Communications in Mathematical Physics

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

An Algebraic Jost-Schroer Theorem for Massive Theories

  • Jens MundEmail author


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.


Single Particle State Massive Theory Universal Covering Group Haag Duality Wedge Region 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araki H.: Von Neumann algebras of local observables for the free scalar field. J. Math. Phys. 5, 1–13 (1964)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Baumgärtel H., Jurke M., Lledo F.: Twisted duality of the CAR-algebra. J. Math. Phys. 43, 4158–4179 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitean scalar field. J. Math. Phys. 16, 985 (1975)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Borchers H.J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Borchers H.J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1–23 (1969)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Federbush P.G., Johnson K.A.: Uniqueness porperty of the twofold vacuum expectation value. Phys. Rev. 120, 1926 (1960)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Garber W.-D.: The connexion of duality and causal properties for generalized free fields. Commun. Math. Phys. 42, 195–208 (1975)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Mund J.: No-go theorem for ‘free’ relativistic anyons in d =  2 + 1. Lett. Math. Phys. 43, 319–328 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Mund J.: The Bisognano-Wichmann theorem for massive theories. Ann. H. Poinc. 2, 907–926 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Mund J., Schroer B., Yngvason J.: String–localized quantum fields from Wigner representations. Phys. Lett. B 596, 156–162 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    Pohlmeyer K.: The Jost-Schroer theorem for zero-mass fields. Commun. Math. Phys. 12, 204 (1969)MathSciNetADSzbMATHCrossRefGoogle 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)MathSciNetADSzbMATHCrossRefGoogle 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

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

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

Personalised recommendations