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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 In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually cont ...
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 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 subsets 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 morphisms. We are given a covariant functor \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: i ...
of this
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
, which is continuous in the norm topology of \mathcal(M) ( Poincaré covariance). 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 m ...
. 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 of the maps :\mathcal(i_) and :\mathcal(i_) commute (spacelike commutativity). If \bar is the causal completion 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 In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (V, \leq) is a linear functional f on V so that for all positive element (ordered group), positive elements v \in V, that is v \geq 0, ...
over it with unit norm. If we have a state over \mathcal(M), we can take the "
partial trace In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in ...
" to get states associated with \mathcal(U) for each open set via the
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
structure. According to the GNS construction, for each state, we can associate a Hilbert space representation of \mathcal(M).
Pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s correspond to irreducible representations 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. 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 di ...
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 have been obtained.


References


Further reading

* * * * * * * * * * *


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

– AQFT resources at the University of Hamburg {{Quantum field theories Axiomatic quantum field theory