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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 perspective.

There are two main challenges in this discipline. First, one must propose a set of axioms which describe the general properties of any mathematical object that deserves to be called a "quantum field theory". Then, one gives rigorous mathematical constructions of examples satisfying these axioms.

Analytic approaches
Wightman axioms

The first set of axioms for quantum field theories, known as the Wightman axioms, were proposed by Arthur Wightman in the early 1950s. These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space. In practice, one often uses the Wightman reconstruction theorem, which guarantees that the operator-valued distributions and the Hilbert space can be recovered from the collection of correlation functions.
Osterwalder–Schrader axioms
Main article: Osterwalder–Schrader theorem

The correlation functions of a QFT satisfying the Wightman axioms often can be analytically continued from Lorentz signature to Euclidean signature. (Crudely, one replaces the time variable t with imaginary time \( \tau =-{\sqrt {-1}}t \); the factors of \( {\sqrt {-1}} \) change the sign of the time-time components of the metric tensor.) The resulting functions are called Schwinger functions. For the Schwinger functions there is a list of conditions—analyticity, permutation symmetry, Euclidean covariance, and reflection positivity—which a set of functions defined on various powers of Euclidean space-time must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.
Haag–Kastler axioms

The Haag–Kastler axioms axiomatize QFT in terms of nets of algebras.
See also

Dirac–von Neumann axioms

References
Streater, R. F.; Wightman, A. S. (1964). PCT, Spin and Statistics, and All That. New York: W. A. Benjamin. OCLC 930068.
Bogoliubov, N.; Logunov, A.; Todorov, I. (1975). Introduction to Axiomatic Quantum Field Theory. Reading, Massachusetts: W. A. Benjamin. OCLC 1527225.
Araki, H. (1999). Mathematical Theory of Quantum Fields. Oxford University Press. ISBN 0-19-851773-4.

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Quantum field theories
Standard
Theories

Chern–Simons Conformal field theory Ginzburg–Landau Kondo effect Local QFT Noncommutative QFT Quantum Yang–Mills Quartic interaction sine-Gordon String theory Toda field Topological QFT Yang–Mills Yang–Mills–Higgs

Models

Chiral Non-linear sigma Schwinger Standard Model Thirring–Wess Wess–Zumino Wess–Zumino–Witten Yukawa

Four-fermion interactions

Theories

BCS theory Fermi's interaction Luttinger liquid Top quark condensate

Models

Gross–Neveu Hubbard Nambu–Jona-Lasinio Thirring Thirring–Wess

Related

History Axiomatic QFT Loop quantum gravity Loop quantum cosmology QFT in curved spacetime Quantum chaos Quantum chromodynamics Quantum dynamics Quantum electrodynamics
links Quantum gravity
links Quantum hadrodynamics Quantum hydrodynamics Quantum information Quantum information science
links Quantum logic Quantum thermodynamics

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