Modular Automorphisms of Local Algebras
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 στ(𝒪) for any such ℜ(𝒪). What is the meaning of these objects? We know that στ(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 στ (𝒪) or J(𝒪) of ℜ( 𝒪1) with 𝒪1 ⊂ 𝒪 is not, in general, some local subalgebra. There are, however, interesting special cases where στ and J do have a direct geometric meaning.
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