The methods of quantum chemistry and solid state theory to solve the
many-body problem are reviewed. We start with the definitions of reduced
density matrices, their properties (contraction sum rules, spectral
resolutions, cumulant expansion, N-representability), and their determining
equations (contracted Schr\"odinger equations) and we summarize recent
extensions and generalizations of the traditional quantum chemical methods, of
the density functional theory, and of the quasi-particle theory: from finite to
extended systems (incremental method), from density to density matrix (density
matrix functional theory), from weak to strong correlation (dynamical mean
field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous
and finite systems. Measures of the correlation strength are discussed. The
cumulant two-body reduced density matrix proves to be a key quantity. Its
spectral resolution contains geminals, being possibly the solutions of an
approximate effective two-body equation, and the idea is sketched of how its
contraction sum rule can be used for a variational treatment.Comment: 27 pages, conference contributio
