Quantum Theory and Unusual Dielectric Functions of Graphene

We address the spatially nonlocal dielectric functions of graphene at any frequency derived starting from the first principles of thermal quantum field theory using the formalism of the polarization tensor. After a brief review of this formalism, the longitudinal and transverse dielectric functions are considered at any relationship between the frequency and the wave vector. The analytic properties of their real and imaginary parts are investigated at low and high frequencies. Emphasis is given to the double pole at zero frequency, which arises in the transverse dielectric function. The role of this unusual property in solving the problem of disagreement between experiment and theory in the Casimir effect is discussed. We believe that a more complete dielectric response of ordinary metals should also be spatially nonlocal and its transverse part may possess the double pole in the region of evanescent waves.