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A Lorentz-Invariant Semi-Relativistic Extension of the de Broglie Hypothesis: Bridging Matter Waves and Slowed-Down Light

Bin Ghanem, Salem Omar

Published Nov 7, 2025• 0 citations
Publisher

Abstract

A Lorentz-invariant semi-relativistic formulation is developed to extend the de Broglie hypothesis and establish a unified connection between matter waves and their photonic limit. Beginning with the classical relation pc = hf, a relativistic momentum expansion p ≈ mv (1 + ½ v²/c² + 3/8 v⁴/c⁴) leads to a generalized energy–frequency relation hf = pc + (v/c) Ek + (1 − γ⁻¹) mc², where Ek = ½ mv² and γ = (1 − v²/c²)⁻½. This provides a continuous transition between the non-relativistic (v ≪ c) and photonic (v → c) domains, describing matter as “slowed-down light” in a Lorentz-consistent sense.To preserve both relativistic and quantum consistency, a modified Dirac equation is introduced:(iħ γ^μ ∂_μ + β_ν ħ² / (2m) γ^ν □) ψ = 0,where β_ν = u_ν / c and □ = ∂_μ ∂^μ.This formulation, derived from a variational Lagrangian, naturally reduces to the Schrödinger and Dirac equations in their respective limits while maintaining Lorentz covariance. Numerical analysis for electrons in the 1–100 keV range reveals wavelength deviations of approximately 0.001–1%, suitable for precision diffraction experiments, and fine-structure variations around 0.7% for light atoms. The model strengthens de Broglie’s original interpretation of matter as a localized manifestation of slowed light within a Lorentz-invariant semi-relativistic framework.

DOI
10.5281/zenodo.17552781

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A Lorentz-invariant semi-relativistic formulation is developed to extend the de Broglie hypothesis and establish a unified connection between matter waves and their photonic limit. Beginning with the classical relation pc = hf, a relativistic momentum expansion p ≈ mv (1 + ½ v²/c² + 3/8 v⁴/c⁴) leads to a generalized energy–frequency relation hf = pc + (v/c) Ek + (1 − γ⁻¹) mc², where Ek = ½ mv² and γ = (1 − v²/c²)⁻½. This provides a continuous transition between the non-relativistic (v ≪ c) and photonic (v → c) domains, describing matter as “slowed-down light” in a Lorentz-consistent sense.To preserve both relativistic and quantum consistency, a modified Dirac equation is introduced:(iħ γ^μ ∂_μ + β_ν ħ² / (2m) γ^ν □) ψ = 0,where β_ν = u_ν / c and □ = ∂_μ ∂^μ.This formulation, derived from a variational Lagrangian, naturally reduces to the Schrödinger and Dirac equations in their respective limits while maintaining Lorentz covariance. Numerical analysis for electrons in the 1–100 keV range reveals wavelength deviations of approximately 0.001–1%, suitable for precision diffraction experiments, and fine-structure variations around 0.7% for light atoms. The model strengthens de Broglie’s original interpretation of matter as a localized manifestation of slowed light within a Lorentz-invariant semi-relativistic framework.
This preprint develops a Lorentz-invariant semi-relativistic framework in which massive particles are modeled as slowed-down photonic excitations described by a modified Dirac equation with a velocity-dependent d’Alembertian operator. The model provides a continuous interpolation between photon-like and massive regimes through a generalized energy–frequency relation and an extended Dirac Lagrangian.The framework is applied to Young’s double-slit experiment, where coherent propagation of the slowed-light field produces interference, while localized perturbations of the four-velocity field u^u by a which-path detector induce entanglement and decoherence without invoking wavefunction collapse.The theory predicts measurable energy-dependent fringe shifts and visibility changes in electron interferometry, offering a unified dynamical perspective on wave–particle duality.
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.
The de Broglie relation serves as a foundational pillar connecting classical linear momentum with quantum mechanical wave properties. For massive particles moving at relativistic speeds, momentum is routinely expressed using the Lorentz factor, yielding a direct identity with the quantum wave parameters. However, when applied to massless particles such as photons traveling at the speed of light, this product encounters an indeterminate mathematical form. This paper demonstrates how the general relativistic energy momentum dispersion relation resolves this breakdown. Through two detailed numerical case studies involving a relativistic electron moving at sixty percent of light speed and a green photon, it shows that while the algebraic expression relying on rest mass fails for massless entities, the underlying physical equivalence between momentum and wavelength remains globally valid across all particle regimes.
Whatever was going to replace classical physics in the description of radiation and atomic phenomena had to confront the difficult task of somehow reconciling the wave-like and particle-like aspects of light in a single structure. An important clue would come from Einstein’s E=mc2. We tend to want to associate mass (and linear momentum) with material particles. But the Planck–Einstein relation E=hν connects energy with frequency, a determinedly wave-like property. So, here are two very simple yet fundamental equations connecting energy to mass and energy to frequency. Can they be combined? Louis de Broglie thought so, and in 1923 he generalized the discovery made by Einstein in 1905 by extending it to all material particles, and most notably to electrons. The de Broglie relation λ=h/p can be derived quite straightforwardly by comparing the Lorentz transformations for energy and momentum with those for a system of plane waves.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
In this article, it is shown that if a particle is at rest and has some harmonic oscillation, then a traveling particle with a constant speed corresponds to a De Broglie Wave.In 1924, De Broglie proposed that particles of matter exhibit wave like behavior, the expression for which became fundamental in the creation of wave mechanics (quantum mechanics).The main equation for this is the Schrödinger equation, which in the case of free motion of a particle becomes the expression for the De Broglie wave.This hypothesis is supported by many early experiments on the diffraction of electrons, and later the development of modern physics.However until now, the origin of this hypothesis remains unclear.We show that the expression for the De Broglie wavelength can be obtained from a more general well-known concept, that any particle at rest has a corresponding harmonic oscillation that transforms into a wave of a moving particle with the help of relativistic change of coordinates (the Lorentz transformations).In this way, a De Broglie wave can be explained as a spacetime wave.