Types of quantum field thories
Algebraic Quantum Field Theory or Axiomatic Quantum Field Theory or AQFT for short is a formalization of quantum field theory that axiomatizes the assignment of algebras of observables to patches of parameter space that one expects a quantum field theory to provide.
As such, the approach of AQFT is roughly dual to that of FQFT, where instead spaces of states are assigned to boundaries of cobordisms and propagation maps between state spaces to cobordisms themselves.
One may roughly think of AQFT as being a formalization of what in basic quantum mechanics textbooks is called the Heisenberg picture of quantum mechanics. On the other hand FQFT axiomatizes the Schrödinger picture .
The axioms of traditional AQFT encode the properties of a local net of observables and are called the Haag-Kastler axioms. They are one of the oldest systems of axioms that seriously attempt to put quantum field theory on a solid conceptual footing.
This is traditionally formulated (implicitly) as a structure in ordinary category theory. More recently, with the proof of the cobordism hypothesis and the corresponding (∞,n)-category-formulation of FQFT also higher categorical versions of systems of local algebras of observables are being put forward and studied. Three structures are curently being studied, that are all conceptually very similar and similar to the Haag-Kastler axioms:
On the other hand, all three of these encode what in physics are called Euclidean quantum field theories, whereas only the notion of local net so far really incorporates crucially the fact that the underlying spacetime of a quantum field theory is a smooth Lorentzian space.
In the context of the Haag-Kastler axioms there is a precise theorem, the Osterwalder-Schrader theorem, relating the Euclidean to the Lorentzian formulation: this is the operation known as Wick rotation.
Sheaves are used explicitly in:
Roberts, John E.: New light on the mathematical structure of algebraic field theory. Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pp. 523–550, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., 1982.
Roberts, John E.: Localization in algebraic field theory. Comm. Math. Phys. 85 (1982), no. 1, 87–98.
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Examples of AQFT local nets of observables that encode interacting quantum field theories are not easy to construct. The construction of free field theories is well understood, see the references below. In perturbation theory also interacting theories can be constructed, see the references here.
A survey of the AQFT description of the free scalar field on Minkowski spacetime is in (Motoya, slides 11-17). Discussion in more general context of AQFT on curved spacetimes in (Brunetti-Fredenhagen, section 5.2)
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
The original article that introduced the Haag-Kastler axioms is
Romeo Brunetti, Klaus Fredenhagen, Quantum Field Theory on Curved Backgrounds , Proceedings of the Kompaktkurs “Quantenfeldtheorie auf gekruemmten Raumzeiten” held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen
See also AQFT on curved spacetimes .
Classical textbooks are
A good account of the mathematical axiomatics of Haag-Kastler AQFT is
This is, among other things, the ideal starting point for pure mathematicians who have always been left puzzled or otherwise unsatisfied by accounts of quantum field theory, even those tagged as being “for mathematicians”. AQFT is truly axiomatic and rigorously formal.
An account written by mathematicians for mathematicians is this:
A classic of the trade is this one:
Recent account of the principle of locality in AQFT from the point of view of traditional school
Talk slides include
Construction of examples is considered for instance in
General discussion of AQFT quantization of free fields is in
which was however mostly ignored and forgotten. It is taken up again in
(a quick survey is in section 8, details are in section 2).
Further developments along these lines are in
(relation to deformation quantization)
(relation to renormalization)
A relation to FQFT is discussed in
The role of von Neumann algebra factors is discussed in