Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964).
The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
be the set of all open and bounded subsets of Minkowski space.
An algebraic quantum field theory is defined via a set
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms.
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).
There exists a pullback of this action, which is continuous in the norm topology of
A state with respect to a C*-algebra is a positive linear functional over it with unit norm.
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
Each irreducible representation (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.
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.