nLab net of C-star-systems

Idea

The notion of a net of $C^*$-systems combines the notion of a C-star system with the notion of local net of observables. In this way, the notion of global gauge groups is introduced into the Haag-Kastler approach to AQFT.

Definition

Let $\mathcal{A}_I$ be a local net of C-star algebras. Let $G$ be a locally compact topological group and $\alpha_G$ a representation of $G$ on the quasi-local algebra $\mathcal{A}$, that is

$$\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{i \in I} \mathcal{A}_i \bigr)$$

so that $(\mathcal{A}, \alpha_G)$ is a C-star system.

Definition

The tuple $(\mathcal{A}_I, \alpha_G)$ is a net of $C^*$-systems if $\alpha_g(\mathcal{A}_i) \subseteq \mathcal{A}_i \; \forall g \in G$.

In the context of Haag-Kastler nets the group $G$ is called the
global gauge group and every automorphism $\alpha_g$ is called a gauge automorphism.

This definition makes sense also if the net consists of star-algebras only, of course.

Related concepts

• local net of observables

• field net

References

Chapter 6 of:

• Hellmut Baumgärtel, Manfred Wollenberg: Causal nets of operator algebras. Berlin: Akademie Verlag 1992 (ZMATH entry)