Relativistic saturation of Coulomb-induced electron decoherence: from eikonal phase noise to Bethe--Salpeter kinetic theory

Abstract

We develop a microscopic theory of the mutual coherence and small-angle scattering of relativistic electron matter waves propagating through a Coulomb-fluctuating medium. Starting from the Dirac equation, we derive a relativistic paraxial wave equation and then obtain, rather than assume, the Bethe--Salpeter and Wigner--Boltzmann kinetic equations for the disorder-averaged two-point coherence. The Coulomb environment enters through the eikonal phase-coupling factor $A_{\rm rel}=1/(\hbar v)$, which saturates at $1/(\hbar c)$ for ultra-relativistic particles. This saturation is the central physical result: increasing the beam voltage suppresses Coulomb phase noise only up to a finite relativistic floor. We first formulate the eikonal approximation, in which the transmitted electron wave accumulates a random longitudinal phase. For a one-component Coulomb medium the electrostatic potential correlator retains a Coulomb tail, producing a logarithmic phase structure function and an algebraic decay of the transverse mutual coherence. We then go beyond the eikonal construction by deriving the Bethe--Salpeter equation for the Cooperon-like two-point coherence propagator directly from the paraxial wave equation. After a longitudinal Markov projection this equation reduces, after a Wigner transform, to a kinetic equation with a Coulomb small-angle scattering kernel. The coordinate-space solution of the same Bethe--Salpeter equation recovers the eikonal mutual coherence function, while the Wigner form additionally describes angular diffusion and transverse broadening caused by multiple small-angle scattering. The resulting theory separates three effects which are often conflated in charged-particle wave propagation: attenuation of the coherent amplitude, decay of mutual coherence, and conservative redistribution of intensity in transverse phase space.