Communications in Mathematical Physics

, Volume 277, Issue 3, pp 821–860 | Cite as

Construction of Quantum Field Theories with Factorizing S-Matrices

Article

Abstract

A new approach to the construction of interacting quantum field theories on two-dimensional Minkowski space is discussed. In this program, models are obtained from a prescribed factorizing S-matrix in two steps. At first, quantum fields which are localized in infinitely extended, wedge-shaped regions of Minkowski space are constructed explicitly. In the second step, local observables are analyzed with operator-algebraic techniques, in particular by using the modular nuclearity condition of Buchholz, d’Antoni and Longo.

Besides a model-independent result regarding the Reeh–Schlieder property of the vacuum in this framework, an infinite class of quantum field theoretic models with non-trivial interaction is constructed. This construction completes a program initiated by Schroer in a large family of theories, a particular example being the Sinh-Gordon model. The crucial problem of establishing the existence of local observables in these models is solved by verifying the modular nuclearity condition, which here amounts to a condition on analytic properties of form factors of observables localized in wedge regions.

It is shown that the constructed models solve the inverse scattering problem for the considered class of S-matrices. Moreover, a proof of asymptotic completeness is obtained by explicitly computing total sets of scattering states. The structure of these collision states is found to be in agreement with the heuristic formulae underlying the Zamolodchikov-Faddeev algebra.

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References

  1. 1.
    Abdalla E., Abdalla M.C.B. and Rothe D. (1991). Non-perturbative Methods in Two-Dimensional Quantum Field Theory. World Scientific Publishing, Singapore Google Scholar
  2. 2.
    Araki, H.: Mathematical Theory of Quantum Fields. Int. Series of Monographs on Physics 101, Oxford: Oxford University Press, 1999Google Scholar
  3. 3.
    Arinshtein A.E., Fateev V.A. and Zamolodchikov A.B. (1979). Quantum S-matrix of the (1 + 1)-dimensional Toda chain. Phys. Lett. B 87: 389–392 ADSGoogle Scholar
  4. 4.
    Babujian H. and Karowski M. (2002). Exact form factors in integrable quantum field theories: the sine-Gordon model (II). Nucl. Phys. B 620: 407–455 CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Babujian H. and Karowski M. (2003). Exact form factors for the scaling ZN-Ising and the affine AN-1-Toda quantum field theories. Phys. Lett. B 575: 144–150 ADSMathSciNetGoogle Scholar
  6. 6.
    Babujian H. and Karowski M. (2004). Towards the construction of Wightman functions of integrable quantum field theories. Int. J. Mod. Phys. A 19S2: 34–49 MathSciNetGoogle Scholar
  7. 7.
    Balog, J., Weisz, P.: Construction and clustering properties of the 2-d non-linear sigma-model form factors: O(3), O(4), large n examples. http://arxiv.org/list/hepth/0701202, 2007
  8. 8.
    Bisognano J.J. and Wichmann E.H. (1975). On the Duality Condition for a Hermitian Scalar Field. J. Math. Phys. 16: 985–1007 MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Bisognano J.J. and Wichmann E.H. (1976). On the Duality Condition for Quantum Fields. J. Math. Phys. 17: 303–321 CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Bochner S. and Martin W.T. (1948). Several Complex Variables. Princeton University Press, Princeton, NJ MATHGoogle Scholar
  11. 11.
    Borchers H.-J. (1992). The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143: 315–332 MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Borchers H.-J. (1998). Half–sided Translations and the Type of von Neumann Algebras. Lett. Math. Phys. 44: 283–290 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Borchers H.-J. (2000). On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41: 3604–3673 MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Borchers H.-J., Buchholz D. and Schroer B. (2001). Polarization-free generators and the S-matrix. Commun. Math. Phys. 219: 125–140 MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Buchholz D., D’Antoni C. and Longo R. (1990). Nuclear Maps and Modular Structures. 1. General Properties. J. Funct. Anal. 88: 233–250 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Buchholz D., D’Antoni C. and Longo R. (1990). Nuclear Maps and Modular Structures. 2. Applications to Quantum Field Theory. Commun. Math. Phys. 129: 115–138 MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Buchholz D. and Lechner G. (2004). Modular nuclearity and localization. Ann. H. Poincaré 5: 1065–1080 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Buchholz D. and Summers S.J. (2004). Stable quantum systems in anti-de Sitter space: Causality, independence and spectral properties. J. Math. Phys. 45: 4810–4831 MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Buchholz, D., Summers, S.J.: Scattering in Relativistic Quantum Field Theory: Fundamental Concepts and Tools. http://arxiv.org/list/math-ph/0509047, 2005
  20. 20.
    Buchholz, D., Summers, S.J.: String– and Brane–Localized Causal Fields in a Strongly Nonlocal Model. http://arxiv.org/list/math-ph/0512060v2, 2005
  21. 21.
    Buchholz D. and Wichmann E.H. (1986). Causal Independence and the Energy Level Density of States in Local Quantum Field Theory. Commun. Math. Phys. 106: 321–344 MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Connes A. (1973). Une classification des facteurs de type III. Ann. Scient. Éc. Norm. Sup. 6: 133–252 MATHMathSciNetGoogle Scholar
  23. 23.
    D’Antoni C. and Longo R. (1983). Interpolation by type I factors and the flip automorphism. J. Funct. Anal. 51: 361–371 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Doplicher S. and Longo R. (1984). Standard and split inclusions of von Neumann algebras. Invent. Math. 75: 493–536 MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Dorey, P.: Exact S-matrices. http://arxiv.org/list/hepth/9810026, 1998
  26. 26.
    Driessler W. (1975). Comments on Lightlike Translations and Applications in Relativistic Quantum Field Theory. Commun. Math. Phys. 44: 133–141 MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Eden R.J., Landshoff P.V., Olive D.I. and Polkinghorne J.C. (1966). The Analytic S-matrix. Cambridge University Press, Cambridge MATHGoogle Scholar
  28. 28.
    Epstein H. (1960). Generalization of the Edge-of-the-Wedge Theorem. J. Math. Phys. 1: 524–531 MATHCrossRefADSGoogle Scholar
  29. 29.
    Epstein, H.: Some Analytic Properties of Scattering Amplitudes in Quantum Field Theory. In: Brandeis University Summer Institute in Theoretical Physics 1965, Axiomatic Field Theory Vol. 1, M. Chretien, S. Deser, eds., New York: Gordon and Breach, 1966Google Scholar
  30. 30.
    Fring A., Mussardo G. and Simonetti P. (1993). Form Factors for Integrable Lagrangian Field Theories, the Sinh-Gordon Model. Nucl. Phys. B 393: 413–441 CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Fröhlich J. (1975). Quantized Sine Gordon Equation with a Non-Vanishing Mass Term in Two Space-Time Dimensions. Phys. Rev. Lett. 34: 833–836 CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Fröhlich, J.: Quantum Sine Gordon Equation and Quantum Solitons in Two Space-Time Dimensions. In: Renormalization Theory, G. Velo, A.S. Wightman, eds., Series C – Math. and Phys. Sciences, Vol 23, Dordrecht-Boston: Reidel, 1976, pp. 371–414Google Scholar
  33. 33.
    Glimm J. and Jaffe A. (1981). Quantum Physics - A Functional Integral Point of View. Springer, New York MATHGoogle Scholar
  34. 34.
    Haag R. (1992). Local Quantum Physics, 2nd edition. Springer, New York MATHGoogle Scholar
  35. 35.
    Hepp, K.: On the connection between Wightman and LSZ quantum field theory. In: Brandeis University Summer Institute in Theoretical Physics 1965, Axiomatic Field Theory Vol. 1, M. Chretien, S. Deser, eds., New York: Gordon and Breach, 1966, pp. 135–246Google Scholar
  36. 36.
    Iagolnitzer D. (1978). Factorization of the multiparticle S-matrix in two-dimensional space-time models. Phys. Rev. D 18: 1275–1285 ADSGoogle Scholar
  37. 37.
    Iagolnitzer D. (1993). Scattering in Quantum Field Theories. Princeton University Press, Princeton, NJ MATHGoogle Scholar
  38. 38.
    Jarchow H. (1981). Locally convex spaces. Teubner, Stuttgart MATHGoogle Scholar
  39. 39.
    Kadison R.V. and Ringrose J.R. (1986). Fundamentals of the Theory of Operator Algebras. Vol. II: Advanced Theory. Academic Press, London-New York Google Scholar
  40. 40.
    Karowski M., Thun H.J., Truong T.T. and Weisz P.H. (1977). On the uniqueness of a purely elastic S-matrix in (1+1) dimensions. Phys. Lett. 67 B: 321–322 ADSGoogle Scholar
  41. 41.
    Kosaki H. (1984). On the continuity of the map \(\varphi\mapsto|\varphi|\) from the predual of a W*-algebra. J. Funct. Anal. 59: 123–131MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Lechner G. (2003). Polarization-Free Quantum Fields and Interaction. Lett. Math. Phys. 64: 137–154 MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Lechner G. (2005). On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38: 3045–3056 ADSMathSciNetGoogle Scholar
  44. 44.
    Lechner, G.: Towards the construction of quantum field theories from a factorizing S-matrix. In: Rigorous Quantum Field Theory, Boutet de Monvel, A., Buchholz, D., Iagolnitzer, D., Moschella, U., eds., Progress in Mathematics Vol. 251, Basel-Boston: Birkhäuser, 2006Google Scholar
  45. 45.
    Lechner, G.: On the Construction of Quantum Field Theories with Factorizing S-Matrices. PhD thesis (advisor: D. Buchholz), Göttingen University, 2006Google Scholar
  46. 46.
    Liguori A. and Mintchev M. (1995). Fock representations of quantum fields with generalized statistics. Commun. Math. Phys. 169: 635–652 MATHCrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Longo R. (1979). Notes on algebraic invariants for noncommutative dynamical systems. Commun. Math. Phys. 69: 195–207 MATHCrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Longo R. and Rehren K.-H. (2004). Local fields in boundary conformal QFT. Rev. Math. Phys. 16: 909–960 MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Müger M. (1998). Superselection Structure of Massive Quantum Field Theory in 1+1 Dimensions. Rev. Math. Phys. 10: 1147–1170 MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Mund J., Schroer B. and Yngvason J. (2006). String-Localized Quantum Fields and Modular Localization. Commun. Math. Phys. 268: 621–672 MATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Pietsch A. (1972). Nuclear Locally Convex Spaces. Springer, New York Google Scholar
  52. 52.
    Reed M. and Simon B. (1975). Methods of Modern Mathematical Physics, Vol II: Fourier Analysis, Self-Adjointness. Academic Press, New York Google Scholar
  53. 53.
    Reed M. and Simon B. (1979). Methods of Modern Mathematical Physics, Vol III: Scattering Theory. Academic Press, New York Google Scholar
  54. 54.
    Sakai S. (1971). C*-Algebras and W*-Algebras. Springer, New York Google Scholar
  55. 55.
    Schroer B. (1997). Modular localization and the bootstrap-formfactor program. Nucl. Phys. B 499: 547–568 CrossRefADSMathSciNetGoogle Scholar
  56. 56.
    Schroer B. (1999). Modular wedge localization and the d = 1+1 formfactor program. Ann. Phys. 275: 190–223 MATHCrossRefADSMathSciNetGoogle Scholar
  57. 57.
    Schroer B. (2005). Constructive proposals for QFT based on the crossing property and on lightfront holography. Ann. Phys. 319: 48–91 MATHCrossRefADSMathSciNetGoogle Scholar
  58. 58.
    Schroer B. and Wiesbrock H.-W. (2000). Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12: 301–326 MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Smirnov F.A. (1992). Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific, Singapore MATHGoogle Scholar
  60. 60.
    Stein E. and Weiss G. (1971). Introduction to Fourier Analysis on Euclidian Spaces. Princeton University Press, Princeton, NJ Google Scholar
  61. 61.
    Streater R.F. and Wightman A.S. (1980). PCT, Spin and Statistics and All That, 3rd edition. Princeton University Press, Princeton, NJ Google Scholar
  62. 62.
    Takesaki M. (1979). Theory of Operator Algebras I. Springer, New York MATHGoogle Scholar
  63. 63.
    Zamolodchikov A.B. and Zamolodchikov A.B. (1979). Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120: 253–291 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Erwin Schrödinger Institute for Mathematical PhysicsViennaAustria

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