Recently it has been a matter of controversy whether inelastically scattered electrons can yield interference fringes so as to obtain holograms, and in particular whether compensation of energy loss in the object by energy gain in the source will maintain coherence [1]. In discussions about coherence (and wave mechanisms in general) it is always dangerous to rely on intuitive arguments (exchange of energy, time of interaction, etc.). In this work we will start from the most general approach, which is inspired by the treatment of inelastic electron diffraction crystals by Yoshioka in 1957 [2]. Energy exchanges are always described quantummechanically by an Hamiltonian. Therefore we can only investigate the balance between energy exchange properly if electron, object, and source are described by one global Hamiltonian. With source we mean the whole electron generating system (emitter, accelerator, condensor).

Consider a global system consisting of an electron, with position vector r, an object with particle vectors ri, and a source with particles at rα.