nLab
net of C-star-systems

Context

AQFT

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

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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

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

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

Definition

The tupel (𝒜 I,α G)(\mathcal{A}_I, \alpha_G) is a net of C *C^*-systems if α g(𝒜 i)𝒜 igG\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 α 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:

Last revised on May 14, 2012 at 15:23:11. See the history of this page for a list of all contributions to it.