Effective-action approach to a trapped Bose gas

The effective-action formalism is applied to a gas of bosons. The equations describing the condensate and the excitations are obtained using the loop expansion for the effective action. For a homogeneous gas, the expansion in terms of the diluteness parameter is identified in terms of the loop expansion. The loop expansion and the limits of validity of the well-known Bogoliubov [J. Phys. (Moscow) 11, 23 (1947)] and Popov, (Zh. Eksp. Teor. Fiz. 47, 1759 (1964) [Sov. Phys. JETP 20, 1185 (1965)]) equations are examined analytically for a homogeneous dilute Bose gas and numerically for a gas trapped in a harmonic-oscillator potential. The expansion to one-loop order, and hence the Bogoliubov equation, is shown to be valid for the zero-temperature trapped gas as long as the characteristic length of the trapping potential exceeds the s-wave scattering length.