Relativistic echo dynamics and the stability of a beam of Landau electrons

We extend the concepts of echo dynamics and fidelity decay to relativistic quantum mechanics, specifically in the context of Klein-Gordon and Dirac equations under external electromagnetic fields. In both cases we define similar expressions for the fidelity amplitude under perturbations of these fields, and a covariant version of the echo operator. Transformation properties under the Lorentz group are established. An alternate expression for fidelity is given in the Dirac case in terms of a 4-current. As an application we study a beam of Landau electrons perturbed by field inhomogeneities.