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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 renormalization procedure is a specific procedure to make these divergent integrals finite and obtain (and predict) finite values for physically measurable quantities. 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 guarantees that computed within the pertrubation expansion Green's functions and matrix elements of the scattering matrix are finite for any renormalized quantum field theory. The theorem specifies a concrete procedure (the Bogoliubov–Parasyuk R-operation) for subtraction of divergencies in any order of pertrubation theory, establishes correctness of this procedure, and guarantees the uniqueness of the obtained results.

The theorem was proved by Nikolay Bogoliubov and Ostap Parasyuk in 1955.[1][2] The proof of the Bogoliubov–Parasyuk theorem was simplified later.[3]
See also

O. I. Zav'yalov (1994). "Bogolyubov's R-operation and the Bogolyubov–Parasyuk theorem", Russian Math. Surveys, 49(5): 67—76 (in English).
D. V. Shirkov (1994): "The Bogoliubov renormalization group", Russian Math. Surveys 49(5): 155—176.


^ Bogoliubov, Nikolay N.; Ostap S. Parasyuk (1955). "[A theory of multiplication of causative singular functions]" (in Russian). Doklady Akademii Nauk SSSR 100: 25–28.
^ N. N. Bogoliubov, O. S. Parasyuk (1957). "Uber die multiplikation der kausalfunktionen in der quantentheorie der felder" (in German). Acta Mathematica 97: 227–266. doi:10.1007/BF02392399.
^ S. A. Anikin, O. I. Zav'yalov, and M. K. Polivanov (1973). "Simple proof of the Bogolyubov–Parasyuk theorem". Acta Mathematica 17 (2): 1082–1088. Bibcode 1973TMP....17.1082A. doi:10.1007/BF01037256.

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