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*.

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### References

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