  The Luttinger-Ward functional was proposed more than five decades ago to
provide a link between static and dynamic quantities in a quantum many-body
system. Despite its widespread usage, the derivation of the Luttinger-Ward
functional remains valid only in the formal sense, and even the very existence
of this functional has been challenged by recent numerical evidence. In a
simpler and yet highly relevant regime, namely the Euclidean lattice field
theory, we rigorously prove that the Luttinger-Ward functional is a
well-defined universal functional over all physical Green's functions. Using
the Luttinger-Ward functional, the free energy can be variationally minimized
with respect to Green's functions in its domain. We then derive the widely used
bold diagrammatic expansion rigorously, without relying on formal arguments
such as partial resummation of bare diagrams to infinite order.
