The Haag-Kastler axiomatic framework for quantum field theory, introduced by Haag and Kastler (1964), is an application to local quantum physics of C*-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 in Minkowski space, and mappings between those.
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a monomorphism in uC*alg (isotony).
The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of (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 of the maps
and
commute (spacelike commutativity). If is the causal completion of an open set U, then 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, we can take the "partial trace" to get states associated with for each open set via the net monomorphism. It's easy to show the states over the open sets form a presheaf structure.
According to the GNS construction, for each state, we can associate a Hilbert space representation of Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible (up to 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 (corresponding to spacetime translations) lies on and in the positive light cone. This is the vacuum sector.
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