On Three Levels pp 53-71 | Cite as

# Weak Coupling Limit: Feynman Diagrams

## 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 study^{1}, joint work with T.G.Ho and A.J.Wilkins. (See reference^{1} for technical details concerning self-adjointness of the Hamiltonian, and control of the perturbative expansion.) The model is a quantum version of the Lorentz gas^{2}, 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 discussion^{3} of the Boltzmann equation for a self-interacting Fermi gas. The difficulties^{1} 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 Emch^{4}, Spohn^{5}, and Dell’Antonio^{6}. Some discussion of their results is given in reference^{1}. 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* _{1} − *x* _{2}) or their Fourier transforms *a**(*p*) and *a*(*p*) satisfying {*a*(*p* _{1}), *a**(*P* _{2})} = *δ*(*p* _{1} − *p* _{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}/2*m*.

## Keywords

Feynman Diagram Random Potential Weak Coupling Limit Momentum Variable Lorentz Gas2## Preview

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## References

- 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.H. Spohn, Kinetic equations from hamiltonian dynamics: markovian limits,
*Rev. Mod. Phys.*,53, 569 (1980).MathSciNetADSCrossRefGoogle Scholar - 3.N.M. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas,
*J. Stat. Phys.*,32, 231 (1983).MathSciNetADSCrossRefGoogle Scholar - 4.P. Martin and G.G. Emch, A rigorous model sustaining van Hove’s phenomenon,
*Helv. Phys. Acta.*,48, 59 (1975).MathSciNetGoogle Scholar - 5.H. Spohn, Derivation of the transport equation for electrons moving through random impurities,
*J. Stat. Phys.*,17, 385 (1977).MathSciNetADSCrossRefGoogle Scholar - 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.O.E. Lanford and D.W. Robinson, Approach to equilibrium of free quantum systems,
*Commun. Math. Phys.*,24, 193 (1972).MathSciNetADSCrossRefGoogle Scholar