Causal perturbation theory is a mathematically rigorous approach to

renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...

theory, which makes it possible to put the theoretical setup of perturbative

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

on a sound mathematical basis. It goes back to a 1973 work by Henri Epstein and Vladimir Jurko Glaser.

Overview

When developing

quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...

in the 1940s, Shin'ichiro Tomonaga,

Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize-winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...

Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...

, and

Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...

discovered that, in perturbative calculations, problems with divergent integrals abounded. The divergences appeared in calculations involving

Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...

with closed loops of virtual particles. It is an important observation that in perturbative

, time-ordered products of

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

s arise in a natural way and may lead to ultraviolet divergences in the corresponding calculations. From the

generalized functions In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...

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 Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of Distribution (mathematics), distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awar ...

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 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 ...

. In their original work, Epstein and Glaser studied only theories involving scalar (spinless) particles. Since then, the causal approach has been applied also to a wide range of

gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

, which represent the most important quantum field theories in modern physics.

References

Additional reading

* * * {{cite journal , last1=Dütsch , first1=Michael , last2=Scharf , first2=Günter , title=Perturbative gauge invariance: the electroweak theory , journal=Annalen der Physik , publisher=Wiley , volume=8 , issue=5 , year=1999 , issn=0003-3804 , doi=10.1002/(sici)1521-3889(199905)8:5<359::aid-andp359>3.0.co;2-m , pages=359–387 , language=de , arxiv=hep-th/9612091, bibcode=1999AnP...511..359D , s2cid=122295550 Renormalization group