Algebraic approach to renormalization

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.