  The order-by-order renormalization of the self-consistent mean-field
potential in many-body perturbation theory for normal Fermi systems is
investigated in detail. Building on previous work mainly by Balian and de
Dominicis, as a key result we derive a thermodynamic perturbation series that
manifests the consistency of the adiabatic zero-temperature formalism with
perturbative statistical mechanics---for both isotropic and anisotropic
systems---and satisfies at each order and for all temperatures the
thermodynamic relations associated with Fermiliquid theory. These properties
are proved to all orders.
