nLab field net

Idea

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.

Reconstruction

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

Definition

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

• $A \in \mathcal{A}$ an observable;

• $\rho : \mathcal{A} \to \mathcal{A}$ a superselection sector, hence an object of the DHR category $\Delta_{DHR}$ of $\mathcal{A}$;

• $\psi \in E(\rho)$ a vector in the space Hilbert space assigned by the fiber functor of the Doplicher-Roberts reconstruction theorem to superselection sector $\rho$;

where the equivalence relation identifies for $T : \rho \to \rho'$ 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 $\mathcal{O}$ the subalgebra of such triples with $A \in \mathcal{A}(\mathcal{O})$ and $\rho$ localized in $\mathcal{O}$.

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

References

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

• Hans Halvorson, Algebraic quantum field theory (pdf)

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

• Michael Müger, Galois Theory for Braided Tensor Categories

  and the Modular Closure_, Adv. Math. 150, 151-201 (2000) (arXiv:math.CT/9812040)