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arXiv:2605.20141

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

Yury A. Budkov

Published May 19, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2605.20141

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In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function h(k)h(k)h(k) and direct correlation function c(k)c(k)c(k). The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.
Point-particle descriptions of electron-electron interaction omit the coherent longitudinal extent of a free-electron quantum wave packet (QEW). A relativistic two-electron wave-packet theory identifies the longitudinal QEW size as a direct control parameter for entanglement generated by mutual electromagnetic coupling. For two electrons in spatially separated paths, the quadratic interaction phase gives a dimensionless entangling parameter $\Gee$ and the logarithmic negativity $\EN=\operatorname{arsinh}(\Gee)/\ln2$. Full time-dependent Schrödinger equation calculations confirm the scaling for narrow QEWs and reveal higher-order Coulomb effects at larger spatial extent. Free drift increases the interaction-point QEW width while preserving the momentum probability distribution, thereby enhancing the subsequently generated entanglement. The results establish a direct connection between free-electron wave-particle duality and bipartite entanglement.
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We study spin effects in twisted-electron propagation through the entrance or exit region of an axially symmetric magnetic coil. Starting from the Foldy-Wouthuysen reduction of the Dirac equation, we derive a paraxial spinor equation in which the longitudinally varying solenoidal field produces, in addition to the usual diagonal Zeeman term, a transverse Pauli coupling proportional to the fringe-field gradient. The scalar transverse dynamics is treated exactly by the Ermakov mapping, which absorbs the longitudinally dependent focusing into a metaplectic scaling transformation and reduces the orbital evolution to that of a stationary two-dimensional oscillator. On this background, the transverse Pauli term is treated perturbatively and yields an explicit first-order correction for arbitrary realistic solenoidal profiles. Axial symmetry implies conservation of the total projection of angular momentum, so each spin flip is accompanied by a compensating one-quantum change of orbital angular momentum. In addition, the linear coordinate structure of the perturbation restricts the first-order dynamics to at most two neighboring radial sidebands for each incoming oscillator component. We derive the corresponding transition amplitudes and show how their phases are governed jointly by the Ermakov accumulation and the diagonal spin-orbital rotation. The resulting framework provides a direct way to quantify spin-orbit mixing of twisted electrons in realistic magnetic lenses and solenoidal beam-line elements, and it identifies a route toward controlled spin-OAM conversion in engineered sequences of magnetic-field edges.