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QUANTUM FIELD THEORY
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unity at the one-particle pole. Define unrenormalized fields ljJo, ljJo, and Ao and cutoff-dependent bare parameters }15, mo, eo according to
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ljJo
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Z!/2(A)ljJ
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m - <5m(A)
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(7-71)
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lifo = Z!/2(A)1if
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Ao = Zj/2(A)A
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}15 = Z3 i (A).u 2
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eo =
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Z 1(A)Z:z1(A)Z3 / 2(A)e
Z3 / 2(A)e
Then the lagrangian (7-70) expressed in terms of bare quantities is similar to the original lagrangian
:e =
1 i 4 Fo2 + }15 Ao - "2 Z3 1(A)(o' Ao) 2 + "2ljJo +tljJo 2 2 A If
moljJoljJo - eoljJo$oljJo
(7-72) Not only do finitely many types of counterterms suffice to render the perturbative series term-by-term convergent (a property which will be preserved to all orders in h, as we shall see) but the very structure of the theory is preserved by renormalization. It is worth emphasizing the role of the Ward identity, a result of which is
eoAo = eA
(7-73)
giving a meaning to the scale of the electromagnetic interaction and preserving the meaning of minimal coupling. Note also that the A 2 and (0' A)2 terms were not affected by counterterms, a special property of electrodynamics. In the same way that we introduced a bare "mass" }10 we can define a bare parameter Ao through
= Z3 i (A)A
The lagrangian (7-72) gives rise to renormalized Green functions defined as follows:
GO(Pb"" P2n, k b ... , k/; }10, mo, eo, Ao, A)
Z3(A)Z~2(A)GR(Pb"" P2n, k b , k/;}1, m, e, A)
(7-74)
Here Pb ... ,P2n (k 10 . ,k/) stand for the external fermion (photon) momenta. Again this holds perturbatively in h, and up to now we may only assert this up to order hi. The relations (7-71) and (7-74) justify the denomination of multiplicative renormalization. The wave function renormalization constants Z!/2 and Zj/2 appear to be infinite when expanded perturbatively. This shows, at least in this sense, that one of the working hypotheses of this approach, namely, the canonical equivalence of the free and interacting picture, is not mathematically well founded. We have succeeded, however, in overcoming this difficulty, and we shall discover later on that there is some interesting physical meaning in these apparently infinite quantities. The extension of this program to all orders will be the subject of Chap. 8.
RADIATIVE CORRECTIONS
7-2 RADIATIVE CORRECTIONS TO THE INTERACTION WITH AN EXTERNAL FIELD
The remainder of this chapter deals with examples of radiative corrections. One of the issues will be to make sense out of the infrared divergenGes arising from the long-range electromagnetic forces. This survey is only indicative and without any pretense to completeness.
7-2-1 Effective Interaction and Anomalous Magnetic Moment
In Chaps. 2 and 4 we have already presented various aspects of the interaction of charged particles with c-number external fields. Here we want to discuss quantum corrections generated when we substitute to the quantum field Aq the sum Aq + A" where Ac is the classical field. To first order in Ac we derive an effective interaction represented by the first three diagrams of Fig. 7-14 including one-loop quantum corrections. This means that the elementary interaction ey pAf has been replaced by (7-75) where r and cO have been computed in the preceding section and G is the free photon propagator. We have assumed the electrons on mass shell, disregarding therefore the self-energy insertions on the external lines which are assumed to be absorbed in the correct definition of the mass and proper normalization of the states. At low momentum transfer q = pi - p, keeping only dominant terms as J1-+ 0, we obtain
eA c yp
[1 +
3nm2
exq2
(In--;;-g-S1)J + 3
i ex 2m 2n O"pvq
(7-76)
When deriving (7-76) the gauge-dependent terms of G proportional to qV q" have not contributed when acting on the vacuum polarization. Also a term in q = P' - Phas disappeared when evaluated on the mass shell. Consequently, the remaining expression remains gauge invariant in the sense that if Ag is replaced by Ag + qPf(q) the added term vanishes on the mass shell. The quasistatic limit q -+ 0 allows a simple interpretation. In configuration space the momentum transfer q can be replaced by the derivative operator io
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