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In contrast with (4-117), the integral in (4-123) has a divergent part proportional to a2 - b2 , which will be cured by a renormalization. The idea is to rewrite
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where C is infinite, and to modify 2 0 in (4-120) into (1 - C)2 0; this modification is undetectable, since only 2 0 + b2 may be observed by assumption. The latter now reads
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(4-124) b2 finite is obtained from b2 by subtracting the divergent term; there is, of course, some arbitrariness in this subtraction. We shall discuss at length this renormalization operation and its arbitrariness in the following chapters. The absorptive part of b2 vanishes with all its derivatives at e = o. Only dispersive effects may be computed by a perturbative calculation. For instance,
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we find, by expanding (4-123) to second order in cx,
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<52'(4)
= 2cx
45m 4
[(E2 _ B2)2
+ 7(E B)2]
(4-125)
The Euler-Heisenberg effective lagrangian (4-123) and (4-124) may be used to discuss various nonlinear effects due to the quantum corrections.
NOTES
The physical interpretation of the infrared catastrophe dates back to the work ofF. Bloch and A. Nordsieck, Phys. Rev., vol. 52, p. 54, 1937. A thorough treatment within perturbation theory is given in D. R. Yennie, S. C. Frautschi, and H. Suura, Ann. Phys. (New York), vol. 13, p. 379, 1961. The use of coherent states to provide a bridge between quantum and classical electromagnetism has been pioneered by R.1. Glauber, Phys. Rev., vol. 131, p. 2766, 1963. Further developments can be found, for instance, in "Quantum Optics and Electronics," edited by C. De Witt, A. Blandin, and C. Cohen-Tannoudji, Les Houches, 1964, Gordon and Breach, New York, 1965. The interaction representation is due to F. J. Dyson, Phys. Rev., vol. 75, p. 486, 1949, while the original work of G. C. Wick is in Phys. Rev., vol. 80, p. 268, 1950. The effective lagrangian for the electromagnetic field arises from the work of W. Heisenberg and H. Euler, Z. Physik, vol. 98, p. 714, 1936, and is worked out in detail by J. Schwinger, Phys. Rev., vol. 82, p. 664, 1951, which is the source for Sec. 4-3. See also J. Schwinger, Phys. Rev., vol. 93, p. 615, and vol. 94, p. 1362, 1954.
CHAPTER
FIVE
ELEMENTARY PROCESSES
We discuss the relation between measurements of cross sections and the dynamical content of field theory provided by Green functions through the reduction formalism of Lehmann, Symanzik, and Zimmermann. This is followed by elementary applications to electromagnetic processes to lowest order. General properties of unitarity and causality are presented and used to introduce partial wave expansions and dispersion relations.
5-1 S MATRIX AND ASYMPTOTIC THEORY
We have to complete our kinematics before embarking on the perturbative evaluation of transition amplitudes. We present in this chapter the general framework which will enable us to relate these calculations to the physical processes of interest. In practice a major tool of investigation is particle scattering off various targets. On a macroscopic scale, interaction times are extremely small. It is therefore out of the question to follow in detail the time evolution during the elementary scattering events. We can only give the following picture. Long before the collision, well-separated wave packets evolve independently and freely. As already discussed in the special instances of Chap. 4, the set of these incoming states builds up a Fock space, the in-space with associated free fields. It is important to note that these in-states must represent exactly the individual characteristics of isolated particles, such as mass and charge. In other words, self-interaction effects must
ELEMENTARY PROCESSES
be absorbed in these measurable parameters. The collision process then follows, involving scattering, absorption, or creation of new particles, submitted to the fundamental conservation laws of energy, momentum, angular momentum, parity, charge conjugation, internal symmetries, etc. Long after the collision, free wave packets separate, representing the outgoing states. They are again described by free particle kinematics and by corresponding free fields. According to the postulates of quantum mechanics the amplitude
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