Identities and Many-Body Approaches in Bose-Einstein Condensates

This paper discusses exact relations in the Bose-Einstein condensates, starting from basic properties of an ideal Bose gas. In particular, focused on are the Hugenholtz-Pines relation, Nepomnyashchii-Nepomnyashchii identity, and identities for the density response function. After introducing these exact relations, a few approaches of many-body approximations are discussed, which satisfy the exact relations in the Bose-Einstein condensates. This paper will serve as a bridge between theories on exact relations and those on approximations in the Bose-Einstein condensates.