The incompressible Navier-Stokes-Fourier limits from Boltzmann–Fermi–Dirac equation for low regularity data

Solitary waves in a two-parameter family of generalized nonlinear Dirac equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg" display="inline" id="d1e718"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> dimensions

The parity-violating asymmetry, accounting for the vector and axial-vector vertex plus self-energy correction as well as for vacuum polarization, is calculated nonperturbatively by solving the corresponding Dirac equation for the electronic scattering states. Investigating the nuclei 27Al, 48Ca and 208Pb at collision energies in the GeV region and at forward scattering angles matching the experimental geometries, it is found that the combined QED effects change the parity-violating asymmetry by less than one percent. The same is true for 12C and 208Pb at an energy of 150 MeV.

Comment on “Quantum phase transitions of Dirac particles in a magnetized rotating curved background: Interplay of geometry, magnetization, and thermodynamics”

On the Nodal Set of Solutions to Dirac Equations

Sampling theory associated with discrete Dirac equations

Infinitely many small-magnitude standing waves of super-linear Dirac equations

Abstract The influence of pressure on finite-nuclear-size (FNS) corrections to atomic energy levels and electron-capture decay rate is investigated in confined hydrogenlike ions. The ions are modeled inside an impenetrable spherical cavity, with a Gaussian distribution used to represent the nuclear charge distribution. For each confinement radius—used to simulate external pressure—the energies and wave functions of the lowest-lying bound states are determined by numerically solving the Dirac equation via the kinetically balanced generalized pseudospectral (KBGPS) method. In contrast to unconfined ions, both the FNS corrections and electron-capture decay rates increase markedly under pressure and exhibit parallel trends with increasing confinement. Pressure also removes level degeneracies and alters the relative magnitudes of FNS corrections across different bound states. Moreover, the nuclear charge radius is found to significantly affect the pressure-enhanced electron-capture decay rate.

We investigate a class of modified quantum field theories based on a fractional-inspired deformation of the action, where nonlocality is introduced through a spacetime-dependent weight rather than fractional derivatives. Using a Riemann–Liouville–type kernel implemented directly in the measure, we derive modified Klein–Gordon and Dirac equations and show that the resulting dynamics acquires additional first-order derivative terms governed by gradients of the weight. These corrections lead to nonconservative evolution and non-Hermitian effective Hamiltonians in the conventional inner product, while preserving a conserved weighted current and a generalized norm. The framework explicitly breaks Lorentz invariance by introducing a preferred spacetime structure, with potential implications for discrete symmetries, including possible violations of parity, time reversal, and CPT. We analyze the structure of the modified Dirac equation, including its Hamiltonian formulation and charge-conjugation properties, and discuss the consistency of the theory in terms of a weighted Hilbert space. Phenomenological implications are explored with emphasis on neutrino propagation, where high-precision oscillation and time-of-flight measurements provide stringent constraints. We estimate that current and future experiments bound deviations from the standard theory, highlighting the challenge of observationally distinguishing the model while identifying potential signatures such as energy-dependent dispersion and CPT asymmetries.

Abstract In this work we prove global well-posedness for the massive Maxwell-Dirac system in the Lorenz gauge in $$\mathbb {R}^{1+3}$$ R 1 + 3 , for small, sufficiently smooth and decaying initial data, as well as modified scattering for the solutions. Heuristically we exploit the close connection between the massive Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel approach which applies directly at the level of the Dirac equations. The modified scattering result follows from a precise description of the asymptotic behavior of the solutions inside the light cone, which we derive via the method of testing with wave packets of Ifrim-Tataru.

We formulate the Dirac oscillator covariantly in the presence of external non-Abelian gauge fields. More precisely, the matter field is written as [Formula: see text], where [Formula: see text] denotes the Dirac index and [Formula: see text] the isospin index, so that the Hamiltonian acts on the tensor-product space [Formula: see text] in the fundamental representation. Starting from the gauge-covariant Dirac equation, we then implement the oscillator interaction through the standard non-minimal substitution and promote the construction to an SU(2) background. In this way, we derive the associated non-Abelian field-strength tensor and isolate the commutator contribution, which has no Abelian analogue. Consequently, the generalized Pauli interaction [Formula: see text] produces matrix-valued spin–isospin couplings. At the same time, the Abelian sector reduces to the conventional Moshinsky–Szczepaniak Dirac oscillator, whose exactly solvable spectrum provides a natural benchmark for the extended theory. For an aligned planar background, the commutator term yields the closed-form isospin splitting [Formula: see text] thereby making the internal-Zeeman mechanism explicit. To render this result visually transparent, we also include a direct spectral representation of the exact branches as functions of the splitting parameter [Formula: see text]. Furthermore, we show that the planar Dirac oscillator admits a direct correspondence with effective graphene Hamiltonians: the same non-minimal confinement rule generates the graphene Dirac-oscillator form once the relativistic mass scale is traded for the appropriate gap parameter. In turn, retaining the internal valley or layer doublet extends this correspondence to the gauge-field case, where matrix-valued effective connections provide a natural implementation of the non-Abelian model and the commutator term controls a possible lifting of internal degeneracies. Overall, the resulting framework separates universal commutator-driven effects from backgrounddependent kinematic shifts and thus provides a controlled setting for studying relativistic bound states in Yang–Mills backgrounds and in graphene-based Dirac materials with effective non-Abelian structures.

Abstract We show that the Schrödinger equation can be solved exactly based only on classical least action. Fundamental postulates of quantum mechanics can in turn be derived directly from this construction. The results extend to the relativistic Klein-Gordon, Pauli, and Dirac equations, and suggest a smooth transition between physics across scales. Most quantum mechanics problems have classical versions which involve multiple least action solutions. The associated classical multipaths stem either from the initial position or momentum distribution, or from branch points, generated, e.g. by a multiply connected manifold (double slit experiment), by spatial inequality constraints (particle in a box), or by a singularity (Coulomb potential). We show that the exact Schrödinger wave function ψ can be constructed by combining this classical multi-valued action ϕ with the classical density ρ, computed analytically from ϕ along each extremal action path. The construction is general and does not involve any semi-classical approximation. Quantum wave collapse at measurement can be derived from the classical density change. Entanglement corresponds to a sum of classical particle actions mapping to a tensor product of spinors. The results also provide a simpler computational alternative to Feynman path integrals, as they use only a minimal subset of classical paths.

Decay of Solutions of Nonlinear Dirac Equations

Magnetic moments in the Poynting theorem, Maxwell equations, Dirac equation, and QED

Resolving the spurious-state problem in the Dirac equation by using the staggered-grid method

We prove the existence of topological solutions to the self-dual Chern-Simons model and the abelian Higgs system on the lattice graphs n for n 2. This extends results of Huang, Lin and Yau (2020) from finite graphs to lattice graphs.M j=1 n j p j(2)with positive integers n 1 , . . ., n M and distinct vortices p 1 , . . ., p M 2 .Here > 0, and p j is the Dirac mass at p j .A solution of (1) or ( 2) is called topological if u(x) 0 as |x| +, and called nontopological if u(x) - as |x| +.For the abelian Higgs system (2), Jaffe and Taubes [1980] proved the existence and uniqueness of general finite energy multivortex solutions to the Bogomol'nyi equations, and there have been many studies on this model since then, such as [Jacobs and Rebbi 1979;Jaffe and Taubes 1980;Wang and Yang 1992].The self-dual Chern-Simons system (1) is the minimal self-dual model containing the Chern-Simons term.The Chern-Simons vortices were discovered in [Jackiw and Weinberg 1990;Hong et al. 1990], which attracted people to investigate the existence problem.The existence of topological solutions in 2 was established in [Wang 1991;Spruck and Yang 1995] by the variational method and iteration argument, and the existence of self-dual doubly periodic vortex solutions was proved in [Caffarelli and Yang 1995].

Higher-order topological photonic systems typically host corner states that are exponentially localized. Here we uncover a distinct regime of uniformly delocalized higher-order topological modes, emerging from the interplay of multiple spatially varying Dirac mass terms under chiral symmetry. These modes exhibit uniform large-area, sublattice-locked profiles that remain pinned at zero energy, independent of system size. Crucially, their internal phase admits a tunable gauge degree of freedom, enabling controlled reconfiguration without loss of topological protection. This dual combination of uniform delocalization and gauge-controlled tunability bridges the gap between scalable optical mode area and robust topological protection, and we further confirm these predictions experimentally in photonic crystals, observing excellent agreement with theory. Our scheme is directly applicable to photonic mode design and enables the realization of robust, large-area topological optical devices.

We present a computational framework for electron–molecule scattering that is intended as a step toward a more accurate and unified description over a broad energy range relevant to applications. The approach combines a spherical complex optical potential (SCOP), constructed from multiconfiguration Dirac–Fock atomic densities via a group-additivity scheme, with a partial-wave solution of the Dirac equation. Methane (CH4) and silane (SiH4) are used as benchmark targets because of their simple tetrahedral structure and well-documented cross sections. The comparison among exchange models is used to highlight the sensitivity of predicted cross sections to the target model potential and the need for improved model descriptions. In addition, the present Dirac-based implementation is benchmarked against our existing [Joshipura et al., Phys. Rev. A, 2004, 69, 022705] nonrelativistic optical-potential treatment that employs the Numerov method to solve the Schrödinger equation, allowing us to quantify both relativistic kinematic and spinor effects. Although relativistic effects on integral cross sections are minimal, the Dirac treatment has a pronounced impact on the phase shifts and large-angle differential cross sections. Among the exchange models tested, the modified Furness–McCarthy exchange shows the most consistent agreement with benchmark data and represents a clear improvement over our earlier group-additivity SCOP results. This enhancement lays the groundwork for extending the method to larger and strongly polar molecules.

Limit Behavior of Multiple Bound States of Nonlinear Dirac Equations

We theoretically investigate the spin-Hall and valley-Hall transport properties of monolayer jacutingaite (Pt2HgSe3) under an off-resonant circularly polarized light field and a perpendicular electric field. Using a low-energy massive Dirac model combined with linear resp Alipourzadeh onse theory, we analyze how spin-orbit coupling, inversion-symmetry breaking, optical driving, and finite temperature jointly influence the Hall conductivities. At zero temperature, the spin-Hall and valley-Hall conductivities exhibit step-like plateau features, reflecting the underlying topological character of the massive Dirac bands. The off-resonant circularly polarized light effectively renormalizes the Dirac mass and modifies the band structure, enabling controllable transitions between distinct Hall response regimes. While the spin-Hall effect arises intrinsically from strong spin-orbit coupling, the valley-Hall effect appears only when inversion symmetry is broken by a staggered sublattice potential, with characteristic critical points associated with gap closing. At finite temperatures, thermal broadening smooths these plateau-like features into continuous Fermi energy dependent responses and reduces the magnitude of both Hall effects. Nevertheless, the corresponding topological signatures remain robust at low and moderate temperatures, particularly near charge neutrality. We further show that circularly polarized light alone cannot generate a finite valley-Hall response at finite temperatures without inversion symmetry breaking. These results demonstrate that the combined action of optical driving, electric field control, and thermal effects provides an effective route to manipulate spin and valley transport in jacutingaite, highlighting its potential for tunable spintronic and valleytronic applications.