Identities and Many-Body Approaches in Bose-Einstein Condensates

This paper discusses exact relations in Bose–Einstein condensates (BECs), 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 BECs. This paper will serve as a bridge between theories on exact relations and those on approximations in BECs.