Covariance and Causality in the Transition Radiation of an Electron Bunch

A theoretical model of the transition radiation (TR) emission of an N electron bunch must comply with the covariance and the temporal-causality principles. A charge-density-like covariance must indeed imprint the formal expression of the TR energy spectrum. A causality relation must constrain the emission phases of the radiation pulse to the temporal sequence of the N electron collisions onto the metallic screen. Covariance and causality are the two faces of the same coin: failing in implementing one of the two constraints into the model necessarily implies betraying the other one. The main formal aspects of a covariance and causality consistent formulation of the TR energy spectrum of an N electron beam will be here described with reference to the case of a radiator surface with an arbitrary size.