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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

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.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.

Nov 7, 20250 citations

XXXIX. A note on M. de Broglie's theory of the photon

Marjorie Williamson

The London Edinburgh and Dublin Philosophical Magazine and Journal of ScienceApr 1, 19480 citations

Reinterpretation of the Schrödinger Equation via Photon and Matter Energy A Quantum-Relativistic Expression of the Hamiltonian

Hakan ÜNAL

Sep 15, 20260 citations

If the Speed of Light is Constant, Then Light is a Wave

Constantinos Ragazas

In this short note we mathematically prove that if we assume that the speed of light is constant, then light propagates as a wave. Introduction: In another paper, What is the Matter With De Broglie Waves? we derived the De Broglie equations for 'wavelength' λ and 'frequency' ν using the 'prime physis' quantity η and the definitions of energy, E t η ∂ = ∂ and momentum x p x η ∂ = ∂ in terms of η . We showed that,

Mar 1, 20111 citation

Big crash of basic concepts of physics of the 20th century

Peter Sujak

This paper analyzes the quantities of energy and momentum in the definitional rela-tionship of relativistic mechanics, in the de Broglie momentum hypothesis and in the Klein-Gordon, Dirac and Schrodinger equation. The results of analysis shows that Planck constant and relativistic relationships on the length contraction and increase in mass are a reflection of the same physical principles in nature, that λ designated in the de Broglie hypothesis λ =h/mv as the wave of matter with rest state value λ = ∞ must be connected with a real dimension of a particle with rest state value λ = lo = moc and that on this ba-sis we can come to the fundamental equations of quantum mechanics that are the Klein-Gordon, Dirac and Schrodinger equation without the necessity of the wave functions.This paper analyzes the quantities of energy and momentum in the definitional rela-tionship of relativistic mechanics, in the de Broglie momentum hypothesis and in the Klein-Gordon, Dirac and Schrodinger equation. The results of analysis shows that Planck constant and relativistic relationships on the length contraction and increase in mass are a reflection of the same physical principles in nature, that λ designated in the de Broglie hypothesis λ =h/mv as the wave of matter with rest state value λ = ∞ must be connected with a real dimension of a particle with rest state value λ = lo = moc and that on this ba-sis we can come to the fundamental equations of quantum mechanics that are the Klein-Gordon, Dirac and Schrodinger equation without the necessity of the wave functions.

Aug 1, 20115 citations

De Broglie Wavelength Defined by the Rest Energy E0 and Its Velocity

K. Orozović, B. Balon

International Journal of Physical and Mathematical SciencesJul 5, 20210 citations

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