Editing Algebraic quantum field theory

{{Short description|Axiomatic approach to quantum field theory}}
'''Algebraic quantum field theory''' ('''AQFT''') is an application to '''local quantum physics''' of [[C*-algebra]] theory. Also referred to as the '''Haag–Kastler [[axiomatic framework]]''' for [[quantum field theory]], because it was introduced by {{harvs|txt|last=Haag|first=Rudolf|authorlink=Rudolf Haag|last2=Kastler|first2=Daniel|author2-link=Daniel Kastler|year=1964}}. The axioms are stated in terms of an algebra given for every open set in [[Minkowski space]], and mappings between those.

== {{anchor|Overview}}Haag–Kastler axioms ==
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Let <math>\mathcal{O}</math> be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set <math>\{\mathcal{A}(O)\}_{O\in\mathcal{O}}</math> of [[von Neumann algebra]]s <math>\mathcal{A}(O)</math> on a common [[Hilbert space]] <math>\mathcal{H}</math> satisfying the following axioms:<ref>{{cite book
 | last1=Baumgärtel | first1=Hellmut | title=Operatoralgebraic Methods in Quantum Field Theory | year=1995 | publisher=Akademie Verlag | location=Berlin | isbn=3-05-501655-6
}}</ref> 
* ''Isotony'': <math>O_1 \subset O_2</math> implies <math>\mathcal{A}(O_1) \subset \mathcal{A}(O_2)</math>.
* ''Causality'': If <math>O_1</math> is space-like separated from <math>O_2</math>, then <math>[\mathcal{A}(O_1),\mathcal{A}(O_2)]=0</math>.
* ''Poincaré covariance'': A strongly continuous unitary representation <math>U(\mathcal{P})</math> of the Poincaré group <math>\mathcal{P}</math> on <math>\mathcal{H}</math> exists such that <math>\mathcal{A}(gO) = U(g) \mathcal{A}(O) U(g)^*,\,\,g \in \mathcal{P}.</math>
* ''Spectrum condition'': The joint spectrum <math>\mathrm{Sp}(P)</math> of the energy-momentum operator <math>P</math> (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 <math>\Omega\in\mathcal{H}</math> exists.

The net algebras <math>\mathcal{A}(O)</math> are called ''local algebras'' and the C* algebra <math>\mathcal{A} := \overline{\bigcup_{O\in\mathcal{O}}\mathcal{A}(O)}</math> is called the ''quasilocal algebra''.

== Category-theoretic formulation <!--'Isotony' redirects here-->==
Let '''Mink''' be the [[category theory|category]] of [[open subset]]s of Minkowski space M with [[inclusion map]]s as [[morphism]]s. We are given a [[covariant functor]] <math>\mathcal{A}</math> from '''Mink''' to '''uC*alg''', the category of [[unital algebra|unital]] C* algebras, such that every morphism in '''Mink''' maps to a [[monomorphism]] in '''uC*alg''' ('''isotony'''<!--boldface per WP:R#PLA-->).

The [[Poincaré group]] acts [[continuity (topology)|continuously]] on '''Mink'''. There exists a [[pullback]] of this [[Group action (mathematics)|action]], which is continuous in the [[norm topology]] of <math>\mathcal{A}(M)</math> ([[Poincaré covariance]]).

Minkowski space has a [[causal structure]]. If an [[open set]] ''V'' lies in the [[causal complement]] of an open set ''U'', then the [[Image (mathematics)|image]] of the maps

:<math>\mathcal{A}(i_{U,U\cup V})</math>

and

:<math>\mathcal{A}(i_{V,U\cup V})</math>

[[Commutative operation|commute]] (spacelike commutativity). If <math>\bar{U}</math> is the [[causal completion]] of an open set ''U'', then <math>\mathcal{A}(i_{U,\bar{U}})</math> is an [[isomorphism]] (primitive causality).

A [[state (functional analysis)|state]] with respect to a C*-algebra is a [[positive linear functional]] over it with unit [[norm (mathematics)|norm]]. If we have a state over <math>\mathcal{A}(M)</math>, we can take the "[[partial trace]]" to get states associated with <math>\mathcal{A}(U)</math> for each open set via the [[net (mathematics)|net]] [[monomorphism]]. The states over the open sets form a [[presheaf]] structure.

According to the [[GNS construction]], for each state, we can associate a [[Hilbert space]] [[group representation|representation]] of <math>\mathcal{A}(M).</math> [[Pure state]]s correspond to [[irreducible representation]]s and [[Mixed state (physics)|mixed state]]s correspond to [[reducible representation]]s. Each irreducible representation (up to [[Equivalence relation|equivalence]]) is called a [[superselection sector]]. We assume there is a pure state called the [[vacuum]] such that the Hilbert space associated with it is a [[unitary representation]] of the [[Poincaré group]] compatible with the Poincaré covariance of the net such that if we look at the [[Poincaré algebra]], the spectrum with respect to [[energy-momentum 4-vector|energy-momentum]] (corresponding to [[Poincare group|spacetime translation]]s) lies on and in the positive [[light cone]]. This is the vacuum sector.

== QFT in curved spacetime ==
More recently, the approach has been further implemented to include an algebraic version of [[quantum field theory in curved spacetime]]. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the [[renormalization]] procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a [[black hole]] have been obtained.{{cn|date=December 2022}}

== References ==
<references/>

== Further reading ==
*{{Citation | last1=Haag | first1=Rudolf | author-link1=Rudolf Haag | last2=Kastler | first2=Daniel | author-link2=Daniel Kastler | title=An Algebraic Approach to Quantum Field Theory | doi=10.1063/1.1704187 | mr=0165864  | year=1964 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=5 | issue=7 | pages=848–861|bibcode = 1964JMP.....5..848H |url=https://aip.scitation.org/doi/10.1063/1.1704187 | url-access=subscription }}
*{{Citation | last1=Haag | first1=Rudolf | author-link1=Rudolf Haag | title=Local Quantum Physics: Fields, Particles, Algebras | orig-year=1992 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Theoretical and Mathematical Physics | isbn=978-3-540-61451-7 | mr=1405610  | year=1996 | url=https://link.springer.com/book/10.1007/978-3-642-61458-3 | doi=10.1007/978-3-642-61458-3 | url-access=subscription }}
*{{cite journal |last1=Brunetti |first1=Romeo |last2=Fredenhagen |first2=Klaus |last3=Verch |first3=Rainer |title=The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory |journal=[[Communications in Mathematical Physics]] |date=2003 |volume=237 |issue=1–2 |pages=31–68 |doi=10.1007/s00220-003-0815-7 |url=https://link.springer.com/article/10.1007/s00220-003-0815-7 |arxiv=math-ph/0112041|bibcode=2003CMaPh.237...31B |s2cid=13950246 }}
*{{cite journal |last1=Brunetti |first1=Romeo |last2=Dütsch |first2=Michael |last3=Fredenhagen |first3=Klaus |title=Perturbative Algebraic Quantum Field Theory and the Renormalization Groups |journal=[[Advances in Theoretical and Mathematical Physics]] |date=2009 |volume=13 |issue=5 |pages=1541–1599 |doi=10.4310/ATMP.2009.v13.n5.a7 |url=https://inspirehep.net/literature/811019 |arxiv=0901.2038|s2cid=15493763 }}
*{{cite book |editor-last1=Bär |editor-first1=Christian |editor-link1=Christian Bär |editor-last2=Fredenhagen |editor-first2=Klaus |editor-link2=Klaus Fredenhagen |title=Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations |publisher=Springer |doi=10.1007/978-3-642-02780-2 |year=2009 |series=Lecture Notes in Physics |volume=786 |isbn=978-3-642-02780-2 |url=https://link.springer.com/book/10.1007/978-3-642-02780-2}}
*{{cite book |editor-last1=Brunetti |editor-first1=Romeo |editor-last2=Dappiaggi |editor-first2=Claudio |editor-last3=Fredenhagen |editor-first3=Klaus |editor-link3=Klaus Fredenhagen |editor-last4=Yngvason |editor-first4=Jakob |editor-link4=Jakob Yngvason |title=Advances in Algebraic Quantum Field Theory |year=2015 |publisher=Springer |doi=10.1007/978-3-319-21353-8 |series=Mathematical Physics Studies |isbn=978-3-319-21353-8 |url=https://link.springer.com/book/10.1007/978-3-319-21353-8}}
*{{cite book |last1=Rejzner |first1=Kasia |author-link1=Kasia Rejzner |title=Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians |year=2016 |publisher=Springer |doi=10.1007/978-3-319-25901-7 |series=Mathematical Physics Studies |arxiv=1208.1428 |bibcode=2016paqf.book.....R |isbn=978-3-319-25901-7 |url=https://link.springer.com/book/10.1007/978-3-319-25901-7}}
*{{cite book |last1=Hack |first1=Thomas-Paul |title=Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes |year=2016 |publisher=Springer |series=SpringerBriefs in Mathematical Physics |volume=6 |doi=10.1007/978-3-319-21894-6 |isbn=978-3-319-21894-6 |arxiv=1506.01869 |bibcode=2016caaq.book.....H |s2cid=119657309 |url=https://link.springer.com/book/10.1007/978-3-319-21894-6}}
*{{cite book |last1=Dütsch |first1=Michael |title=From Classical Field Theory to Perturbative Quantum Field Theory |year=2019 |publisher=Birkhäuser |series=Progress in Mathematical Physics |volume=74 |doi=10.1007/978-3-030-04738-2 |isbn=978-3-030-04738-2 |s2cid=126907045 |url=https://link.springer.com/book/10.1007/978-3-030-04738-2}}
*{{cite book |last1=Yau |first1=Donald |title=Homotopical Quantum Field Theory |year=2019 |publisher=World Scientific |doi=10.1142/11626|arxiv=1802.08101 |url=https://www.worldscientific.com/worldscibooks/10.1142/11626 |isbn=978-981-121-287-1|s2cid=119168109 }}
*{{cite journal |last1=Dedushenko |first1=Mykola |title=Snowmass white paper: The quest to define QFT |journal=International Journal of Modern Physics A |arxiv=2203.08053 |date=2023 |volume=38 |issue=4n05 |doi=10.1142/S0217751X23300028 |s2cid=247450696 }}

== External links ==
* [https://www.lqp2.org Local Quantum Physics Crossroads 2.0] – A network of scientists working on local quantum physics
* [https://www.lqp2.org/faceted-paper-view Papers] – A database of preprints on algebraic QFT
* [https://www.physik.uni-hamburg.de/en/th2/ag-fredenhagen.html Algebraic Quantum Field Theory] – AQFT resources at the University of Hamburg

{{Quantum field theories}}

[[Category:Axiomatic quantum field theory]]