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

Abstract

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 $k^{\mu }=\omega (1, \hat{k})$, $\omega \ge 0$. This class includes test functions whose low-frequency behavior is given by ${\phi }^{\mu }\mathrm{}\left(k\right)\mathrm{}\sim \frac{e{p}^{\mu }}{p}·k$, for which the usual inner product $\int d^{3}k{\left(2\mathrm{}\omega \right)}^{-1\mathrm{}}{{\phi }_{\mu }}^{*}\mathrm{}\left(k\right)(-g^{\mu \nu }){\phi }_{\nu }\mathrm{}\left(k\right)\mathrm{}$ 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 $\psi$. It is used as a starting point in the accompanying article, for the derivation of reduction formulas for the quantum electrodynamical $S$ matrix.