nLab
C-star-system

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

Contents

Idea

A C *C^*-dynamical system, or only C *C^*-system is a C-star-algebra together with an action of a group of automorphisms. In quantum mechanics as well as in AQFT the observables of the theory are self-adjoint operators of (a local net of) C-star-algebras, in this context the global gauge group of the theory is the maximal group of unitary operators that leave all observables invariant, the algebra and the gauge group form a C *C^*-system.

Definition

Definition

A C *C^*-system (𝒜,α G)(\mathcal{A}, \alpha_G) consists of a C *C^*-algebra 𝒜\mathcal{A}, a locally compact group GG and a continuous homomorphism α\alpha of GG into the group aut(𝒜)aut(\mathcal{A}) of **-automorphisms of 𝒜\mathcal{A} equipped with the topology of pointwise convergence.

If the algebra is a **-algebra only, then some authors call it a **-system.

Sometimes the continuity condition is dropped entirely or replaced by some weaker assumption, therefore one should always check what – if any – continuity assumption an author makes.

Definition

The fixed point algebra of a C *C^*-system (𝒜,α G)(\mathcal{A}, \alpha_G) is {A𝒜:a gA=AgG}\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G \}. If the fixed point algebra is trivial then α G\alpha_G acts ergodically.

Definition

A state ρ\rho of the algebra 𝒜\mathcal{A} is an invariant state if

ρ(A)=ρ(α gA)A𝒜,gG. \rho (A) = \rho(\alpha_g A) \; \forall A \in \mathcal{A}, \; \forall g \in G.

Properties

Lemma

The set of invariant states is convex, weak-** closed and weak-** compact. (see operator topology).

References

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

  • chapter 6 in Gerd Petersen, Pullback and pushout constructions in C *C^\ast-algebra theory, J. Funct. Analysis 167, 243–344 (1999) pdf

Created on December 23, 2013 at 08:06:54. See the history of this page for a list of all contributions to it.