nLab field net




algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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.


Given a local net of observables 𝒜\mathcal{A} the corresponding field algebra, according to Doplicher-Roberts the the following.


The field algebra 0\mathcal{F}_0 corresponding to 𝒜\mathcal{A} is the collection of equivalence classes of triples (A,ρ,ψ)(A, \rho, \psi) consisting

where the equivalence relation identifies for T:ρρT : \rho \to \rho' an intertwiner two such triples by the rule

(AT,ρ,ψ)(A,ρ,E(T)ψ). (A T, \rho, \psi) \sim (A, \rho', E(T)\psi) \,.

This becomes an algebra by defining the product on representatives as

(A 1,ρ 1,ψ 1)(A 2,ρ 2,ψ 2)=(A 1ρ 1(A 2),ρ 1ρ 2,ψ 1ψ 2). (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 𝒪\mathcal{O} the subalgebra of such triples with A𝒜(𝒪)A \in \mathcal{A}(\mathcal{O}) and ρ\rho localized in 𝒪\mathcal{O}.

Finally, one can construct a representation of this in bounded operators, and taking the norm closure there gives the field net \mathcal{F}.


The construction of a field net for every local net of observables in DHR superselection theory is due to

  • Sergio Doplicher, John Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Comm. Math. Phys. 131 (1) (1990)

A detailed review is in sections 9 and 10 of

A purely category theoretic construction of the field-net analog of the DHR category is given in

Last revised on May 15, 2012 at 10:32:42. See the history of this page for a list of all contributions to it.