Effective action for Dirac spinors in the presence of general uniform electromagnetic fields

Some new expressions are found, concerning the one-loop effective action of four dimensional massive and massless Dirac fermions in the presence of general uniform electric and magnetic fields, with ~ E · ~ H 6= 0 and ~ E2 6= ~ H2. The rate of pairproduction is computed and briefly discussed. Non-renormalizable effective field theories turn out to be quite useful in the description of physics below some specific momentum scale. One of the early known examples is provided, taking one-loop corrections into account, by the (nonlinear) effective lagrangean for the electromagnetic field in the presence of virtual fermions, which enables to describe light-light scattering at low momenta of the order of the electron mass. As it is well known, the use of path-integral techniques allows to obtain the correction to the classical action in terms of the evaluation of the determinant of the Dirac operator. The first efforts in this direction date back to Euler and Heisenberg [1], who worked out an implicit expression of the effective Lagrange function for Maxwell theory in the context of electron-hole theory. Later on, Schwinger [2] derived a gauge-invariant integral representation of the effective lagrangean by means of the so-called ”proper-time” technique. Then, during the next three-four decades, no step was made towards an explicit expression of an effective theory of electromagnetism. The introduction of Hawking’s ζ-function technique [3] renewed the interest in the subject and some further progress in this direction was put forward: Blau, Visser and Wipf [4] obtained an analytic form 1 E-mail address: [email protected] 2 E-mail address: [email protected] Preprint submitted to Elsevier Preprint 1 February 2008 for the effective action of a uniform electromagnetic field in any number of space-time dimensions, both for massive and for massless fermions. Unfortunately, none of these expressions is fully satisfactory: on the one hand, Schwinger’s one, besides being implicit, is not valid in a theory with massless fermions. On the other hand, the expressions given in ref.[4] are explicit and valid also for massless Fermi particles, but they are established only for some particular configurations of the uniform electric and magnetic fields. In this paper a completion will be given to the work of Blau et al., in order to obtain a general and explicit expression for the effective lagrangean of massive and massless QED in the presence of uniform general electromagnetic fields. The use of the path-integral method forces us to work in the euclidean framework: only at the very end of our calculations we shall operate a Wick rotation to Minkowski space-time. The effective euclidean action, in the one-loop approximation, is given by S Eff [Aμ] = S E Cl[Aμ]− logDet( / D[Aμ] + im) (1) where S Cl[Aμ] is the classical euclidean action ∫ dxFμνFμν/4 and / D[Aμ] ≡ γμ(∂μ − ieAμ) is the euclidean Dirac operator, m being the fermion mass. As the Dirac operator is normal ([ / D + im, / D − im] = 0), we have |Det(/ D + im)| = Det(/ D/ D +m) (2) and consequently the effective lagrangean is defined (by means of the ζfunction regularization) to be S Eff [Aμ] = S E Cl[Aμ]− 1 2 ∂ ∂s ζ(s; [Aμ]) ∣