The Bogoliubov–Parasyuk theorem in quantum field theory states that renormalized Green's functions and matrix elements of the scattering matrix (S-matrix) are free of ultraviolet divergencies.
Green's functions and scattering matrix are the fundamental objects in quantum field theory which determine basic physically measurable quantities.
Formal expressions for Green's functions and S-matrix in any physical quantum field theory contain divergent integrals (i.e., integrals which take infinite values) and therefore formally these expressions are meaningless.
The Bogoliubov–Parasyuk theorem states that for a wide class of quantum field theories, called renormalizable field theories, these divergent integrals can be made finite in a regular way using a finite (and small) set of certain elementary subtractions of divergencies.
The theorem specifies a concrete procedure (the Bogoliubov–Parasyuk R-operation) for subtraction of divergences in any order of perturbation theory, establishes correctness of this procedure, and guarantees the uniqueness of the obtained results.