# Contents

## Idea

The notion of a net ${C}^{*}$-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 ${\mathrm{\pi }}_{I}$ be a local net of C-star algebras. Let $G$ be a locally compact topological group and ${\mathrm{\Xi ±}}_{G}$ a representation of $G$ on the quasi-local algebra $\mathrm{\pi }$, that is

$\mathrm{\pi }:={\mathrm{clo}}_{\beta ₯\beta  \beta ₯}\left(\underset{i\beta I}{\beta }{\mathrm{\pi }}_{i}\right)$\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{i \in I} \mathcal{A}_i \bigr)

so that $\left(\mathrm{\pi },{\mathrm{\Xi ±}}_{G}\right)$ is a C-star system.

###### Definition

The tupel $\left({\mathrm{\pi }}_{I},{\mathrm{\Xi ±}}_{G}\right)$ is a net of ${C}^{*}$-systems if ${\mathrm{\Xi ±}}_{g}\left({\mathrm{\pi }}_{i}\right)\beta {\mathrm{\pi }}_{i}\phantom{\rule{thickmathspace}{0ex}}\beta g\beta G$.

In the context of Haag-Kastler nets the group $G$ is called the
global gauge group and every automorphism ${\mathrm{\Xi ±}}_{g}$ is called a gauge automorphism.

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

## References

Chapter 6 of:

Revised on May 14, 2012 15:23:11 by Urs Schreiber (82.113.99.198)