- Local quantum field theory
The

**Haag-Kastler**axiomatic framework forquantum field theory , named afterRudolf Haag andDaniel Kastler , is an application to "local" quantum physics ofC*-algebra theory. It is therefore also known as**Algebraic Quantum Field Theory**(**AQFT**). The axioms are stated in terms of an algebra given for every open set inMinkowski space , and mappings between those.Let

**Mink**be the category ofopen subset s of Minkowski space M withinclusion map s asmorphism s. We are given acovariant functor $mathcal\{A\}$ from**Mink**to**uC*alg**, the category ofunital C* algebras, such that every morphism in**Mink**maps to amonomorphism in**uC*alg**("isotony").The

Poincaré group acts continuously on**Mink**. There exists apullback of this action, which is continuous in thenorm topology of $mathcal\{A\}(M)$ (Poincaré covariance ).Minkowski space has a

causal structure . If anopen set "V" lies in thecausal complement of an open set "U", then the image of the maps:$mathcal\{A\}(i\_\{U,Ucup\; V\})$

and

:$mathcal\{A\}(i\_\{V,Ucup\; V\})$

commute (spacelike commutativity). If $ar\{U\}$ is the causal completion of an open set "U", then $mathcal\{A\}(i\_\{U,ar\{U)$ is an

isomorphism (primitive causality).A state with respect to a C*-algebra is a

positive linear functional over it with unit norm. If we have a state over $mathcal\{A\}(M)$, we can take the "partial trace " to get states associated with $mathcal\{A\}(U)$ for each open set via the net monomorphism. It's easy to show the states over the open sets form apresheaf structure.According to the

GNS construction , for each state, we can associate aHilbert space representation of $mathcal\{A\}(M).$Pure state s correspond toirreducible representation s andmixed state s correspond toreducible representation s. Each irreducible (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 translation s) lies on and in the positivelight cone . This is the vacuum sector.**External links*** [

*http://www.lqp.uni-goettingen.de/ Local Quantum Physics Crossroads*] - A network of scientists working on Local Quantum Physics

* [*http://unith.desy.de/research/aqft/ Algebraic Quantum Field Theory*] - AQFT resources at the University of Hamburg**Suggested reading*** Haag, Rudolf (1992). "Local Quantum Physics: Fields, Particles, Algebras." Springer.

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