Causal perturbation theory is a mathematically rigorous approach to

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 va ...

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 Henri is an Estonian, Finnish, French, German and Luxembourgish form of the masculine given name Henry. People with this given name ; French noblemen :'' See the ' List of rulers named Henry' for Kings of France named Henri.'' * Henri I de Mon ...

and Vladimir Jurko Glaser.

Overview

When developing

quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...

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, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...

, and

Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...

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 introduce ...

with closed loops of virtual particles. It is an important observation that in perturbative quantum field theory,

time-ordered In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). H ...

products of distributions arise in a natural way and may lead to

ultraviolet divergence In physics, an ultraviolet divergence or UV divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with unbounded energy, or, equivalently, because of physical phenomena at infi ...

s in the corresponding calculations. From the

generalized functions In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous function ...

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 distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...

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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...

, 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