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Emergent Quantum Formalism from Infrared Information Flux: De Broglie Matter Waves and Heisenberg Uncertainty within HIGM

Samir, Ahmed, Hamouda

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.29779.34087

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De Broglie Wave Analysis of the Heisenberg Uncertainty Minimum Limit under the Lorentz Transformation

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A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.

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1 Wave-Particle Duality: de Broglie Waves and Uncertainty

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Abstract In 1925, de Broglie hypothesized that any material particle has an associated wave with /h p . Electron diffraction seems to support that Hypothesis. But then, the electron at rest will have infinite wavelength, and infinite wave phase velocity. This says that for a material particle, the de Broglie relation does not hold. Failed attempts to save the postulate, kept the flawed relation, and modified the waves into train waves, pilot waves, probability waves, … to name a few. We keep the de Broglie waves unchanged, and modify the relation. First, we observe that the Planck energy E h used by de Broglie defines virtual electromagnetic waves. Consequently, for any particle, the virtual electromagnetic wavelength is / /c h mc , and / /h m c . Refinement of de Broglie argument, indicates that /h mmay be x , Heisenberg’s uncertainty in the particle location. de Broglie’s later analysis supports this interpretation, and we offer an explanation to particle diffraction as a consequence of Heisenberg’s uncertainty. We apply / /h m c to obtain the dispersion relations for the de Broglie virtual waves.

•Jan 1, 2006•0 citations
A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.
Abstract In 1925, de Broglie hypothesized that any material particle has an associated wave with /h p . Electron diffraction seems to support that Hypothesis. But then, the electron at rest will have infinite wavelength, and infinite wave phase velocity. This says that for a material particle, the de Broglie relation does not hold. Failed attempts to save the postulate, kept the flawed relation, and modified the waves into train waves, pilot waves, probability waves, … to name a few. We keep the de Broglie waves unchanged, and modify the relation. First, we observe that the Planck energy E h used by de Broglie defines virtual electromagnetic waves. Consequently, for any particle, the virtual electromagnetic wavelength is / /c h mc , and / /h m c . Refinement of de Broglie argument, indicates that /h mmay be x , Heisenberg’s uncertainty in the particle location. de Broglie’s later analysis supports this interpretation, and we offer an explanation to particle diffraction as a consequence of Heisenberg’s uncertainty. We apply / /h m c to obtain the dispersion relations for the de Broglie virtual waves.