Algebraic quantum field theory

Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those. Let be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a net of von Neumann algebras on a common Hilbert space satisfying the following axioms: Isotony: implies . Causality: If is space-like separated from , then . Poincaré covariance: A strongly continuous unitary representation of the Poincaré group on exists such that , . Spectrum condition: The joint spectrum of the energy-momentum operator (i.e. the generator of space-time translations) is contained in the closed forward lightcone. Existence of a vacuum vector: A cyclic and Poincaré-invariant vector exists. The net algebras are called local algebras and the C* algebra is called the quasilocal algebra. Let Mink be the of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor from Mink to uCalg, the category of unital C algebras, such that every morphism in Mink maps to a monomorphism in uCalg (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 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. The states over the open sets form a presheaf structure.