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The self-consistent Geld method in which a many-electron system is described by a time-dependent interaction of a single electron with a self-consistent electromagnetic field is shown to be equivalent for many purposes to the treatment given by Sawada and Brout. Starting with the correct many-electron Hamiltonian, it is found, when the approximations characteristic of the Sawada-Brout scheme are made, that the equation of motion for the pair creation operators is the same as that for the one-particle density matrix in the self-consistent 6eld framework. These approximations are seen to correspond to (1) factorization of the two-particle density matrix, and (2) linearization with respect to off-diagonal components of the one-particle density matrix. The complex, frequency-dependent dielectric constant is obtained straightforwardly from the self-consistent 6eld approach both for a free-electron gas and a real solid. It is found to be the same as that obtained by Nozieres and Pines in the random phase approximation. The resulting plasma dispersion relation for the solid in the limit of long wavelengths is discussed.HE electromagnetic properties of crystals have long been studied by considering the timedependent interaction of a single particle with a selfconsistent electromagnetic field. ' This procedure seems plausible for studying the response of electrons to any external perturbation, and Bardeen, ' WolG, ' Lindhard, ' Frolich and Pelzer, ' Ferrell, ' and others' have used this or a closely related approach with considerable success in discussing such phenomena as the electronphonon interaction, the frequency and wave-number dependence of the dielectric constant, plasma oscillations, and characteristic energy losses in solids. These, and similar phenomena, have also been studied on the basis of more sophisticated treatments of the manybody problem' " with largely identical results. The explicit relationship of the self-consistent field approach (e.g. , Lindhard') to the many-body approach (e.g. , Sawada and Brout" ") has not been stated. It is the purpose of this note to examine this relationship and to show that for many problems the two approaches may be regarded as rigorously equivalent. We do so by showing that the approximations introduced by Sawada and Brout are in fact sufhcient to deduce the equation of the self-consistent field approach.

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