Communications in Mathematical Physics

, Volume 312, Issue 1, pp 265–302 | Cite as

Deformations of Quantum Field Theories and Integrable Models

  • Gandalf Lechner


Deformations of quantum field theories which preserve Poincaré covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free field theories to integrable models with non-trivial S-matrix.


Invariant State Spectrum Condition Double Cone Opposite Deformation Wedge Algebra 
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. AAR91.
    Abdalla E., Abdalla C., Rothe K.D.: Non-perturbative methods in 2-dimensional quantum field theory. River Edge, NJ, World Scientific (1991)CrossRefGoogle Scholar
  2. Åks65.
    Åks S.: Proof that scattering implies production in quantum field theory. J. Math. Phys. 6, 516–532 (1965)ADSCrossRefGoogle Scholar
  3. Ara99.
    Araki H.: Mathematical Theory of Quantum Fields. Int. Series of Monographs on Physics. Oxford University Press, Oxford (1999)Google Scholar
  4. BBS01.
    Borchers H.-J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  5. BFK06.
    Babujian H.M., Foerster A., Karowski M.: The Form Factor Program: a Review and New Results - the Nested SU(N) Off-Shell Bethe Ansatz. SIGMA 2, 082 (2006)MathSciNetGoogle Scholar
  6. BGL02.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. BL04.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Annales Henri Poincaré 5, 1065–1080 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  8. BLS10.
    Buchholz D., Lechner G., Summers S.J.: Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories. Commun. Math. Phys. 304(1), 95–123 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  9. Bor62.
    Borchers H.-J.: On Structure of the Algebra of Field Operators. Nuovo Cimento 24, 214–236 (1962)MathSciNetMATHCrossRefGoogle Scholar
  10. Bor92.
    Borchers H.-J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  11. BS07.
    Buchholz D., Summers S.J.: String- and brane-localized fields in a strongly nonlocal model. J. Phys. A 40, 2147–2163 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  12. BS08.
    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, pp. 107–121. River Edge NJ, World Scientific (2008)CrossRefGoogle Scholar
  13. Buc90.
    Buchholz D.: On quantum fields that generate local algebras. J. Math. Phys. 31, 1839–1846 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  14. BW75.
    Bisognano J.J., Wichmann E.H.: On the Duality Condition for a Hermitian Scalar Field. J. Math. Phys. 16, 985–1007 (1975)MathSciNetADSMATHCrossRefGoogle Scholar
  15. BW92.
    Baumgärtel H., Wollenberg M.: Causal Nets of Operator Algebras. Akademie Verlag, Berlin (1992)MATHGoogle Scholar
  16. BY90.
    Borchers H.-J., Yngvason J.: Positivity of Wightman functionals and the existence of local nets. Commun. Math. Phys. 127, 607 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  17. BZ63.
    Borchers H.-J., Zimmermann W.: On the Self-Adjointness of Field Operators. Nuovo Cimento 31, 1047–1059 (1963)MathSciNetGoogle Scholar
  18. CA01.
    Castro-Alvaredo, O.: Bootstrap Methods in 1+1-Dimensional Quantum Field Theories: the Homogeneous Sine-Gordon Models. PhD thesis, Santiago de Compostela, 2001Google Scholar
  19. DLM11.
    Dappiaggi C., Lechner G., Morfa-Morales E.: Deformations of quantum field theories on spacetimes with Killing fields. Commun. Math. Phys. 305, 99–130 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  20. DSW86.
    Driessler W., Summers S.J., Wichmann E.H.: On the Connection between Quantum Fields and von Neumann Algebras of Local Operators. Commun. Math. Phys. 105, 49–84 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  21. DT10.
    Dybalski W., Tanimoto Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Commun. Math. Phys. 305, 427–440 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  22. Fad84.
    Faddeev, L.D.: Quantum completely integrable models in field theory. In: Novikov, S.P. (ed.) Mathematical Physics Reviews, Vol. 1, pp. 107–155 (1984)Google Scholar
  23. Ger64.
    Gerstenhaber M.: On the deformation of rings and algebras. Ann. Math. 79, 59–103 (1964)MathSciNetMATHCrossRefGoogle Scholar
  24. GL07.
    Grosse H., Lechner G.: Wedge-Local Quantum Fields and Noncommutative Minkowski Space. JHEP 11, 012 (2007)MathSciNetADSCrossRefGoogle Scholar
  25. GL08.
    Grosse, H., Lechner, G.: Noncommutative Deformations of Wightman Quantum Field Theories. JHEP 09, 131 (2008)MathSciNetADSCrossRefGoogle Scholar
  26. Haa96.
    Haag R.: Local Quantum Physics - Fields, Particles, Algebras. 2nd edition. Springer, Berlin-Heildelberg-New York (1996)MATHGoogle Scholar
  27. Hep65.
    Hepp K.: On the connection between Wightman and LSZ quantum field theory. Commun. Math. Phys. 1, 95–111 (1965)MathSciNetADSMATHCrossRefGoogle Scholar
  28. Jos65.
    Jost R.: The General Theory of Quantized Fields. Amer. Math. Soc., Providence, RI (1965)MATHGoogle Scholar
  29. Lec03.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003)MathSciNetMATHCrossRefGoogle Scholar
  30. Lec05.
    Lechner G.: On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38, 3045–3056 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  31. Lec06.
    Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. PhD thesis, University of Göttingen, 2006Google Scholar
  32. Lec06.
    Lechner G.: Construction of Quantum Field Theories with Factorizing S-Matrices. Commun. Math. Phys. 277, 821–860 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  33. LMW00.
    Leitz-Martini M., Wollenberg M.: Notes on Modular Conjugations of von Neumann Factors. Z. Anal. Anw. 19, 13–22 (2000)MathSciNetMATHGoogle Scholar
  34. LR04.
    Longo R., Rehren K.-H.: Local fields in boundary conformal QFT. Rev. Math. Phys. 16, 909 (2004)MathSciNetMATHCrossRefGoogle Scholar
  35. LW10.
    Longo R., Witten E.: An Algebraic Construction of Boundary Quantum Field Theory. Commun. Math. Phys. 303, 213–232 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  36. Mor11.
    Morfa-Morales E.: Deformations of quantum field theories on de Sitter spacetime. J. Math. Phys. 52, 102304 (2011)ADSCrossRefGoogle Scholar
  37. MSY06.
    Mund J., Schroer B., Yngvason J.: String-localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  38. Mun10.
    Mund, J.: An Algebraic Jost-Schroer Theorem for Massive Theories. [hep-th] 2010
  39. Rie92.
    Rieffel M.A.: Deformation Quantization for Actions of R d. Volume 106 of Memoirs of the Amerian Mathematical Society. Amer. Math. Soc., Providence, RI (1992)Google Scholar
  40. Rie93.
    Rieffel M.A.: Compact quantum groups associated with toral subgroups. Cont. Math. 145, 465–491 (1993)MathSciNetCrossRefGoogle Scholar
  41. RS75.
    Reed M., Simon B.: Methods of Modern Mathematical Physics II - Fourier Analysis. Academic Press, New York (1975)MATHGoogle Scholar
  42. Sch97.
    Schroer B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499, 547–568 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  43. SW64.
    Streater R.F., Wightman A.: PCT, Spin and Statistics, and All That. Reading, MA, Benjamin-Cummings (1964)MATHGoogle Scholar
  44. SW00.
    Schroer B., Wiesbrock H.W.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301–326 (2000)MathSciNetMATHCrossRefGoogle Scholar
  45. Tre67.
    Treves F.: Topological vector spaces, distributions, and kernels. Academic Press, London-New York (1967)MATHGoogle Scholar
  46. TW97.
    Thomas L.J., Wichmann E.H.: On the causal structure of Minkowski space-time. J. Math. Phys. 38, 5044–5086 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  47. Uhl62.
    Uhlmann A.: Über die Definition der Quantenfelder nach Wightman und Haag. Wissenschaftliche Zeitschrift der Karl-Marx-Universität Leipzig 2, 213–217 (1962)MathSciNetGoogle Scholar
  48. Wol92.
    Wollenberg M.: Notes on Perturbations of Causal Nets of Operator Algebras. SFB 288 Preprint, N2. 36, 1992, unpublishedGoogle Scholar
  49. Yng81.
    Yngvason J.: Translationally invariant states and the spectrum ideal in the algebra of test functions for quantum fields. Commun. Math. Phys. 81, 401 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
  50. Yng84.
    Yngvason J.: On the Locality Ideal in the Algebra of Test Functions for Quantum Fields. Publ. RIMS, Kyoto University 20, 1063–1081 (1984)MathSciNetMATHCrossRefGoogle Scholar
  51. ZZ79.
    Zamolodchikov A.B., Zamolodchikov Al.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Annals Phys. 120, 253–291 (1979)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of ViennaViennaAustria

Personalised recommendations