Derivative expansion of the effective action for quantum electrodynamics in 2+1 and 3+1 dimensions

The derivative expansion of the one-loop effective action in QED₃ and QED₄ (quantum electrodynamics) is considered. The first term in such an expansion is the effective action for a constant electromagnetic field. An explicit expression for the next term containing two derivatives of the field strength F_μv, but exact in the magnitude of the field strength, is obtained. The general results for both fermion and scalar electrodynamics are presented. The cases of pure electric and pure magnetic external fields are considered in detail. The Feynman technique for the perturbative expansion of the one-loop effective action in the number of derivatives is developed.