- Local quantum field theory
axiomatic frameworkfor quantum field theory, named after Rudolf Haagand Daniel Kastler, is an application to "local" quantum physics of C*-algebratheory. 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 functorfrom Mink to uC*alg, the category of unitalC* algebras, such that every morphism in Mink maps to a monomorphismin uC*alg ("isotony").
Poincaré groupacts continuously on Mink. There exists a pullbackof this action, which is continuous in the norm topologyof ( Poincaré covariance).
Minkowski space has a
causal structure. If an open set"V" lies in the causal complementof an open set "U", then the image of the maps
commute (spacelike commutativity). If is the causal completion of an open set "U", then is an
A state with respect to a C*-algebra is a
positive linear functionalover 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 presheafstructure.
According to the
GNS construction, for each state, we can associate a Hilbert spacerepresentation 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 vacuumsuch that the Hilbert space associated with it is a unitary representationof the Poincaré groupcompatible 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.
* [http://www.lqp.uni-goettingen.de/ Local Quantum Physics Crossroads] - A network of scientists working on Local Quantum Physics
* [http://unith.desy.de/research/aqft/ Algebraic Quantum Field Theory] - AQFT resources at the University of Hamburg
* Haag, Rudolf (1992). "Local Quantum Physics: Fields, Particles, Algebras." Springer.
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