Contents

Idea

The Haag–Kastler axioms (sometimes also called Araki–Haag–Kastler axioms) try to define in a mathematically precise way the notion of quantum field theory (QFT), by axiomatizing how its local algebras of observables should behave.

The approch to quantum field theory based on these axioms is often called AQFT: algebraic quantum field theory .

Although they are called axioms, one should keep in mind that the Haag–Kastler approach to QFT has not reached its final state, so that different versions of the axioms are used by practitioners of the field.

From the nPOV, the Haag-Kastler axioms descibe a coflabby presheaf of algebras. This kind of structure is similar to the notion of a factorization algebra, which plays a role in other approaches to formalize local algebras of observables.

Definition

1. Axiom A:

   To every “allowed” (e.g. bounded open) region $O$ in spacetime there is associated a C*-algebra; this association fulfils isotony. The basic assumption of the Haag–Kastler approach is that everything that can be measured in certain regions of spacetime $O$ (like temperature or particle count) is described by an algebra $A(O)$, so that the theory consists of a net of operator algebras and a state? $p$ that describes the physical system. There are different approaches to define what regions are “allowed”, one common approach is to take all bounded? open? subsets of Minkowski spacetime?.

2. Axiom B: locality

   Locality Algebras assigned to regions that are spacelike separated commute.

3. Axiom C: Transformation Properties

   The geometric symmetry operations map the algebra of a region onto the algebra of the transformed region.

   In Minkowski spacetime the geometric symmetry group is usually be 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”).

4. Axiom D: 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.

Remarks

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.

Example: Causal Nets of Operator Algebras

We will lay down a specific set of axioms knowing that this set is not the set of Haag-Kastler axioms, but one specific choice. This will allow us to state and prove important general properties. It is possible to construct examples that fulfill the axioms, to show that they are not empty, but we will not engage in this task here, at least not now. Note however that up to now there was no success in the task to construct systems in 4 dimensions with interactions, which has led to some doubts about the usefulness of this approach in the physics community: It has yet to be shown if the approach does or does not capture the essential features that makes possible the tremendous success of the standard model of particle physics.

Notation

The Minkowski Spacetime

We talk about 4-dimensional Minkowski spacetime $\mathrm{ℳ𝒾𝓃}$ only, i.e. $\mathrm{ℳ𝒾𝓃}$ is the vector space $R^4=R\times R^3$ equipped with the scalar product $<x,y>:=x_0{y}_0-(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{y})$ with $(\cdot ,\cdot )$ being the Euclidean scalar product on $R^3$. Open bounded subsets of $\mathrm{ℳ𝒾𝓃}$ will be denoted by $𝒪$. The union of these $𝒪$ form an index set $𝒥$, that is partially ordered by inclusion.

If two sets are spacelike seperated, this will be denoted by ${𝒪}_1\perp {𝒪}_2$.

We denote the open forward cone at x by $V_{+}(x)$, similar $V_{-}(x)$ is the open backward cone at $x$, if $x=0$ we simply write $V_{+}$ and $V_{-}$.

The Poincaré Group

The universal covering $\mathrm{SL}(2,C)⋉R^4$ of the restriced Poincaré group ${𝒫}_{+}^{†}$ will be denoted by $𝒫$, the abelian subgroup of all translations by $𝒯$.

Operator Algebras

Von Neumann algebras $ℳ$ will always be concrete operator algebras acting on a given Hilbert space $\mathscr{H}$, as is the rule in the literature (see also von Neumann algebra here on the nLab). The commutant of $ℳ$ will be denoted by $ℳ\prime$, the positive cone by ${ℳ}^{+}$. The minimal von Neumann algebra that contains two given ones ${ℳ}_1$ and ${ℳ}_2$ will be denoted by:

An automorphismus of an algebra $\alpha ℳ\to ℳ$ is called an inner automorphismus if there is an invertible element $u\in ℳ$ such that $\alpha$ is given by conjugation with $u:\alpha (m)=um{u}^{-1}\forall m\in ℳ$ (note that our convention here differs from that used by Wikipedia).

Definition of Vacuum Representations

A net of von Neumann algebras $ℳ(𝒪)$ on a common Hilbert space $(\mathscr{H})$, indexed by $𝒪\in 𝒥$, is called a vacuum respresentation (on the 4-dimensional Minkowski spacetime) if it satisfies the following axioms:

• (V1) isotony: ${𝒪}_1\subset {𝒪}_2\mathrm{implies}ℳ({𝒪}_1)\subseteq ℳ({𝒪}_2)$

• (V2) additivity: $𝒪={\bigcup }_j{𝒪}_j$ implies $ℳ(𝒪)=({\bigcup }_jℳ({𝒪}_j))″$

• (V3) locality, see local net: ${𝒪}_1\perp {𝒪}_2\mathrm{implies}ℳ({𝒪}_1)\subseteq (ℳ({𝒪}_2))\prime$

• (V4) covariance: There is a strongly continuous unitary representation $𝒰(𝒫)$ of $𝒫$ such that for each $g\in 𝒫$ the following holds: $ℳ(g𝒪)=𝒰(g)ℳ(𝒪)𝒰(g{)}^{-1}$

• (V5) spectrum condition: $\mathrm{spec}𝒰(𝒯)\subseteq \mathrm{clo}{𝒱}_{+}$

Note: $𝒯$ is the abelian subgroup of translations, and ${𝒱}_{+}$ is the (open) forward cone at 0, see above. For the definiton of the spectrum of the representation $𝒰(𝒯)$ see spectral measure.

• (V6) existence of a vacuum vector: There exists a vector $\Omega \in \mathscr{H},\parallel \Omega \parallel =1$, such that $({\bigcup }_{𝒪\in 𝒥}ℳ(𝒪)\Omega$ is dense in $\mathscr{H}$ and $𝒰(g)\Omega =\Omega$ for all $g\in 𝒫$

Note: The uniqueness is sometimes part of the axioms, but not here. Instead we will cite theorems that will specify necessary and sufficient conditions to ensure that there is a unique vacuum vector.

Additional Notations and Notions of Vacuum Representations

A short hand notation for vacuum representations will be $ℳ(𝒥)$ in the following.

The algebras $ℳ(𝒪)$ are sometimes called local algebras.

The $C^{*}-$algebra $𝒜:={\mathrm{clo}}_{\parallel \cdot \parallel }({\bigcup }_{𝒪\in 𝒥}ℳ(𝒪)$ is called quasilocal algebra, the smallest von Neuman algebra that contains $𝒜$ is called the global algebra and denoted by $ℛ$.

A vacuum representation is called irreducible if $ℛ=ℒ(\mathscr{H})$ (the global algebra is the whole algebra of all bounded linear operators on the given Hilbert space), it is called factorial if $ℛ$ is a factor.

• Wikipedia on factors of von Neumann algebras.

Classical Theorems

…will be cited here

• Theorem 1 (Borchers): The representatives of the translations are elements of the global algebra $ℛ$, i.e. they are inner automorphisms of $ℛ$: $𝒰(𝒯)\subseteq ℛ$.

References

… should go here. See also AQFT. Since on that page there are already some references to sources that stress the mathematical aspects, we will cite some that are more oriented to the physical interpretations:

The classic references are of course:

• Rudolf Haag: Local quantum physics. Fields, particles, algebras. 2nd., rev. and enlarged ed. Springer 1996 ZMATH entry.

and:

• Huzihiro Araki: Mathematical theory of quantum fields. Oxford University Press 1999 ZMATH entry.

An online reference page by is here:

• Stephen J. Summers: AQFT online