Advertisement

Towards the Construction of Quantum Field Theories from a Factorizing S-Matrix

  • Gandalf Lechner
Part of the Progress in Mathematics book series (PM, volume 251)

Summary

Starting from a given factorizing S-matrix S in two space-time dimensions, we review a novel strategy to rigorously construct quantum field theories describing particles whose interaction is governed by S. The construction procedure is divided into two main steps: Firstly certain semi-local Wightman fields are introduced by means of Zamolodchikov’s algebra. The second step consists in proving the existence of local observables in these models. As a new result, an intermediate step in the existence problem is taken by proving the modular compactness condition for wedge algebras.

Keywords

Double Cone Local Observable Norm Topology Local Algebra Nuclear Norm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Araki: Mathematical Theory of Quantum Fields. Oxford University Press, New York, 1999.MATHGoogle Scholar
  2. 2.
    H. Babujian and M. Karowski: The “Bootstrap Program” for Integrable Quantum Field Theories in 1 + 1 Dim. Preprint (2001). [arXiv: hep-th/0110261].Google Scholar
  3. 3.
    B. Berg, M. Karowski and P. Weisz: Construction of Green’s functions from an exact S matrix. Phys. Rev. D 19:2477 (1979).ADSCrossRefGoogle Scholar
  4. 4.
    J.J. Bisognano and E.H. Wichmann: On the duality condition for a hermitian scalar field. J. Math. Phys. 16:985 (1975).ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    H.-J. Borchers: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143:315 (1992).ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H.-J. Borchers, D. Buchholz and B. Schroer: Polarization-Free Generators and the S-Matrix. Commun. Math. Phys. 219:125 (2001). [arXiv: hep-th/0003243].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Bros: A Proof of Haag-Swieca’s Compactness Property for Elastic Scattering States. Commun. Math. Phys. 237:289 (2003).ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    R. Brunetti, D. Guido and R. Longo: Modular localization and Wigner particles. Rev. Math. Phys. 14:759 (2002). [arXiv: math-ph/0203021].MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    D. Buchholz: Product States for Local Algebras. Commun. Math. Phys. 36:287 (1974).ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D. Buchholz, C. D’Antoni and R. Longo: Nuclear Maps and Modular Structures I: General Properties. J. Funct. Anal. 88:233 (1990).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    D. Buchholz, C. D’Antoni and R. Longo: Nuclear Maps and Modular Structures II: Applications to Quantum Field Theory. Commun. Math. Phys. 129:115 (1990).ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    D. Buchholz and P. Jacobi: On the nuclearity condition for massless fields. Lett. Math. Phys. 13:313 (1987).ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    D. Buchholz and P. Junglas: On The Existence of Equilibrium States in Local Quantum Field Theory. Commun. Math. Phys. 121:255 (1989).ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D. Buchholz and G. Lechner: Modular Nuclearity and Localization. Ann. H. Poincaré 5:1065 (2004). [arXiv: math-ph/0402072].MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    D. Buchholz and E. H. Wichmann: Causal Independence and the Energy-Level Density of States in Local Quantum Field Theory. Comm. Math. Phys. 106:321 (1986).ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    O.A. Castro-Alvaredo: Bootstrap Methods in 1+1-Dimensional Quantum Field Theories: The Homogeneous Sine-Gordon Models. PhD thesis, 2001. [arXiv: hep-th/0109212].Google Scholar
  17. 17.
    S. Doplicher and R. Longo: Standard and split inclusions of von Neumann algebras. Commun. Math. Phys 75:493 (1984).MathSciNetMATHGoogle Scholar
  18. 18.
    H. Epstein: Some analytic properties of scattering amplitudes in quantum field theory. In: Particle Symmetries and Axiomatic Field Theory, Brandeis Summer School 195, Gordon and Breach, New York, 1966.Google Scholar
  19. 19.
    R. Haag: Local Quantum Physics. Springer Verlag, Berlin, 2nd ed., 1996.CrossRefMATHGoogle Scholar
  20. 20.
    R. Haag and J.A. Swieca: When does a quantum field theory describe particles?. Commun. Math. Phys. 1:308 (1965).ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    R.V. Kadison and J.R. Ringrose: Fundamentals of the Theory of Operator Algebras, Vol. II. Academic Press, Orlando, 1986.MATHGoogle Scholar
  22. 22.
    G. Lechner: Polarization-Free Quantum Fields and Interaction. Lett. Math. Phys. 64:137 (2003). [arXiv: hep-th/0303062].MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    G. Lechner: On the existence of local observables in theories with a factorizing S-matrix. To appear in J. of Phys. A (2005). [arXiv: math-ph/0405062].Google Scholar
  24. 24.
    A. Liguori and M. Mintchev: Fock spaces with generalized statistics. Commun. Math. Phys. 169:635 (1995). [arXiv: hep-th/9403039].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    R. Longo: Notes on algebraic invariants for noncommutative dynamical systems. Commun. Math. Phys. 69:195 (1979).ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    B.M. McCoy, C.A. Tracy and T.T. Wu: Two Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n-Point Functions. Phys. Rev. Lett. 38:793 (1977).ADSCrossRefGoogle Scholar
  27. 27.
    P. Mitra: Elasticity, Factorization and S-Matrices in (1 + 1)-Dimensions. Phys. Lett. B 72:62 (1977).ADSCrossRefGoogle Scholar
  28. 28.
    M. Müger: Superselection structure of massive quantum field theories in (1+1)-dimensions. Rev. Math. Phys. 10:1147 (1998). [arXiv: hep-th/9705019].MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    J. Mund: The Bisognano-Wichmann theorem for massive theories. Annales Henri Poincaré 2:907 (2001). [arXiv: hep-th/0101227].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    A. Pietsch: Nuclear locally convex spaces. Springer Verlag, Berlin, Heidelberg, New York, 1972.CrossRefMATHGoogle Scholar
  31. 31.
    W. Rudin: Real and complex analysis. McGraw-Hill Book Company, 1987.Google Scholar
  32. 32.
    S. Sakai: C*-Algebras and W*-Algebras. Springer Verlag, 1971.Google Scholar
  33. 33.
    B. Schroer: Modular localization and the bootstrap form-factor program. Nucl. Phys. B 499:547 (1997). [arXiv: hep-th/9702145].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    B. Schroer and H.W. Wiesbrock: Modular constructions of quantum field theories with interactions. Rev. Math. Phys 12:301 (2000). [arXiv: hep-th/9812251].MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    F.A. Smirnov: Formfactors in completely integrable models in quantum field theory. Advanced Series in Mathematical Physics 14, World Scientific, 1992.Google Scholar
  36. 36.
    M. Reed and B. Simon: Methods of modern mathematical physics I: Functional Analysis. Revised and enlarged edition, Academic Press, 1980.Google Scholar
  37. 37.
    M. Reed and B. Simon: Methods of modern mathematical physics II: Fourier Analysis, Self-Adjointness. Academic Press, 1975.Google Scholar
  38. 38.
    M. Reed and B. Simon: Methods of modern mathematical physics III: Scattering Theory. Academic Press, 1979.Google Scholar
  39. 39.
    S.J. Summers: On the Independence of Local Algebras in Quantum Field Theory. Rev. Math. Phys. 2:201 (1990).MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    A. Zamolodchikov: Factorized S-matrices as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120:253 (1979).ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag 2007

Authors and Affiliations

  • Gandalf Lechner
    • 1
  1. 1.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany

Personalised recommendations