Coherent Soft-Photon States and Infrared Divergences. IV. The Scattering Operator

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Abstract
In Paper III of this series, the asymptotic states of quantum electrodynamics were defined in terms of the mass-shell singularity structure of the Green's functions. In this paper the reduction formulas obtained are used to derive simple expressions for the matrix elements of the scattering operator. This operator is defined on the space of asymptotic states, which is the direct product of the Fock space of the particles (massive particles and hard photons) with the nonseparable Hilbert space, defined in Paper I, which is spanned by the soft-photon coherent states. It is shown that the scattering operator so defined is gauge-invariant, Lorentz-invariant, unitary, crossing-symmetric, and independent of the choice of the small parameter that defines the separation between hard and soft photons. For a given initial state, the only nonvanishing scattering matrix elements are those to final states in a specific equivalence class, and conditions for states to be equivalent in this sense are obtained. The relationship between these matrix elements and physically measurable cross sections is discussed. In this way, results obtained by conventional methods are reproduced, but in addition questions inaccessible to such methods, such as the effect of an infinite number of soft photons in the initial state, may be investigated.