Communications in Mathematical Physics

, Volume 354, Issue 3, pp 913–956 | Cite as

Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories



We present a solution method for the inverse scattering problem for integrable two-dimensional relativistic quantum field theories, specified in terms of a given massive single particle spectrum and a factorizing S-matrix. An arbitrary number of massive particles transforming under an arbitrary compact global gauge group is allowed, thereby generalizing previous constructions of scalar theories. The two-particle S-matrix S is assumed to be an analytic solution of the Yang–Baxter equation with standard properties, including unitarity, TCP invariance, and crossing symmetry. Using methods from operator algebras and complex analysis, we identify sufficient criteria on S that imply the solution of the inverse scattering problem. These conditions are shown to be satisfied in particular by so-called diagonal S-matrices, but presumably also in other cases such as the O(N)-invariant nonlinear \({\sigma}\)-models.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdalla E., Abdalla C.B., Rothe K.D.: Non-Perturbative Methods in 2-Dimensional Quantum Field Theory. World Scientific, Singapore (2001)MATHGoogle Scholar
  2. 2.
    Alazzawi, S.: Deformations of Quantum Field Theories and the Construction of Interacting Models. Ph.D. Thesis, University of Vienna. arXiv:1503.00897 (2014)
  3. 3.
    Araki H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999)MATHGoogle Scholar
  4. 4.
    Babujian, H.M., Foerster, A., Karowski, M.: SU(N) and O(N) off-shell nested Bethe ansatz and form factors. Low Dimens. Phys. Gauge Princ. 46 (2011). doi: 10.1142/9789814440349_0005
  5. 5.
    Babujian, H.M., Foerster, A., Karowski, M. et al.: The form factor program: a review and new results—the nested SU(N) off-shell Bethe ansatz. SIGMA 2, 082 (2006)Google Scholar
  6. 6.
    Baumgärtel H., Wollenberg M.: Causal Nets of Operator Algebras. Akademie Verlag, Berlin (1992)MATHGoogle Scholar
  7. 7.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, Cambridge (1982)MATHGoogle Scholar
  8. 8.
    Benfatto G., Falco P., Mastropietro V.: Functional integral construction of the thirring model: axioms verification and massless limit. Commun. Math. Phys. 273, 67–118 (2007)ADSMATHGoogle Scholar
  9. 9.
    Benfatto G., Falco P., Mastropietro V.: Massless Sine-Gordon and massive Thirring models: proof of the Coleman’s equivalence. Commun. Math. Phys. 285, 713–762 (2009)ADSMATHGoogle Scholar
  10. 10.
    Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. H. Poincaré 16(2), 569–608 (2015)MATHGoogle Scholar
  11. 11.
    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975)ADSMATHGoogle Scholar
  12. 12.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17(3), 303–321 (1976)ADSGoogle Scholar
  13. 13.
    Borchers H.-J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219(1), 125–140 (2001)ADSMATHGoogle Scholar
  14. 14.
    Borchers H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)ADSMATHGoogle Scholar
  15. 15.
    Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46(9), 095401 (2013)ADSMATHGoogle Scholar
  16. 16.
    Bostelmann H., Cadamuro D.: Characterization of local observables in integrable quantum field theories. Commun. Math. Phys. 337(3), 1199–1240 (2015)ADSMATHGoogle Scholar
  17. 17.
    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (1987)MATHGoogle Scholar
  18. 18.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)MATHGoogle Scholar
  19. 19.
    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88, 223–250 (1990)MATHGoogle Scholar
  20. 20.
    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures II: applications to quantum field theory. Commun. Math. Phys. 129(1), 115–138 (1990)ADSMATHGoogle Scholar
  21. 21.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. H. Poincaré 5, 1065–1080 (2004)MATHGoogle Scholar
  22. 22.
    Cadamuro D., Tanimoto Y.: Wedge-local fields in integrable models with bound states. Commun. Math. Phys. 340(2), 661–697 (2015)ADSMATHGoogle Scholar
  23. 23.
    Doplicher S., Longo R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)ADSMATHGoogle Scholar
  24. 24.
    Epstein H.: Generalization of the “Edge-of-the-Wedge” Theorem. J. Math. Phys. 1(6), 524–531 (1960)ADSMATHGoogle Scholar
  25. 25.
    Epstein, H.: Some analytic properties of scattering amplitudes in quantum field theory. In: Axiomatic Field Theory, vol. 1, p. 1. (1966)Google Scholar
  26. 26.
    Fröhlich J.: Quantized “Sine-Gordon” equation with a non-vanishing mass term in two space-time dimensions. Phys. Rev. Lett. 34(13), 833–836 (1975)ADSGoogle Scholar
  27. 27.
    Glimm J., Jaffe A.: Quantum Physics. Springer, Berlin (1981)MATHGoogle Scholar
  28. 28.
    Grosse, H., Wulkenhaar, R.: Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity. arXiv:1406.7755 (2014)
  29. 29.
    Haag R.: Local Quantum Physics: Fields, Particles, Algebras. Springer, Berlin (1996)MATHGoogle Scholar
  30. 30.
    Hollands, S., Lechner, G.: SO(d,1)-invariant Yang–Baxter operators and the dS/CFT-correspondence. arXiv:1603.05987 (2016)
  31. 31.
    Iagolnitzer D.: Scattering in Quantum Field Theories: The Axiomatic and Constructive Approaches. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  32. 32.
    Jarchow H.: Locally Convex Spaces. Teubner, Stuttgart (1981)MATHGoogle Scholar
  33. 33.
    Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102(4), 537–547 (1986)ADSMATHGoogle Scholar
  34. 34.
    Kauffman L.: Knots and Physics. World Scientific, Singapore (1993)MATHGoogle Scholar
  35. 35.
    Ketov S.V.: Quantum Non-Linear Sigma-Models. Springer, Berlin (2000)MATHGoogle Scholar
  36. 36.
    Kosaki H.: On the continuity of the map \({\varphi\rightarrow|\varphi|}\) from the predual of a W*-algebra. J. Funct. Anal. 59(1), 123–131 (1984)Google Scholar
  37. 37.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64(2), 137–154 (2003)MATHGoogle Scholar
  38. 38.
    Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D Thesis, University of Göttingen. arXiv: math-ph/0611050 (2006)
  39. 39.
    Lechner G.: Towards the construction of quantum field theories from a factorizing S-matrix. Prog. Math. 251, 175–198 (2007)MATHGoogle Scholar
  40. 40.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)ADSMATHGoogle Scholar
  41. 41.
    Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312, 265–302 (2012)ADSMATHGoogle Scholar
  42. 42.
    Lechner G.: Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques, pp. 397–449. Springer, Berlin (2015)MATHGoogle Scholar
  43. 43.
    Lechner G., Sanders K.: Modular nuclearity: a generally covariant perspective. Axioms 5(1), 5 (2016)Google Scholar
  44. 44.
    Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. H. Poincaré 15, 645–678 (2014)MATHGoogle Scholar
  45. 45.
    Liguori A., Mintchev M.: Fock representations of quantum fields with generalized statistics. Commun. Math. Phys. 169, 635–652 (1995)ADSMATHGoogle Scholar
  46. 46.
    Liguori A., Mintchev M.: Fock spaces with generalized statistics. Lett. Math. Phys. 33, 283–295 (1995)ADSMATHGoogle Scholar
  47. 47.
    Mund J.: The Bisognano–Wichmann theorem for massive theories. Ann. H. Poincaré 2, 907–926 (2001)MATHGoogle Scholar
  48. 48.
    Pietsch A.: Nuclear locally convex spaces. Cambridge University Press, Cambridge (1972)MATHGoogle Scholar
  49. 49.
    Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1975)MATHGoogle Scholar
  50. 50.
    Reed M., Simon B.: Methods of modern mathematical physics III: scattering theory. Academic Press, Academic Press (1980)MATHGoogle Scholar
  51. 51.
    Schroer B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499(3), 547–568 (1997)ADSMATHGoogle Scholar
  52. 52.
    Schroer B., Wiesbrock H.-W.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12(02), 301–326 (2000)MATHGoogle Scholar
  53. 53.
    Schützenhofer, C.: Multi-particle S-matrix models in \({1+1}\)-dimensions and associated Quantum Field theories. Diploma thesis, University of Vienna (2011)Google Scholar
  54. 54.
    Simon B.: Trace Ideals and Their Applications. American Mathematical Society, Providence, RI (2005)MATHGoogle Scholar
  55. 55.
    Smirnov F.A.: Form-factors in completely integrable models of quantum field theory. Adv. Ser. Math. Phys. 14, 1–208 (1992)MATHGoogle Scholar
  56. 56.
    Stein E.M., Weiss G.L.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  57. 57.
    Streater R.F., Wightman A.S.: PCT, spin and statistics, and all that. Princeton University Press, Princeton (1964)MATHGoogle Scholar
  58. 58.
    Tanimoto Y.: Bound state operators and wedge-locality in integrable quantum field theories. SIGMA 12, 100 (2016)ADSMATHGoogle Scholar
  59. 59.
    Zamolodchikov A.B., Zamolodchikov A.B.: Relativistic factorized S-matrix in two-dimensions having O(N) isotopic symmetry. Nucl. Phys. B 133, 525 (1978)ADSGoogle Scholar
  60. 60.
    Zamolodchikov A.B., Zamolodchikov A.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120(2), 253–291 (1979)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany
  2. 2.School of MathematicsCardiff UniversityCardiffUK

Personalised recommendations