  On the basis of the Luttinger-Ward functional for interacting many-body
systems given in terms of full Green's function $G$ and the bare interaction
vertex $\Gamma^{(0)}$, we develop a novel Legendre transformation to express
the grand thermodynamic potential $\Omega$ as a functional of $G$ and the
renormalized interaction vertex $\Gamma$ so that (i) $G$ and $\Gamma$ obey the
stationarity conditions $\delta\Omega/\delta G=0$ and $\delta\Omega
/\delta\Gamma=0$ and (ii) $\Gamma$ reduces to $\Gamma^{(0)}$ in the
weak-coupling limit. The formalism enables us to perform microscopic studies of
thermodynamic, single-particle, and two-particle properties in a unified
self-consistent conserving framework.
