Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT

Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication



Algebraic Quantum Field Theory or Axiomatic Quantum Field Theory or AQFT for short is a formalization of quantum field theory (and specifically full, hence non-perturbative quantum field theory) that axiomatizes the assignment of algebras of observables to patches of parameter space (spacetime, worldvolume) 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.

From the nPOV we may think of a local net as a co-flabby copresheaf of algebras on spacetime which satisfies a certain locality axiom with respect to the Lorentzian structure of spacetime:

  • locality: algebras assigned to spacelike separated regions commute with each other when embedded into any joint superalgebra.

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:

Initially, all three of these encoded what in physics are called Euclidean quantum field theories, whereas only the notion of local net incorporated the fact that the underlying spacetime of a quantum field theory is a smooth Lorentzian space. Recent developments in the formalism of factorization algebras have extended their theory to globally hyperbolic Lorentzian manifolds.

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:

— much information to be filled in —




Generically the algebra of a relativistic AQFT turns out to be a (the) hyperfinite type III 1III_1 von Neumann algebra factor. See (Yngvason)


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.

Free scalar field / Klein Gordon field

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)

Free fermion / Dirac field

The free Dirac field and its deformations is discussed for instance in (DLM, section 3.2), (Dimock 83).

Electromagnetic field

The quantized electromagnetic field is discussed for instance in (Dimock 92).

Proca field


duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}



The original article that introduced the Haag-Kastler axioms is


  • Rudolf Haag, Discussion des “axiomes” et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151.

translated to English as:

The generalization of the spacetime site from open in Minkowski space to more general and curved spacetimes (see AQFT on curved spacetimes) is due to

See also AQFT on curved spacetimes .

Lecture notes and Textbooks

Introductory lecture notes:

and for just quantum mechanics in the algebraic perspective:

Textbook accounts:

An account written by mathematicians for mathematicians:


More on the role of von Neumann algebra factors in AQFT


Construction of examples is considered for instance in

  • Jonathan Dimock, Dirac quantum fields on a manifold, Trans. Amer. Math. Soc. 269 (1982), 133-147. (web)
  • Jonathan Dimock, Quantized electromagnetic field on a manifdold, Reviews in mathematical physics, Volume 4, Issue 2 (1992) (web)
  • Edward Furliani, Quantization of massive vector fields in curved space–time, J. Math. Phys. 40, 2611 (1999) (web)

General discussion of AQFT quantization of free quantum fields is in

Examples of non-perturbative interacting scalar field theory in any spacetime dimension (in particular in d4d \geq 4) are claimed in

Local gauge theory

Discussion of aspects of gauge theory includes

Construction and axiomatization of gauge field AQFT via homotopy theory and homotopical algebra (see also at field bundle) is being developed in

The issue of the tension between local gauge invariance and locality and the need to pass to stacks/higher geometry is made explicit in

Further development of this homotopical algebraic quantum field theory includes

Perturbation theory and renormalization

Perturbation theory and renormalization in the context of AQFT and is discussed in the following articles.

The observation that in perturbation theory the Stückelberg-Bogoliubov-Epstein-Glaser local S-matrices yield a local net of observables was first made in

  • V. Il’in, D. Slavnov, Observable algebras in the S-matrix approach Theor. Math. Phys. 36 , 32 (1978)

which was however mostly ignored and forgotten. It is taken up again in

  • Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds Commun.Math.Phys.208:623-661 (2000) (arXiv)

(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)

(relation to gauge theory and QED)

Lecture notes are in

and a textbook acount is in

  • Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)

Further developments

Relation to factorization algebras:

Relation to smooth stacks:

Relation to holographic entanglement entropy

Discussion of local nets of observables in AQFT as the natural language for grasping holographic entanglement entropy:

Relation between algebraic and functorial field theory

On the relation between functorial quantum field theory (axiomatizing the Schrödinger picture of quantum field theory) and algebraic quantum field theory (axiomatizing the Heisenberg picture):

Last revised on April 16, 2024 at 09:02:10. See the history of this page for a list of all contributions to it.