Derivation of the Kelbg potential/functional

Abstract

The density matrix for a system of particles interacting via the Coulomb potential is obtained in the high--temperature limit following almost entirely the original work by Kelbg. For this purpose the Bl\"och equation is solved in the first order of perturbation theory. We tried to explain all the transformations in the derivation in order to simplify the understanding of this non-trivial theory. The solution of Kelbg is widely used in path integral simulations of Coulomb systems.