Bogoliubov-Parasyuk theorem
Encyclopedia
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. The proof of the Bogoliubov–Parasyuk theorem was simplified later.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
states that renormalized Green's functions and matrix elements of the scattering matrix (S-matrix) are free of ultraviolet divergencies
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
. 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
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
(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. The proof of the Bogoliubov–Parasyuk theorem was simplified later.
See also
- RenormalizationRenormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
- 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. ShirkovDmitry ShirkovDmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.-Biography:...
(1994): "The Bogoliubov renormalization group", Russian Math. Surveys 49(5): 155—176.