Communications in Mathematical Physics

, Volume 315, Issue 2, pp 445–464 | Cite as

An Algebraic Jost-Schroer Theorem for Massive Theories



We consider a purely massive local relativistic quantum theory specified by a family of von Neumann algebras indexed by the space-time regions. We assume that, affiliated with the algebras associated to wedge regions, there are operators which create only single particle states from the vacuum (so-called polarization-free generators) and are well-behaved under the space-time translations. Strengthening a result of Borchers, Buchholz and Schroer, we show that then the theory is unitarily equivalent to that of a free field for the corresponding particle type. We admit particles with any spin and localization of the charge in space-like cones, thereby covering the case of string-localized covariant quantum fields.


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  1. 1.
    Araki H.: Von Neumann algebras of local observables for the free scalar field. J. Math. Phys. 5, 1–13 (1964)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Araki, H.: Mathematical theory of quantum fields. Int. Series of Monographs in Physics, no. 101, Oxford: Oxford University Press, 1999Google Scholar
  3. 3.
    Baumgärtel H., Jurke M., Lledo F.: On free nets over Minkowski space. Rep. Math. Phys. 35, 101–127 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Baumgärtel H., Jurke M., Lledo F.: Twisted duality of the CAR-algebra. J. Math. Phys. 43, 4158–4179 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitean scalar field. J. Math. Phys. 16, 985 (1975)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Borchers H.J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Borchers H.J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Borchers H.J., Yngvason J.: Positivity of Wightman functionals and the existence of local nets. Commun. Math. Phys. 127, 607–615 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Bros, J., Epstein, H.: Charged physical states and analyticity of scattering amplitutdes in the Buchholz Fredenhagen framework. In: 11th International Conference on Mathematical Physics, 1994, pp. 330–341Google Scholar
  10. 10.
    Bros J., Epstein H., Glaser V.: Some rigorous analyticity properties of the four-point function in momentum space. Nuovo Cim. 31, 1265–1302 (1964)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bros, J., Mund, J.: Braid group statistics implies scattering in three-dimensional local quantum physics. Commun. Math. Phys. (2012). [hep-th]
  12. 12.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Buchholz D., Epstein H.: Spin and statistics of quantum topological charges. Fysica 17, 329–343 (1985)Google Scholar
  14. 14.
    Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Buchholz D., Lechner G., Summers S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Buchholz D., Summers S.J.: String- and brane- localized causal fields in a strongly nonlocal model. J. Phys. A 40, 2147–2163 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Buchholz, D., Summers, S.J.: Warped convolutions: A novel tool in the construction of quantum field theories, In: Quantum Field Theory and Beyond, E. Seiler, K. Sibold, eds., Singapore: World Scientific 2008, pp. 107–121Google Scholar
  18. 18.
    Dell’Antonio G.F.: Structure of the algebras of some free systems. Commun. Math. Phys. 9, 81–117 (1968)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1–23 (1969)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Federbush P.G., Johnson K.A.: Uniqueness porperty of the twofold vacuum expectation value. Phys. Rev. 120, 1926 (1960)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Garber W.-D.: The connexion of duality and causal properties for generalized free fields. Commun. Math. Phys. 42, 195–208 (1975)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Grosse H., Lechner G.: Wedge-local quantum fields and noncommutative Minkowski space. JHEP 0711, 012 (2007)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Grosse H., Lechner G.: Noncommutative deformations of Wightman quantum field theories. JHEP 0809, 131 (2008)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Jost, R.: Properties of Wightman functions, In: Lectures on Field Theory: The many Body Problem, E.R. Caianello, ed., New York: Academic Press, 1961Google Scholar
  25. 25.
    Jost, R.: The general theory of quantized fields. Providence, RI: Amer. Math. Soc., 1965Google Scholar
  26. 26.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras. Providence, RI: Amer. Math. Soc., 1997Google Scholar
  27. 27.
    Leyland, P., Roberts, J., Testard, D.: Duality for quantum free fields. Unpublished notes, 1978Google Scholar
  28. 28.
    Mandelstam S.: Quantum electrodynamics without potentials. Ann. Phys. 19, 1–24 (1962)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Mund J.: No-go theorem for ‘free’ relativistic anyons in d =  2 + 1. Lett. Math. Phys. 43, 319–328 (1998)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Mund J.: The Bisognano-Wichmann theorem for massive theories. Ann. H. Poinc. 2, 907–926 (2001)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Mund J., Schroer B., Yngvason J.: String–localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  32. 32.
    Pohlmeyer K.: The Jost-Schroer theorem for zero-mass fields. Commun. Math. Phys. 12, 204 (1969)MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Reed M., Simon B.: Methods of modern mathematical physics I, II. Academic Press, New York (1975/1980)Google Scholar
  34. 34.
    Schroer, B.: Master’s thesis, University Hamburg, 1958Google Scholar
  35. 35.
    Steinmann O.: A Jost-Schroer theorem for string fields. Commun. Math. Phys. 87, 259–264 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  36. 36.
    Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. New York: W. A. Benjamin Inc., 1964Google Scholar
  37. 37.
    Weinberg S.: The quantum theory of fields I. Cambridge University Press, Cambridge (1995)Google Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de Juiz de ForaJuiz de ForaBrazil

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