Scattering theory for quantum electrodynamics. I. Infrared renormalization and asymptotic fields
Daniel Zwanziger · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · 1975
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 ${k}^{\ensuremath{\mu}}=\ensuremath{\omega}(1, \stackrel{^}{k})$, $\ensuremath{\omega}\ensuremath{\ge}0$. This class includes test functions whose low-frequency behavior is given by ${\ensuremath{\varphi}}^{\ensuremath{\mu}}(k)\ensuremath{\sim}\frac{e{p}^{\ensuremath{\mu}}}{p}\ifmmode\cdot\else\textperiodcentered\fi{}k$, for which the usual inner product $\ensuremath{\int}{d}^{3}k{(2\ensuremath{\omega})}^{\ensuremath{-}1}{{\ensuremath{\varphi}}_{\ensuremath{\mu}}}^{*}(k)(\ensuremath{-}{g}^{\ensuremath{\mu}\ensuremath{ u}}){\ensuremath{\varphi}}_{\ensuremath{ u}}(k)$ 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 $\ensuremath{\psi}$. It is used as a starting point in the accompanying article, for the derivation of reduction formulas for the quantum electrodynamical $S$ matrix.