Recent Developments in Mathematical Physics pp 167-198 | Cite as

# The Infrared Problem in Electron Scattering

Conference paper

## Abstract

The subject of these lectures is the problem of the field theoretical description of electron scattering. We know that scattering and production processes are usually described with the help of an S-matrix, but we know also that this description is fraught with difficulties if applied to electrons, or more generally any charged particles, as long as their electromagnetic interactions cannot be forgotten. If we try, for example, to define an S-matrix in QED (= quantum electrodynamics) in the canonical way as
H = H

$$ S = \mathop{{\lim }}\limits_{{t \to \infty }} {e^{{i{H_{0}}t}}}{e^{{ - iH(t - {t^{,}})}}}{e^{{ - i{H_{0}}{t^{,}}}}} $$

(1)

_{0}+ H_{int}the Hamiltonian of the theory, we find that this limit does not exist, even if the ultraviolet divergences are duly removed by some renormalization or regularization procedure. There still remain what is known as “infrared divergences”. The axiomatic way of defining S also does not work. In the known proofs of asymptotic conditions it is assumed that the particles under consideration belong to isolated one-particle hyperboloids in the energy-momentum spectrum of the relevant superselection sector. This is not the case for electrons. A state with one electron and any number of photons with arbitrarily small momenta can have the same quantum numbers as a single electron: the continuum starts right at the one-particle hyperboloid. Hence the asymptotic conditions cannot be proved and the axiomatic definition of S breaks down. From perturbation theory we learn that this is not merely a problem of unsuitable mathematical techniques, the asymptotic conditions are indeed not satisfied.## Keywords

Normal Case Mass Shell Vertex Function Electron Field Soft Photon
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## References

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## Copyright information

© Springer-Verlag 1973