Scattering theory for quantum electrodynamics. I. Infrared renormalization and asymptotic fields

The present article lays the theoretical foundation for a scattering theory of quantum electrodynamics, which is completed into a practical calculational scheme in the accompanying article. In order to circumvent infrared divergences, an infrared renormalization procedure is instituted whereby a Lorentz-invariant, but indefinite, inner product is defined for a class of photon test functions defined on the future light cone k=(1, k^), 0. This class includes test functions whose low-frequency behavior is given by (k)eppk, for which the usual inner product d3k(2)-1*(k)(-g)(k) is infrared-divergent. The Fock space of such test functions provides a representation space for the asymptotic fields of quantum electrodynamics. It contains subspaces in which the indefinite metric is non-negative which, when completed in the norm, yield physical Hilbert spaces. This Fock space of test functions thereby replaces the nonphysical Hilbert space of the usual Gupta-Bleuler method and its positive-definite but noncovariant metric. As an application the S matrix and finite transition probabilities are found for the bremsstrahlung emitted by the classical external current of a scattered charged particle. A final result is a simple weak asymptotic limit of the charged field. It is used as a starting point in the accompanying article, for the derivation of reduction formulas for the quantum electrodynamical S matrix.