This chapter examines a simple example of a quantum liquid that is a weakly nonideal gas. In this case, the interaction among gas particles is relatively small. The chapter examines a weakly interacting Bose gas at almost zero temperature. Interactions in atomic gases are usually very weak, but they can have important effects. It shows that the interactions among gas particles are repulsive. This implies that the scattering amplitude is positive. It is important to note that for the case of a Bose gas, the assumption is connected with the fact that even for infinitesimal attractive forces, a Bose gas cannot remain dilute at low temperatures. All particles in the condensate are in the ground state of an ideal Bose gas. This is for the zero-state energy. The gas is often said to be in the condensate or the condensed state. The diagonalization is done by introducing the canonical transformation known as a Bogoliubov transformation.
Bogoliubov-Zubarev's exact transformation, specific to a weakly interacting Bose gas having a condensate, yields a Hamiltonian describing a gas of interacting excitations. The non-Hermiticity of the Hamiltonian arises from the nonorthogonality of the physical states; it has some significant consequences. The excitation spectrum is gapless in every approximation; the lowest-order correction is evaluated numerically and shown to be small. It is concluded that the approach presents a prospective method for a quantitative description of the weakly interacting Bose gas.