The dual particle-wave nature of matter is a guiding principle of modern quantum theory, formally arising from Planck’s energy e = hν, de Broglie’s momentum p = h/λ and Einstein’s energy from special relativity, where h is the Planck constant, ν and λ are the frequency and wavelength respectively of the wave with wave speed w = νλ. Motivated by the symmetry in special relativity between sub-luminal and superluminal motions, these fundamental relations can be generalised to a family that is characterised by a second fundamental constant h′ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. By Lorentz invariance, we refer to invariance under the combined special relativistic space-time and energymomentum transformations. The power-law expressions are important since they share the same relationship with the extension of the Planck-de Broglie energy-momentum relations as does Einstein’s energy expression with the Planck-de Broglie relations which are included through the special case h′= 0. For h′̸= 0, the new relations allow the determination of the stationary wave energy and momentum states and leading to a table for the critical particle and wave velocities in terms of the power law parameter. We further develop these new relations to provide an explicit profile for a Lorentz invariant wave associated with an elementary particle. Overall, we formulate a particlewave model allowing two distinct energy and momentum states, corresponding to particles and waves, and we propose a mechanism for the particle-wave transition.
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Particle-Wave Critical Velocities and Wave Profiles for Power-Law Energies
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Received: 22 Jul 2026 | Revised: 28 Aug 2026 | Accepted: 03 Sep 2026 | Published: 08 Sep 2026
Abstract
Keywords
special relativity | Lorentz invariance | power-law energy momenta | critical particle and wave velocities | Lorentz invariant wave profile
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