Contents

Idea

Local nets are a structure used in AQFT in order to axiomatize local algebras of observables in quantum field theory.

Definition

Let $X$ be a topological space. A co-presheaf on the category of open subsets of $X$

is a net if it is “co-flabby”, i.e. if it sends every inclusion to a monomorphism.

If

• the net takes values in algebras;

• there is given the structure of a Minkowski manifold on $X$

the net $A$ is called local precisely if for all open subset $O_1,O_2\subset O$ the images of the algebras $A(O_1)$ and $A(O_2)$ in $A(O)$ commute whenever $O_1$ and $O_2$ are completely spacelike seperated

This 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 like.

It is to be noted that many auxiliary operators in usual quantum field theory do not satisfy this axioms, 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.

Related concepts

Factorization algebras

There is a version of the notion of local nets for Euclidean spaces. This is closely related to the notion of factorization algebra.

Conformal nets

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

References

for the moment see the references at AQFT.