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The Hidden First-Order Structure of the Lorentz Invariant: A Phase Slotting Interpretation of de Broglie Phase Harmony

Ping Zhang

Published Jul 1, 2026• 0 citations
Publisher

Abstract

The Lorentz-invariant relation between frequency and wave number providesthe standard relativistic foundation of de Broglie's phase harmony.While this quadratic invariant preserves the proper phase norm for allinertial observers, it does not explicitly reveal how temporal and spatialphase evolution are locally organized after a Lorentz transformation.In particular, the mechanism by which the observed de Broglie frequencymaintains phase harmony remains implicit. In this work, we revisit this question from a distinguished referenceconfiguration in which the phase field possesses purely temporal evolution.Treating the phase as a Lorentz scalar and following the phase evolution ofindividual medium elements, we derive an exact first-order relation, \[\omega=\frac{\omega_0}{\gamma}+vk,\] which we call the Phase Slotting Principle. This relation separatesthe observed coordinate frequency into a temporal projection associated withproper phase evolution and a spatial contribution generated by sweepingacross the Lorentz-induced phase gradient. The familiar quadratic invariantis recovered exactly, showing that the conventional Lorentz relationpreserves the total phase budget, whereas the first-order formulationretains the explicit allocation of phase evolution between the temporal andspatial sectors. Using the standard quantum identifications between frequency, wave number,energy, and momentum, the same first-order structure naturally yields anenergy allocation relation. In the non-relativistic limit, Newtoniankinetic energy emerges as the observable remainder of a continuous balancebetween temporal reduction and spatial accumulation, while the completerelativistic energy expansion is recovered order by order. Rather thanmodifying special relativity, the present work reveals an explicitfirst-order description already implicit in the Lorentz transformation,providing a complementary geometric interpretation of de Broglie phaseharmony and of the energetic organization of relativistic phase evolution.

DOI
10.5281/zenodo.21245622

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