Letters in Mathematical Physics

, Volume 77, Issue 1, pp 63–81 | Cite as

Point Interactions in One Dimension and Holonomic Quantum Fields

  • Oleg LisovyyEmail author


We introduce and study a family of quantum fields, associated to δ-interactions in one dimension. These fields are analogous to holonomic quantum fields of Sato et al. in Holonomic quantum fields I–V (Publ. RIMS, Kyoto University, 14: 223–267, 1978; 15: 201–278, 1979; 15: 577–629, 1979; 15: 871-972, 1979; 16: 531–584, 1979). Corresponding field operators belong to an infinite-dimensional representation of the group \(SL(2,\mathbb{R})\) in the Fock space of ordinary harmonic oscillator. We compute form factors of such fields and their correlation functions, which are related to the determinants of Schroedinger operators with a finite number of point interactions. It is also shown that these determinants coincide with tau functions, obtained through the trivialization of the det*-bundle over a Grassmannian associated to a family of Schroedinger operators.


point interactions Schroedinger operators tau functions 

Mathematics Subject Classification (2000)

34B10 34M55 


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  1. 1.
    Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. (1988). Solvable models in quantum mechanics. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  2. 2.
    Berezin F.A. (1965). Method of secondary quantization. Nauka, MoscowGoogle Scholar
  3. 3.
    Bernard D., Leclair A. (1994). Differential equations for sine-Gordon correlation functions at the free fermion point. Nucl. Phys. B426:534–558; Erratum ibid. B498, 619–621 (1997)zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Coleman S. (1975). Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D11:2088–2097CrossRefADSGoogle Scholar
  5. 5.
    Itzykson C. Zuber J.B. (1977). Quantum field theory and the two-dimensional Ising model. Phys. Rev. D15:2875–2884CrossRefADSGoogle Scholar
  6. 6.
    Kadanoff L.P., Ceva H. (1971). Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B3:3918–3938CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Mason L.J., Singer M.A., Woodhouse N.M.J. (2000). Tau functions and the twistor theory of integrable systems. J. Geom. Phys. 32:397–430zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Palmer J. (1990). Determinants of Cauchy–Riemann operators as τ-functions. Acta Appl. Math. 18:199–223zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Palmer J. (1993). Tau functions for the Dirac operator in the Euclidean plane. Pacific J. Math. 160:259–342zbMATHMathSciNetGoogle Scholar
  10. 10.
    Sato M., Miwa T., Jimbo M. (1978). Holonomic quantum fields I. Publ. RIMS, Kyoto University 14:223–267MathSciNetGoogle Scholar
  11. 11.
    Sato M., Miwa T., Jimbo M. (1979). Holonomic quantum fields II. Publ. RIMS, Kyoto University 15:201–278MathSciNetGoogle Scholar
  12. 12.
    Sato M., Miwa T., Jimbo M. (1979). Holonomic quantum fields III. Publ. RIMS, Kyoto University 15:577–629zbMATHMathSciNetGoogle Scholar
  13. 13.
    Sato M., Miwa T., Jimbo M. (1979). Holonomic quantum fields IV. Publ. RIMS, Kyoto University 15:871–972zbMATHMathSciNetGoogle Scholar
  14. 14.
    Sato M., Miwa T., Jimbo M. (1980). Holonomic quantum fields V. Publ. RIMS, Kyoto University 16:531–584zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Schroer B., Truong T.T. (1978). The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit. Nucl. Phys. B144:80–122CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Segal G., Wilson G. (1985). Loop groups and equations of KdV type. Publ. Math. I.H.E.S. 63:1–64Google Scholar
  17. 17.
    Sitenko Yu.A. (1992). The Aharonov–Bohm effect and inducing of vacuum charge by a singular magnetic string. Nucl. Phys. B372:622–634CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  2. 2.School of Theoretical PhysicsDublin Institute for Advanced StudiesDublin 4Ireland

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