Communications in Mathematical Physics

, Volume 61, Issue 3, pp 267–273 | Cite as

Remarks on the modular operator and local observables

  • Carlo Rigotti


In this paper we give a characterization of the modular group of a von Neumann algebra ℛ, with a cyclic and separating vector, which provides at the same time a necessary and sufficient condition so that two von Neumann algebras ℛ1 and ℛ2, such that ℛ1⊂ℛ′2, are the mutual commutants, i.e. ℛ1=ℛ′2.

An application is made to the duality property in Quantum Field Theory, and we give a sufficient condition for PCT invariance in a theory of local observables.


Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
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  1. 1.
    Bisognano, J., Wichmann, E.: On the duality condition for a Hermitian scalar field. J. Math. Phys.16, 985 (1975)CrossRefGoogle Scholar
  2. 2.
    Borchers, J.: On the vacuum state in quantum field theory. Commun. math. Phys.1, 57 (1965)CrossRefGoogle Scholar
  3. 3.
    Dixmier, J.: Les algebres d'operateurs dans l'espace hilbertien (les algebres de von Neumann). Paris: Gautier-Villars 1969Google Scholar
  4. 4.
    Doplicher, S., Haag, R., Roberts, J.: Local observables and particles statistics. I. Commun. math. Phys.23, 199 (1971)CrossRefGoogle Scholar
  5. 5.
    Reed, M., Simon, B.: Fourier analysis. Selfadjointness. New York-London: Academic Press 1975Google Scholar
  6. 6.
    Rieffel, M., Van Daele, A.: The commutation theorem for tensor products of von Neumann algebras. Bull. Lond. Math. Soc.7, 257–260 (1975)Google Scholar
  7. 7.
    Roberts, J.: Local cohomology and its structural implications for field theory. Preprint (1977)Google Scholar
  8. 8.
    Takesaki, M.: Tomita theory of modular Hilbert algebras and its applications. Lecture notes in mathematics, Vol. 128. Berlin-Heidelberg-New York: Springer 1970Google Scholar
  9. 9.
    Wightman, A., Streater, R.: PCT, spin and statics and all that. New York: Benjamin 1964Google Scholar
  10. 10.
    Araki, H.: Positive cone, Radon Nicodym theorems, relative hamiltonian. Proceedings of the International School of Physics “Enrico Fermi”, Varenna 1973 (Course LX)Google Scholar
  11. 11.
    Rieffel, M.: A communtation theorem and duality for free Bose fields. Commun. math. Phys.39, 153–164 (1974)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Carlo Rigotti
    • 1
  1. 1.Istituto Matematico “G. Castelnuovo”Università di RomaRomaItaly

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