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Weak Coupling Limit: Feynman Diagrams

  • L. J. Landau
Part of the NATO ASI Series book series (NSSB, volume 324)

Abstract

An elementary discussion is given of the derivation of irreversible behavior in the weak coupling limit for a Fermi gas in a random potential based on the Feynman- diagrammatic description of the individual terms in the Dyson perturbative expansion, which is one aspect of the study1, joint work with T.G.Ho and A.J.Wilkins. (See reference1 for technical details concerning self-adjointness of the Hamiltonian, and control of the perturbative expansion.) The model is a quantum version of the Lorentz gas2, proposed by Lorentz in 1905 as a model for electron conduction in metals, and describes a gas of non-interacting particles in the presence of randomly distributed static impurities on which the gas particles scatter elastically. The diagrammatic analysis is presented in the same spirit as Hugenholtz’s discussion3 of the Boltzmann equation for a self-interacting Fermi gas. The difficulties1 with Hugenholtz’s treatment are not present in the simpler model considered here. The impurities are represented as a random potential. The weak coupling limit for a single quantum particle in a random potential has been considered by Martin and Emch4, Spohn5, and Dell’Antonio6. Some discussion of their results is given in reference1. The Fermi gas is described by the canonical anticommutation relations generated by the creation operators a*(x) and the destruction operators a(x) satisfying {a(x 1),a*(x 2)} = δ(x 1x 2) or their Fourier transforms a*(p) and a(p) satisfying {a(p 1), a*(P 2)} = δ(p 1p 2)- The free Hamiltonian is H o = dp ε(p)a*(p)a(p), where for the non-relativistic gas considered here ε(p) = p 2/2m.

Keywords

Feynman Diagram Random Potential Weak Coupling Limit Momentum Variable Lorentz Gas2 
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

  1. 1.
    T.G. Ho, L.J. Landau and A.J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential, Rev. Math. Phys. ,(to appear).Google Scholar
  2. 2.
    H. Spohn, Kinetic equations from hamiltonian dynamics: markovian limits, Rev. Mod. Phys. ,53, 569 (1980).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    N.M. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas, J. Stat. Phys. ,32, 231 (1983).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    P. Martin and G.G. Emch, A rigorous model sustaining van Hove’s phenomenon, Helv. Phys. Acta. ,48, 59 (1975).MathSciNetGoogle Scholar
  5. 5.
    H. Spohn, Derivation of the transport equation for electrons moving through random impurities, J. Stat. Phys. ,17, 385 (1977).MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    G.F. Dell’Antonio, Large time, small coupling behaviour of a quantum particle in a random potential, Ann. Inst. Henri Poincar é A ,39, 339 (1983).MathSciNetMATHGoogle Scholar
  7. 7.
    O.E. Lanford and D.W. Robinson, Approach to equilibrium of free quantum systems, Commun. Math. Phys. ,24, 193 (1972).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • L. J. Landau
    • 1
  1. 1.Mathematics DepartmentKing’s CollegeLondonUK

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