The Bogoliubov–Parasyuk theorem in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
states that renormalized
Green's functions
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
and matrix elements of the
scattering matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
(''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 perturbation 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 divergences in any order of perturbation theory, establishes correctness of this procedure, and guarantees the uniqueness of the obtained results.
The theorem was proved by
Nikolay Bogoliubov
Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretic ...
and
Ostap Parasyuk Ostap ( uk, Остап) is a Ukrainian male given name. Its Russian counterpart is Evstafiy. It derives from the Greek name Eustathius.
People with this name include:
*Ostap Bender, a fictional character from the Russian novel ''The Twelve Chai ...
in 1955. The proof of the Bogoliubov–Parasyuk theorem was simplified later.
See also
*
Renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
*
Krylov-Bogolyubov theorem on the existence of invariant measures in dynamics.
References
* 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.
{{DEFAULTSORT:Bogoliubov-Parasyuk Theorem
Quantum field theory
Theorems in quantum mechanics