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arXiv:2602.17429

Self-Consistent Dynamics of Electron Radiation Reaction via Structure-Preserving Geometric Algorithms for Coupled Schrödinger-Maxwell Systems

Jacob M. Molina, Hong Qin

Published Feb 19, 2026• 0 citations
arXivHTMLPDF

Abstract

Classically, a charged particle in a magnetic field emits radiation, losing momentum and experiencing the Abraham-Lorentz (AL) / Landau-Lifshitz (LL) radiation reaction (RR) force. However, at atomic scales and outside the range of their applicability, the AL/LL equations fail and RR destroys the coherent state of an electron-undermining the very concept of a RR force. This process can be described by the coupled Schrödinger-Maxwell (SM) system under appropriate limits, but the system's nonlinear complexity has long limited purely analytical studies. We present geometric structure-preserving algorithms for the SM system that preserve gauge invariance, symplecticity, and unitarity on the discrete space-time lattice, which are implemented in our Structure-Preserving scHrodINger maXwell (SPHINX) code. By constructing coherent states from the Landau levels, SPHINX simulates the fully-coupled nonlinear dynamics of an electron coherent state, the energy partition evolution, and decoherence/relaxation of the electron wave packet in time due to RR. These simulations indicate that, in an external magnetic field, an electron prepared in an atomic-scale coherent state can radiate strongly, rapidly losing coherence and dispersing into a decoherent wave packet. Additionally, we also present the fully-coupled nonlinear evolution of the non-degenerate ground- and first-excited Landau levels themselves to understand how the coupled SM system modifies the well-known ideal (i.e., Schrödinger-only) dynamics of the Landau Levels. With appropriate boundary conditions, simulations show that the Landau levels are renormalized into stationary dressed eigenstates with constant electromagnetic and kinetic energies. This opens a new computational window into RR physics and advances modeling of extreme-field phenomena in fusion plasmas, astrophysics, and next-generation laser experiments

DOI
10.48550/arxiv.2602.17429

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Classically, a charged particle in a magnetic field emits radiation, losing momentum and experiencing the Abraham-Lorentz (AL) / Landau-Lifshitz (LL) radiation reaction (RR) force. However, at atomic scales and outside the range of their applicability, the AL/LL equations fail and RR destroys the coherent state of an electron-undermining the very concept of a RR force. This process can be described by the coupled Schrödinger-Maxwell (SM) system under appropriate limits, but the system's nonlinear complexity has long limited purely analytical studies. We present geometric structure-preserving algorithms for the SM system that preserve gauge invariance, symplecticity, and unitarity on the discrete space-time lattice, which are implemented in our Structure-Preserving scHrodINger maXwell (SPHINX) code. By constructing coherent states from the Landau levels, SPHINX simulates the fully-coupled nonlinear dynamics of an electron coherent state, the energy partition evolution, and decoherence/relaxation of the electron wave packet in time due to RR. These simulations indicate that, in an external magnetic field, an electron prepared in an atomic-scale coherent state can radiate strongly, rapidly losing coherence and dispersing into a decoherent wave packet. Additionally, we also present the fully-coupled nonlinear evolution of the non-degenerate ground- and first-excited Landau levels themselves to understand how the coupled SM system modifies the well-known ideal (i.e., Schrödinger-only) dynamics of the Landau Levels. With appropriate boundary conditions, simulations show that the Landau levels are renormalized into stationary dressed eigenstates with constant electromagnetic and kinetic energies. This opens a new computational window into RR physics and advances modeling of extreme-field phenomena in fusion plasmas, astrophysics, and next-generation laser experiments
An effective model for describing the relativistic quantum dynamics of a radiating electron is developed via a relativistic generalization of the Lindblad master equation. By incorporating both radiation reaction and vacuum fluctuations into the Dirac equation within an open quantum system framework, our approach captures the Zitterbewegung of the electron, ensuing noncommutativity of its effective spatial coordinates, and provides the quantum analogue of the Landau-Lifshitz (LL) classical equation of motion with radiation reaction. We develop the corresponding phase-space representation via the relativistic Wigner function and derive the semiclassical limit through a Foldy-Wouthuysen transformation. The latter elucidates the signature of quantum vacuum fluctuations in the LL equation, and shows its relationship with the corrected Sokolov equation. Our results offer a robust framework for investigating quantum radiation reaction effects in ultrastrong laser fields.
We investigate the effects of quantized radiation reaction fields on the motion of a charged particle using the gauge-independent Wigner operator (GIWO) and gauge-independent Wigner function (GIWF) introduced earlier [Phys. Rev. A 33, 2913 (1986)]. To complement the equation of motion of the GIWO, the Heisenberg equations of motion of the quantized electromagnetic fields are solved within the Markov approximation. After considering the operator orderings and orders of magnitude of the radiation reaction terms, we eliminate the quantum fields from the evolution equation of the GIWO, and obtain for the GIWF a closed equation containing relaxation terms. As an example of the formalism we derive a Fokker-Planck equation (FPE) for the GIWF of a particle in a constant magnetic field. To the order \ensuremath{\Elzxh} 0^{0}0 the classical radiation damping ensues, and the first quantum correction proportional to \ensuremath{\Elzxh} emerges as diffusion. The diffusion operator turns out to be indefinite and the FPE consequently defies our attempts at a complete analysis, but we demonstrate that at least the coherent states constructed from the Landau levels exhibit a manifestly physical time evolution under the FPE. We point out that the GIWF calculated with quantized electromagnetic fields is divergent even if the fields are in the vacuum state, and suggest that the GIWF should be associated with the particle state by ignoring the quantized fields altogether.
Electromagnetic waves are an inherent part of all plasmas—laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation—a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrödinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, and magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space.
This paper presents a unified pedagogical derivation of the hydrogen atom spectrum and radiative transition structure by organizing three historically separate frameworks into a single mathematical chain: Schrödinger’s non-relativistic wave mechanics, Sommerfeld’s relativistic fine-structure theory, and Maxwell’s classical electrodynamics. The analysis shows how Bohr energy levels arise as the non-relativistic bound-state limit, how Sommerfeld-type fine structure appears when relativistic corrections are included, and how Maxwell radiation theory connects naturally to photon emission through transition dipole moments and spontaneous decay constants. The framework is then advanced by adding the necessary modern ingredients: spin, Lorentz covariance, minimal electromagnetic coupling, gauge invariance, and electromagnetic field quantization. With these additions, the derivational chain reaches the Dirac equation coupled to the quantized electromagnetic field, which forms the standard foundation of quantum electrodynamics. The resulting Scretching Master Equation is presented as a structured bridge from introductory quantum mechanics to QED. It links energy eigenvalues, angular-frequency scaling, transition dipole strength μ², spontaneous emission Γ = A₂₁, lifetime τ, and photon radiation into one coherent derivational system. No new physical law is claimed. The contribution is a corrected synthesis, notation system, and teaching route through established physics.
The goal of this thesis is to provide a mathematical rigorous derivation of the Schrödinger-Klein-Gordon equations, the Maxwell-Schrödinger equations and the defocusing cubic nonlinear Schrödinger equation in two dimensions. We study the time evolution of the Nelson model (with ultraviolet cutoff) in a limit where the number N of charged particles gets large while the coupling of each particle to the radiation field is of order N^{−1/2}. At time zero it is assumed that almost all charges are in the same one-body state (a Bose-Einstein condensate) and that the radiation field is close to a coherent state. We show the persistence of condensation over time and prove that the time evolution is approximately described by the Schrödinger-Klein-Gordon system of equations in the large N limit. Subsequently, we consider the spinless Pauli-Fierz Hamiltonian which models the interaction between charged bosons and the quantized electromagnetic field. We discuss the limit previously described and prove that the time evolution is approximated by the Maxwell-Schrödinger equations. To our knowledge, this is the first rigorous result concerning a mean-field limit of the Pauli-Fierz Hamiltonian. We then turn to the evolution of Bose-Einstein condensates in two dimensions and consider N bosons which interact by a repulsive two-body potential. The interaction is given either by N^{−1+2β}V(N^{β}x) with β∈R^{+}_{0} or by e^{2N}V(e^{N}x), for some spherical symmetric, positive and compactly supported V∈L_{∞}(R^{2},R). We prove that the dynamics is approximated by the defocusing two-dimensional cubic nonlinear Schrödinger equation in the large N limit. In case of the exponential scaling, we show that a short-scale correlation structure affects the dynamics of the condensate. This is the first rigorous derivation that considers an exponential scaling of the interaction. All derivations rely on a method developed by Pickl in [Lett. Math. Phys. 97(2), 151–164 (2011)]. The first two results are obtained by an extension of the method to systems which interact with quantized radiation fields. The latter is derived by an appropriate adaption of the proof in three space dimensions [Rev. Math. Phys., 27, 1550005 (2015)]. The crucial insight to derive the Maxwell-Schrödinger equations is to restrict the class of many-body wave functions to a subspace of states whose energy per particle only fluctuates little around the energy functional of the Maxwell-Schrödinger system. To derive the two-dimensional Gross-Pitaevskii equation it is essential to define a measure of condensation which properly incorporates the correlations that arise from the exponential scaling of the interaction. This thesis is based on the preprints [54, 47].
Phase space as an arena for atom optics in a quantized light field, D.S. Krahmer et al photon statistics for multimode squeezed Schrodinger cat states, V.I. Man'ko quantum dissipation and coherence, G. Vitiello recent results on the Abraham-Lorentz-Dirac equation, A. Carati et al shift operators and coherent states in nonlinear dynamics, E.C.G. Sudarashan quantumlike manifestation in a classical mechanical system, R.K. Varma quantum-like model for charged particle beam dynamics, R. Fedele coherent instabilties in particle accelerators - conventional and novel approaches, M. Migliorati and L. Palumbo crystalline beams - theory, experiments and proposals, A. Ruggiero coherent structures in magnetized plasmas, P.K. Shukla dynamics of interaction of relativistically intense electromagnetic waves with a relativistic hot electron-positron plasmas, N.L. Tsintsadze coherent and incoherent soliton interactions - a variational approach, D. Anderson and M. Lisak stationary and radiating solitons in high order dispersive systems, V.I. Karpman.