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Aspects of Holonomic quantum fields

Isomonodromic deformation and ising model
  • Michio Sato
  • Tetsuji Miwa
  • Michio Jimbo
Part IV Two-Dimensional Models and Related Developments
Part of the Lecture Notes in Physics book series (LNP, volume 126)

Keywords

Free Fermion Linear Ordinary Differential Equation Deformation Equation Regular Singularity Irregular Singularity 
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.

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References

The general framework of monodromy preserving deformation with irregular singularities was given by

  1. [1]
    [1] K. Ueno, Master's thesis, Kyoto University (1979).Google Scholar
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    See also B. Klares, Sur une classe de connexions relatives, C. R. Acad. Sc. Paris, 288 (1979), 205–208.Google Scholar
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    H. Flaschka and A. C. Newell, Monodromy and spectral preserving deformatoins I, preprint Clarkson college of tech. (1979).Google Scholar

The proof of complete integrability and the general definition of T function was given by

  1. [4]
    M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I, preprint RIMS 315, Kyoto Univ. (1980).Google Scholar

As for Schlesinger transformations and T-quotients, see

  1. [5]
    M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential euqations with rational coefficients II, preprint RIMS 316, Kyoto Univ. (1980).Google Scholar

In this chapter, we have not mentioned special examples such as Schlesinger equation and Painlevé equations. Schlesinger equation is discussed in chapter 1. As for Painlevé equations, see

  1. [6]
    K. Okamoto, Polynomial Hamiltonians associated to the Painlevé equations, preprint Tokyo Univ. (1980).Google Scholar

References There are an enormous number of literatures concerning the Ising model. see e.g.

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    B.M. McCoy and T.T. Wu, The two-dimensional Ising model, Harvard University Press, 1973.Google Scholar

Explicit formulas of the spin operators and correlation functions on the lattice are given from the viewpoint of the Clifford group in

  1. [2]
    M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, preprint RIMS 267, Kyoto Univ. (1978).Google Scholar

Some recent developments in this branch are quoted in the references of [2]. Deformation theory and closed expression of correlation functions were introduced by

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    M. Sato, T. Miwa and M. Jimbo, Proc. Japan Acad. 53A (1977), 147–152, 153-158, 183-185.Google Scholar

This work is a generalization of the monumental papers

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    E. Barouch, B.M. McCoy and T.T. Wu, Phys. Rev. Lett. 31 (1973) 1409–1411, T.T. Wu, B.M. McCoy, C.A.,Tracy and E. Barouch, Phys. Rev. B13 (1976) 316-374. in which the 2 point functions are obtained in terms of a Painlevé transcendent of the third kind. Details along with some generalizations of [3] are found inCrossRefGoogle Scholar
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    M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto Univ. 15 (1979) 577–629. or[[5a] —, Holonomic quantum fields IV, to appear in Publ. RIMS, Kyoto Univ. 15 (1980).Google Scholar
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    —, Supplement to Holonomic quantum fields IV, preprint RIMS 304, Kyoto Univ. (1979).Google Scholar

Extension of the above sheme to higher dimensions is briefly touched in

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    M. Jimbo, T. Miwa, M. Sato and Y. Môri, Holonomic quantum fields-The unanticipated link between deformation theory of differential equations and quantum fields-, preprint RIMS 305, Kyoto Univ. (1979) and the references cited therein.Google Scholar

References This review is based on

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    M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields V, RIMS preprint 267 (1978), to appear in Publ. RIMS, Kyoto Univ.Google Scholar
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    M. Jimbo, T. Miwa and M. Sato, Studies on holonomic quantum fields XV, Proc. Japan Acad. 55 A (1979) 267–272.Google Scholar
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    M. Jimbo, T. Miwa, Y. Mori and M. Sato, Density matrix of impenetrable bose gas and the fifth Painlevé transcendent, RIMS preprint 303 (1979), to appear in Physics D.Google Scholar

In [2] the correlation function for the non-linear Schrödinger equation with finite particle density at the infinite coupling is studied in detail. See also

  1. [4]
    H. G. Vaidya and C. A. Tracy, One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature, J. Math. Phys. 20 (1979), 2291–2312.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Michio Sato
    • 1
  • Tetsuji Miwa
    • 1
  • Michio Jimbo
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto 606JAPAN

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