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14 - BOUND STATES IN EXTERNAL FIELDS

Published online by Cambridge University Press:  05 May 2013

Steven Weinberg
Affiliation:
University of Texas, Austin
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Summary

In our calculations of radiative corrections in Chapter 11 we went just one step beyond the lowest order in perturbation theory. However, there is a very important class of problems where even the simplest calculation requires that from the beginning we consider classes of Feynman diagrams of arbitrarily high order in coupling constants like e. These problems are those involving bound states – in electrodynamics, either ordinary atoms and molecules, or such exotic atoms as positronium or muonium.

It is easy to see that such problems necessarily involve a breakdown of ordinary perturbation theory. Consider for instance the amplitude for electron–proton scattering as a function of the center-of-mass energy E. As shown in Section 10.3, the existence of a bound state like the ground state of hydrogen implies the existence of a pole in this amplitude at E = mp + me—13.6 eV. However, no single term in the perturbation series for electron-proton scattering has such a pole. The pole therefore can only arise from a divergence of the sum over all diagrams at center-of-mass energies near mp + me.

The reason for this divergence of the perturbation series is also easy to see, especially if for the moment we consider the time-ordered diagrams of old-fashioned perturbation theory instead of Feynman diagrams. Suppose that in the center-of-mass system the electron and proton both have momenta of magnitude qme, and consider an intermediate state in which the electron and proton momenta are different but also of order q.

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Publisher: Cambridge University Press
Print publication year: 1995

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