Communications in Mathematical Physics

, Volume 304, Issue 1, pp 95–123 | Cite as

Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories

  • Detlev BuchholzEmail author
  • Gandalf Lechner
  • Stephen J. Summers
Open Access


Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C *–dynamical systems with automorphic actions of \({\mathbb R^n}\) , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita–Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.


Minkowski Space Spacetime Dimension Adjoint Action Covariant Representation Strong Operator Topology 
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.



GL wishes to thank S. Waldmann for interesting discussions about Rieffel deformations. Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Araki H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  2. 2.
    Baumgärtel H., Wollenberg M.: Causal Nets of Operator Algebras. Akademie Verlag, Berlin (1992)zbMATHGoogle Scholar
  3. 3.
    Borchers H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 15–332 (1992)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Borchers H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41, 3604–3673 (2000)CrossRefzbMATHADSMathSciNetGoogle Scholar
  5. 5.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Buchholz D., D’Antoni C., Fredenhagen K.: The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. 7.
    Buchholz D., Dreyer O., Florig M., Summers S.J.: Geometric modular action and spacetime symmetry groups. Rev. Math. Phys. 12, 475–560 (2000)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)CrossRefzbMATHADSMathSciNetGoogle Scholar
  9. 9.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  10. 10.
    Buchholz D., Summers S.J.: An Algebraic characterization of vacuum states in Minkowski space. 3. Reflection maps. Commun. Math. Phys. 246, 625–641 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  11. 11.
    Buchholz D., Summers S.J.: Stable quantum systems in Anti-de Sitter space: Causality, independence and spectral properties. J. Math. Phys. 45, 4810–4831 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  12. 12.
    Buchholz D., Summers S.J.: String– and brane–localized causal fields in a strongly nonlocal model. J. Phys. A 40, 2147–2163 (2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  13. 13.
    Buchholz, D., Summers, S.J.: Warped convolutions: A novel tool in the construction of quantum field theories. In: Quantum Field Theory and Beyond, edited by Seiler, E., Sibold, K. Singapore: World Scientific, 2008, pp. 107–121Google Scholar
  14. 14.
    Dappiaggi, C., Lechner, G., Morfa-Morales, E.: Deformations of quantum field theories on spacetimes with Killing vector fields. Commun. Math. Phys. (2010). arXiv:1006.3548 (to appear)Google Scholar
  15. 15.
    Florig M.: On Borchers’ theorem. Lett. Math. Phys. 46, 289–293 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Glimm J., Jaffe A.: Quantum Physics. A Functional Integral Point of View. Springer Verlag, Berlin-Heidelberg-New York (1987)Google Scholar
  17. 17.
    Grosse H., Lechner G.: Wedge–local quantum fields and noncommutative Minkowski space. JHEP 0711, 012 (2007)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Grosse H., Lechner G.: Noncommutative deformations of Wightman quantum field theories. JHEP 0809, 131 (2008)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Guido D.: Modular covariance, PCT, Spin and Statistics. Ann. Inst. Henri Poincaré 63, 383–398 (1995)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Haag, R.: Local Quantum Physics. Berlin, Heidelberg and New York: Springer Verlag, 1992Google Scholar
  21. 21.
    Kaschek D., Neumaier N., Waldmann S.: Complete positivity of Rieffel’s quantization by actions of \({\mathbb R^d}\). J. Noncommut. Geom. 3, 361–375 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lechner G.: On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38, 3045–3056 (2005)CrossRefzbMATHADSMathSciNetGoogle Scholar
  24. 24.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)CrossRefzbMATHADSMathSciNetGoogle Scholar
  25. 25.
    Lechner, G.: Article in preparationGoogle Scholar
  26. 26.
    Mund J., Schroer B., Yngvason J.: String–localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006)CrossRefzbMATHADSMathSciNetGoogle Scholar
  27. 27.
    Pedersen G.K.: C*–Algebras and Their Automorphism Groups. Academic Press, London-New York-San Francisco (1979)zbMATHGoogle Scholar
  28. 28.
    Rieffel M.A.: Deformation quantization for actions of \({\mathbb R^d}\). Memoirs A.M.S. 506, 1–96 (1993)Google Scholar
  29. 29.
    Schroer B.: Modular localization and the bootstrap–formfactor program. Nucl. Phys. B 499, 547–568 (1997)CrossRefzbMATHADSMathSciNetGoogle Scholar
  30. 30.
    Takesaki M.: Tomita’s Theory of Modular Hilbert Algebras and Its Applications. Springer Verlag, Berlin-Heidelberg-New York (1970)zbMATHGoogle Scholar
  31. 31.
    Takesaki M.: Theory of Operator Algebras. Volume II. Springer Verlag, Berlin-Heidelberg-New York (2003)Google Scholar

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© The Author(s) 2010

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Detlev Buchholz
    • 1
    Email author
  • Gandalf Lechner
    • 2
  • Stephen J. Summers
    • 3
  1. 1.Institut für Theoretische Physik and Courant Centre “Higher Order Structures in Mathematics”Universität GöttingenGöttingenGermany
  2. 2.Fakultät für PhysikUniversität WienViennaAustria
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA

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