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The Haag–Kastler axiomatic framework for
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, introduced by , is an application to local quantum physics of
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
theory. Because of this it is 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, and mappings between those.


Haag–Kastler axioms

Let \mathcal 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 \mathcal(O) on a common Hilbert space \mathcal satisfying the following axioms: * ''Isotony'': O_1 \subset O_2 implies \mathcal(O_1) \subset \mathcal(O_2). * ''Causality'': If O_1 is space-like separated from O_2, then mathcal(O_1),\mathcal(O_2)0. * ''Poincaré covariance'': A strongly continuous unitary representation U(\mathcal) of the Poincaré group \mathcal on \mathcal exists such that \mathcal(gO) = U(g) \mathcal(O) U(g)^*, g \in \mathcal. * ''Spectrum condition'': The joint spectrum \mathrm(P) of the energy-momentum operator P (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 \Omega\in\mathcal exists. The net algebras \mathcal(O) are called ''local algebras'' and the C* algebra \mathcal := \overline is called the ''quasilocal algebra''.


Category-theoretic formulation

Let Mink be the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
s of Minkowski space M with
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...
s as
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s. We are given a
covariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
\mathcal from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
in uC*alg (isotony). The
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
acts continuously on Mink. There exists a
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
of this action, which is continuous in the
norm topology In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Int ...
of \mathcal(M) (
Poincaré covariance Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luc ...
). Minkowski space has a
causal structure In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
. If an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
''V'' lies in the causal complement of an open set ''U'', then the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of the maps :\mathcal(i_) and :\mathcal(i_)
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
(spacelike commutativity). If \bar is the
causal completion Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
of an open set ''U'', then \mathcal(i_) is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(primitive causality). A
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over \mathcal(M), we can take the " partial trace" to get states associated with \mathcal(U) for each open set via the net
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
. The states over the open sets form a presheaf structure. According to the
GNS construction GNS may refer to: Places * Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia * Gainesville station (Georgia), an Amtrak station in Georgia, United States Companies and organizations * Gesellschaft für Nuklear-Service, a German nuclear-wa ...
, for each state, we can associate a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
representation of \mathcal(M). Pure states correspond to
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _ ...
s and mixed states correspond to
reducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s. Each irreducible representation (up to equivalence) is called a
superselection sector In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was origina ...
. We assume there is a pure state called the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
such that the Hilbert space associated with it is a
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
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 In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
. This is the vacuum sector.


QFT in curved spacetime

More recently, the approach has been further implemented to include an algebraic version of
quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while givi ...
. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
have been obtained.


References


Further reading

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External links


Local Quantum Physics Crossroads 2.0
– A network of scientists working on local quantum physics
Papers
– A database of preprints on algebraic QFT
Algebraic Quantum Field Theory
– AQFT resources at the University of Hamburg {{Quantum field theories Axiomatic quantum field theory