Goldstone Theorem, Hugenholtz-Pines Theorem and Ward-Takahashi Relation in Finite Volume Bose-Einstein Condensed Gases

We construct an approximate scheme based on the concept of the spontaneous symmetry breakdown, satisfying the Goldstone theorem, for finite volume Bose-Einstein condensed gases in both zero and finite temperature cases. In this paper, we discuss the Bose-Einstein condensation in a box with periodic boundary condition and do not assume the thermodynamic limit. When energy spectrum is discrete, we found that it is necessary to deal with the Nambu-Goldstone mode explicitly without the Bogoliubov's prescription, in which zero-mode creation- and annihilation-operators are replaced with a {\it c}-number by hand, for satisfying the Goldstone theorem. Furthermore, we confirm that the unitary inequivalence of vacua in the spontaneous symmetry breakdown is true for the finite volume system.