In the AQFT formalization of quantum field theory a local net of observables assigns to each region of spacetime the algebra of observables localized in that region. But in typical constructions of quantum field theories this algebra is obtained from an algebra of quantum fields that are not all observable by quotienting out a gauge group action. A field net corresponding to a local net of observables is a net of C-star-systems which formalizes this idea, notably so that the quotient by the group action reproduces the given local net of observables.
The field algebra corresponding to is the collection of equivalence classes of triples consisting
where the equivalence relation identifies for an intertwiner two such triples by the rule
(A T, \rho, \psi) \sim (A, \rho', E(T)\psi) \,.
This becomes an algebra by defining the product on representatives as
(A_1, \rho_1, \psi_1) \cdot (A_2, \rho_2, \psi_2) = (A_1 \rho_1(A_2), \rho_1 \otimes \rho_2, \psi_1 \otimes \psi_2) \,.
With a bit more work a star algebra structure is defined.
This becomes a net by assigning to an open the subalgebra of such triples with and localized in .
The construction of a field net for every local net of observables in DHR superselection theory is due to
A detailed review is in sections 9 and 10 of