Generalized Langevin equation and recurrence relations

The generalized Langevin equation (GLE) is a reformulation of the Heisenberg equation of motion, and hence, an exact equation. It is the basis of the memory function approach, a very widely used method for studying dynamics of classical and quantum fluids. The GLE was first derived by Mori in a very formal way. A much simpler and more physically motivated derivation was given by us some years later. In this work we provide perhaps the simplest possible derivation of the GLE. The simplicity of the derivation helps to bring out the subtleties present in this important dynamical relationship.