algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
quantum mechanical system, quantum probability
interacting field quantization
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
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 (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:
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.
— much information to be filled in —
Generically the algebra of a relativistic AQFT turns out to be a (the) hyperfinite type $III_1$ 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.
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)
The free Dirac field and its deformations is discussed for instance in (DLM, section 3.2), (Dimock 83).
The quantized electromagnetic field is discussed for instance in (Dimock 92).
(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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\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{<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$\phantom{A}$ |
The original article that introduced the Haag-Kastler axioms is
following
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 .
Introductory lecture notes include
and for just quantum mechanics in the algebraic perspective:
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:
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
Construction of examples is considered for instance in
General discussion of AQFT quantization of free 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)
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
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
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
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)
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)
(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
A relation to FQFT is discussed in
The role of von Neumann algebra factors is discussed in
Last revised on January 17, 2018 at 13:24:19. See the history of this page for a list of all contributions to it.