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Modular Automorphisms of Local Algebras

  • Rudolf Haag
Part of the Texts and Monographs in Physics book series (TMP)

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 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.

Keywords

Local Algebra Classical Phase Space Vacuum Sector Modular Automorphism Modular Conjugation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Rudolf Haag
    • 1
  1. 1.IL Institut für Theoretische PhysikUniversität HamburgHamburg 50Fed. Rep. of Germany

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