  Quantum-field-theoretic descriptions of interacting condensed bosons have
suffered from the lack of self-consistent approximation schemes satisfying
Goldstone's theorem and dynamical conservation laws simultaneously. We present
a procedure to construct such approximations systematically by using either an
exact relation for the interaction energy or the Hugenholtz-Pines relation to
express the thermodynamic potential in a Luttinger-Ward form. Inspection of the
self-consistent perturbation expansion up to the third order with respect to
the interaction shows that the two relations yield a unique identical result at
each order, reproducing the conserving-gapless mean-field theory [T. Kita, J.
Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The
uniqueness implies that the series becomes exact when infinite terms are
retained. We also derive useful expressions for the entropy and superfluid
density in terms of Green's function and a set of real-time dynamical equations
to describe thermalization of the condensate.
