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 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 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 ²/2m.