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Photon and Motion of a Material Body as Wave Processes: A Relativistic Phase Interpretation

Igor Balamut

Published Jun 4, 2026• 0 citations
Publisher

Abstract

A unified wave-phase description of the photon and of the motion of a material body is developed. The construction is based on the description of wave processes and on the preservation of their phase state as a fundamental principle, leading to Lorentz transformations and the structure of Minkowski spacetime without introducing them a priori. It is shown that the physically significant characteristic of a wave process is its phase state, which is preserved during evolution and acquires physical meaning only through the comparison of different processes. The wave invariant associated with the motion of a material body is factorized into two oppositely directed light-like components Q± = Ω ± cK = 2Ω_{l,±}, whose product, after a natural rescaling, takes the normalized form Q₊Q₋ = 1. In this representation, the parameters (Ω, K) are not independent quantities but are determined by the two light-like components: Ω characterizes their combined frequency content, whereas cK characterizes the asymmetry between them. Within the proposed framework, the Doppler shift is interpreted as a phase relation between wave processes, while relative velocity in special relativity corresponds to a Doppler asymmetry between the two light-like components of the wave process associated with motion. The resulting construction reproduces the kinematic structure of special relativity and provides a unified wave-phase description of the photon and of the motion of a material body.

DOI
10.5281/zenodo.20544822

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A unified wave-phase description of the photon and of the motion of a material body is developed. The construction is based on the description of wave processes and on the preservation of their phase state as a fundamental principle, leading to Lorentz transformations and the structure of Minkowski spacetime without introducing them a priori. It is shown that the physically significant characteristic of a wave process is its phase state, which is preserved during evolution and acquires physical meaning only through the comparison of different processes. The wave invariant associated with the motion of a material body is factorized into two oppositely directed light-like components Q± = Ω ± cK = 2Ω_{l,±}, whose product, after a natural rescaling, takes the normalized form Q₊Q₋ = 1. In this representation, the parameters (Ω, K) are not independent quantities but are determined by the two light-like components: Ω characterizes their combined frequency content, whereas cK characterizes the asymmetry between them. Within the proposed framework, the Doppler shift is interpreted as a phase relation between wave processes, while relative velocity in special relativity corresponds to a Doppler asymmetry between the two light-like components of the wave process associated with motion. The resulting construction reproduces the kinematic structure of special relativity and provides a unified wave-phase description of the photon and of the motion of a material body.
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.
We introduce physical phase invariance as a minimal, observer-independent principle capable of unifying quantum theory and relativity. By identifying the physical phase with the classical action measured in units of Planck’s constant, the framework assigns phase a direct operational meaning. Spacetime structure, Lorentz invariance, and quantum probabilistic behavior emerge as effective descriptions of underlying phase dynamics rather than fundamental postulates. Time is defined relationally through phase evolution, and classical motion arises from stationary phase conditions. A simple dynamical equation for the phase field is proposed, while quantum probabilities are shown to result from statistical coarse-graining over unresolved phase degrees of freedom. This approach provides a coherent conceptual foundation linking quantum mechanics, special relativity, and gravitation.
A simple covariant derivation of the de Broglie relations breaks down when one considers propagation of light in an anisotropic medium. This breakdown results in two distasteful conclusions. The first is that the de Broglie relations are not completely universal and their covariance is restricted. The second is that wave-particle duality is not universal. In this paper a resolution of these problems is suggested, a resolution which is an extrapolation of ideas of de Broglie and Andrade e Silva. The relations are generalized for propagation in anisotropic media by defining a new phase wave with wave vector l in the direction of energy transport and distinct from the conventional electromagnetic wave. The introduction of l follows the method by which de Broglie associated a wave with an electron. The magnitude of l is omega u/c-squared where omega is the angular frequency and u the velocity of energy transport. The physical meaning of l and the covariance of physical laws in the presence of matter are discussed. An experiment is described which would test the existence of the phase wave associated with the new wave vector l. This experiment is essentially a Young two-slit experiment carried out within an anisotropic crystal. Keywords: Scattering; Anisotropic materials.
Correction June 28, 2022: The classical wave-equation should read d/dt d/dt partial W(x,t) = vv d/dx d/dx partial W(x,t) Historically two types of “free” mechanical motions were considered, that of Newtonian mechanics x=vt (v=constant) and classical waves such as waves on a string which were described in terms of Newton’s laws for a continuous medium. It seems these two were considered as fundamentally different phenomena. Equations of motion were developed for classical mechanics as well as electromagnetic phenomena. Maxwell discovered that electromagnetic equations allowed for a classical wavelike solution which he identified as light. In the early 1900s it was realized that to retain electromagnetic equations invariance as seen from a constant moving frame moving, Lorentz transformations were needed. In 1905 Einstein obtained these transformations in a different manner considering that light has the same vacuum speed when viewed from different constant moving frames. Later in the 1920s quantum mechanics was developed using the idea of a probability wave exp(-iEt+ipx) for a free particle beginning with the work of deBroglie who made use of ideas of special relativity. In this note we suggest that there seems to be one fundamental “free” mechanical motion, namely that associated with a wave exp(-iEt+ipx) whether one considers a classical wave or a quantum free particle. Classical mechanics is a mapping into a continuous x-t space, we argue, and describes energy and momentum motion in this continuous space. This same description also occurs for classical waves. The medium undergoes a somewhat complicated nonlocal change, but the “velocity” of the wave maps to a continuous local space time i.e. v=x/t=dx/dt where E= pv. E=pv, however, is only a description of energy momentum propagation given in a simple form. It does not describe the overall picture. We argue the same is true for the simple x=vt (v=constant). It seems that an overall wave structure is associated with energy and momentum and for free motion this is linked with exp(-iEt+ipx). If this wave is fundamental one may argue that its phase may remain constant when viewed from different frames moving at constant v. Special relativity would then follow from this requirement and so is in a sense a consequence of wave behaviour. We investigate these ideas.
This paper presents a conceptual framework for reinterpreting the Lorentz transformation through the lens of quantum mechanics. By treating spacetime as a quantum structure, the author explores the eigenvalue nature of the speed of light (±c) and derives relativistic effects from the probabilistic superposition of lightlike intervals. Key highlights: - **Quantum Spacetime:** Proposes that space and time are not independent backgrounds but emerge from the quantum states of spacetime quanta. - **Derivation of Lorentz Transformation:** Demonstrates how the transformation arises from the statistical correlation of quantum states rather than classical kinematics. - **Philosophical Implications:** Offers a novel perspective on the ontology of motion, suggesting that reference frames are representations of quantum states. This work aims to provide an intuitive understanding of special relativity and stimulate discussion on the quantum nature of spacetime.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. 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Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.