Derivative expansion of the one-loop effective action.

The one-loop eeective action in QED at zero and nite temperature is obtained by using the worldline approach. The Feynman rules for the perturbative expansion of the action in the number of derivatives are derived. The general structure of the temperature dependent part of the eeective action in an arbitrary external inhomogeneous magnetic eld is established. The two-derivative term in the eeective action for spinor and scalar QED in a static magnetic background at T 6 = 0 is calculated. The problem of calculating the eeective action in QED is an old one. Its history starts with the well known papers by Heisenberg and Euler 2], and Weisskopf 3]. Later, some results were obtained by Schwinger 4] who, by using the proper time technique, rederived the one-loop eeective action for the case of a constant electromagnetic eld. Perhaps, the next most natural step in solving the general problem is to take into account the eeect of small deviations from a constant connguration of the eld. It turns out, however, that the latter is very diicult to realize 5,6,7,8]. Here we present our recent work 1] that generalizes the previously known results on the eeective action in QED. There, in particular, the derivative expansions up to two derivatives of the eld strength with respect to space-time coordinates in scalar and spinor QED at zero and at nite temperature are obtained (for some partial results in a non-Abelian gauge theory see 9,10]). Our method is heavily based on the very elegant and now widely developed Here we follow a self-contained approach of Ref. 1], so that even a non-expert in the eld should understand all the details. As is known, the one-loop eeective action in QED reduces to computing the fermion determinant W (1)