Algebraic quantum field theory (AQFT) is an application tolocal quantum physics ofC*-algebra theory. Also referred to as theHaag–Kastleraxiomatic framework forquantum field theory, because it was introduced byRudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set inMinkowski space, and mappings between those.

Let${\displaystyle \mathcal{O}}$ be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set${\displaystyle \{\mathcal{A}(O){\}}_{O\in \mathcal{O}}}$ ofvon Neumann algebras${\displaystyle \mathcal{A}(O)}$ on a commonHilbert space${\displaystyle \mathcal{H}}$ satisfying the following axioms:^[1]

• Isotony:${\displaystyle O_1\subset O_2}$ implies${\displaystyle \mathcal{A}(O_1)\subset \mathcal{A}(O_2)}$.
• Causality: If${\displaystyle O_1}$ is space-like separated from${\displaystyle O_2}$, then${\displaystyle [\mathcal{A}(O_1),\mathcal{A}(O_2)]=0}$.
• Poincaré covariance: A strongly continuous unitary representation${\displaystyle U(\mathcal{P})}$ of the Poincaré group${\displaystyle \mathcal{P}}$ on${\displaystyle \mathcal{H}}$ exists such that${\displaystyle \mathcal{A}(gO)=U(g)\mathcal{A}(O)U(g{)}^{\ast },g\in \mathcal{P}.}$
• Spectrum condition: The joint spectrum${\displaystyle \mathrm{S}\mathrm{p}(P)}$ of the energy-momentum operator${\displaystyle P}$ (i.e. the generator of space-time translations) is contained in the closed forward lightcone.
• Existence of a vacuum vector: A cyclic and Poincaré-invariant vector${\displaystyle \mathrm{\Omega }\in \mathcal{H}}$ exists.

The net algebras${\displaystyle \mathcal{A}(O)}$ are calledlocal algebras and the C* algebra${\displaystyle \mathcal{A}:=\overline{\bigcup _{O\in \mathcal{O}}\mathcal{A}(O)}}$ is called thequasilocal algebra.

LetMink be thecategory ofopen subsets of Minkowski space M withinclusion maps asmorphisms. We are given acovariant functor${\displaystyle \mathcal{A}}$ fromMink touC*alg, the category ofunital C* algebras, such that every morphism inMink maps to amonomorphism inuC*alg (isotony).

ThePoincaré group actscontinuously onMink. There exists apullback of thisaction, which is continuous in thenorm topology of${\displaystyle \mathcal{A}(M)}$ (Poincaré covariance).

Minkowski space has acausal structure. If anopen setV lies in thecausal complement of an open setU, then theimage of the maps

${\displaystyle \mathcal{A}(i_{U,U\cup V})}$

and

${\displaystyle \mathcal{A}(i_{V,U\cup V})}$

commute (spacelike commutativity). If${\displaystyle \overline{U}}$ is thecausal completion of an open setU, then${\displaystyle \mathcal{A}(i_{U,\overline{U}})}$ is anisomorphism (primitive causality).

Astate with respect to a C*-algebra is apositive linear functional over it with unitnorm. If we have a state over${\displaystyle \mathcal{A}(M)}$, we can take the "partial trace" to get states associated with${\displaystyle \mathcal{A}(U)}$ for each open set via thenetmonomorphism. The states over the open sets form apresheaf structure.

According to theGNS construction, for each state, we can associate aHilbert spacerepresentation of${\displaystyle \mathcal{A}(M).}$Pure states correspond toirreducible representations andmixed states correspond toreducible representations. Each irreducible representation (up toequivalence) is called asuperselection sector. We assume there is a pure state called thevacuum such that the Hilbert space associated with it is aunitary representation of thePoincaré group compatible with the Poincaré covariance of the net such that if we look at thePoincaré algebra, the spectrum with respect toenergy-momentum (corresponding tospacetime translations) lies on and in the positivelight cone. This is the vacuum sector.

More recently, the approach has been further implemented to include an algebraic version ofquantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize therenormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of ablack hole have been obtained.^{[citation needed]}

• Haag, Rudolf;Kastler, Daniel (1964),"An Algebraic Approach to Quantum Field Theory",Journal of Mathematical Physics,5 (7):848–861,Bibcode:1964JMP.....5..848H,doi:10.1063/1.1704187,ISSN 0022-2488,MR 0165864
• Haag, Rudolf (1996) [1992],Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics (2nd ed.), Berlin, New York:Springer-Verlag,doi:10.1007/978-3-642-61458-3,ISBN 978-3-540-61451-7,MR 1405610
• Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003)."The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory".Communications in Mathematical Physics.237 (1–2):31–68.arXiv:math-ph/0112041.Bibcode:2003CMaPh.237...31B.doi:10.1007/s00220-003-0815-7.S2CID 13950246.
• Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009)."Perturbative Algebraic Quantum Field Theory and the Renormalization Groups".Advances in Theoretical and Mathematical Physics.13 (5):1541–1599.arXiv:0901.2038.doi:10.4310/ATMP.2009.v13.n5.a7.S2CID 15493763.
• Bär, Christian;Fredenhagen, Klaus, eds. (2009).Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Lecture Notes in Physics. Vol. 786. Springer.doi:10.1007/978-3-642-02780-2.ISBN 978-3-642-02780-2.
• Brunetti, Romeo; Dappiaggi, Claudio;Fredenhagen, Klaus;Yngvason, Jakob, eds. (2015).Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer.doi:10.1007/978-3-319-21353-8.ISBN 978-3-319-21353-8.
• Rejzner, Kasia (2016).Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer.arXiv:1208.1428.Bibcode:2016paqf.book.....R.doi:10.1007/978-3-319-25901-7.ISBN 978-3-319-25901-7.
• Hack, Thomas-Paul (2016).Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer.arXiv:1506.01869.Bibcode:2016caaq.book.....H.doi:10.1007/978-3-319-21894-6.ISBN 978-3-319-21894-6.S2CID 119657309.
• Dütsch, Michael (2019).From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser.doi:10.1007/978-3-030-04738-2.ISBN 978-3-030-04738-2.S2CID 126907045.
• Yau, Donald (2019).Homotopical Quantum Field Theory. World Scientific.arXiv:1802.08101.doi:10.1142/11626.ISBN 978-981-121-287-1.S2CID 119168109.
• Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT".International Journal of Modern Physics A.38 (4n05).arXiv:2203.08053.doi:10.1142/S0217751X23300028.S2CID 247450696.

• Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
• Papers – A database of preprints on algebraic QFT
• Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg

• Axiomatic quantum field theory

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from December 2022