Mean-field approximation is the basic tool for getting overall insight into physical properties of many-body systems. The archetypal Landau mean-field theory of the critical behavior uses an expansion in the static, local order parameter around the critical point. A comprehensive mean-field approximation, however, aims at a global picture of equilibrium states in the whole range of the input parameters. This is usually achieved by an exact solution in a specific limit, such as long-range interaction or infinite lattice dimension. I will review in this talk various approaches to the construction of mean-field approximations.
For pedagogical reasons I will start with constructions of the mean-field approximation for the Ising model from which we derive its fundamental general characteristics and flaws: a self-consistent analytic, static, and local solution neglecting spatial fluctuations.
In the second part I will use the Hubbard model to explain the differences in the construction of mean-field approximations quantum dynamics brings in. I will derive the dynamical mean-field theory as the limit to infinite dimensions within the Baym and Kadanoff scheme of renormalizations the perturbation expansion. In the last part I will discuss options how to remove spurious critical behavior of static mean-field approximations by introducing renormalizations of the perturbation expansion for two-particle Green and response functions. The key element missed in the standard mean-field theories is a two-particle self-consistency. This will be introduced via the parquet approach within which, after suitable simplifications, one ends up with a static, local, semi-analytic theory free of unphysical behavior even in low dimensions.