Advertisement

Communications in Mathematical Physics

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

Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories

  • Sabina Alazzawi
  • Gandalf Lechner
Article

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abdalla E., Abdalla C.B., Rothe K.D.: Non-Perturbative Methods in 2-Dimensional Quantum Field Theory. World Scientific, Singapore (2001)zbMATHCrossRefGoogle 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)zbMATHGoogle 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)zbMATHGoogle Scholar
  7. 7.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, Cambridge (1982)zbMATHGoogle 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)ADSzbMATHCrossRefGoogle 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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. H. Poincaré 16(2), 569–608 (2015)zbMATHMathSciNetCrossRefGoogle 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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17(3), 303–321 (1976)ADSMathSciNetCrossRefGoogle 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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Borchers H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46(9), 095401 (2013)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Bostelmann H., Cadamuro D.: Characterization of local observables in integrable quantum field theories. Commun. Math. Phys. 337(3), 1199–1240 (2015)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin (1987)zbMATHCrossRefGoogle Scholar
  18. 18.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle 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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. H. Poincaré 5, 1065–1080 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Cadamuro D., Tanimoto Y.: Wedge-local fields in integrable models with bound states. Commun. Math. Phys. 340(2), 661–697 (2015)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Doplicher S., Longo R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Epstein H.: Generalization of the “Edge-of-the-Wedge” Theorem. J. Math. Phys. 1(6), 524–531 (1960)ADSzbMATHMathSciNetCrossRefGoogle 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)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Glimm J., Jaffe A.: Quantum Physics. Springer, Berlin (1981)zbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle Scholar
  32. 32.
    Jarchow H.: Locally Convex Spaces. Teubner, Stuttgart (1981)zbMATHCrossRefGoogle Scholar
  33. 33.
    Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102(4), 537–547 (1986)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Kauffman L.: Knots and Physics. World Scientific, Singapore (1993)zbMATHGoogle Scholar
  35. 35.
    Ketov S.V.: Quantum Non-Linear Sigma-Models. Springer, Berlin (2000)zbMATHCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64(2), 137–154 (2003)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 312, 265–302 (2012)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Lechner G.: Algebraic Constructive Quantum Field Theory: Integrable Models and Deformation Techniques, pp. 397–449. Springer, Berlin (2015)zbMATHCrossRefGoogle Scholar
  43. 43.
    Lechner G., Sanders K.: Modular nuclearity: a generally covariant perspective. Axioms 5(1), 5 (2016)CrossRefGoogle Scholar
  44. 44.
    Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. H. Poincaré 15, 645–678 (2014)zbMATHCrossRefGoogle Scholar
  45. 45.
    Liguori A., Mintchev M.: Fock representations of quantum fields with generalized statistics. Commun. Math. Phys. 169, 635–652 (1995)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Liguori A., Mintchev M.: Fock spaces with generalized statistics. Lett. Math. Phys. 33, 283–295 (1995)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Mund J.: The Bisognano–Wichmann theorem for massive theories. Ann. H. Poincaré 2, 907–926 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Pietsch A.: Nuclear locally convex spaces. Cambridge University Press, Cambridge (1972)zbMATHCrossRefGoogle Scholar
  49. 49.
    Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, Cambridge (1975)zbMATHGoogle Scholar
  50. 50.
    Reed M., Simon B.: Methods of modern mathematical physics III: scattering theory. Academic Press, Academic Press (1980)zbMATHGoogle Scholar
  51. 51.
    Schroer B.: Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499(3), 547–568 (1997)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Schroer B., Wiesbrock H.-W.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12(02), 301–326 (2000)zbMATHMathSciNetCrossRefGoogle 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)zbMATHGoogle Scholar
  55. 55.
    Smirnov F.A.: Form-factors in completely integrable models of quantum field theory. Adv. Ser. Math. Phys. 14, 1–208 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Stein E.M., Weiss G.L.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  57. 57.
    Streater R.F., Wightman A.S.: PCT, spin and statistics, and all that. Princeton University Press, Princeton (1964)zbMATHGoogle Scholar
  58. 58.
    Tanimoto Y.: Bound state operators and wedge-locality in integrable quantum field theories. SIGMA 12, 100 (2016)ADSzbMATHMathSciNetGoogle 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)ADSMathSciNetCrossRefGoogle 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)ADSMathSciNetCrossRefGoogle 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