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The enlargement of the analyticity domain is such that for fixed in VS .NET
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ELEMENTARY PROCESSES
the series for the amplitude in the physical region where IP 1( cos 8) I : :; 1 is bounded by 16n I~" (21 + 1) I / Iand again from unitarity I / I < 12q ~ 1. Thus I!T(s, cos 8) I <
16n/~ax
= constant s(ln S)2 (5214) An accurate estimate of the constant may be obtained in terms of the maximum number of subtractions n. The latter can, in fact, be reduced to two using a similar bound as (5214) in the u channel and the PhragmenLindelof theorem to show that the bound s(ln sf holds in the complex directions; thus n = 2. The final estimates of the Froissart bounds are then I!T(s, cos 8) I < 16n ~ (In S)2 to
and from the optical theorem
a!o! (5215) <  (In s) 1 to
(5216) Even though the scale of the logarithm is not ascertained, it is striking that the trend exhibited by the highenergy data obtained by the latest generation of accelerators is in qualitative agreement with the prediction of this bound. It is interesting to interpret the angular momentum cutoff (5213) in terms of an effective opacity radius p = Imaxlq ~ In slFo. On the grounds of a potential analogy, p would be of the order liFo (112m" for pionnucleon scattering). It only differs from this naive prediction by a logarithmic increase with energy. Another property predicted on the basis of analytic properties is the asymptotic equality of particle and antiparticle total cross sections from a given target. The theoretical observation is originally due to Pomeranchuk. The statement of his theorem needs some qualification in view of the fact that total cross sections may increase as (In S)2 for large s. If a and a denote the two cross sections, then under mild assumptions we can show that ala + 1 as s goes to infinity. Much more could, and should, be said on the whole subject of the interplay of locality and unitarity as dictating the framework of scattering. The reader will undoubtedly find a more substantial treatment in the large literature devoted to this subject. NOTES
For the KallenLehmann representation see G. Kallen, "Quantum Electrodynamics," SpringerVerlag, Berlin, 1972. The asymptotic theory is covered in the work of H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento, vol. 1, p. 205, 1955. Additional terms in the current commutation relations were discussed by J. Schwinger, Phys. Rev. Lett., vol. 3, p. 296, 1959. The perturbative calculations are by now classroom exercises. A wealth of information on the Compton effect can be found in R. D. Evans, H andbuch der Physik, vol. XXXIV, p. 218, 1958. QUANTUM FIELD THEORY
On the foundations of dispersion relations the reader may consult some of the following references according to his or her taste: N. N. Bogoliubov and D. V. Shirkov, "Introduction to the Theory of Quantized Fields," Interscience, New York, 1959; Proceedings of the 1960 Les Houches Summer School, "Dispersion Relations and Elementary Particles," edited by C. de Witt and R. Omnes, Hermann, Paris, 1960, in particular the lectures of M. L. Goldberger, A. S. Wightman, and R. Omnes; 1960 Scottish Universities' Summer School, "Dispersion Relations," edited by G. R. Screaton, Oliver and Boyd, Edinburgh, 1961; and Proceedings of the International 1964 School of Physics Enrico Fermi, "Dispersion Relations and Their Connection with Causality," edited by E. P. Wigner, Academic Press, New York, 1964, in particular the lectures by M. Froissart. On pionnucleon scattering see, for instance, the lectures of J. Hamilton in "Strong Interactions and High Energy Physics," 1963 Scottish Universities' Summer School, edited by R. G. Moorhouse, Oliver and Boyd, Edinburgh, 1964. For a summary of the work on analyticity and its physical consequences derived from first principles, including a discussion of various highenergy bounds, see the lectures of A. Martin in "Physique des Particules," Les Houches 1971 Summer School, edited by C. de Witt and C. Itzykson, Gordon and Breach, New York, 1973. The foundations of S matrix theory are studied in 'The SMatrix," by D. Iagolnitzer, North Holland, Amsterdam, 1978.

