nLab AQFT

Idea

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:

• factorization algebras

• topological chiral homology

• blob homology.

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:

• 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.

— much information to be filled in —

Axioms

• Wightman axioms

• Haag-Kastler axioms

Theorems

• Reeh-Schlieder theorem

• Osterwalder-Schrader theorem

• PCT theorem

• Bisognano-Wichmann theorem

• spin-statistics theorem

Properties

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

Examples

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

(Furliani)

Related concepts

• quantum mechanics, quantum field theory, perturbative quantum field theory

  • AQFT

    • Haag-Kastler axioms, Wightman axioms

    • local net of observables, field net

    • time slice axiom, split property

    • quantum lattice systems, string-localized quantum field

    • locally covariant AQFT

    • perturbative AQFT

    • homotopical algebraic quantum field theory

    • Haag-Ruelle scattering theory

  • FQFT

• constructive quantum field theory

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\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 }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

Axioms

The original article that introduced the Haag-Kastler axioms is

• Rudolf Haag, Daniel Kastler, An algebraic approach to quantum field theory, Journal of Mathematical Physics, 5 (1964) 848-861 [doi:10.1063/1.1704187, spire:9124]

following

• 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:

• Rudolf Haag, Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles, EPJ H 35, 243–253 (2010) (doi:10.1140/epjh/e2010-10041-3)

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

• Romeo Brunetti, Klaus Fredenhagen, Quantum field theory on curved spacetimes arXiv:0901.2063

• Romeo Brunetti, Klaus Fredenhagen, Rainer Verch, The generally covariant locality principle – A new paradigm for local quantum physics Commun. Math. Phys. 237:31-68 (2003) (arXiv:math-ph/0112041)

• 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 .

Lecture notes and Textbooks

Introductory lecture notes:

• Garth Warner: Quantum Field Theory Seminar (School of Haag-Kastler et al.), seminar notes, University of Washington [pdf, pdf]

• Klaus Fredenhagen, Algebraische Quantenfeldtheorie, lecture notes, 2003 (pdf)

• Christopher Fewster, Kasia Rejzner, Algebraic Quantum Field Theory - an introduction (arXiv:1904.04051)

• Jonathan Sorce: Bootstrap 2024: Lectures on “The algebraic approach: when, how, and why?” [arXiv:2408.07994]

  (aimed at non-mathematically inclined field theorists)

and for just quantum mechanics in the algebraic perspective:

• Jonathan Gleason, The $C^*$-algebraic formalism of quantum mechanics (2009) [pdf, pdf]

• Jonathan Gleason, From Classical to Quantum: The $F^\ast$-Algebraic Approach, contribution to VIGRE REU 2011, Chicago (2011) [pdf, pdf]

Textbook accounts:

• Ola Bratteli, Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics – vol 1: $C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States., Springer (1979, 1987, 2002) [doi:10.1007/978-3-662-02520-8]

• Raymond F. Streater, Arthur S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press (1989, 2000) [ISBN:9780691070629, jstor:j.ctt1cx3vcq]

• Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]

• Rudolf Haag, Local Quantum Physics – Fields, Particles, Algebras, Texts and Monographs in Physics, Springer (1992)

• Huzihiro Araki, Mathematical Theory of Quantum Fields (1999)

• Franco Strocchi, An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [doi:10.1093/acprof:oso/9780199671571.001.0001]

• Hans Halvorson, Michael Müger, Algebraic Quantum Field Theory [arXiv:math-ph/0602036]

An account written by mathematicians for mathematicians:

• Hellmut Baumgärtel, Manfred Wollenberg, Causal nets of operator algebras. Berlin: Akademie Verlag 1992 (ZMATH entry)

• Hellmut Baumgärtel, Operator algebraic Methods in Quantum Field Theory. A series of lectures. Akademie Verlag 1995 (ZMATH entry)

Reviews

• Franco Strocchi, Relativistic Quantum Mechanics and Field Theory, Found. Phys. 34 (2004) 501-527 [arXiv:hep-th/0401143]

• Sergio Doplicher, The principle of locality: Effectiveness, fate, and challenges, J. Math. Phys. 51, 015218 (2010), doi

• Edison Montoya, Algebraic quantum field theory (2009) (pdf)

• Detlev Buchholz, Klaus Fredenhagen, Algebraic quantum field theory: objectives, methods, and results, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2305.12923]

More on the role of von Neumann algebra factors in AQFT

• J. Yngvason, The role of type III factors in quantum field theory [arXiv:math-ph/0411058]

Examples

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

• Christian Bär, N. Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, (EMS, 2007) (arXiv:0806.1036)

• Christian Bär, N. Ginoux, Classical and quantum fields on lorentzian manifolds (2011) (arXiv:1104.1158)

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

• Detlev Buchholz, Klaus Fredenhagen, A $C^\ast$-algebraic approach to interacting quantum field theories, Commun. Math. Phys. 377 (2020) 947–969 [arXiv:1902.06062, doi:10.1007/s00220-020-03700-9]

Local gauge theory

Discussion of aspects of gauge theory includes

• Fabio Ciolli, Giuseppe Ruzzi, Ezio Vasselli, Causal posets, loops and the construction of nets of local algebras for QFT (arXiv:1109.4824)

• Fabio Ciolli, Giuseppe Ruzzi, Ezio Vasselli, QED Representation for the Net of Causal Loops (arXiv:1305.7059)

• Giuseppe Ruzzi, Nets of local algebras and gauge theories, 2014 (pdf slides)

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

• Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)

• Marco Benini, Alexander Schenkel, Richard Szabo, Homotopy colimits and global observables in Abelian gauge theory (arXiv:1503.08839)

• Marco Benini, Alexander Schenkel, Quantum field theories on categories fibered in groupoids (arXiv:1610.06071)

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

• Alexander Schenkel, On the problem of gauge theories

  in locally covariant QFT_, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (pdf) (with further emphasis on this point in the companion talk Schreiber 14)

Further development of this homotopical algebraic quantum field theory includes

• Marco Benini, Alexander Schenkel, Quantum field theories on categories fibered in groupoids (arXiv:1610.06071)

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

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic quantum field theory and deformation quantization, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (arXiv:hep-th/0101079)

(relation to deformation quantization)

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

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

(relation to renormalization)

• Michael Dütsch, Klaus Fredenhagen, A local (perturbative) construction of observables in gauge theores: the example of qed , Commun. Math. Phys. 203 (1999), no.1, 71-105, (arXiv:hep-th/9807078).

(relation to gauge theory and QED)

Lecture notes are in

• Klaus Fredenhagen, Katarzyna Rejzner, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)

• Klaus Fredenhagen, Katarzyna Rejzner, Perturbative Construction of Models of Algebraic Quantum Field Theory (arXiv:1503.07814)

and a textbook acount is in

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

Further developments

• Claudio Dappiaggi, Gandalf Lechner, E. Morfa-Morales, Deformations of quantum field theories on spacetimes with Killing vector fields, Commun.Math.Phys.305:99-130, (2011), (arXiv:1006.3548)

• Angelos Anastopoulos, Marco Benini. Gluing algebraic quantum field theories on manifolds (2024). (arXiv:2404.09638).

Relation to factorization algebras:

• Marco Benini, Marco Perin, Alexander Schenkel, Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds, Communications in Mathematical Physics volume 377, pages 971–997 (2020) (arXiv:1903.03396v2, doi:10.1007/s00220-019-03561-x)

Relation to smooth stacks:

• Marco Benini, Marco Perin, Alexander Schenkel, Smooth 1-dimensional algebraic quantum field theories (arXiv:2010.13808)

Relation to holographic entanglement entropy

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

• Edward Witten, Notes on Some Entanglement Properties of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018) (arXiv:1803.04993)

• Thomas Faulkner, The holographic map as a conditional expectation (arXiv:2008.04810)

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):

• Urs Schreiber, AQFT from n-Functorial QFT, Comm. Math. Phys. 291 2 (2009) 357-401 [arXiv:0806.1079, doi:10.1007/s00220-009-0840-2]

• Theo Johnson-Freyd, Heisenberg-picture quantum field theory, in Representation Theory, Mathematical Physics, and Integrable Systems, Progress in Mathematics 340 (2021) [arXiv:1508.05908, doi:10.1007/978-3-030-78148-4_13]

• Severin Bunk, James MacManus, Alexander Schenkel, Lorentzian bordisms in algebraic quantum field theory [arXiv:2308.01026]