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. 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 functor 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).
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 image 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. It's easy to show 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 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 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. This is the vacuum sector.
Famous quotes containing the words local, quantum, field and/or theory:
“The country is fed up with children and their problems. For the first time in history, the differences in outlook between people raising children and those who are not are beginning to assume some political significance. This difference is already a part of the conflicts in local school politics. It may spread to other levels of government. Society has less time for the concerns of those who raise the young or try to teach them.”
—Joseph Featherstone (20th century)
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“The snow had begun in the gloaming,
And busily all the night
Had been heaping field and highway
With a silence deep and white.”
—James Russell Lowell (1819–1891)
“... liberal intellectuals ... tend to have a classical theory of politics, in which the state has a monopoly of power; hoping that those in positions of authority may prove to be enlightened men, wielding power justly, they are natural, if cautious, allies of the “establishment.””
—Susan Sontag (b. 1933)