each strong field diagram is the sum of an infinite SA = i~cγ ∂ mc2 eγ A(x) , · − − · number of weak field diagrams, and each SCF diagram is the sum of an infinite number of strong field diagrams. and the equation for the electron Green function becomes
S(x, y)= SA(x, y) 5 The initial Green function ′ A δS(z,y) νµ ′ +ie~c dzdz S (x, z)γµ D (z ,z). (21) Z δAν (z′) Once the functional equation is know, it remains to specify This equation, together with Eq.(19) and Eq.(14) are a the boundary conditions. For example, we can specify the complete system of equations for the determination of the Green function without interaction. In the case of a static bare Green functions and the potential. external field aµ(r) or a static self-consistent field Aµ(r), As compared to the case of a strong external poten- this Green function is well-known and can be written tial, the SCF potential has the advantage of taking into 0 0 0 account the electrons in the system. Thus, the nuclear po- G (x, y)= iθ(y x ) ψn(x)ψ¯n(y) N − tentials are screened by the electrons in a self-consistent EnX≤EF way. All the tadpoles of strong field QED have been re- iθ(x0 y0) ψ (x)ψ¯ (y), (22) summed. − − n n EnX>EF This formulation of QED is closer to the standard methods of solid state physics or quantum chemistry. It where E is the Fermi energy, and where ψ (x) is the was used in nuclear physics under the name Hartree QED F n solution of the Dirac equation at energy En. Note that when the current is calculated by a single electron loop 0 ~ a 0 ~ A GN (x, y) = cSN (x, y) or GN (x, y) = cSN (x, y) where [23], see also [24]. a A SN (x, y) and SN (x, y) are the Feynman electron Green functions in the external potential aµ(x) or Aµ(x) (with N occupied bound states). In the present section, we want 4.3.1 Electron Green function for self-consistent field to discuss and justify this expression. The notation follows standard textbooks on the Dirac 0 equation [10],[27]. The index N means that GN (x, y) is For the electron Green functions, all electron loops are the Green function for a total charge of Ne. The Green oriented anticlockwise and the propagator is oriented from function G0(x, y) represents the vacuum in the presence right to left. 0 of aµ(x) (or Aµ(x)). This vacuum Green function is given 0 by Eq.(22) for EF = 0. The difference between GN (x, y) 0 and G0(x, y) is 0 A
e = S → G0 (x, y) G0(x, y)= iθ(y0 x0) ψ (x)ψ¯ (y) N − 0 − n n 0 e + ¯ → = i ψn(x)ψn(y). 0 + + The Fermi energy EF is determined by the condition that the sum is over N states: The number of Feynman diagrams at each order was 1= N. calculated in Ref.[25] (see also [26]). 0 5.1 Early works is less singular. The physical idea developed by Dirac is to measure the charge density with respect to charge den- To understand the origin of the Green function given by sity of the vacuum ρ0(x). The induced charge density ρ(x) = 0 in the presence of an external potential was in- Eq.(22), it is useful to describe how it appeared histori- 6 cally. vestigated in the thirties [28], [29], [30], [31], [32]. The po- In the vacuum, Dirac [28],[29] assumed that all the tential corresponding to this induced charge is called the Uehling potential [33]. Physically, ρ(x) = 0 is a reaction of negative-energy states are filled. This gives the density 6 matrix the vacuum which is polarized by the external potential. The induced charge has observable consequences. For in- 0 0 † stance, its effect on the energy levels of hydrogen atoms is ρ(x, y)= ψn(x)ψn (y), EXn<0 well documented [22]. Computer programs are available to evaluate the Uehling potential for general nuclear charge 0 models [34]. Expansions in (Zα)n where investigated in where ψn(x) is the free solution of the Dirac equation for energy En. However, such a density matrix would be [35]. highly impractical, since it would lead to an infinite charge density. In a strike of genius, Dirac made the following observation [28],[29]. Since 5.2 Checking the Green function † ψ0 (x)ψ0 (y)= δ(x y), In this section, we check that Eq.(22) gives the charge n n − XEn density proposed by Dirac. In Eq.(22) we write 0 the “renormalized” density matrix ψn(x) = exp( iEnx /c~)ψn(x), − 1 and we define the frequency dependent Green function ρ0(x, y) = ρ(x, y) δ(x y) − 2 − ∞ 1 0 0 † 1 0 0 † G0 (x, y; ω)= dt eiωtG0 (x, y), = ψn(x)ψn (y) ψn(x)ψn (y) N N 2 − 2 Z−∞ EXn<0 EXn>0 0 0 1 where, on the right hand side, x = y + ct. Since the is much better behaved. In particular, the charge density external potential is time independent, the Green function is 0 depends only on t. To determine GN (x, y; ω) we use the distribution identity (Ref.[10], p.92) ρ0(x) = tr[ρ0(x, x)] 1 1 ∞ 0 0 = ψ0 (x) 2 ψ0 (x) 2 =0, 0 0 dω exp [ iω(x y )] 2 | n | − 2 | n | θ(x y )= i − − EXn<0 EXn>0 − Z−∞ 2π ω + iǫ where the last equality is due to charge conjugation sym- and we obtain metry. ψn(x)ψ¯n(y) In the presence of a static potential aµ(x) or Aµ(x), 0 GN (x, y; ω)= and without bound states, the analogous density matrix ω En/~ iǫ EnX≤EF − − is ψ (x)ψ¯ (y) + n n . 1 0 † 1 0 † ω En/~ + iǫ ρ(x, y)= ψ (x)ψ (y) ψ (x)ψ (y), EnX>EF 2 n n − 2 n n − EXn<0 EXn>0 This is the standard expression for the non-interacting where ψn(x) is now the solution of the Dirac equation with electron Green function in the relativistic case (see Ref.[22] a potential aµ(x) or Aµ(x) and energy En. Of course, the p. 32 for EF = 0) and as well as in the nonrelativistic case charge conjugation symmetry is broken by the external (up to a factor γ0, see Ref. [36], p.124). 0 0 potential, but we still have that The induced current is ietr[γ GN (x, x)]. But, this formal expression is infinite and− must be regularized. The ρ(x) =tr ρ(x, x) = tr ρ(x, x) ρ (x) limit involved in the expression G0 (x, x) is discussed in − 0 N 1 detail in section 6: = ( ψ (x) 2 ψ0 (x) 2) 2 | n | − | n | EXn<0 0 1 0 GN (x, x)= lim lim GN (x + z, x z) 2 z→0 z0→0+ − 1 2 0 2 ( ψn(x) ψn(x) ) 0 −2 | | − | | +GN (x z, x + z) . EXn>0 − 1 In fact, Dirac defined ρ0(x, y) as the double of the right- This definition agrees with experiments in nuclear physics hand side. This was immediately corrected by Heisenberg [30]. [7] and atomic physics (Ref.[22] p. 275). Christian Brouder: Renormalization of QED in an external field 9 With this definition, we obtain phenomena such as chiral anomalies [48] and fractional fermion numbers [49]. 0 i The case when the Fermi energy is degenerate will not G (x, x)= ψn(x)ψ¯n(x) ψn(x)ψ¯n(x) . N 2 − be considered in this paper. EnX≤EF EnX>EF In particular, we obtain the charge density: 6 The induced current 0 0 0 ρ(x)= J (x)= ietr[γ GN (x, x)] e − = ψ (x) 2 ψ (x) 2 . It is important to derive the proper expression of the four- 2 | n | − | n | current as a function of the electron Green function. As a EnX≤EF EnX>EF field operator, the current must be self-adjoint (it is mea- This is indeed the density obtained by Dirac. One surable), conserved, gauge invariant and should change might wonder how this charge density is related to the sign under charge conjugation. many-body charge density (or the non-relativistic one), which is obtained by summing over the occupied states with positive energy. The answer is that we must subtract 6.1 Definition 0 the contribution of the Green function G0(x, x) (for which E = 0). Then we obtain We follow Refs.[15], [50], [51] and define the bilocal field F operator 0 0 0 2 ietr[γ GN (x, x) G0(x, x) ]= e ψn(x) . µ e µ ¯ ¯ − − | | J (x, y)= γαβ ψα(x)ψβ(y) ψβ(x)ψα(y) . 0 From definition (23), for z0 > 0 we have Moreover, if x and y are space-like separated, we can rewrite the bilocal current in terms of field commutators: i µ µ Sαβ(x + z, x z)= ψ (x + z)ψ¯ (x z) , (26) J (x, y)+ J (y, x) − −~ch α β − i Jµ(x, y)= i 2 ¯ e Sαβ(x z, x + z)= ψβ(x + z)ψα(x z) . (27) µ ¯ ¯ ~c = γ [ψα(x), ψβ(y)] + [ψα(y), ψβ(x)] . − h − i 4 αβ Xαβ When we compare this with the definition of the bilocal (29) current field operator, we obtain µ µ µ The fact that the current J (x) defined by Eq.(24) J (x) = lim lim J (x + z, x z) . is real can also be obtained directly from a property of z→0 z0→0+h − i S(x, y). If z is a space-like vector, causality ensures that On the other hand, if z0 < 0 † † † ψα(x + z)ψβ(x z) = ψβ(x z)ψα(x + z) µ Jµ − − J (x) = lim lim− (x z, x + z) . † z→0 z0→0 h − i = ψα(x + z)ψβ(x z). − − Thus, when the definition of J µ(x) is written in terms of Now Eq.(26) gives the bilocal operator, the first argument of the bilocal oper- † ~ 0 ator is always later than x. It is not clear that the current ψα(x + z)ψ (x z) = i c Sαλ(x + z, x z)γ , h β − i − λβ defined by Eq.(24) is real, as a measurable quantity should Xλ be. and Eq.(27) gives Now we prove that the current is real and transforms properly under charge conjugation. ψ† (x + z)ψ (x z) = i~c S (x + z, x z)γ0 . h α β − i − βµ − µα Xµ 6.2 Self-adjointness Thus, we obtain, for space-like z † 0 0 † S (x + z, x z)= γ S(x z, x + z)γ . (30) Using the definition ψ¯ = ψ γ0, the bilocal operator can − − − be rewritten Here the dagger operator acts only on the spin-variables: † ∗ µ e † 0 µ 0 µ T † S αβ(x+z, x z)= Sβα(x+z, x z). From Eq.(30) and the J (x, y)= ψ(x) γ γ ψ(y) ψ(x)(γ γ ) ψ(y) , − − ∗ 2 − identity γ0γµγ0 = 㵆 we can check that J µ(x) = J µ(x). where AT is the transpose of A. From the identity (γ0γµ)† = γ0γµ (Ref.[10] p.693) we obtain immediately 6.3 Charge conjugation Jµ(x, y)† = Jµ(y, x). (28) Let be the charge conjugation operator acting on fields and CC = iγ2γ0 the charge conjugation matrix. We have 2 Now we can make an important remark concerning the the identities ([10] p.152 and [53] p.70) bilocal current. If x and y are separated by a space-like † ¯ interval (i.e. if (x y) (x y) < 0), then ψ¯ (x) and ψ(y) ψ(x; e) = Cψ(x; e), − · − C C − anticommute. This property is called causality and means ψ¯ (x; e) † = ψ(x; e)C. that two events that are too far to be linked by a light- C C − ray are independent. Causality is a basis of all axiomatic In the presence of an external field, the sign of the charge approaches to quantum field theory [52]. in the field operators is reversed under charge conjugation Thus, if (x y) (x y) < 0 we have ([15] p.19). − · − The action of the charge operator on the bilocal cur- µ e µ ¯ ¯ rent is J (x, y)= γ ψβ(y)ψα(x)+ ψα(y)ψβ(x) 2 αβ − e Xαβ Jµ(x, y; e) † = ψ(x; e)CγµCψ¯ (y; e) µ C C 2 − − = J (y, x). ψ¯ (x; e)(CγµC)T ψ(y; e) . − − − Combined with Eq.(28), this means that the bilocal cur- T rent is self-adjoint for space-like separated points. From the identities ([10] p.693) CγµC† = γµ and CT = † −1 − The symmetric definition (25) of the current in terms C = C = C we obtain 0 + − of the Green function was chosen such that first z 0 µ † µ 0 → J (x, y; e) = J (x, y; e). with z = 0. This is to ensure that, for z small enough C C − − (i.e. z|0 |< 6 z ), the interval z becomes space-like and the 2 Their misprinted definition of C is corrected on page 693 current is real.| | of [10]. Christian Brouder: Renormalization of QED in an external field 11 Therefore, the sign of the bilocal current is reversed under Why symmetric? If a bilocal operator F (x, y) is antisym- charge conjugation. metric (i.e. F (x, y)= F (y, x)), then its value at x = y is Now we investigate the behaviour of the current un- zero, except at divergences,− such as (x y)/(x y)2. Since der charge conjugation, when it is defined in terms of the the regular part of F (x, y) is zero on− the diagonal,− the Green function. To do this, we need an equation for the use of a symmetric operator eliminates the antisymmet- Green function of the charge conjugated problem. We ob- ric divergences without changing the value of the regular tain it by transposing both sides of Eq.(17). This trans- operator on the diagonal. T ′ x y µ position is meant for the 4x4 matrices only: S ss (x, y)= An obvious candidate is (∂µ +∂µ)J (x, y)/2. From the Ss′s(x, y). Then we multiply both sides of the equation by field equations [7] p.195 the charge conjugation matrix C on the left and C−1 on the right. We use the property CγµT C−1 = γµ and we (i~c∂ eA )γµψ mc2ψ =0, − µ − µ − obtain µ 2 (i~c∂µ + eAµ)ψ¯ γ + mc ψ¯ =0, δ(x y) = (i~cγ ∂ mc2 + eγ A(x))Sc(x, y) − · − · we check that (∂x + ∂y)Jµ(x, y) = 0. δSc(x, y) µ µ +ie~cγµ , In terms of the electron Green function, this becomes δjµ(x) ~ ie c x µ µ c T −1 ∂ tr[γ S(x + z, x z)+ γ S(x + z, x z)], where we have defined S (x, y) = CS(y, x) C . If we − 4 µ − − compare this equation with the second basic equation (13), x+z x−z x we see that they become identical if we make the trans- where we have used ∂µ + ∂µ = ∂µ. and where z is formation e e, or equivalently Aµ Aµ and jµ space-like. Charge conservation can then be directly veri- jµ. But, using→ − Eq.(10) and the charge→− conjugation prop-→ fied from Eqs.(13) and (17). erty− of the induced current J µ(x), we see that jµ jµ implies Aµ Aµ. In other words, we have the symmetry→− →− 6.5 Mass-reversal symmetry Sc(x, y)= S(x, y; j)= CS(y, x; j)T C−1. − Therefore The concept of mass-reversal symmetry was discussed by Peaslee [55] and Tiomno [56]. For massless particles, mass- tr[γµS(x z, x + z; j)] = tr[CγµC−1S(x + z, x z; j)T ] reversal symmetry is known as chiral symmetry. − − − µT T The mass-reversed fermion field ψ( m) satisfies the = tr[γ S(x + z, x z; j) ] same equations as the usual fermion field−ψ(m), but with − µ − − = tr[γ S(x + z, x z; j)]. a reversed mass: − − − And the current can be rewritten as the antisymmetric (i~cγ ∂ eγ A + mc2)ψ( m)=0. limit · − · − Since γµγ = γ γµ we can take ψ(x; m) = γ5ψ(x; m) 1 5 − 5 − S(x, x; j)= lim lim S(x + z, x z; j) (up to a phase factor ηm which disappears in the Green 2 z→0 z0→0+ − function). To see how the Green function transforms under S(x + z, x z; j) . − − − mass reversal, we multiply both sides of Eq.(13) by γ5 on the left and on the right. The commutation rules between This equation shows that the current is an odd function γ and γµ yield of the external current: J (x; j) = J (x; j). Changing 5 µ − µ − the sign of j amounts to changing the sign of A, and we 2 δ(x y) = ( i~cγ ∂ mc + eγ A(x))γ5S(x, y; m)γ5 can also write Jµ(x; A) = Jµ(x; A): the current is an − − · − · − − δS(x, y; m) odd functions of A. Now, since the electron loops of QED +ie~cγµγ5 γ5, (31) are generated by the current, and since the x in S(x, x; A) δjµ(x) is linked to another photon propagator, we see that the electron loops with an odd number of photon lines are which is the equation for S(x, y; m). Therefore, − − zero. This is a version of Furry’s theorem for QED with S(x, y; m)= γ S(x, y; m)γ , external field. The relevance of charge conjugation for the − − 5 5 QED current was already noticed by Kramers in 1937 [54]. Related results can be found in Ref.[7], sections 4.2, 9.4 and the mass-reversed current is and 15.1. J µ(x; m)= ie~ctr[γµS(x, x; m)] − − µ − µ = ie~ctr[γ5γ γ5S(x, x; m)] = J (x; m). 6.4 Charge conservation Thus, the induced current is even under mass-conjugation, µ Charge conservation is described by the equation ∂µJ (x)= and the vector potential also, as expected [56]. 0. How does this translate for our bilocal operator? We In other words, the induced current is an even function µ want a real and symmetric expression for ∂µJ (x) = 0. of m. 12 Christian Brouder: Renormalization of QED in an external field 6.6 Gauge invariance attempts exist [1], [59], [39], [51], but none of them was fully completed. We start now a renormalization of QED The Lagrangian of QED is invariant under the gauge trans- in an external field which closely follows the renormal- formations ([10] p.64) ization of vacuum QED. The first step is to determine Dyson’s relations, which are non perturbative expressions Λ −ieΛ(x) ψ (x)=e ψ(x), that link renormalized and non renormalized propagators Λ ~ Aµ (x)= Aµ(x)+ c∂µΛ(x). and fields. Then we give a non perturbative relation be- tween the renormalized potential and the renormalized After a gauge transformation, the bilocal current for space- vacuum polarization. Finally, we describe the renormal- like separated points Eq.(29) becomes ization rules that enable us to obtain finite quantities for any strong field or SCF Feynman diagram. JΛ(x, y) = cos(eΛ(x) eΛ(y))J(x, y) − ie + sin(eΛ(x) eΛ(y)) 4 − 7.1 Dyson relations for strong field QED γµ [ψ¯ (x), ψ (y)] [ψ¯ (y), ψ (x)] .(32) × αβ α β − α β The first point is to determine the Dyson relations between Xαβ renormalized and unrenormalized propagators. This rela- In terms of the electron Green function, we must consider tion is given by Sterman[60] for scalar fields and we derive the transformed Green functions it for QED. If we denote the renormalized quantities with a overlined symbol, in the presence of an external poten- Λ −ieΛ(x)+ieΛ(y) S (x, y)=e S(x, y). (33) tial aµ(x), Dyson’s relations are [61] Hence, S(x, y; a; e0,m0)= Z2S¯(x, y; Z3a; e,m), Λ Λ S (x + z, x z)+ S (x z, x + z)= D(x, y; a; e ,m )= Z D¯(x, y;p Z a; e,m), − − 0 0 3 3 cos(eΛ(x) eΛ(y)) S(x + z, x z)+ S(x z, x + z) ¯ p − − − A(x; a; e0,m0)= Z3A(x; Z3a; e,m). i sin(eΛ(x) eΛ(y )) S(x + z, x z) S(x z, x +z) . p p − − − − − To derive these relations, we expand S over a: A priori, the singularities of the Green function are not weak enough to make the second term tend to zero when δS(x, y; 0) S(x, y; a)= S(x, y;0)+ dz1 aµ(z1)+ z 0. Therefore, our definition of the current is not ob- Z δaµ(z1) ··· viously→ gauge invariant. This problem can be solved by using a Green function The Dyson relation for the vacuum propagator S(x, y; 0) multiplied by Dirac’s phase [29] is S(x, y;0) = Z2S¯(x, y; 0). For the other terms, the func- tional derivative with respect to the external potential 1 ′ ie µ brings a factor A(x) in the path integral. Therefore, S (x + z, x z) = exp ~ dλz Aµ(x + λz) − c Z−1 δnS(x, y; 0) δnS¯(x, y; 0) S(x + z, x z). = Z Zn/2 . × − δan 2 3 δan A generalization of this phase was used recently for the renormalization of infrared divergences in QED [57], [58]. Hence, The modified Green function S′(x + z, x z) is gauge − δS¯(x, y; 0) invariant because, according to Eq.(53) in the appendix, S(x, y; a)= Z2S¯(x, y;0)+ Z2 dz1 Z3aµ(z1) Z δaµ(z1) 1 p µ + dλ∂µΛ(x + λz)z = Λ(x + z) Λ(x z). ··· Z−1 − − = Z2S¯(x, y; Z3a). p The right hand side compensates for the gauge transfor- Or, more precisely, mation phase in Eq.(33), so that S′Λ(x, y)= S′(x, y). If we substitute S′(x+z, x z) for S(x+z, x z) in the S(x, y; a; e ,m )= Z S¯(x, y; Z a; e,m). − − 0 0 2 3 definition of the current (24), we obtain a current which p is real, odd under charge conjugation and gauge invariant, Because of this relation, we also define the renormal- but charge conservation becomes doubtful. Thus, we stick ized external potential asa ¯µ(x)= √Z3aµ(x). It may seem to the original definition Eq.(24). strange that the external potential aµ(x) and the full po- tential Aµ(x) are not renormalized with the same formula. However, we must recall that aµ(x) was defined in terms 7 Renormalization of an external current jµ(x), which was used to generate Green functions by functional derivatives. The bare Green µ The renormalization of QED in the presence of an external functions are generated by A (x)jµ(x) and the renormal- ¯µ field does not seem to have been treated in detail. Several ized ones by A (x)¯µ(x). Since we want these two terms to Christian Brouder: Renormalization of QED in an external field 13 ¯ ~ ν ¯ be equal, the relation √Z3Aµ(x)= Aµ(x) implies ¯µ(x)= Now ie ctr[γ S(s,s; 0)] is the current in the presence − ¯ √Z3jµ(x) (see Refs.[60] p. 297 and [19] p.288). More phys- of a zero external field A = 0. Thus, it is the vacuum cur- ically, we can say that an external field is made by prepar- rent, which is zero (more precisely, tr[γνS¯(s,s; 0)] is made ing a certain density of (charged) matter n(x). Before of fermion loops with one external photon line, which are renormalization, this density corresponds to a charge den- zero by Furry’s theorem). 0 sity j (x) = e0n(x). The renormalization modifies the The second term is also well known from the renor- charge, but not the density of matter. Thus the renor- malization of the vacuum polarization (see e.g. Eq.(22) of 0 0 malized charge density is now ¯ (x)= en(x)= √Z3j (x), Ref.[62]) and the corresponding relation for the external potential ¯ ¯ µ µ δS(s,s; λA) is againa ¯ (x)= √Z3a (x). ie~cZ tr γ = 2 ν ¯µ In terms of the renormalized propagators, the func- − h δA (y) i −1 tional equation becomes (using e0 = e/√Z3 and m0 = ¯ ¯ 0 Πνµ(s,y; λA) + (Z3 1)D µν (s,y). (39) m + δm) − Thus, we obtain (i~cγ ∂ mc2 δmc2 eγ A¯(x))Z S¯(x, y)= δ(x y) · − − − · 2 − ¯ ~ ¯ ¯ ~ µ δS(x, y) 0 ie cZ2 dytr[γν S(s,s; A)] = +ie cZ2γ ds D (s, x). − Z δa¯ (s) λµ Z λ 1 ¯ ¯ ¯µ and dλ dyΠνµ(s,y; λA)A (y) Z0 Z e 0−1 ¯µ Z A¯ (x)=¯a (x)/ Z i ~cZ +(Z3 1) dyD µν (s,y)A (y). 3 µ µ 3 √ 2 − Z p p − Z3 After this reorganization of the induced current, Eq.(38) dsD0 (x, y)tr[γν S¯(y,y)], × Z µν becomes or Z3 1 Aµ(x)= aµ(x)+ − A¯µ(x) √Z3 a¯µ(x) eZ2 0 ν A¯µ(x)= i ~c dsD (x, y)tr[γ S¯(y,y)], Z − Z µν 1 0 ¯ νρ ¯ ¯ 3 3 Z + dsdydλDµν (x, s)Π (s,y; λA)Aρ(y). (34) √Z3 Z Now we know that the relation between the bare and renormalized potentials is A (x)= √Z A¯ (x). This yields 7.2 Dyson relations for SCF-QED µ 3 µ 1 ¯ 0 Notice that, in the case of SCF-QED, Dyson’s relations Aµ(x)=¯aµ(x)+ dλ dsdyDµν (x, s) Z0 Z are much simpler, because e0A = eA¯: νρ Π¯ (s,y; λA¯)A¯ρ(y), (40) S(x, y; e0A; e0,m0)= Z2S¯(x, y; eA¯; e,m), (35) × where the true external potential aµ(x) is renormalized by D(x, y; e0A; e0,m0)= Z3D¯(x, y; eA¯; e,m), (36) a¯ (x)= √Z a (x). ¯ ¯ µ 3 µ A(x; e0A; e0,m0)= Z3A(x; eA; e,m). (37) It should be noticed that, in all our manipulations, p the renormalization factors Z2, Z3 and δm where taken The main difference between vacuum QED and SCF- ¯ QED is the renormalization of the induced vector poten- at Aµ = 0. In other words, the renormalization factors of tial. The total bare potential is QED with external field are the same as the renormaliza- tion factors of vacuum QED. More precisely, the vacuum 0 ν renormalization factors removes the divergences of QED Aµ(x)= aµ(x) ie0~c dsD (x, s)tr[γ S(s,s)],(38) − Z µν in external field. However, renormalization conditions may ν introduce finite differences between the renormalization and the problem is the renormalization of tr[γ S(s,s)]. To factors of vacuum QED and QED in an external field. indicate that the electron Green function is calculated in From Eq. (40), it can be checked that Π¯ is indeed the the presence of the full potential Aµ(x) we denote it by renormalized vacuum polarization. If we write Xµ(x) for S(x, y; A). the second term on the right hand side of Eq. (40) we find Dyson’s relation (35) gives us ¯ ν ν δAµ(x) δa¯µ(x) δXµ(x) tr[γ S(s,s; A)] = Z2tr[γ S¯(s,s; A¯)]. D¯ (x, y)= = + µν δ¯ν (y) δ¯ν (y) δ¯ν (y) We rewrite the right hand side using Eq.(51): δX (x) δA¯ (x) = D0 (x, y)+ dz µ ρ 1 µν ν Z δA¯ρ(z) δ¯ (y) tr[γν S¯(s,s; A¯)] = tr[γν S¯(s,s;0)]+ dλ dx Z0 Z = D0 (x, y)+ ds dzD0 (x, s) ¯ ¯ µν Z Z µσ ν δS(s,s; λA) µ tr γ A¯ (x). σρ ¯µ Π¯ (s,z)D¯ (z,y), (41) × h δA (x) i × ρν 14 Christian Brouder: Renormalization of QED in an external field where we used subdiagram of Γ , then in the term Γ/γ(0), the 1PI diagram γ is replaced by a point and in the term Γ/γ the 1PI δX (x) (2) µ 0 ¯ σρ diagram γ is replaced by a free electron line. ¯ = dsDµσ(x, s)Π (s,z). δAρ(z) Z If γ is a vacuum polarization 1PI diagram, then i is 3. If the complete diagram is Γ and γ is a vacuum polarization This last equation is proved by using Eq.(39) to rewrite 1PI subdiagram of Γ , then in the term Γ/γ , the 1PI X (x) in terms of a functional derivative of S¯, and by (3) µ diagram γ is replaced by a free photon propagator. using Eq.(54) to carry out the integral over λ. If γ is a reduced vertex 1PI diagram, then i is 1. If Equation (41) expresses the usual relation between the the complete diagram is Γ and γ is a reduced vertex photon Green function and the vacuum polarization. 1PI diagram, then in the term Γ/γ(1), the vertex dia- ′ gram is replaced by a point. The terms Γ/ γ ,γ ′ ,... { (i) (i ) } are then defined recursively. For instance, to define Γ ′′ = 8 Weak field renormalization ′ ′ ′′ Γ/ γ(i),γ(i′) , we first put Γ = Γ/γ(i), so that Γ = ′ { ′ } In this section, we give the renormalization rules of vac- Γ /γ(i′). uum QED, which are valid for weak field renormalization. We insist here on the fact that the removal of subdi- We repeat some of the rules given in [62] for complete- vergences given by Eq.(42) is valid for any (connected or ness, and because we work now in the direct space (and disconnected) Feynman diagram. It is the expression, for not in the Fourier space). The first set of rules is equivalent each diagram, of Dyson’s relations. The reason why the to Zimmermann’s forest formula for the removal of subdi- counterterms C1(γ) are required, although Z2 = Z1, is vergences. The second set of rules subtracts the superficial explained in [62]. divergences. In Eq.(42) all counterterms Ci(γ) are assumed to be known from the renormalization of the superficial diver- gence of the subdiagram γ. 8.1 Removal of subdivergences If Γ is a Feynman diagram, and U(Γ ) is the regularized 8.2 Superficial divergences value of the diagram as a function of external momenta and masses and regularization parameters, then the value of the diagram with subdivergences subtracted is R¯(Γ ) In the second step, we determine the counterterms Ci(Γ ) defined by [10], [63] of the divergent graph Γ . If the diagram Γ is not 1PI, the counterterms Ci(Γ ) are zero. If Γ is 1PI, we must ′ distinguish three cases. A self-energy diagram is linearly R¯(Γ )= U(Γ )+ Ci(γ)Ci′ (γ ) ··· divergent, thus we must remove two terms. From Lorentz {γi,γX′i′,... } ′ covariance, we can write the renormalized value of the U(Γ/ γ ,γ ′ ,... ). (42) × { (i) (i ) } diagram Γ as In this expression, the sum runs over all sets of disjoint 2 R(Γ ; x, y)= R¯(Γ ; x, y)+ C0(Γ )c δ(x y) renormalization parts of Γ . A renormalization part γ of Γ − 0−1 is a one-particle irreducible (1PI) subgraph of Γ , different +C2(Γ )S (x, y). (43) from Γ itself, such that γ has two or three (amputated) external lines. A diagram is 1PI when it is connected and A vacuum polarization diagram is quadratically diver- cannot be disconnected by cutting through any of its inter- gent, thus we should have to remove three terms. However, nal lines. Two renormalization parts γ and γ′ are disjoint Lorentz covariance and the Ward identities for the pho- if they have no vertex in common. ton propagators cancel the first two counterterms, and the There are three types of renormalization part: self- renormalized value of Γ is energy 1PI diagrams (i.e. 1PI diagrams with two am- ¯ 0−1 puted external electron lines), vacuum polarization 1PI Rµν (Γ ; x, y)= Rµν (Γ ; x, y)+ C3(Γ )D µν (x, y). (44) diagrams (i.e. 1PI diagrams with two amputed external photon lines) and reduced vertex 1PI diagrams (i.e. 1PI Finally, a reduced vertex diagram is logarithmically diver- diagrams with two amputed external electron lines and gent, and its renormalized value is one amputed external photon line). The index (i) depends ¯ on the type of the renormalization part. If γ is a self energy R(Γ ; x, y; λ, z)= R(Γ ; x, y; λ, z) λ 1PI diagram then i is 0 or 2, if γ is a vacuum polariza- +C1(Γ )γ δ(x y)δ(y z). (45) tion 1PI diagram then i is 3, if γ is a reduced vertex 1PI − − diagram then i is 1. The infinite constants Ci(Γ ) are determined from the renor- ′ Finally let us define Γ/ γ(i),γ ′ ,... . We start by the malization conditions. { (i ) } definition of Γ/γ(i). It varies with the type of renormal- It was noticed in Ref.[62] that ization part and with the index (i). If γ is a self-energy 1PI diagram, then i can be 0 or C (Γ )= C (Γ ), (46) 2 − 1 j 2. If the complete diagram is Γ and γ a self-energy 1PI Xj Christian Brouder: Renormalization of QED in an external field 15 where Γj is the vertex diagram obtained by branching a Using Eq.(45), the counterterm for the second diagram is photon line on the j-th free electron propagator of the self-energy diagram Γ . ′ ′ dzR(Γ ; x, y; λ, z)¯aλ(z)= dzR¯(Γ ; x, y; λ, z)¯aλ(z) Since the renormalization of weak field QED is exactly Z 1 Z 1 the same as the renormalization of vacuum QED, we refer ′ λ +C1(Γ1)γ δ(x y)¯aλ(y). the reader to Refs.[10] and [62] for examples. − The total counterterms for the strong field diagram Γ are ′ ′ obtained by adding the counterterms for Γ and Γ1. From ′ ′ 9 Strong field renormalization the relation (46) between C1(Γ1) and C2(Γ ) we obtain for the strong field self-energy diagrams For the strong field renormalization, the removal of sub- 2 divergences is again given by Eq.(42), which is a purely R(Γ ; x, y)= R¯(Γ ; x, y)+ C0(Γ )c δ(x y) − algebraic identity related to the Hopf algebra structure of −1 +C (Γ )(S0 (x, y) eγ a¯(y)δ(x y)) renormalization [64], [65]. 2 − · − = R¯(Γ ; x, y)+ C (Γ )c2δ(x y) According to the general strategy of renormalization, 0 − a¯−1 the non-renormalized diagrams are evaluated in terms of +C2(Γ )S (x, y), (47) renormalized electron propagators. In vacuum QED, the ′ ′ mass m used in the free electron propagator is finite, it is where C0(Γ ) = C0(Γ ) and C2(Γ ) = C2(Γ ). Therefore, not the infinite mass m0 of the bare free electron propa- the strong field self-energy diagrams are renormalized with gator. Similarly, the charge and the potential used in the the same formula as the vacuum diagrams, the countert- basic electron propagator of strong field QED are finite. erms C0(Γ ) and C1(Γ ) are the same as for the vacuum In other words we take case, the only difference is that the free propagator S0 is a −1 replaced by the strong field propagator S . More precisely, Sa¯ = i~cγ ∂ mc2 eγ a¯(x) . · − − · the counterterms are the same as those of vacuum QED if Therefore, the external potential diagram means now the renormalization conditions are compatible. Otherwise, their difference is finite. For a general strong field self-energy diagram Γ , the U =a ¯ (x). µ proof is similar. The subdivergences are removed with for- mula (42). The same Born expansion is made. The first To remove the superficial divergences we start by ex- diagram Γ ′ is now the diagram Γ (with subtracted sub- amining all possible superficially divergent diagrams (Chap- divergences) where all strong field electron propagators ter 8 of Ref.[10]). A priori, the superficially divergent 1PI a 0 ′ S are replaced by free electron propagators S , and Γj diagrams are the self-energy and vertex diagrams, and the is the diagram obtained by adding a photon line to the 1PI diagrams with one external photon (tadpole), two ex- j-th electron line of Γ ′. The remaining diagrams are fi- ternal photon (vacuum polarization), three external pho- nite by power counting. The superficial divergences of Γ ′ tons and four external photons (scattering of light by ′ and Γj are removed by the vacuum QED renormalization light). prescription, and we find Eq.(47) again. For a strong field vertex diagram, we have a logarith- 9.1 Self-energy and vertex diagrams mic divergence and the result is exactly the same as for vacuum QED: To renormalize self-energy diagrams, we take the exam- R(Γ ; x, y; λ, z)= R¯(Γ ; x, y; λ, z) ple of the one-loop diagram, and we expand the electron λ propagator using the Born expansion ([66] p.572) +C1(Γ )γ δ(x y)δ(y z), (48) − − a¯ 0 0 0 2 0 0 a¯ S = S + eS a¯ γS + e S a¯ γS a¯ γS . where C1(Γ ) is the same as for vacuum QED, up to pos- · · · sible finite terms. Thus = + e + e2 . 9.2 Vacuum polarization and tadpoles We still have the subdivergence-subtracted 1PI diagrams The strong field diagram on the left hand side is denoted with one, two, three or four external photon lines (and no ′ Γ . The first diagram on the right hand side (denoted Γ ) is external electron line). If the diagram has three or four a self-energy diagram of vacuum QED, the second diagram external photon lines, then it is finite. To prove this, we ′ (denoted Γ1) is a vertex diagram of vacuum QED and the expand the strong field electron propagator using the Born third diagram is finite by power counting. Using Eq.(43), expansion. Then we are back to the case of vacuum QED, ′ the counterterms for Γ are where we know that a 1PI diagram with three external photon lines is zero and a 1PI with four external photon R(Γ ′; x, y)= R¯(Γ ′; x, y)+ C (Γ ′)c2δ(x y) 0 − lines is finite [10] (once subdivergences are subracted). A ′ 0−1 +C2(Γ )S (x, y), 1PI diagram with two external photon lines is a vacuum 16 Christian Brouder: Renormalization of QED in an external field polarization diagram, which is renormalized as for vacuum Similar equations can be found in [66] p. 552, [59]. QED with Eq.(44) Therefore, the counterterm for the tadpole diagram Γ is ¯ 0−1 Rµν (Γ ; x, y)= Rµν (Γ ; x, y)+ C3(Γ )D µν (x, y), (49) Rλ(Γ ; x)= R¯λ(Γ ; x)+ eC4(Γ )¯aλ(x), where C3(Γ ) has the same value as for the corresponding vacuum QED diagram, up to a possible finite quantity. ′ where C4(Γ )= C3(Γ1). Now comes the diagram which is really typical of QED For a general tadpole diagram Γ , the result is similar. with external field: the tadpole. The induced vector po- If R¯µ(Γ ; x) is the value of Γ when all subdivergences are tential Aµ(x) a¯µ(x) is given by the sum of all tadpole − removed, the superficial divergence is removed with the diagrams Γ . In vacuum QED, the induced vector poten- counterterm tial is zero and the tadpoles do not intervene. For the case µ µ µ of QED with an external field, once subdivergences are R (Γ ; x)= R¯ (Γ ; x)+ eC4(Γ )¯a (x), (50) removed, the tadpole has a cubic superficial divergence. But because of Furry’s theorem (i.e. of the fact that the where current is odd under charge conjugation), the only diver- gence that remains is the same as for vacuum polarization. C4(Γ )= C3(Γi). We illustrate the general procedure with the simplest ex- Xi ample (see also [22]). By using the Born expansion for Sa¯ In the last relation, Γ is the vacuum polarization obtained we obtain i from the tadpole Γ by adding a photon line to the i-th electron line of Γ and by transforming all electron lines = + e Sa into free electron lines S0 in the resulting diagram. In other words, Γi is a vacuum QED vacuum polarization diagram, and C3(Γi) the corresponding counterterm. After this analysis, we obtain the following addition +e2 + e3 . to the description of the renormalization of vacuum QED. There is now a new index i = 4 and new 1PI diagrams γ which are tadpoles. In the term Γ/γ , the tadpole sub- Furry’s theorem eliminates electron loops with one and (4) three external photon lines (the first and the third dia- diagram γ(4) becomes an external potential line . grams on the right hand side). More precisely, since the induced current is an odd function of the external poten- tial, the first and the third diagrams are not present on 9.3 Summary the right hand side. The last diagram is finite, because a¯ if we make a further Born expansion of S in it, we ob- In this section, we summarize the renormalization rules for tain a vacuum photon-photon scattering diagram, which strong field QED. For any strong field Feynman diagram is finite, plus a diagram with five external photon lines, Γ , the subdivergences are subtracted by applying the for- which is finite by power counting. Therefore, the only di- est formula (42). The only difference with the weak field vergent diagram on the rhs is the second one, which is a ′ case is that the vacuum electron propagators are replaced vacuum polarization diagram. We denote Γ1 this vacuum by strong field electron propagators. polarization diagram and Γ the strong field tadpole on the When the subdivergences are subtracted, the superfi- lhs. cial divergence is treated as follows. If Γ is not 1PI, the We can write the bare potential as a sum of Feynman superficial counterterms are zero. In particular, if Γ con- diagrams tains tadpoles, then the counterterms Ci(Γ ) are zero. This A (x)= a (x)+ e|Γ |U (x; Γ ), is because a tadpole inside Γ can always be disconnected λ λ 0 λ from Γ by cutting a photon line. XΓ If Γ is 1PI, then the superficial divergence is removed where Γ is the number of vertices of the tadpole diagram using Eq.(47) for a self-energy and the indices are i = 0 | | Γ . If Γ is the simple tadpole of the above example, our and i = 2, Eq.(48) for a vertex and the index is i = 1, discussion shows that its value is Eq.(49) for a vacuum polarization with the index i = 3 and Eq.(50) for a tadpole with the index i = 4. 0 µν ′ Uλ(x; Γ )= e dydzD (x, y)R¯ (Γ ; y,z)¯aν (z) Z λµ 1 Now we give a few example to illustrate the general rules. +finite terms. ′ Now, we introduce the counterterm of Γ1 using Eq.(44), and we obtain the renormalized induced potential 9.4 Example 1: the potential 0 ¯µν ′ As far as we know, the renormalization rules for strong Rλ(x; Γ )= e dydzDλµ(x, y)R (Γ1; y,z)¯aν(z) Z field QED are new. Therefore, a number of examples are ′ +eC3(Γ1)¯aλ(x) + finite terms. required to see how they work in practice. Christian Brouder: Renormalization of QED in an external field 17 We start with the renormalization of the potential for Z is also the sum of all C (Γ ) where Γ runs over the 2,n − 2 the diagrams given in section 4.2.1. The first diagram has (1PI) self-energy diagrams with n loops, Z3,n is the sum no subdivergence. It is renormalized with Eq.(50). of all C3(Γ ) where Γ runs over the (1PI) vacuum polar- ization diagrams with n loops, Z3,n is also the sum of all C4(Γ ) where Γ runs over the (1PI) tadpole diagrams with R = U + eC4 U . n−1 n loops. Finally, δmn + δmkZ2,n−k is the sum of all k=1 C0(Γ ) where Γ runs overP the (1PI) self-energy diagrams The second diagram is not 1PI. To renormalize it, we with n loops. renormalize its two disjoint divergent subdiagrams: the For our example, we have vacuum polarization diagram on the left (index i = 3) and the tadpole on the right (index i = 4), and we multi- Z = C = C , 2,1 1 − 2 ply the renormalized diagrams. The general rule for a non 1PI diagram Γ is the following: write Γ as a product of Z = C = C , 1PI diagrams, renormalized each 1PI diagram, and multi- 3,1 3 4 ply the renormalized diagrams [18]. This rule is equivalent to the forest formula (42) for non 1PI diagrams, because δm = C . renormalization parts belonging to different 1PI subdia- 1 0 grams are disjoint. Thus, the renormalized induced potential up to two loops is R = U eR + e3R + e3R = +C U + eC U 3 4 eU + e3U + e3U +eC C U . 2 4 2 3 4 + e Z3,1 + e (Z3,2 + Z3,1) U For the last diagram, there are two subdivergences: on 3 3 +e Z3,1U + e Z3,1U the left, a self-energy γ (indices i = 0 and i = 2) and on the right, a vertex γ′ (index i = 1). Since these subdia- grams are not disjoint we obtain, adding the superficial 3 +e δm1U . divergence counterterm To check this result, we take Dyson’s relation between R = U + eC4 U renormalized and bare electron Green functions ea¯ e Z2(e,m)S¯(x, y; ea¯; e,m)= S(x, y; ; ,m + δm). Z3 √Z3 +C0 U + C2 U We expand the right hand side +C1 U . 2 ¯ Z2(e,m)S(x, y; ea¯; e,m)= U + e Z3,1U To check these results, we make use of the renormal- 2 2 +e δm1U + e U ization factors. ∞ 2n Z2 =1+ e Z2,n, +e2U + nX=1 ··· ∞ Z =1 e2nZ , 3 − 3,n nX=1 where the electron propagator is calculated in the field ∞ 2n δm = e δmn. of the renormalized external potentialsa ¯(x). If we use this nX=1 expansion in Eq.(34) we obtain the renormalized potential up to two loops. We recall that The relation between the renormalization factors and the a counterterms is as follows [62]. Z2,n is the sum of all C1(Γ ) a a ∂S (x, y) U = dzS (x, z)S (z,y)= . where Γ runs over the (1PI) vertex diagrams with n loops, Z ∂m 18 Christian Brouder: Renormalization of QED in an external field Obviously, it is much simpler to renormalize directly 11 Renormalization conditions from the Dyson relation. However, this does not renormal- ized all Feynman diagrams. Only the sum of them is finite, as was discussed in [62]. In practice, all the Feynman dia- The renormalization theory is an unambiguous method to grams of a certain order cannot always be calculated, and remove the subdivergences of a Feynman diagram. How- it is necessary to renormalize each diagram separately. ever, once all the subdivergences are removed, we must still specify the value of the superficial divergence. This is done by giving renormalization conditions. In the case 9.5 Example 2: the self-energy of QED without external field, these conditions are well- known ([10] p. 413). However, for QED with external field, For QED in an external field, the bare and renormalized they have never been stated precisely. It is the lack of self-energies are defined by proper renormalization conditions that yields the ambi- guities in the results of Dosch and M¨uller [51]. There are −1 S = i~cγ ∂ m0 e0γ A Σ, three kinds of renormalization conditions: for the current, · − − · − for the photon propagator and for the electron propaga- S¯−1 = i~cγ ∂ m eγ A¯ Σ.¯ · − − · − tor. Some of them were investigated in the early days of From Dyson’s relations between S and S¯ we obtain the QED [33]. following relation between Σ and Σ¯. ¯ Σ(ea¯; e,m)= Z2Σ(ea/Z¯ 3; e/ Z3,m + δm)+ Z2δm 11.1 The current +(Z 1)( i~pcγ ∂ + m + eγ A¯). 2 − − · · In the self-energy, the only new diagram with respect The renormalization condition for the current is deduced to vacuum QED is from the neutrality of matter. The total charge of a piece of matter is the sum of the proton charges and the elec- tron charges. Thus, the induced charge density due to the . vacuum polarization integrates to zero. If this were not true, a piece of matter with an equal number of electrons and protons would have a net charge. This diagram is not 1PI. In vacuum QED, all self-energy In the case of strong field QED, we operate the renor- 0−1 diagrams are 1PI, but this is not the case in the presence malization rule for tadpoles (50) with D µν (x, y) and we of an external field. This diagram is easily renormalized obtain as a product of two 1PI diagrams: −1 −1 dyD0 (x, y)R(Γ ; y,ν)= dyD0 (x, y)R¯(Γ ; y,ν) Z µν Z µν R = U + eC4 U ℄ +eC (Γ )¯ (x). 4 µ +C1 U + eC1 C4 U . The left hand side represents the renormalized current in- duced by the external current ¯µ(x). Two points must be noticed here. Firstly, the superficial divergence of the cur- rent diagrams (tadpoles) are removed by a counterterm 10 Self-consistent field renormalization proportional to the external current, and any induced cur- rent proportional to the external current cannot be dis- The case of self-consistent QED is simpler, since we have tinguished from a renormalization of the charge (see [66] the same diagrams as for vacuum QED (with S0 replaced A¯ p.553). With some hindsight, this point can be recognized by S ). Therefore, the renormalization rules are the same, in Ref.[67]. Secondly, the integral of the induced current except that the free fermion lines become the fermion lines must be zero to ensure matter neutrality. Thus in the SCF potential A¯(x). Since there is no tadpole in the fermion propagator of 0−1 self-consistent QED, the only point which is delicate is the dx dyD µν (x, y)R(Γ ; y,ν)=0 renormalization of tadpoles to calculate the self-consistent Z Z potential. When all subdivergences are removed, the su- perficial counterterm is obtained by expanding the SCF This second point was clearly made by Uehling [33]. When potential in terms of strong field diagrams and by renor- the external current is not neutral, this is enough to de- malizing those. The result is termine C4(Γ ). In the case of self-consistent QED, the counterterm is µ µ µ ¯ R (x; Γ )= R¯ (x; Γ )+ eC4(Γ )A¯ (x). proportional to A(x), and for a neutral atom the corre- sponding J¯(x) integrates to zero. Therefore, the criterium Thus, the counterterm is proportional to the renormalized of matter neutrality is not enough to determine the renor- potential itself. malized current of self-consistent QED. Christian Brouder: Renormalization of QED in an external field 19 11.2 The photon Green function Finally, it would be interesting for the spectroscopic applications to generalize the Schwinger equations to a To specify a renormalization condition for the photon Green degenerate “vacuum”. When the unperturbed state is de- function, we use the requirement that, very far from the generate, we must consider not only a single matrix ele- external field, low frequency Compton scattering should ment Φ Aµ(x) Φ but a matrix Φi Aµ(x) Φj . The modi- h | | i h | | i be given by the Thomson formula [66] p.640 and two fications induced by the presence of a degenerate vacuum close charges should interact with a Coulomb potential will be presented in a forthcoming publication. [66] p.670 and [10] p.325. In other words, for very large x and y and small x y, ¯ 0 − Dµν (x, y) should tend to Dµν (x, y). 13 Acknowledgements It is a pleasure to thank Paul Indelicato and Eric-Olivier 11.3 The electron Green function Le Bigot for helpful discussions. I also thank Alessandra Frabetti and Eric-Olivier Le Bigot for their thorough read- For the electron Green function, the renormalization con- ing of the manuscript. This is IPGP contribution #0000. ditions do not seem to have been studied beyond vacuum QED. One could probably assume that, for very large x and y and small x y, S¯(x, y) should tend to SA(x, y). 14 Appendix We intend to study− the validity of these renormalization conditions in the specific case of atomic physics. In this appendix, we give the proof of equations that are used in the text. 12 Conclusion 0 14.1 Derivative of SN (A) In this paper, the equations of self-consistent quantum We saw in section 5 that S0 (A)= S0(A)+ P with electrodynamics were derived from the Schwinger approach. N 0 N These equations are valid for any number of electrons in i P (x, y)= ψ (x)ψ¯ (y). the system and the functional derivative of the Green func- N ~c n n tion in the presence of bound electrons was used to calcu- 0 ′ where RN is a solution of the Dirac and adjoint Dirac 14.2 Integral equations. Finally, we calculate the functional derivative by taking V = δ δ(x z) so that In this section we show the following identity λ,µ − 1 δF (λϕ) δPN (x, y) 0 λ F (ϕ)= F (0) + dλ dx ϕ(x). (51) = S0 (x, z)γ PN (z,y) Z0 Z δϕ(x) eδAλ(z) λ 0 ′ This is a functional form of the classical Taylor formula +PN (x, z)γ S0 (z,y)+ RN . 1 ′ ′ f(x)= f(0) + dλf (λt)t, (52) To determine RN we require that the perturbation does Z0 not change the number of bound electrons. In other words where f ′(t) is the derivative of f(t). The n-dimensional δP (x, x) generalization of Eq.(52) is x 0 N ∆N = d tr γ =0. 1 Z h δAλ(y) i d f(x)= f(0) + dλ f(λx) Z dλ To evaluate this, we first calculate 0 1 n i = f(0) + dλ ∂if(λx)x . (53) 0 0 λ Z0 dx tr[γ S (x, y)γ PN (y, x)]. Xi=1 Z 0 Eq.(51) can be derived as the above n-dimensional We use Eq.(22) for EF = 0, because of the cyclic property case, but wee choose to prove it by an explicit calcula- of the trace we can bring the last ψ¯m(x) in the front. tion which shows that the purpose of the integral over λ then we use the orthogonality of the solutions of the Dirac is to change the multiplicity of some terms. 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