Editing Constructive quantum field theory

{{Short description|Formalization of quantum field theory}}
{{No footnotes|date=June 2022}}
In [[mathematical physics]], '''[[Constructive proof|constructive]] quantum field theory''' is the field devoted to showing that [[quantum field theory]] can be defined in terms of precise mathematical structures.  This demonstration requires new [[mathematics]], in a sense analogous to classical [[real analysis]], putting [[calculus]] on a [[mathematical rigor|mathematically rigorous]] foundation. [[weak force|Weak]], [[strong force|strong]], and [[Electromagnetism|electromagnetic]] [[Fundamental interaction|forces of nature]] are believed to have their natural description in terms of [[Field (physics)#Quantum fields|quantum fields]].

Attempts to put [[quantum field theory]] on a basis of completely defined concepts have involved most branches of mathematics, including [[functional analysis]], [[differential equation]]s, [[probability theory]], [[representation theory]], [[geometry]], and [[topology]]. It is known that a ''quantum field'' is inherently hard to handle using conventional mathematical techniques like explicit estimates.  This is because a quantum field has the general nature of an [[Wightman axioms|operator-valued distribution]], a type of object from [[mathematical analysis]].  The [[existence theorem]]s for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. 

One discovery of the theory that can be related in non-technical terms, is that the dimension ''d'' of the [[spacetime]] involved is crucial. Notable work in the field by [[James Glimm]] and [[Arthur Jaffe]] showed that with ''d'' < 4 many examples can be found. Along with work of their students, coworkers, and others, constructive field theory resulted in a mathematical foundation and exact interpretation to what previously was only a set of [[Algorithm|recipes]], also in the case ''d'' < 4. 

[[Theoretical physicist]]s had given these rules the name "[[renormalization]]," but most physicists had been skeptical about whether they could be turned into a [[Theory#Mathematics|mathematical theory]].  Today one of the most important open problems, both in theoretical physics and in mathematics, is to establish similar results for [[gauge theory]] in the realistic case ''d'' = 4.

The traditional basis of constructive quantum field theory is the set of [[Wightman axioms]].  [[Konrad Osterwalder]] and [[Robert Schrader]] showed that there is an equivalent problem in mathematical probability theory. The examples with ''d'' < 4 satisfy the Wightman axioms as well as the [[Schwinger function|Osterwalder–Schrader axioms]] .  They also fall in the related framework introduced by [[Rudolf Haag]] and [[Daniel Kastler]], called [[Local quantum field theory|algebraic quantum field theory]].  There is a firm belief in the physics community that the gauge theory of [[Yang Chen-Ning|C.N. Yang]] and [[Robert Mills (physicist)|Robert Mills]] (the [[Yang–Mills theory]]) can lead to a tractable theory, but new ideas and new methods will be required to actually establish this, and this could take many years.

==External links==
*{{Cite journal|last=Jaffe|first=Arthur|date=2000|title=Constructive Quantum Field Theory|pages=111–127|journal=Mathematical Physics 2000|url=http://www.arthurjaffe.com/Assets/pdf/CQFT.pdf|doi=10.1142/9781848160224_0007|isbn=978-1-86094-230-3}}
* {{cite book | last=Baez | first=John | title=Introduction to algebraic and constructive quantum field theory | publisher=Princeton University Press | publication-place=Princeton, New Jersey | year=1992 | isbn=978-0-691-60512-8 | oclc=889252663 |url=http://math.ucr.edu/home/baez/bsz.html}}

[[Category:Axiomatic quantum field theory]]
[[Category:Functional analysis]]