Letters in Mathematical Physics

, Volume 103, Issue 4, pp 421–437 | Cite as

On the Equivalence of Two Deformation Schemes in Quantum Field Theory

Open Access


Two recent deformation schemes for quantum field theories on two-dimensional Minkowski space, making use of deformed field operators and Longo–Witten endomorphisms, respectively, are shown to be equivalent.

Mathematics Subject Classification (2010)

81T05 81T40 


deformations of quantum field theories two-dimensional models modular theory 


  1. 1.
    Alazzawi, S.: Deformations of Fermionic Quantum Field Theories and Integrable Models. Lett. Math. Phys. (2012). doi: 10.1007/s11005-012-0576-3
  2. 2.
    Bischoff, M.:Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods. Ph.D. Thesis, Università à di Roma “Tor Vergata” (2012)Google Scholar
  3. 3.
    Bischoff, M., Tanimoto, Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. II. Commun. Math. Phys. (2012). doi: 10.1007/s00220-012-1593-x
  4. 4.
    Borchers H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)ADSMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bostelmann H., Lechner G., Morsella G.: Scaling limits of integrable quantum field theories. Rev. Math. Phys. 23(10), 1115–1156 (2011)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buchholz D.: Collision theory for massless bosons. Commun. Math. Phys. 52, 147 (1977)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004)ADSMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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)ADSMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Buchholz, D., Summers, S.J.: Warped convolutions: a novel tool in the construction of quantum field theories. In: Seiler, E., Sibold, K. (eds.) Quantum Field Theory and Beyond: Essays in Honor of Wolfhart Zimmermann. World Scientific, pp. 107–121 (2008)Google Scholar
  10. 10.
    Dybalski W., Tanimoto Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Commun. Math. Phys. 305, 427–440 (2011)ADSMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fendley, P., Saleur, H.: Massless integrable quantum field theories and massless scattering in 1+1 dimensions. Tech. Report USC-93-022, October (1993)Google Scholar
  12. 12.
    Grosse H., Lechner G.: Wedge-local quantum fields and noncommutative Minkowski Space. JHEP 11, 012 (2007)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Guichardet A.: Symmetric Hilbert Spaces and Related Topics. Springer, Berlin (1972)MATHGoogle Scholar
  14. 14.
    Haag R.: Local Quantum Physics—Fields, Particles Algebras, 2 edn. Springer, Berlin (1996)Google Scholar
  15. 15.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)ADSMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312(1), 265–302 (2012)ADSMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Longo R., Rehren K.H.: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys. 311(3), 769–785 (2012)ADSMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Longo R., Witten E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303(1), 213–232 (2011)ADSMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Morfa-Morales E.: Deformations of quantum field theories on de Sitter spacetime. J. Math. Phys. 52, 102304 (2011)ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    Much A.: Wedge-local quantum fields on a nonconstant noncommutative spacetime. J. Math. Phys. 53, 082303 (2012)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Plaschke, M.: Wedge local deformations of charged fields leading to anyonic commutation relations (2012). http://arxiv.org/abs/1208.6141v1
  23. 23.
    Takesaki, M.:Theory of Operator Algebras II. Springer, Berlin (2003). http://www.springer.com/mathematics/analysis/book/978-3-540-42914-2
  24. 24.
    Tanimoto Y.: Construction of wedge-local nets of observables through Longo–Witten endomorphisms. Commun. Math. Phys. 314(2), 443–469 (2012)ADSMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tanimoto Y.: Noninteraction of waves in two-dimensional conformal field theory. Commun. Math. Phys. 314(2), 419–441 (2012)ADSMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wollenberg, M.: Notes on Perturbations of Causal Nets of Operator Algebras. SFB 288 Preprint, N2. 36 (1992, unpublished)Google Scholar

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© The Author(s) 2012

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

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany
  2. 2.Department of PhysicsUniversity of ViennaViennaAustria
  3. 3.Institute for Theoretical PhysicsUniversity of GöttingenGöttingenGermany

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