Ward-Takahashi relation at finite temperature in Bose-Einstein condensation of trapped neutral atoms

We prove that the Ward-Takahashi relations at finite temperature are satisfied at the tree level for a system of a trapped Bose-Einstein condensate when the zero mode is included in the quasi-particle picture properly. This implies that the zero mode represents the Nambu-Goldstone mode and that it is introduced in a way consistent with the Goldstone theorem. In the proof, we employ thermo field dynamics which is a real-time operator formalism of quantum field theory at finite temperature.