Letters in Mathematical Physics

, Volume 77, Issue 3, pp 235–248 | Cite as

On the Generator of Massive Modular Groups



The purpose of this paper is to shed more light on the transition from the known massless modular action to the wanted massive one in the case of forward light cones and double cones. The infinitesimal generator δm of the modular automorphism group \(\big(\sigma_{\rm m}^t\big)_{t\in\mathbb R}\) is investigated, in particular, some assumptions on its structure are verified explicitly for two concrete examples.

Mathematics Subject Classification (2000)

81T05 46L40 46L60 47G30 


algebraic quantum field theory modular automorphism group pseudo-differential operator 


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  1. 1.
    Araki H., (1999) Mathematical Theory of Quantum Fields. Univ. Pr., Oxford, 236 p.MATHGoogle Scholar
  2. 2.
    Bisognano J.J., Wichmann E.H. (1975) On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007CrossRefADSMathSciNetMATHGoogle Scholar
  3. 3.
    Bisognano J.J., Wichmann E.H. (1976) On the duality condition for Quantum fields. J. Math. Phys. 17, 303–321CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Borchers H.J. (1992) The CPT Theorem in Two-dimensional Theories of local observables. Commun. Math. Phys. 143, 315–332CrossRefADSMathSciNetMATHGoogle Scholar
  5. 5.
    Borchers H.J., Yngvason J. (1999) Modular groups of Quantum fields in Thermal states. J. Math. Phys. 40, 601–624CrossRefADSMathSciNetMATHGoogle Scholar
  6. 6.
    Brunetti R., Guido D., Longo R. (1994) Group cohomology, modular Theory and space–Time symmetries. Rev. Math. Phys. 7, 57–71CrossRefMathSciNetGoogle Scholar
  7. 7.
    Buchholz D.: On the structure of local quantum fields with non-trivial interpretation Buchholz D (REC.OCT 77) 10 p(preprint)Google Scholar
  8. 8.
    Buchholz D., D’Antoni C., Longo R. (1990) Nuclear maps and modular structure 2. Applications to quantum field Theory. Commun. Math. Phys. 129, 115–138ADSMathSciNetMATHGoogle Scholar
  9. 9.
    Connes A. (1973) Une Classification des Facteurs de Type III. Ann. Sci. Ecole Norm. Sup. 6, 133–252MathSciNetMATHGoogle Scholar
  10. 10.
    Figliolini F., Guido D. (1989) The Tomita operator for the free scalar field. Ann. Inst. H. Poincaré 51, 419–435MathSciNetMATHGoogle Scholar
  11. 11.
    Gaier J., Yngvason J. (2000) Geometric modular action, wedge duality, and Lorentz covariance are equivalent for generalized free fields. J. Math. Phys. 41, 5910–5919CrossRefADSMathSciNetMATHGoogle Scholar
  12. 12.
    Guido D., Longo R. (1995) An algebraic spin and statistics Theorem. Commun. Math. Phys. 172, 517–533CrossRefADSMathSciNetMATHGoogle Scholar
  13. 13.
    Haag R.: Local Quantum Physics: Fields, Particles, Algebras, 356 p. Springer, Berlin Heidelberg Newyork (Texts and monographs in physics, 1992)Google Scholar
  14. 14.
    Haag R., Kastler D. (1964) An algebraic approach to quantum field Theory. J. Math. Phys. 5, 848–861CrossRefADSMathSciNetMATHGoogle Scholar
  15. 15.
    Hislop P.D., Longo R. (1982) Modular structure of the local algebras associated with the free massless scalar field Theory. Commun. Math. Phys. 84, 71–85CrossRefADSMathSciNetMATHGoogle Scholar
  16. 16.
    Hörmander L. (1971) Fourier integral operators 1. Acta Math. 127, 79–183CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Hörmander L. (1990) The Analysis of Linear Partial Differential Operators, vol. 1, 3, 2nd edn. Springer, Berlin Heidelberg NewyorkGoogle Scholar
  18. 18.
    Jones V.F.R. (1983) Index for subfactors. Invent. Math. 72, 1–25CrossRefADSMathSciNetMATHGoogle Scholar
  19. 19.
    Kawahigashi Y., Longo R. Classification of local conformal nets: case c < 1. math-ph/0201015 (2002)Google Scholar
  20. 20.
    Kosaki H. (1986) Extension of Jones’ Theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Saffary T.: Modular action on the massive algebra. DESY-THESIS-2005-039Google Scholar
  22. 22.
    Schroer B., Wiesbrock H.W. (2000) Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301–326CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Schroer B., Wiesbrock H.W. (2000) Modular Theory and geometry. Rev. Math. Phys. 12, 139–158CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Takesaki M. (1970) Tomita’s Theory of modular Hilbert algebras and its applications. Lecture Notes in Math. Springer, Berlin Heidelberg Newyork 128, 273–286Google Scholar
  25. 25.
    Trebels S. (1997) Über die geometrische Wirkung modularer Automorphismen Thesis, GöttingenGoogle Scholar
  26. 26.
    Yngvason J. (1994) A note on essential duality. Lett. Math. Phys. 31, 127–141CrossRefADSMathSciNetMATHGoogle Scholar

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© Springer 2006

Authors and Affiliations

  1. 1.Fachbereich Wirtschafts- und OrganisationswissenschaftenHelmut-Schmidt-Universität, Universität der Bundeswehr HamburgHamburgGermany

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