Communications in Mathematical Physics

, Volume 357, Issue 1, pp 421–446 | Cite as

The Quantum Sine-Gordon Model in Perturbative AQFT

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Article

Abstract

We study the Sine-Gordon model with Minkowski signature in the framework of perturbative algebraic quantum field theory. We calculate the vertex operator algebra braiding property. We prove that in the finite regime of the model, the expectation value—with respect to the vacuum or a Hadamard state—of the Epstein Glaser S-matrix and the interacting current or the field respectively converge, both given as formal power series.

References

  1. Bas64.
    Bastiani A.: Applications différentiables et variétés différentiables de dimension infinie. Journal d’Analyse mathématique. 13(1), 1–114 (1964)MathSciNetCrossRefMATHGoogle Scholar
  2. BDF09.
    Brunetti R., Dütsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. BDH14.
    Brouder, C., Dang, N.V., Hélein, F.: Boundedness and continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker). arXiv preprint arXiv:1409.7662 (2014)
  4. BF00.
    Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories. Commun. Math. Phys. 208(3), 623–661 (2000)ADSCrossRefMATHGoogle Scholar
  5. BFR13.
    Brunetti R., Fredenhagen K., Rejzner K.: Quantum gravity from the point of view of locally covariant quantum field theory. Commun. Math. Phys. 345(3), 741–779 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. BFV03.
    Brunetti R., Fredenhagen K., Verch R.: The generally covariant locality principle—A new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. BRZ14.
    Bahns D., Rejzner K., Zahn J.: The effective theory of strings. Commun. Math. Phys 327(3), 779–814 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. BS59.
    Bogoliubov N., Shirkov D.: Introduction to the Theory of Quantized Fields. Interscience, New York (1959)Google Scholar
  9. Col75.
    Coleman S.: Quantum Sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11(8), 2088 (1975)ADSCrossRefGoogle Scholar
  10. Dab14a.
    Dabrowski, Y.: Functional properties of generalized Hörmander spaces of distributions I: duality theory, completions and bornologifications. arXiv preprint arXiv:1411.3012 (2014)
  11. Dab14b.
    Dabrowski, Y.: Functional properties of generalized Hörmander spaces of distributions II: multilinear maps and applications to spaces of functionals with wave front set conditions. arXiv:1412.1749 (2014)
  12. DB14.
    Dabrowski Y., Brouder C.: Functional properties of Hörmander’s space of distributions having a specified wavefront set. Commun. Math. Phys 332(3), 1345–1380 (2014)ADSCrossRefMATHGoogle Scholar
  13. DF01a.
    Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219(1), 5–30 (2001)Google Scholar
  14. DF01b.
    Dütsch M., Fredenhagen K.: Perturbative algebraic field theory, and deformation quantization. Math. Phys. Math. Phys: Quantum Oper. Algebr. Asp. 30, 1–10 (2001)MathSciNetMATHGoogle Scholar
  15. DH93.
    Dimock J., Hurd T.: Construction of the two-dimensional Sine-Gordon model for \({\beta < 8\pi}\). Commun. Math. Phys. 156(3), 547–580 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. DM06.
    Dereziński J., Meissner K.A.: Quantum massless field in 1 + 1 dimensions. In: Asch, J., Joye, A. (eds) Mathematical Physics of Quantum Mechanics, pp. 107–127. Springer, Berlin (2006)CrossRefGoogle Scholar
  17. EG73.
    Epstein H., Glaser V.: The role of locality in perturbation theory. AHP 19(3), 211–295 (1973)MathSciNetMATHGoogle Scholar
  18. Fol92.
    Folland, G.B.: Fourier Analysis and its Applications. American Mathematical Soc., Providence (1992)Google Scholar
  19. FR15a.
    Fredenhagen K., Rejzner K.: Perturbative algebraic quantum field theory. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories, pp. 17–55. Springer, Berlin (2015)Google Scholar
  20. FR15b.
    Fredenhagen, K., Rejzner, K. : Perturbative construction of models of algebraic quantum field theory. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K.,Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory, pp. 31–74. Springer, Berlin (2015)Google Scholar
  21. Fro76.
    Fröhlich J.: Classical and quantum statistical mechanics in one and two dimensions: two-component Yukawa—and Coulomb systems. Commun. Math. Phys. 47(3), 233–268 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Haa93.
    Haag R.: Local quantum physics. Springer, Berlin (1993)MATHGoogle Scholar
  23. Ham82.
    Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7, 65–222 (1982)MathSciNetCrossRefMATHGoogle Scholar
  24. HK64.
    Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5(7), 848–861 (1964)Google Scholar
  25. Hor03.
    Hörmander, L.: The analysis of the linear partial differential operators I: Distribution theory and Fourier analysis. Classics in Mathematics, Springer, Berlin, (2003)Google Scholar
  26. HW02.
    Hollands S., Wald M.R.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231(2), 309–345 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. Kac94.
    Kac G.V.: Infinite-dimensional Lie algebras, vol. 44. Cambridge University Press, Cambridge (1994)Google Scholar
  28. Mil84.
    Milnor, J.: Remarks on infinite-dimensional Lie groups, (1984)Google Scholar
  29. MPS90.
    Morchio G., Pierotti D., Strocchi F.: Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field. J. Math. Phys. 31(6), 1467–1477 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. Nee06.
    Neeb, K.H.: Monastir summer school. Infinite-dimensional Lie groups, TU Darmstadt Preprint 2433 (2006)Google Scholar
  31. NST13.
    Nikolov M.N., Stora R., Todorov I.: Renormalization of massless Feynman amplitudes in configuration space. Rev. Math. Phys. 26, 1430002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. Pie88.
    Pierotti D.: The exponential of the two-dimensional massless scalar field as an infrared Jaffe field. Lett. Math. Phys. 15(3), 219–230 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. Rad96.
    Radzikowski J.M.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. Rej16.
    Rejzner, K.: Perturbative algebraic quantum field theory. An Introduction for Mathematicians, Mathematical Physics Studies, Springer, Berlin (2016)Google Scholar
  35. Sch95.
    Scharf, G.: Finite QED: the Causal Approach, (1995)Google Scholar
  36. Sch12.
    Schubert, S.: On the characterization of states regarding expectation values of quadratic operators, Diploma Thesis, Hamburg (2012)Google Scholar
  37. Ste71.
    Steinmann, O.: Perturbation expansions in axiomatic field theory. Cambridge University Press, Cambridge (1971)Google Scholar
  38. Sum12.
    Summers S.J.: A perspective on constructive quantum field theory (2012). arXiv:1203.3991
  39. Wig67.
    Wightman A.S.: Introduction to Some Aspects of the Relativistic Dynamics of Quantized Fields, Cargèse Lectures in Theoretical Physics. Gordon and Breach Science Publishers, New York (1967)Google Scholar
  40. Zah16.
    Zahn, J.: The semi-classical energy of open Nambu–Goto strings (2016). arXiv:1605.07928

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of GöttingenGöttingenGermany
  2. 2.York UniversityYorkUK

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