QED electrical conductivity using the two-particle-irreducible effective action

In this article we calculate the electrical conductivity in QED using the two-particle-irreducible (2PI) effective action. We use a resummed version of the usual 2PI effective action which is defined with respect to self-consistent solutions of the 2-point functions. We show that the Green functions obtained from this 2PI resummed effective action satisfy Ward identities and that the conductivity obtained from the Kubo relation is gauge-invariant. We work to 3-loop order in the 2PI resummed effective action and show explicitly that the resulting expression for the conductivity contains the square of the amplitude that corresponds to all binary collision and production processes