Covariance and Causality in the Transition Radiation of an Electron Bunch

TL;DR

This paper develops a theoretical model for the transition radiation emitted by an electron bunch, ensuring it adheres to covariance and causality principles, which are essential for accurate physical description.

Contribution

It introduces a formal framework that incorporates both covariance and causality constraints into the transition radiation energy spectrum of an electron bunch.

Findings

Formal expression of TR energy spectrum consistent with covariance

Causality constrains emission phases to electron collision sequence

Model applicable to arbitrary radiator surface size

Abstract

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.