  In close analogy to the Bloch-Feshbach formalism known from the theory of
nuclear dynamics, I develop a mathematical framework that allows one to
understand renormalization in terms of purely algebraic operations
(projections, dilatations) in Hilbert space. This algebraic approach is put to
the test in the study of the low-energy dynamics of interacting quantum gases,
and proves to be efficient in deriving such diverse results as the
renormalization group equation for an interacting Bose gas, the $\beta$
function of $\phi^4$ theory, the screening of fermion-fermion interactions or
the BCS instability.
