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The covariant 5-matrix formalism of Dyson has been applied to the calculation of the fourth order non-linear polarization of the vacuum, which is related to the lowest order non-linear interaction between electromagnetic fields. The finiteness and the gauge invariance of the interaction are exhibited explicitly by an expression for the fourth-rank vacuum polarization tensor in momentum space.

The behavior of an ionized plasma is discussed in an approximation in which an individual particle is assumed to obey a Fokker-Planck equation, and where its interaction with the environment is incorporated in the coefficients of the partial differential equation. It is found that if the interaction of the test particle with the medium is divided into a "nearest neighbor" interaction (which manifests itself in "large-angle collisions") and an interaction with the rest of the medium, then the latter can be adequately treated by a perturbation method. If the nearest neighbor interaction is neglected, the coefficients of successive derivatives form a rapidly decreasing sequence, provided the average kinetic energy greatly exceeds the mean potential energy (which is usually the case). Within the framework of this approximation the coefficients of damping (dynamical friction) and diffusion in velocity space are calculated and the higher (small) coefficients are estimated.

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671test of the cascade nature of the process. A detailed report to be published shortly will show that all the results of the 400-Mev proton interactions can be satisfactorily explained by the model.A similar investigation at 150 Mev is now in progress.The authors wish to thank Mr. Leon Landowitz and Mr. Jack Leitner for their excellent work on the Monte Carlo calculations.