Exact solution of two-dimensional Dirac equation for Coulomb potential (Dirac–Kepler problem) is presented. Linear and nonlinear screening of the Coulomb potential is discused. The main focus is on the phenomenon of relativistic collapse of supercritical charges, which was discussed for many years in high-energy physics (this is the process that determines the end of the periodic table) and was at last discovered in graphene. We introduce Hartree–Fock theory for massless Dirac electrons and show that their Coulomb interaction essentially renormalizes Fermi velocity in such a way that Dirac cone is, strictly speaking, no more cone.

Optical properties of massless Dirac electrons are considered. In particular, it is shown that they provide a universal, frequency-independent adsorption coefficient determined by fine structure constant. The possible effect of interelectron interaction on this property is discussed. Using a perturbation theory for density matrix, we derive Kubo formula for various response functions and use it to consider optics, magnetooptics, charge screening and diamagnetism of massless Dirac electrons. Graphene plasmonics is briefly reviewed.