Foundations of Physics

, Volume 15, Issue 11, pp 1079–1177 | Cite as

Some approaches to polaron theory

  • N. N. Bogolubov
  • N. N. BogolubovJr.


Here, in our approximation of polaron theory, we examine the importance of introducing theT product, which turn out to be a very convenient theoretical approach for the calculation of thermodynamical averages.

We focus attention on the investigation of the so-called linear polaron Hamiltonian and present in detail the calculation of the correlation function, spectral function, and Green function for such a linear system.

It is shown that the linear polaron Hamiltonian provides an exactly solvable model of our system, and the result obtained with this approach holds true for an arbitrary coupling constant which describes the strength of interaction between the electron and the lattice vibrations. Then, with the help of a variational technique, we show the possibility of reducing the real polaron Hamiltonian to a socalled trial or approximate linear model Hamiltonian.

We also consider the exact calculation of free energy with a special technique that reduces calculations with the help of the T product, which, in our opinion, works much better and is easier than other analogous considerations, for example, the path-integral or Feynman-integral method.(1,2)

Here we furthermore recall our own work,(4) where it was shown that the results of Refs. 7 and 8 concerning the impedance calculation in the polaron model may be obtained directly without the use of the path-integral method.

The study of the polaron system's thermodynamics is carried out by us in the framework of the functional method. A calculation of the free energy and the momentum distribution function is proposed.

Note also that the polaron systems with strong coupling(9) proved to be useful in different quantum field models in connection with the construction of dynamical models of composite particles. A rigorous solution of the special strong-coupling polaron problem, describing the interaction of a nonrelativistic particle with a quantum field, was given by Bogolubov.(3) The works of Tavkhelidze, Fedyanin, Khrustalev, and others(10–13) are dedicated to the further development and generalization of the Bogolubov method.

Notice, too, that the electron-photon interaction effects play an important part in many problems of modern solid state theory (see, e.g., Refs. 7 and 14–19).

The present paper summarizes a set of lectures delivered as a special course in the physics department of Moscow State University.


Analogous Consideration Polaron Model Arbitrary Coupling Solid State Theory Nonrelativistic Particle 
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  1. 1.
    R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).Google Scholar
  2. 2.
    R. P. Feynman,Statistical Mechanics (Benjamin, Reading, Massachusets, 1972).Google Scholar
  3. 3.
    N. N. Bogolubov, “On the New Form of Adiabatic Perturbation Theory in the Problem of Interaction of Particle with Quantum Field,”Ukr. Mat. Zh. 11(2), (1950).Google Scholar
  4. 4.
    N. N. Bogolubov and N. N. Bogolubov, Jr.,Fiz. Elem. Chastits At. Yadra 11(2), (1980).Google Scholar
  5. 5.
    N. N. Bogolubov, Jr. and B. I. Sadovnikov,Zh. Eksp. Teor. Fiz. 13(8), (1962).Google Scholar
  6. 6.
    N. N. Bogolubov, Jr. and B. I. Sadovnikov,Some Problems of Statistical Mechanics (Vysshaya Shkola Press, Moscow, 1975).Google Scholar
  7. 7.
    J. T. Devreese and R. Evard,Linear and Nonlinear Transport in Solids (Plenum Press, New York, 1976), p. 91.Google Scholar
  8. 8.
    K. K. Thornber and R. P. Feynman,Phys. Rev. B 1, 4099 (1970).Google Scholar
  9. 9.
    S. I. Pekar,Unterschungen Über Die Elektronen Theorie der Kristalle (Akademie-Verlag, Berlin, 1954).Google Scholar
  10. 10.
    E. P. Solodovnikova, A. N. Tavkhelidze, and O. A. Khrustalev, “Oscillatory levels of particle as a consequence of strong interaction with a field,”Teor. Mat. Fiz. 10, 162–181 (1972).Google Scholar
  11. 11.
    E. P. Solodovnikova, A. N. Tavkhelidze, and O. A. Khrustalev, “N. N. Bogolubov transformation in the strong coupling theory. II.”Teor. Mat. Fiz. 11, 317–330 (1972).Google Scholar
  12. 12.
    V. K. Fedyanin, B. V. Mochinsky, and C. Rodrigues, “Generating Functional and Functional Analog of the Variational Principle of N. N. Bogolubov,” Joint Institute for Nuclear Research, E17-12850, Dubna, 1979).Google Scholar
  13. 13.
    E. A. Kochetov, S. P. Kuleshov, and M. A. Smondyrev, “Investigation of the polaron model by path-integral method,”Teor. Mat. Fiz. 25, 30–36 (1975).Google Scholar
  14. 14.
    Ahn VoHong, “A quantum approach to the parametric excitation problem in solids,”Phys. Rep. 64(1), 1–45 (1980).Google Scholar
  15. 15.
    T. D. Schultz, “Slow electrons in polar crystals: Self-energy, mass, and mobility,”Phys. Rev. 116(3), 526–543 (1959).Google Scholar
  16. 16.
    Y. Osaka, “Polaron state at a finite temperature,”Prog. Theor. Phys. 22(3), 437–446 (1959).Google Scholar
  17. 17.
    R. Abe and K. Okamoto, “An improvement of the Feynman action in the theory of polaron: I,”J. Phys. Soc. Jpn. 31(5), 1337–1343 (1971); II.J. Phys. Soc. Jpn. 33(2), 343–347 (1972).Google Scholar
  18. 18.
    M. Saitoh, “Theory of a polaron at finite temperatures,”J. Phys. Soc. Jpn. 49(3), 878–885 (1980).Google Scholar
  19. 19.
    J. M. Luttinger and C.-Y. Lu, “Generalized path-integral formalism of the polaron problem and its second-order semi-invariant corrections to the ground-state energy,”Phys. Rev. B21(10), 4251–4263 (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • N. N. Bogolubov
    • 1
  • N. N. BogolubovJr.
    • 2
  1. 1.Joint Institute for Nuclear ResearchDubnaUSSR
  2. 2.Department of Theoretical PhysicsMoscow State UniversityMoscowUSSR

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