Effective Action for Bose-Einstein Condensates

We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field psi itself acquires a finite thermodynamic average <psi > besides two-point Green's function (G) over cap to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of <psi > and (G) over cap so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particlereducible structure due to <psi > not equal 0 to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.