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 for the category whose
Here we say a morphism is a causal embedding if for every two points we have that is in the future of in only if is in the future of in .
A causally local net of observables is a functor
such that whenever is a causal embedding, def. 1, we have that commutes with .
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 satisfies Einstein locality if for every causal embedding the subsystems
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 a causal embedding, we have
A net of observables is strongly local if it is microlocal in that algebras and associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras and the algebra satisfy
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 an isomorphism.
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
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
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).