A Collective Description of Electron Interactions. I. Magnetic Interactions

TLDR

The collective description captures the organized behavior of a high‑density electron gas, producing plasma oscillations and naturally describing long‑range correlations that the single‑particle picture misses. The paper introduces a collective description specifically for magnetic interactions among electrons, setting aside Coulomb forces for a subsequent study. By applying a canonical transformation, the authors rewrite the full Hamiltonian as a sum of a collective field term, kinetic energy, and a short‑range screened residual interaction, and provide both classical and quantum treatments of this decomposition. The analysis yields criteria that delineate when the collective description is valid and shows that the residual interaction reduces to a short‑range screened force.

Abstract

A new approach to the treatment of the interactions in a collection of electrons is developed, which we call the collective description. The collective description is based on the organized behavior produced by the interactions in an electron gas of high density; this organized behavior results in oscillations of the system as a whole, the so-called "plasma oscillations." The collective description, in contrast to the usual individual particle description, describes in a natural way the long-range correlations in electron positions brought about by their mutual interaction. In this paper we confine our attention to the magnetic interactions between the electrons; the coulomb interactions will be discussed in a subsequent paper.The transition from the usual single-particle description to the collective description of the electron motion in terms of organized oscillations is obtained by a suitable canonical transformation. The complete hamiltonian for a collection of charges interacting with the transverse electromagnetic field is re-expressed as a sum of three terms. One involves the collective field coordinates and expresses the degree of excitation of organized oscillations. The others represent the kinetic energy of the electrons and the residual particle interaction, which is not describable in terms of the organized oscillations, and corresponds to a screened interparticle force of short range.Both a classical and a quantum-mechanical treatment are given, and the criteria for the validity of the collective description are discussed.