Contents

# Contents

## 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:

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:

— much information to be filled in —

## 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)

Isbell 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{Gelfand-Kolmogorov}}{\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}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\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}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$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}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\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}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\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, Bd.5, 1964, S.848-861

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.

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

### Lecture notes and Textbooks

Introductory lecture notes include

and for just quantum mechanics in the algebraic perspective:

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

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:

and this:

A classic of the trade is this one:

• Nikolay Bogolyubov, Logunov, Oksak, Todorov: General principles of quantum field theory (Mathematical Physics and Applied Mathematics, 10. Dordrecht etc.: Kluwer Academic Publishers, 1990)

### Reviews

Recent account of the principle of locality in AQFT from the point of view of traditional school

• 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

• Franco Strocchi, An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press, 2013

Talk slides include

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

### 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 fields is 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)

### Relation to functorial QFT

A relation to FQFT is discussed in

• Urs Schreiber, AQFT from $n$-functorial QFT , Comm. Math. Phys., Volume 291, Issue 2, pp.357-401 (pdf)

The role of von Neumann algebra factors is discussed in

Last revised on September 8, 2018 at 03:38:51. See the history of this page for a list of all contributions to it.