Algebraic quantum field theory
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.An algebraic quantum field theory is defined via a setLet Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms.There exists a pullback of this action, which is continuous in the norm topology ofA state with respect to a C*-algebra is a positive linear functional over it with unit norm.According to the GNS construction, for each state, we can associate a Hilbert space representation ofEach 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.Several rigorous results concerning QFT in presence of a black hole have been obtained.