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. Because of this it is 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.
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Overview
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor
The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of
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
A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over
According to the GNS construction, for each state, we can associate a Hilbert space representation of
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.