Remarks on Luttingers Derivation of Kubos Conductivity Formula

Luttinger has given a derivation of Kuboʹs static conductivity formula in a famous paper some thirty years ago. In a subsequent analysis to obtain the Einstein relation, he introduced an assumption. We show that this assumption is not consistent with linear response theory, on the basis of which the conductivity formula is derived. This inconsistency can be removed. At the same time, one can extend Luttingerʹs treatment to include the dynamic conductivity. This generalization is allowed by the recurrence relations method, which solves the Heisenberg equation of motion formally as well as explicitly.