nLab
net of C-star-systems

Context

AQFT

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

The notion of a net C *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 π’œ I\mathcal{A}_I be a local net of C-star algebras. Let GG be a locally compact topological group and Ξ± G\alpha_G a representation of GG on the quasi-local algebra π’œ\mathcal{A}, that is

π’œ:=clo βˆ₯β‹…βˆ₯(⋃ i∈Iπ’œ i) \mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{i \in I} \mathcal{A}_i \bigr)

so that (π’œ,Ξ± G)(\mathcal{A}, \alpha_G) is a C-star system.

Definition

The tupel (π’œ I,Ξ± G)(\mathcal{A}_I, \alpha_G) is a net of C *C^*-systems if Ξ± g(π’œ i)βŠ†π’œ iβˆ€g∈G\alpha_g(\mathcal{A}_i) \subseteq \mathcal{A}_i \; \forall g \in G.

In the context of Haag-Kastler nets the group GG is called the
global gauge group and every automorphism Ξ± g\alpha_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)