Heisenberg, langevin, and current equations via the recurrence relations approach

Some years ago the Heisenberg equation of motion was formally solved by the recurrence relations approach. It is shown here that the Langevin equation represents a structural property of the recurrence relations. The Langevin equation is useful for studying the time evolution of the current. The resulting current-current correlation function is compared with Luttinger's phenomenological theory. Geometric interpretations are made for the conductivity and the dielectric function.