Algebraic quantum field theory explained
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 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, 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 algebras
on a common
Hilbert space
satisfying the following axioms:
[1]
implies
.
is space-like separated from
, then
.
- Poincaré covariance: A strongly continuous unitary representation
of the Poincaré group
on
exists such that
l{A}(gO)=U(g)l{A}(O)U(g)*,g\inl{P}.
- 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
l{A}:=\overline{cupO\inl{O
}\mathcal(O)} is called the
quasilocal algebra.
Category-theoretic formulation
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. 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 representation (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.
QFT in curved spacetime
More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization 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
- Book: Baumgärtel . Hellmut . Operatoralgebraic Methods in Quantum Field Theory . 1995 . Akademie Verlag . Berlin . 3-05-501655-6.
Further reading
External links
