Editing Quantum field theory (section)

==Mathematical rigor==
In spite of its overwhelming success in particle physics and condensed matter physics, QFT itself lacks a formal mathematical foundation. For example, according to [[Haag's theorem]], there does not exist a well-defined [[interaction picture]] for QFT, which implies that [[perturbation theory (quantum mechanics)|perturbation theory]] of QFT, which underlies the entire [[Feynman diagram]] method, is fundamentally ill-defined.<ref>{{cite journal |last=Haag |first=Rudolf |author-link=Rudolf Haag |date=1955 |title=On Quantum Field Theories |url=http://cdsweb.cern.ch/record/212242/files/p1.pdf |journal=Dan Mat Fys Medd |volume=29 |issue=12 }}</ref>

However, ''perturbative'' quantum field theory, which only requires that quantities be computable as a formal power series without any convergence requirements, can be given a rigorous mathematical treatment. In particular, [[Kevin Costello]]'s monograph ''Renormalization and Effective Field Theory''<ref name=costello>Kevin Costello, ''Renormalization and Effective Field Theory'', Mathematical Surveys and Monographs Volume 170, American Mathematical Society, 2011, {{ISBN|978-0-8218-5288-0}}</ref> provides a rigorous formulation of perturbative renormalization that combines both the effective-field theory approaches of [[Leo Kadanoff|Kadanoff]], [[Kenneth G. Wilson|Wilson]], and [[Joseph Polchinski|Polchinski]], together with the [[Batalin-Vilkovisky]] approach to quantizing gauge theories. Furthermore,  perturbative path-integral methods, typically understood as formal computational methods inspired from finite-dimensional integration theory,<ref name=ren>Gerald B. Folland, ''Quantum Field Theory: A Tourist Guide for Mathematicians'', Mathematical Surveys and Monographs Volume 149, American Mathematical Society, 2008, {{ISBN|0821847058}} | chapter=8</ref> can be given a sound mathematical interpretation from their finite-dimensional analogues.<ref name="nguyen">{{Cite journal |last=Nguyen |first=Timothy |arxiv=1505.04809 |title=The perturbative approach to path integrals: A succinct mathematical treatment |journal=J. Math. Phys.|volume=57|year=2016 |issue=9 |page=092301 |doi=10.1063/1.4962800|bibcode=2016JMP....57i2301N |s2cid=54813572 }}</ref>

Since the 1950s,<ref name="buchholz">{{Cite book |last=Buchholz |first=Detlev |chapter=Current Trends in Axiomatic Quantum Field Theory |author-link=Detlev Buchholz |arxiv=hep-th/9811233 |title=Quantum Field Theory |series=Lecture Notes in Physics |volume=558 |pages=43–64 |year=2000 |doi=10.1007/3-540-44482-3_4 |bibcode=2000LNP...558...43B |isbn=978-3-540-67972-1 |s2cid=5052535 }}</ref> theoretical physicists and mathematicians have attempted to organize all QFTs into a set of [[axiom]]s, in order to establish the existence of concrete models of relativistic QFT in a mathematically rigorous way and to study their properties. This line of study is called [[constructive quantum field theory]], a subfield of [[mathematical physics]],<ref name="summers">{{cite arXiv |last=Summers |first=Stephen J. |eprint=1203.3991v2 |title=A Perspective on Constructive Quantum Field Theory |class=math-ph |date=2016-03-31 }}</ref>{{rp|2}} which has led to such results as [[CPT theorem]], [[spin–statistics theorem]], and [[Goldstone's theorem]],<ref name="buchholz" /> and also to mathematically rigorous constructions of many interacting QFTs in two and three spacetime dimensions, e.g. two-dimensional scalar field theories with arbitrary polynomial interactions,<ref name="Simon">{{cite book|last=Simon|first=Barry|title=The P(phi)_2 Euclidean (quantum) field theory|publisher=Princeton University Press|year=1974|isbn=0-691-08144-1|publication-place=Princeton, New Jersey|page=|oclc=905864308}}</ref> the three-dimensional scalar field theories with a quartic interaction, etc.<ref name="Glimm1987">{{cite book|last1=Glimm|first1=James|title=Quantum Physics : a Functional Integral Point of View|last2=Jaffe|first2=Arthur|publisher=Springer New York|year=1987|isbn=978-1-4612-4728-9|publication-place=New York, NY|page=|oclc=852790676}}</ref>

Compared to ordinary QFT, [[topological quantum field theory]] and [[conformal field theory]] are better supported mathematically — both can be classified in the framework of [[representation (mathematics)|representation]]s of [[cobordism]]s.<ref>{{cite arXiv |last1=Sati |first1=Hisham |last2=Schreiber |first2=Urs |author-link2=Urs Schreiber |eprint=1109.0955v2 |title=Survey of mathematical foundations of QFT and perturbative string theory |class=math-ph |date=2012-01-06 }}</ref>

[[Algebraic quantum field theory]] is another approach to the axiomatization of QFT, in which the fundamental objects are local operators and the algebraic relations between them. Axiomatic systems following this approach include [[Wightman axioms]] and [[Haag–Kastler axioms]].{{r|summers|page1=2–3}} One way to construct theories satisfying Wightman axioms is to use [[Osterwalder–Schrader axioms]], which give the necessary and sufficient conditions for a real time theory to be obtained from an [[imaginary time]] theory by [[analytic continuation]] ([[Wick rotation]]).{{r|summers|page1=10}}

[[Yang–Mills existence and mass gap]], one of the [[Millennium Prize Problems]], concerns the well-defined existence of [[Yang–Mills theory|Yang–Mills theories]] as set out by the above axioms. The full problem statement is as follows.<ref>{{cite web |url=http://www.claymath.org/sites/default/files/yangmills.pdf |title=Quantum Yang–Mills Theory |last1=Jaffe |first1=Arthur |last2=Witten |first2=Edward |author-link1=Arthur Jaffe |author-link2=Edward Witten |publisher=[[Clay Mathematics Institute]] |access-date=2018-07-18 |archive-date=2015-03-30 |archive-url=https://web.archive.org/web/20150330003812/http://www.claymath.org/sites/default/files/yangmills.pdf |url-status=dead }}</ref>
{{Blockquote|
Prove that for any [[compact space|compact]] [[simple group|simple]] [[gauge group]] {{math|''G''}}, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a [[mass gap]] {{math|Δ > 0}}. Existence includes establishing axiomatic properties at least as strong as those cited in {{Harvtxt|Streater|Wightman|1964}}, {{Harvtxt|Osterwalder|Schrader|1973}} and {{Harvtxt|Osterwalder|Schrader|1975}}.
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