Local Quantum Physics pp 244-253 | Cite as

# Modular Automorphisms of Local Algebras

## Abstract

Let us return to the consideration of the vacuum sector of a relativistic local theory. We use the concepts and notation of section III.4. The theorem of Reeh and Schlieder (subsection II.5.3) says that the state vector Ω of the vacuum is cyclic and separating for any algebra ℜ(𝒪) as long as the causal complement of 𝒪 contains a non-void open set. Thus we can apply the Tomita-Takesaki theory. The vacuum defines a modular operator Δ(𝒪), a modular conjugation *J*(𝒪) and a modular automorphism group σ_{τ}(𝒪) for any such ℜ(𝒪). What is the meaning of these objects? We know that σ_{τ}(𝒪) maps ℝ(𝒪) onto itself, *J*(𝒪) maps ℜ(𝒪) onto ℜ (𝒪) . But beyond this we have, in general, no geometric interpretation; the image under σ_{τ} (𝒪) or *J*(𝒪) of ℜ( 𝒪_{1}) with 𝒪_{1} ⊂ 𝒪 is not, in general, some local subalgebra. There are, however, interesting special cases where σ_{τ} and *J* do have a direct geometric meaning.

## Keywords

Local Algebra Classical Phase Space Vacuum Sector Modular Automorphism Modular Conjugation## Preview

Unable to display preview. Download preview PDF.