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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bastiani A.: Applications différentiables et variétés différentiables de dimension infinie. Journal d’Analyse mathématique. 13(1), 1–114 (1964)
Brunetti R., Dütsch M., Fredenhagen K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541–1599 (2009)
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)
Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories. Commun. Math. Phys. 208(3), 623–661 (2000)
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)
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)
Bahns D., Rejzner K., Zahn J.: The effective theory of strings. Commun. Math. Phys 327(3), 779–814 (2014)
Bogoliubov N., Shirkov D.: Introduction to the Theory of Quantized Fields. Interscience, New York (1959)
Coleman S.: Quantum Sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11(8), 2088 (1975)
Dabrowski, Y.: Functional properties of generalized Hörmander spaces of distributions I: duality theory, completions and bornologifications. arXiv preprint arXiv:1411.3012 (2014)
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)
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)
Dütsch, M., Fredenhagen, K.: Algebraic quantum field theory, perturbation theory, and the loop expansion. Commun. Math. Phys. 219(1), 5–30 (2001)
Dütsch M., Fredenhagen K.: Perturbative algebraic field theory, and deformation quantization. Math. Phys. Math. Phys: Quantum Oper. Algebr. Asp. 30, 1–10 (2001)
Dimock J., Hurd T.: Construction of the two-dimensional Sine-Gordon model for \({\beta < 8\pi}\). Commun. Math. Phys. 156(3), 547–580 (1993)
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)
Epstein H., Glaser V.: The role of locality in perturbation theory. AHP 19(3), 211–295 (1973)
Folland, G.B.: Fourier Analysis and its Applications. American Mathematical Soc., Providence (1992)
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)
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)
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)
Haag R.: Local quantum physics. Springer, Berlin (1993)
Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. 7, 65–222 (1982)
Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5(7), 848–861 (1964)
Hörmander, L.: The analysis of the linear partial differential operators I: Distribution theory and Fourier analysis. Classics in Mathematics, Springer, Berlin, (2003)
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)
Kac G.V.: Infinite-dimensional Lie algebras, vol. 44. Cambridge University Press, Cambridge (1994)
Milnor, J.: Remarks on infinite-dimensional Lie groups, (1984)
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)
Neeb, K.H.: Monastir summer school. Infinite-dimensional Lie groups, TU Darmstadt Preprint 2433 (2006)
Nikolov M.N., Stora R., Todorov I.: Renormalization of massless Feynman amplitudes in configuration space. Rev. Math. Phys. 26, 1430002 (2013)
Pierotti D.: The exponential of the two-dimensional massless scalar field as an infrared Jaffe field. Lett. Math. Phys. 15(3), 219–230 (1988)
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)
Rejzner, K.: Perturbative algebraic quantum field theory. An Introduction for Mathematicians, Mathematical Physics Studies, Springer, Berlin (2016)
Scharf, G.: Finite QED: the Causal Approach, (1995)
Schubert, S.: On the characterization of states regarding expectation values of quadratic operators, Diploma Thesis, Hamburg (2012)
Steinmann, O.: Perturbation expansions in axiomatic field theory. Cambridge University Press, Cambridge (1971)
Summers S.J.: A perspective on constructive quantum field theory (2012). arXiv:1203.3991
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)
Zahn, J.: The semi-classical energy of open Nambu–Goto strings (2016). arXiv:1605.07928
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by D. Buchholz, K. Fredenhagen, Y. Kawahigashi.
Rights and permissions
Open Access This 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.
About this article
Cite this article
Bahns, D., Rejzner, K. The Quantum Sine-Gordon Model in Perturbative AQFT. Commun. Math. Phys. 357, 421–446 (2018). https://doi.org/10.1007/s00220-017-2944-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2944-4