AQFT and operator algebra
A local net of observables is the assignment associated to a quantum field theory of algebras of local observables to pieces of spacetime.
In the context of AQFT the structure of local nets is used as the very axiomatization of what a quantum field theory is (as opposed to the context of FQFT, where instead the state-propagation is used as the basic axiom).
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.
Write $LorSp$ for the category whose
objects are spacetime manifolds; (Lorentzian manifolds equipped with a time-orientation);
Here we say a morphism $f : X \hookrightarrow Y$ is a causal embedding if for every two points $x_1,x_2 \in X$ we have that $f(x_1)$ is in the future of $f(x_2)$ in $Y$ only if $x_1$ is in the future of $x_2$ in $X$.
Write $Alg$ 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
for the subcategory on the monomorphisms.
A causally local net of observables is a functor
such that whenever $X_1 \coprod X_2 \hookrightarrow X$ is a causal embedding, def. 1, we have that $\mathcal{A}(X_1) \subset \mathcal{A}(X)$ commutes with $\mathcal{A}(X_2) \subset \mathcal{A}(X)$.
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.
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 $\mathcal{A}$ satisfies Einstein locality if for every causal embedding $X_1 \coprod X_2 \to X$ the subsystems
and
are independent in that the algebra $\mathcal{A}(X_1) \vee \mathcal{A}(X_2) \in \mathcal{A}(X)$ which they generate is isomorphic to the tensor product $\mathcal{A}(X_1) \otimes \mathcal{A}(X_2)$.
This appears as (BrunettiFredenhagen, 5.3.1, axiom 4).
A local net is Einstein local precisely if it is a monoidal functor
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 $X_1 \coprod X_2 \to X$ a causal embedding, we have
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
A net of observables is strongly local if it is microlocal in that algebras $A_1 = A(O_1)$ and $A_2 = A(O_2)$ associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras $C_1 \subset A_1$ and $C_2 \subset A_2$ the algebra $C_1 \vee C_2 \subset A(O_1 \vee O_2)$ satisfy
$(C_1 \vee C_2) \cap A_1 = C_1$
$(C_1 \vee C_2) \cap A_2 = C_2$.
This is (Nuiten 11, def 14).
It is clear that Einstein locality implies strong locality, def. 3
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).
A local net is said to satisfy the time slice axiom if whenever
is a causal embedding of globally hyperbolic spacetimes such that $X_1$ contains a Cauchy surface of $X_2$, then
is an isomorphism.
See dual net of von Neumann algebras
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The notion of local net in the context of conformal field theory is a conformal net.
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
Discussion of properly co-stacky nets of local observables in gauge theory is in
The observation that in perturbation theory the StΓΌckelberg-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).