causally local net of observables



In quantum field theory (or classical field theory) a causally local net of observables is a system (a co-presheaf) of algebras of observables assigned to regions of spacetime, such that this satisfies a few basic properties, such as notably causal locality, saying that observables whose spacetime support is spacelike-separated (graded-)commute with each other (Poisson-commute, in the case of classical field theory).

(Beware that the meaning of “net” here is vaguely similar to, but different from, the concept of net (as in: generalized sequences of points) as used in topology. Better terminology might be “causally local system of spacetime-localized observables”. But “local net” is traditional and has become standard.)

In the context of algebraic quantum field theory the structure of the local net of quantum observables is used as the very axiomatization of what a quantum field theory actually is (“Haag-Kastler axioms”). This may be thought of as a formalization of a spacetime-localized form of the Heisenberg picture of quantum physics (or rather the interaction picture in the case of perturbative AQFT); as opposed to the formalization of the Schrödinger picture in FQFT, where instead the state-propagation is used as the basic axiom.

Traditionally local nets of observables are assumed to take values in C*-algebras, but the basic form of the axioms does not actually refer to topological structure on the algebras, and makes sense more generally.

In particular in perturbative quantum field theory made precise via causal perturbation theory, the algebras of quantum observables are taken to be formal power series algebras (reflecting the infinitesimal nature of perturbation theory) and one derives from causal additivity of the S-matrix that the perturbative quantum observables form a local not of formal power series algebras (see at S-matrix – Causal locality and Quantum observables). Accordingly, this infinitesimal/pertrubative version of AQFT is called perturbative AQFT.

Other variants may be considered. For AQFT on curved spacetimes one generalizes from observables associated with regions of Minkowski spacetime toobsrvables associated with more general globally hyperbolic spacetimes. Combining this with perturbation theory is then called locally covariant perturbative AQFT.

Moreover, if gauge theory with nontrivial global gauge field configurations is to be considered (instantons) then one may show that one needs to consider some kind of homotopy theoretic local nets of homotopical algebras. See at homotopical AQFT.


In the literature there is a certain variance and flexibility of what precisely the axioms on a local net of observables are, though the core aspects are always the same: it is a copresheaf of (C-star algebra s) on pieces of spacetime such that algebras assigned to causally disconnected regions commute inside the algebra assigned to any joint neighbourhood.

Historically this was first formulated for Minkowski spacetime only, where it is known as the Haag-Kastler axioms. Later it was pointed out (BrunettiFredenhagen) that the axioms easily and usefully generalize to arbitrary spacetimes.

We give the modern general formulation first, and then comment on its restriction to special situations.

Basic general definition


Write LorSpLorSp for the category whose

Here we say a morphism f:XYf : X \hookrightarrow Y is a causal embedding if for every two points x 1,x 2Xx_1,x_2 \in X we have that f(x 1)f(x_1) is in the future of f(x 2)f(x_2) in YY only if x 1x_1 is in the future of x 2x_2 in XX.

Write AlgAlg for a suitable category of associative algebras. Usually this is taken to be the category of C-star algebras or that of von Neumann algebras. Write

Alg incAlg Alg_{inc} \hookrightarrow Alg

for the subcategory on the monomorphisms.


A causally local net of observables is a functor

𝒜:LorSpAlg incAlg \mathcal{A} : LorSp \to Alg_{inc} \to Alg

such that

  • (causal locality) whenever X 1X 2XX_1 \coprod X_2 \hookrightarrow X is a causal embedding, def. 1, we have that 𝒜(X 1)𝒜(X)\mathcal{A}(X_1) \subset \mathcal{A}(X) commutes with 𝒜(X 2)𝒜(X)\mathcal{A}(X_2) \subset \mathcal{A}(X).

The locality axiom encodes the the physical property known as Einstein-causality or micro-causality, which states that physical effects do not propagate faster that the speed of light.


Many auxiliary operators in quantum field theory do not satisfy causal locality: for instance operators associate to currents in gauge theory. The idea is that those operators that actually do qualify as observables do satisfy the axiom, however, i.e. in particular those that are gauge invariant.

Extra axioms

Einstein locality

Commutativity of spacelike separated observables can be argued to capture only part of causal locality.

A natural stronger requirement is that spacelike separated regions of spacetime are literally independent quantum subsystems of any larger region. By the formalization of independent subsystem in quantum mechanics this means the following:


A local net 𝒜\mathcal{A} satisfies Einstein locality if for every causal embedding X 1X 2XX_1 \coprod X_2 \to X the subsystems

𝒜(X 1)𝒜(X) \mathcal{A}(X_1) \hookrightarrow \mathcal{A}(X)


𝒜(X 2)𝒜(X) \mathcal{A}(X_2) \hookrightarrow \mathcal{A}(X)

are independent in that the algebra 𝒜(X 1)𝒜(X 2)𝒜(X)\mathcal{A}(X_1) \vee \mathcal{A}(X_2) \in \mathcal{A}(X) which they generate is isomorphic to the tensor product 𝒜(X 1)𝒜(X 2)\mathcal{A}(X_1) \otimes \mathcal{A}(X_2).

This appears as (BrunettiFredenhagen, 5.3.1, axiom 4).


A local net is Einstein local precisely if it is a monoidal functor

𝒜:(LorSp,)(Alg,). \mathcal{A} : (LorSp, \coprod) \to (Alg, \otimes) \,.

This appears as (BrunettiFredenhagen, 5.3.1, theorem 1).


Einstein locality implies causal locality, but is stronger.


Other properties implied by Einstein locality are sometimes extracted as separate axioms. For instance the condition that for X 1X 2XX_1 \coprod X_2 \to X a causal embedding, we have

𝒜(X 1)𝒜(X 2)=. \mathcal{A}(X_1) \cap \mathcal{A}(X_2) = \mathbb{C} \,.

Strong locality

In (Nuiten 11) the following variant of causal locality was considered and shown to be equivalent to a descent condition for the system of Bohr toposes associated with a local net of observables


(strong causal locality?)

A net of observables is strongly (causally) local? if it is microlocal (i.e. causally local) in that algebras A 1=A(O 1)A_1 = A(O_1) and A 2=A(O 2)A_2 = A(O_2) associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras C 1A 1C_1 \subset A_1 and C 2A 2C_2 \subset A_2 the algebra C 1C 2A(O 1O 2)C_1 \vee C_2 \subset A(O_1 \vee O_2) satisfies

  1. (C 1C 2)A 1=C 1(C_1 \vee C_2) \cap A_1 = C_1

  2. (C 1C 2)A 2=C 2(C_1 \vee C_2) \cap A_2 = C_2.

This is (Nuiten 11, def. 14).


It is clear that Einstein locality implies strong locality, def. 3

EinsteinlocalityStronglocality. Einstein\;locality \;\;\Rightarrow \;\; Strong\;locality \,.

In fact strong locality is strictly weaker than Einstein locality in that there are strongly locally embedded subalgebras which are not Einstein locally embedded. More discussion of this is in (Wolters 13, section 6.3.3).

Time-slice axiom


A local net is said to satisfy the time slice axiom if whenever

i:X 1X 2 i : X_1 \to X_2

is a causal embedding of globally hyperbolic spacetimes such that X 1X_1 contains a Cauchy surface of X 2X_2, then

𝒜(i):𝒜(X 1)𝒜(X 2) \mathcal{A}(i) : \mathcal{A}(X_1) \stackrel{\simeq}{\to} \mathcal{A}(X_2)

is an isomorphism.


See dual net of von Neumann algebras

Positive energy condition


Spectrum condition


Special cases and variants

Minkowski nets / Vacuum representation

Conformal nets

The notion of local net in the context of conformal field theory is a conformal net.


see for instance (Bär-Ginoux 11)


For more details see the references at AQFT.


The axioms of local nets on general spacetimes were first articulated in

A comprehensive review, with plenty of background information, is in

Discussion of Einstein locality of a net of observables equivalently as a descent condition on the system of Bohr toposes induced by the algebras of observables is in

A review of this with some further discussion is in section 6 of

  • Sander Wolters, Quantum toposophy, PhD Thesis 2013 (pdf)

Discussion of properly co-stacky nets of local observables in gauge theory is in


Discussion of standard example includes

  • Christian Bär, N. Ginoux, Classical and quantum fields on lorentzian manifolds, in Global Differential Geometry, Springer Proceedings in Math. 17, Springer-Verlag (2011), 359-400 (arXiv:1104.1158)

In perturbation theory

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).

Revised on September 18, 2017 13:46:50 by Urs Schreiber (