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Izvestiya of Altai State University

An Interpretation of Relativistic Kinematics with the Standing Waves Method. Part 2

Alexander Goncharov

Published Jan 1, 2015• 1 citation
Publisher
DOI
10.14258/izvasu(2015)1.2-02

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Solutions of Dirac equation are well known in terms of bi-spinors, but the complete determination of these spinors is always derived using the assumption that the wave function is a plane wave. We investigate in this paper the search of general solutions under the form of standing wave modes, using methods issued from electromagnetism. It is shown that such solutions exist as solutions of a system of 64 equations and 64 unknowns, and that they must be associated with a relativistic relation of energy conservation. Complete exact solutions are presented. It is shown that these solutions highlight in a new manner the duality wave-particle, and the uncertainty principle.
We present a phase-based formulation of special relativity in which the kinematical structure of the theory is reconstructed from the requirement of phase coherence of localized wave states. Starting from the assumption that physical propagation is associated with surfaces of constant phase and that matter admits an intrinsic rest-frame oscillation, we show that the relativistic relations for time dilation, energy, and momentum follow from the invariant accumulation of phase along particle trajectories. In this framework, proper time is identified operationally as the phase count of an internal clock, and the Minkowski interval arises as the quadratic form consistent with phase invariance across inertial observers. The relation between mass and rest-frame frequency emerges naturally, providing a unified interpretation of relativistic kinematics in wave-mechanical terms. This formulation does not introduce new dynamics but offers a coherent structural perspective that bridges relativistic kinematics and wave propagation.
In optics and acoustics, in structured beams, non-diffracting polarization measures in clearly diffracting beams, and spin direction distributions in the core of these waves that have Skyrmionic behavior have been found. We here study the equivalents of these for fermions and show that in corresponding circumstances the non-diffractive spin textures persist independently of spin, statistics, or kinematics (or propagation speed of the structured wave). As a part of the study, we present LG solutions for Dirac particles valid for both relativistic and non-relativistic kinematics.