Classical theory of boson wave fields

Abstract

The subject of discussion is the Hamiltonian for a system of bosons interacting by two body forces, as expressed in the formalism of second quantization. In this paper, we examine properties of the classical wave field governed by the Hamiltonian. For a general potential there is always an exact solution representing a uniform density. Exact solutions are exhibited, which represent disturbances of a definite velocity and of arbitrary amplitude. For small amplitudes the disturbances obey Bogolyubov's dispersion relation. Corresponding solutions are found for disturbances when the system moves as a whole. For suitably attractive potentials we find a class of exact solutions, degenerate in energy, with spatially periodic density. These solutions have a lower energy than the uniform type. Small amplitude excitations are investigated for the periodic case. They are phonons for long wavelengths, but show a band character at shorter wavelengths. A theory of the motion of foreign atoms in the boson fluid is formulated.