Algebraic quantum field theory (AQFT) 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 contin ...
theory. Also referred to as the Haag–Kastler
axiomatic framework for
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, because it was introduced by . The axioms are stated in terms of an algebra given for every open set in
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
, and mappings between those.
Haag–Kastler axioms
Let
be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set
of
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
s
on a common
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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''.
Category-theoretic formulation
Let Mink be the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of
open subset
In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
s of Minkowski space M with
inclusion map
In mathematics, if A is a subset of B, then the inclusion map 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(x)=x.
An inclusion map may also be referred to as an inclu ...
s as
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 ...
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 monomorphis ...
in uC*alg (isotony).
The
Poincaré group
The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
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
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
, 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. Informal ...
of
(
Poincaré covariance).
Minkowski space has a
causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''c ...
. If an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''V'' lies in the
causal complement of an open set ''U'', then the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of the maps
:
and
:
commute (spacelike commutativity). If
is the
causal completion
Causality is an 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 cause is at least partly responsible for the effect, ...
of an open set ''U'', then
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
(primitive causality).
A
state
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
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 elements v \in V, that is v \geq 0, it holds that
f(v) \geq 0.
In oth ...
over it with unit
norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
. If we have a state over
, we can take the "
partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace (linear algebra), trace. Whereas the trace is a scalar (mathematics), scalar-valued function on operators, the partial trace is an operator (mathemati ...
" to get states associated with
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 monomorphis ...
. The states over the open sets form a
presheaf
In mathematics, a sheaf (: sheaves) 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 d ...
structure.
According to the
GNS construction, for each state, we can associate a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
representation of
Pure state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s 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, _W, ...
s and
mixed states correspond to
reducible representations. 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 (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
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 the ca ...
of the
Poincaré group
The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
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 (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
. 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 uses a semi-classical approach; it treats spacetime as a fixed ...
. 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, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
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 massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
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
– AQFT resources at the University of Hamburg
{{Quantum field theories
Axiomatic quantum field theory