  Quantum field theory of equilibrium and nonequilibrium Bose-Einstein
condensates is formulated so as to satisfy three basic requirements: the
Hugenholtz-Pines relation; conservation laws; identities among vertices
originating from Goldstone's theorem I. The key inputs are irreducible
four-point vertices, in terms of which we derive a closed system of equations
for Green's functions, three- and four-point vertices, and two-particle Green's
functions. It enables us to study correlated Bose-Einstein condensates with a
gapless branch of single-particle excitations without encountering any infrared
divergence. The single- and two-particle Green's functions are found to share
poles, i.e., the structure of the two-particle Green's functions predicted by
Gavoret and Nozi\`eres for a homogeneous condensate at $T=0$ is also shown to
persist at finite temperatures, in the presence of inhomogeneity, and also in
nonequilibrium situations.
