Causal perturbation theory
Encyclopedia
Causal perturbation theory is a mathematically rigorous approach to renormalization
theory, which makes
it possible to put the theoretical setup of perturbative quantum field theory
on a sound mathematical basis.
It goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser
("The role of locality in perturbation
theory", published in Annales Poincaré Phys. Theor. A19, p. 211, 1973).
When developing quantum electrodynamics
in the 1940s, Shin'ichiro Tomonaga, Julian Schwinger
,
Richard Feynman
, and Freeman Dyson
discovered that, in perturbative calculations, problems with divergent
integrals abounded. The divergences appeared in calculations involving Feynman diagrams with closed loops of virtual particles. It is an important observation that in perturbative quantum field theory
, time-ordered products of distribution
s arise in a natural way and may lead to ultraviolet divergence
s in the corresponding calculations.
From the mathematical point of view, the problem of divergences is rooted in the fact that the theory of
distributions
is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as was proved by Laurent Schwartz
in the 1950s.
Epstein and Glaser solved this problem for a special class of distributions that
fulfill a causality
condition, which itself is a basic requirement in axiomatic quantum field theory
. In their original work, Epstein and Glaser studied only theories involving scalar (spinless)
particles. In the meantime, the causal approach has been applied also to a wide range of gauge theories,
which represent the most important quantum field theories in modern physics.
Renormalization
In 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....
theory, which makes
it possible to put the theoretical setup of perturbative quantum field theory
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...
on a sound mathematical basis.
It goes back to a seminal work by Henri Epstein and Vladimir Jurko Glaser
Vladimir Jurko Glaser
Vladimir Jurko Glaser was a theoretical physicist working on quantum field theory and the canonization of the analytic S-matrix....
("The role of locality in perturbation
theory", published in Annales Poincaré Phys. Theor. A19, p. 211, 1973).
When developing quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
in the 1940s, Shin'ichiro Tomonaga, Julian Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
,
Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
, and Freeman Dyson
Freeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
discovered that, in perturbative calculations, problems with divergent
integrals abounded. The divergences appeared in calculations involving Feynman diagrams with closed loops of virtual particles. It is an important observation that in perturbative quantum field theory
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...
, time-ordered products of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s arise in a natural way and may lead to ultraviolet divergence
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...
s in the corresponding calculations.
From the mathematical point of view, the problem of divergences is rooted in the fact that the theory of
distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as was proved by Laurent Schwartz
Laurent Schwartz
Laurent-Moïse Schwartz was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work...
in the 1950s.
Epstein and Glaser solved this problem for a special class of distributions that
fulfill a causality
Causality (physics)
Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....
condition, which itself is a basic requirement in axiomatic quantum field theory
Axiomatic quantum field theory
Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial...
. In their original work, Epstein and Glaser studied only theories involving scalar (spinless)
particles. In the meantime, the causal approach has been applied also to a wide range of gauge theories,
which represent the most important quantum field theories in modern physics.