Linearized Many-Body Problem

Abstract

The theory of an electron gas at high density is generalized so as to include the exchange scattering of a particle and hole in the singlet as well as in the triplet spin state. This approximation is the maximum possible linearization of a quantum-mechanical many-body problem, and corresponds to the theory of small-amplitude oscillations of a classical many-body system. The linearized Hamiltonian is that of Wentzel's meson pair theory, in which now two kinds of mesons are involved, one with spin zero corresponding to the singlet state of a particle and hole and the other with spin one corresponding to the triplet state. The correlation energy is shown to be the sum of the change in zero-point energies of the two-meson fields with correction only in the second-order term. The theory is then formulated in terms of Green's function, and it is proved explicitly that the linearized theory (or the random-phase approximation with exchange effects taken into account) is equivalent to the "ladder approximation" for the particle-hole scattering. The particle-particle scattering by means of the screened interaction is also discussed. The case of the $\delta$-function potential is explicitly solved, and it is shown that the correlation energy becomes complex in the region where the instability of the paramagnetic state occurs.