algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
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The Haag–Kastler axioms (Haag-Kastler 64) (sometimes also called Araki–Haag–Kastler axioms) try to capture in a mathematically precise way the notion of quantum field theory (QFT), by axiomatizing how its algebras of quantum observables should depend on spacetime regions, namely as local nets of observables.
The main point of these axioms is to say that
to every causally closed subset $\mathcal{O} \subset X$ of spacetime $X$ there is associated a C*-algebra $\mathcal{A}(\mathcal{O})$;
for every inclusion $\mathcal{O}_1 \hookrightarrow \mathcal{O}_2$ of such spacetime regions there is a corresponding inclusion $\mathcal{A}(\mathcal{O}_1) \hookrightarrow \mathcal{A}(\mathcal{O}_2)$
such that this respects composition of inclusions (hence such that $\mathcal{A}$ is a functor from the poset of causally closed subsets to C*-algebras)
(causal locality) and such that whenever $\mathcal{O}_1, \mathcal{O}_2 \subset \mathcal{O} \subset X$ are spacelike separated, then the elements of the corresponding algebras of observables (graded-)commute with each other:
Moreover one wants these assignments to behave well with spacetime symmetry.
The formulation of quantum field theory via these axioms has come to be known as algebraic quantum field theory or AQFT, for short. There are various further axioms in the list such as the time slice axiom. The precise details of the list of axioms is in flux as the theory develops.
The Haag-Kastler axioms in their original form aim for the description of non-perturbative quantum field theory on Minkowski spacetime. They could be used to give rigorous proof of structural statements of QFT such as the spin-statistics theorem or the PCT theorem, but examples of interacting field theories in dimensions $\geq 4$ are missing.
There are however variants:
In perturbative AQFT one drops the requirement of C*-algebras and instead considers just formal power series algebras (in Planck's constant $\hbar$) in order to describe perturbative quantum field theory and connect to the traditional construction of pQFTs via the Feynman perturbation series.
The observation that causal perturbation theory yields quantum observables that satisfy the Haag-Kastler axioms, except that C*-algebras are replaced by formal power series algebras, is due to (Il’in-Slavnov 78, Brunetti-Fredenhagen 00, Brunetti-Dütsch-Fredenhagen 09). See at S-Matrix – Causal Locality and Quantum Observables.
In locally covariant perturbative AQFT one generalizes from Minkowski spacetime to more general globally hyperbolic spacetimes in order to describe QFT on curved spacetimes.
In homotopical AQFT one consider homotopical algebras and commutativity only up to coherent homotopy in order to discuss gauge theory with non-trivial topological (instanton) sectors.
The approch to quantum field theory based on these axioms is often called AQFT : either for axiomatic quantum field theory (since it was among the first attempts to put the edifice of QFT on solid axiomatic grounds) or algebraic quantum field theory (since it amplifies the algebras of local observables over the spaces of states). Neither of these terms is very descriptive. First there is another, dual, axiomatization which does axiomatize the propagation of states – see FQFT – which, second, is also “algebraic” in some sense, even though algebras of observables to not appear directly.
Another common term for these axioms is local quantum field theory (see the title of the standard textbook (Haag)) since, as becomes clear below, they are focused on encoding the locality properties of QFT in terms of the algebras of observables. However, also the core aspect of extended FQFT is all about the notion of locality of QFT.
Therefore neither of the traditional terms for QFT as axiomatized by Haag-Kastler is truly descriptive in that it genuinely distinguishes from the other, the Atiyah-Segal axiomatization by FQFT. What does distinguish the two approaches may be characterized in traditional terminology of quantum theory as follows (Schreiber, SatiSchreiber):
FQFT axiomatizes the Schrödinger picture of QFT, which encodes the propagation of states through spacetimes;
AQFT axiomatizes the Heisenberg picture of QFT, which encodes the way that observables depend on spacetime.
A central difference between the Haag-Kastler axioms and traditionally more widespread formulations of QFT (usually far from being formalized in any way) is the emphasis of the algebra of observables of a QFT (Heisenberg picture) and the de-emphasis of the (Hilbert) spaces of states (Schrödinger picture). This emphasis receives motivation from the the fact that many technical problems of QFT simply disappear when one is not trying to form its spaces of states, while at the same time no real information about the theory is lost.
Examples of technical problems that formulation in terms of spaces of states bring with them are the following:
In quantum field theory as opposed to quantum mechanics, the Stone-von Neumann theorem fails, making the unitary representation of the Heisenberg group on the spaces of states non-unique, hence requiring an explicit choice of representation. There is no generally good theory available for how to make this choice.
More seriously, Haag's theorem says that at a crucial step in perturbation theory where one wants to pass from the representation “free fields” to that of “interacting fields”, the two representations are necessarily inequivalent, contrary to what is (silently or explicitly) assumed in much traditional QFT literature (see EarmanFraser).
In the renormalization or perturbation theory the formulation in terms of states brings with it infrared problems that are simply absent when formulating renormalization just in terms of observables (DuetschFredenhagen).
We formulate the ideas of the core axioms of Haag-Kastler, and their intended physical meaning. For more details see local net of observables.
spacetime locality
Since the fields in quantum field theory (such as the electromagnetic field) exhibit and are characterized by their local excitations (for instance the value of the electric/magnetic field strength at any point) having effects only locally (the field excitations at two points a finite distance apart do not directly influence each other) the fields over any region of spacetime form a subsystem of the fields of any larger region and in particular of the total system.
If “quantum mechanical system” is formalized as “C-star algebra” (of observables) then “subsystem” translates to “sub-$C^*$-algebra”. Therefore the above sentence translates into: quantum fields form a copresheaf of C-star algebras on spacetimes whose co-restriction morphisms are monomorphisms.
In AQFT such is called an isotonic net of algebras .
There are different approaches to define what kind of spacetime regions the algebras of observables are assigned to, hence different approaches as to what exactly the site is on which the co-presheaf is defined. A common approach is to take all bounded open subsets of Minkowski spacetime. For more general setups see AQFT on curved spacetimes.
causal locality
If two regions of spacetime are spacelike separated, then there can be no influence between them whatsoever. Not only do the field excitations in one of the two regions not directly influence those in the other region (as per item 1), but they do not even influence indirectly : no waves of excitations (for instance electromagnetic waves: light) can run from one region to a spacelike separated region. Therefore the two subsystems constituted by these two regions accordording to the first point are even independent subsystems . This is called causal locality.
The formalization of “two independent subsystems” in quantum mechanics is: two subalgebras that commute with each other inside the larger C-star algebra. (And usually one adds: and such that the algebra they generate in the larger algebra is isomorphic to their tensor product.)
Therefore this translates into the axiom: quantum fields on a spacetime form an isotonic copresheaf of algebras such that the algebras assigned to any two spacelike separated regions commute with each other inside the algebra assigned to any larger region containing these two regions.
spacetime covariance
The geometric symmetry operations map the algebra of a region onto the algebra of the transformed region.
(this is not an extra axiom if one defines the site of spacetime regions general enough…)
In Minkowski spacetime the geometric symmetry group is usually taken to be the Poincaré group, but note that some authors consider subgroups of the full Poincaré group, like the subgroup of translations (Borchers: “Translation group and particle representations in quantum field theory”).
positivity of energy
An axiom is needed to ensure that only nonnegative energies occur – one possibility is the “spectrum condition”, which says that the spectrum (to be more precise: the support of the spectral measure) of the operator associated with a translation is contained in the closed forward light cone, for all translations.
It is possible to prove both a spin-statistics theorem and a PCT theorem in the Haag-Kastler approach. The mathematically precise, model independent statements and their proofs are considered to be a major breakthrough of the theory.
Unlike the Wightman axioms, the Haag–Kastler axioms do not need the notion of “field”: the fields in the Wightman axioms are – from the Haag–Kastler point of view – only necessary to describe how the algebras of observables are constructed; any way to consistently construct the net of algebras would suffice.
The original article that introduced these axioms is
Textbook accounts include
Rudolf Haag, Local Quantum Physics -- Fields, Particles, Algebras
Huzihiro Araki: Mathematical Theory of Quantum Fields Oxford University Press 1999 ZMATH entry.
precursors include
An online reference page is here:
See also the references at AQFT and at perturbative AQFT.
The observation that causal perturbation theory yields quantum observables that satisfy the Haag-Kastler axioms, except that C*-algebras are replaced by formal power series algebras is due to
V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (spire, doi)
Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)
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)
An introduction into Tomita-Takesaki modular theory is here:
…while a paper that puts it to serious work is this:
A discussion of how the Haag-Kastler axioms (those concerning locality) follow from an extended FQFT with Lorentzian structure is in
A discussion putting the state of the art of the AQFT-axiomatization in context with that of the FQFT-axiomatization is in
A discussion of perturbation theory and renormalization in terms of the Haag-Kastler axioms is in
Haag's theorem and its meaning and implication is discussed thoroughly in