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Although the conceptual difficulties associated with infrared divergences in quantum electrodynamics have long since been resolved, a convenient technique for identifying the appropriate infrared-finite part of Feynman graphs has been lacking. Such a technique is presented here. The polarization sums for both real and virtual photons are rearranged into two parts. One of these (called the $K$-polarization sum) resembles a gauge transformation whose structure permits a simple demonstration of infrared factorization and exponentiation. It is easy to see by inspection that the residual factors (with $G$-polarization sums) are infrared free. This is done for each separate graph, in contrast to earlier treatments where contributions from sets of graphs had to be considered together. The technique is illustrated by a detailed treatment of the radiative corrections to lowest-order potential scattering, and generalizations to other processes are indicated.

Massless particle decay and recombination is treated from the viewpoint of pure probability theory, using as a guide the empirical success of the exponential decay law for massive systems. The result is a linked set of linear integro-differential equations. Lorentz invariance is shown to be easily accomodated. As derived, the equations contain two arbitrary functions, one governing decays and the other recombinations. These equations are analyzed in certain special cases. Forms for the decay and recombination functions are proposed on the basis of Lorentz invariance and simplicity. Using phase space considerations, an argument is given that verifies these forms of the decay and recombination functions, and in addition specifies their relative strengths. When this input is applied to photon instability it is found that, first, there is no infrared divergence and secondly a photon emitted at time t=0 will have a probability of 24 25 to be observed in its original state as t→∞.

Thermal photons are present in every initial state, and nonthermal photons are present in many systems of interest. Furthermore, stimulated emission promotes the infrared divergence of quantum electrodynamics from logarithmic to an inverse power in a Planck distribution. I have calculated to order ${e}^{2}$ the background-photon corrections to the cross section for scattering a charged particle off an external Coulomb potential. I demonstrate the cancelation of infrared divergences, and derive a formal shift in the masses of charged particles. I interpret this shift, and discuss atomic spectra.

The one-photon radiative correction to the high-energy electron scattering by a nuclear field is analyzed to all orders of Born approximation for the nuclear potential. The leading term of the fractional decrease of the elastic scattering cross section is shown to be exactly the same as that given by the first Born approximation. A detailed analysis of the effect of the long Coulomb tail on the infrared divergence is made. Also, an analysis is made on the origin of the $logm$ term which appears in the radiative correction.

The infrared divergences of quantum electrodynamics are eliminated to all orders of perturbation theory in the matrix elements by an appropriate choice of initial and final soft photon states. The condition for this cancellation restricts these states to representations of the canonical commutation rules which are unitarily inequivalent to the usual Fock representation.

correlation function are studied in the Born approximation. A new formulation is developed which relates the hole correlation function to the retarded polarization propagator of the electron gas. Previously obtained results for the free-electron gas at finite temperature are recovered. The Migdal approximation is employed to find an expression for the imaginary part of the polarization propagator and it is seen that although the strength of the singularity is potentially modified by a constant factor the time dependence is not changed. Inclusion of the electron-phonon interaction via the Migdal form for the electron spectral weight func­ tion has no effect on the hole correlation function if the one-electron density of states varies slowly on the scale of the width of the spectral weight function. Similarity of this behavior to that of the nuclear spin relaxation in electron-phonon system is pointed out.

An exact analytic calculation is presented for the cross section of the nonrelativistic Compton-type inelastic scattering of photons by electrons bound in the ground state of a hydrogenlike atom. The matrix element of the process was taken to be of the Kramers-Heisenberg-Waller form. It was integrated by the Coulomb-Green's-function method, developed previously by the author in connection with Rayleigh scattering. This is based on replacing the sum over intermediate states by the Green's function which is expressed in terms of the integral representation in momentum space given by Schwinger. The final continuum-state wave function was also described by an integral representation. Thus the evaluation of the matrix element requires the carrying out of momentum-space integrations, followed by two contour integrations. Its final form was expressed in terms of hypergeometric functions of four variables of Lauricella's type ${F}_{D}$. At small photon energies, where the dipole approximation is valid, the result simplifies considerably. The low-energy end of the scattered-photon spectrum is considered in detail in connection with the infrared divergence problem. The shape of the spectrum is discussed and a comparison with previous results is given.

We develop an S-matrix method for calculating the effect of small perturbations on a partial-wave amplitude, and in particular, on the positions and residues of bound states. The method is applicable to both nonrelativistic and relativistic problems. It has, as a particular virtue, rapid convergence of the dispersion integrals. Electromagnetic corrections to strong interactions are the main application we have in mind, and modifications useful for handling the infrared divergence that occurs in this case are described in detail.

By way of extension of a previous study of the bound-state problem within the framework of the covariant Bethe-Salpeter formalism, the scattering of nucleons by nucleons has been investigated in the ladder approximation, assuming scalar nucleons and zero-mass scalar mesons. The solution of the scattering problem can be effected by using an integral-transform method which is similar to that developed for the bound-state problem, subject to the assumption that the nucleon mass is large compared to the kinetic energy in the center-of-mass system. An essential complication which is now encountered is the appearance of an infrared divergence. Two methods of circumventing this difficulty are discussed. The cross section can be calculated from the Green's function by a limiting process; the usual $S$-matrix formalism leads to incorrect results in this case, and an amplitude renormalization is required. In this connection, it is instructive to re-examine, in detail, the Coulomb-scattering problem in momentum space, since this is very closely related to the Bethe-Salpeter scattering problem. The Green's-function method turns out to be unsuitable for the calculation of higher-order corrections to the nucleon-nucleon scattering cross section, so that here a cutoff procedure must be used.

The perturbation formulas for Compton scattering with radiative corrections are written as dispersion relations. The purpose is to learn something about (1) the number of subtractions that are needed in the relations, (2) the analytic continuation of the scattering amplitude into the unphysical region, and (3) the role of the infrared divergence. It is found that one subtraction is needed to obtain dispersion relations but that these equations still depend on the fictitious mass of the photon. To obtain relations which contain only physical quantities another subtraction is made. It is suggested that the imaginary part of the amplitude has fewer singularities than the amplitude itself, and consequently may, in some cases, be continued by means of the phase-shift expansion. Finally, it is shown that the Born approximation is equivalent to the so-called one-electron-intermediate-state contribution to the dispersion relation.

In a private communication, Biedenharn pointed out that the statistical independence of photon emission in the infrared limit in a bremsstrahlung scattering process may lead to a logarithmically divergent angular momentum for the radiated photons and the scattering system. It is shown that the expectation values of these angular momentum vectors are free of infrared divergence, but the expectation values of their squares are infinite due to an infrared divergence. The physical significance of this result is that an incident electron of arbitrarily large impact parameter can scatter from a finite range potertial. This may be ascribed to the physical structure of the electron, namely its photon cloud. (auth)

A formalism intended to evaluate the expectation value of physical quantities directly is applied to processes involving multiple production of photons in quantum electrodynamics. This application is achieved by constructing the Green's function, which is accomplished by expanding it in terms of hard and soft photon parts of the electromagnetic field. A general treatment of infrared divergence is given. The cross section for n-photon production in Coulomb scattering and pair annihilation is evaluated. The limitation of the expansion of the Green's function in terms of hard and soft photon parts is examined. A plausible generalization regarding the cross section of any primary process in quantum electrodynamics at high energies with n-photon emission in the final state is enunclated, (auth)

We have computed thee6 corrections to the differential cross-section for Moller scattering. The divergences for high values of the momentum of the virtual photons have been removed by the usual techniques of renormalization. The infrared divergence has been eliminated by adding the cross-section for the bremsstrahlung process associated with Moller scattering. Numerical computations give raisonable results for the corrections.

The problem of maintaining the locality of the bound-state wave function is discussed in the $S$-matrix approach to the first-order energy shift. It is shown first that the nearby-singularity prescription in the Dashen-Frautschi formalism leads to the inclusion of unphysical terms coming from the nonlocal component of the wave function. It is pointed out that the infrared divergence of Dashen and Frautschi comes from this unphysical term. We then present an alternative method based on the Gelfand-Levitan formalism on the inverse-scattering problem. We discuss here an approximate solution of the Gelfand-Levitan equation that leads to a localized bound-state wave function. It is pointed out that this wave function gives the first-order energy shift without the infrared difficulty. Using soluble exponential potentials, we discuss numerical accuracies of this approach.

This chapter represents an intermediate step between classical theory and quantumelectrodynamical treatment. Soft and hard photons will be separated in a rigorous way; the contribution of soft photons (classical current) as a whole can be taken into account as in the previous chapter (here we shall employ the derivation by means of the perturbation expansion of the Smatrix), the application of the rules of QED to the hard-photon contribution introduces no principal difficulties (infrared divergences).

It is shown that the Lamb shift can be calculated by dispersion theory. In this context no infrared divergence appears. This approach leads to a calculation which is different from the conventional one, of which it can be an independent check (hopefully at least as accurate). Only the leading contributions are discussed here.

A great deal of insight can be obtained into the infrared divergence problem in quantum electrodynamics by consideration of a simple model in which the electromagnetic field is quantized but the electric current is treated classically. It is not necessary to use box normalization.