LElTER TO THE EDITOR Generalized Kramers-Kronig formula for spatially dispersive media

Abstract. For a spatially dispersive medium the requirements of causality and special relativity are used to develop a generalized dispersion formula (a generalization of the Kramers-Kronig formula). An error is pointed out in earlier versions of this formula. The vacuum polarization tensor is shown to satisfy the generalized dispersion formula explicitly. Causality (no effect precedes its cause) implies that any response function is analytic in the upper half of the complex frequency plane (Toll 1952,1956, Hilgevoord 1962). For spatially dispersive media (when the response depends on k as well as U) causality and special relativity lead to the requirement that the response be non-zero only inside the forward light cone. The associated restriction on the properties of the response function will be called the generalized dispersion formula. Here we point out an error in existing discussions of the generalized dispersion formula (Leontovich 1961, Silin and Rukhadze 1961, appendix). In addition we show explicitly that the vacuum polarization tensor satisfies the generalized dispersion formula. Considering the generalized dispersion formula, the linear response of a medium (e.g. a plasma or the vacuum) to an electromagnetic disturbance may be described i