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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 5(353), Pages 163–196 (Mi umn668)  

This article is cited in 13 scientific papers (total in 13 papers)

Boson-fermion correspondence and quantum integrable and dispersionless models

A. K. Pogrebkov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper is devoted to a detailed description of the notion of boson-fermion correspondence introduced by Coleman and Mandelstam and to applications of this correspondence to integrable and related models. An explicit formulation of this correspondence in terms of massless fermionic fields is given, and properties of the resulting scalar field are studied. It is shown that this field is a well-defined operator-valued distribution on the fermionic Fock space. At the same time, this is a non-Weyl field, and its correlation functions do not exist. Further, realizing a bosonic field as a current of massless (chiral) fermions, we derive a hierarchy of quantum polynomial self-interactions of this field determined by the condition that the corresponding evolution equations of the fermionic fields are linear. It is proved that all the equations of this hierarchy are completely integrable and admit unique global solutions; however, in the classical limit this hierarchy reduces to the dispersionless KdV hierarchy. An application of our construction to the quantization of generic completely integrable interactions is shown by examples of the KdV and mKdV equations for which the quantization procedure of the Gardner–Zakharov–Faddeev bracket is carried out. It is shown that in both cases the corresponding Hamiltonians are sums of two well-defined operators which are bilinear and diagonal with respect to either fermion or boson (current) creation–annihilation operators. As a result, the quantization procedure needs no spatial cut-off and can be carried out on the whole axis of the spatial variable. It is shown that, in the framework of our approach, soliton states exist in the Hilbert space, and the soliton parameters are quantized.


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English version:
Russian Mathematical Surveys, 2003, 58:5, 1003–1037

Bibliographic databases:

UDC: 517.958+530.145.83
MSC: Primary 81T10, 35Q53, 37K10; Secondary 35Q51, 81T40, 81U30, 81U40
Received: 05.08.2003

Citation: A. K. Pogrebkov, “Boson-fermion correspondence and quantum integrable and dispersionless models”, Uspekhi Mat. Nauk, 58:5(353) (2003), 163–196; Russian Math. Surveys, 58:5 (2003), 1003–1037

Citation in format AMSBIB
\by A.~K.~Pogrebkov
\paper Boson-fermion correspondence and quantum integrable and dispersionless models
\jour Uspekhi Mat. Nauk
\yr 2003
\vol 58
\issue 5(353)
\pages 163--196
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 5
\pages 1003--1037

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    This publication is cited in the following articles:
    1. P. P. Kulish, A. M. Zeitlin, “Quantum inverse scattering method and (super)conformal field theory”, Theoret. and Math. Phys., 142:2 (2005), 211–221  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Yair Zarmi, “Quantized representation of some nonlinear integrable evolution equations on the soliton sector”, Phys. Rev. E, 83:5 (2011)  crossref  isi  scopus  scopus
    3. Talalov S.V., “The Anyon Model: An Example Inspired By String Theory”, Internat J Modern Phys A, 26:16 (2011), 2757–2772  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. S V Talalov, “Planar string as an anyon model: cusps, braids and soliton exitations”, J. Phys.: Conf. Ser, 343 (2012), 012121  crossref  adsnasa  isi  scopus  scopus
    5. Yair Zarmi, “Nonlinear quantum-dynamical system based on the Kadomtsev-Petviashvili II equation”, J. Math. Phys, 54:6 (2013), 063515  crossref  mathscinet  zmath  isi  scopus  scopus
    6. M. L. Nazarov, E. K. Sklyanin, “Sekiguchi-Debiard Operators at Infinity”, Commun. Math. Phys, 2013  crossref  mathscinet  isi  scopus  scopus
    7. P. G. Gavrilenko, A. V. Marshakov, “Free fermions, $W$-algebras, and isomonodromic deformations”, Theoret. and Math. Phys., 187:2 (2016), 649–677  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. Ristivojevic Z., Matveev K.A., “Decay of Bogoliubov excitations in one-dimensional Bose gases”, Phys. Rev. B, 94:2 (2016), 024506  crossref  isi  elib  scopus
    9. Dubrovin B., “Symplectic Field Theory of a Disk, Quantum Integrable Systems, and Schur Polynomials”, Ann. Henri Poincare, 17:7 (2016), 1595–1613  crossref  mathscinet  zmath  isi  elib  scopus
    10. Koroteev P., Sciarappa A., “On elliptic algebras and large- n supersymmetric gauge theories”, J. Math. Phys., 57:11 (2016), 112302  crossref  mathscinet  zmath  isi  elib  scopus
    11. Sotiriadis S., “Equilibration in One-Dimensional Quantum Hydrodynamic Systems”, J. Phys. A-Math. Theor., 50:42 (2017), 424004  crossref  mathscinet  zmath  isi  scopus
    12. Koroteev P., Sciarappa A., “Quantum Hydrodynamics From Large-N Supersymmetric Gauge Theories”, Lett. Math. Phys., 108:1 (2018), 45–95  crossref  mathscinet  zmath  isi  scopus
    13. M. G. Matushko, “Calogero–Sutherland system at a free fermion point”, Theoret. and Math. Phys., 205:3 (2020), 1593–1610  mathnet  crossref  crossref  mathscinet  isi  elib
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