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B = curl A
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which imply the first set of homogeneous Maxwell equations curlE+B=O div B = 0
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The variation with respect to A yields the second pair of equations div E = P curlB-E=j
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(9-168)
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8:(X)
d y B(y) - curl A(y)
curl B(x)
Among these equations B = curl A and div E = P appear as constraints. The first one may be simply solved by replacing B everywhere by curl A which amounts after a partial integration to rewriting the action as
If this is compared to (9-138) we realize that A plays, as did A, the role of a
{ . f
d4 x
-E-A- [E2 + (curl A)2 +j-A ] +Ao(divE-p)} 2
(9-169)
Lagrange multiplier without any conjugate variable. We are thus led to identify the canonical variables (Pi, qJ with A(x) and E(x) respectively. Poisson brackets are defined through (9-170)
It is necessary to generalize the compatibility equations (9-152) to cope with the case where p depends on time. Those are now written aF F '" 0 => -
at + {h' F} '" 0
(9-171)
where the time derivative operates on the explicit time dependence. In the present case, F = div E - P and
~ (div E _ p) =
The condition (9-171) reduces to
_ ap + {fd 3y [E2 + (curl A + j _A] div E _ p} = _ ap _ div j = 0 (9-172)
an identity by virtue of current conservation. Thus, apart from this slight generalization, the case at hand can be cast into the framework discussed in the preceding subsection. It remains to choose auxiliary conditions g(E, A) = O. This is, of course, arbitrary to a large extent. A serious simplification will follow if g is linear in the dynamical variables as is the constraint div E - P = O. Under such circumstances det {g, f} will be independent of the variables and hence may be absorbed in the normalization of the path integral. A possible condition is
al'AI'
== AO + div A = c(x, t)
(9-173)
f~(F,
involving an arbitrary function c. We return to covariant notations and write the transition amplitude as
1] 8(a- A -
c)exp
{i fd x [iFl'vPV- ~Fl'v(aI'AV - aVAI') - j- A]}
(9-174)
FUNCTIONAL METHODS
The gaussian integral over B realizes automatically the substitution B -+ curl A. Similarly, the integral over E substitutes for the electric field its value -(VAO + A). When this is done we have for the amplitude
E&(A)
IJ c5(0 A -
c)exp
{-i f
d4 x [!(oI'A v - ovAI')(oI'A V- oVAI') + j. A]}
(9-175)
This is not quite the expression used in the previous chapters. However, since (9-175) is independent of the arbitrary function c(x, t) we may complete this identification by integrating on these functions c with a weight exp [ - (iAI2) Sd4 x c2 ]. If we denote by F I'V the quantity ol'Av - ovAI" the final version of the path integral is the familiar one
E&(A) exp { - i
f [:2 + ~
(0 A
+ j. A ]}
(9-176)
This presentation illustrates the arbitrariness implied by gauge invariance. It is also worth comparing it to the operator formalism.
9-4 LARGE ORDERS IN PERTURBATION THEORY 9-4-1 Introduction
Functional integrals provide us with new tools to investigate numerous aspects of field theory. As an illustration we close this chapter with a discussion of the behavior of perturbation theory at large orders. One of the goals of this study is of course an attempt at a better understanding of the amazing accuracy of the successive approximations in quantum electrodynamics and the other promising weak coupling models. Another motivation lies in the hope of overcoming the very limitations of the perturbative series and of coping with strong coupling phenomena. Even though present-day knowledge is far from being satisfactory, interesting developments have occurred which justify this endeavor. This will also afford us the occasion to introduce some useful techniques in dealing with path integrals. The nature of the perturbative series is related to the analytic properties of Green functions in terms of the coupling constant in the vicinity of zero. Such a study is possible but extremely difficult. Fortunately there exists a less rigorous, but manageable approach which leads to similar conclusions, namely, that the perturbative series, in all cases of interest, is strongly divergent. In spite of this fact it may be very useful, as wiII be explained below. Thus the problem may be divided into two distinct parts. First, we look for an estimate of large orders in the expansion of Green functions as given by a well-defined set of Feynman rules and renormalization prescriptions. Some aspects of this program have been completed. It is noteworthy that this is independent
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