Abstract Various attempts at a thoroughly wave-theoretic explanation of matter have taken as their fundamental ingredient the de Broglie or matter wave. But that wave is superluminal, whereas it is implicit in the Lorentz transformation that influences propagate ultimately at the velocity c of light. It is shown that if the de Broglie wave is understood, not as a wave in its own right, but as the relativistically induced modulation of an underlying standing wave comprising counter-propagating ináuences of velocity c, the energy, momentum, mass and inertia of a massive particle can be explained from the manner in which the modulated wave structure must adapt to a change of inertial frame. With those properties of the particle explained entirely from wave structure, nothing remains to be apportioned to anything discrete or ìsolidîwithin the wave. Consideration may thus be given to the possibility of wave-theoretic explanations of particle trajectories, and to a deeper understanding of the Klein-Gordon, Schrˆdinger and Dirac equations, all of which were conceived as equations for the de Broglie wave.
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave function
Ψ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,
∂t∂R2+∇⋅[R2m∇S−qA]=0,∂t∂S+2m(∇S−qA)2+qϕ+U+Q=0,where
Q=−2mℏ2R∇2R. The gauge-invariant kinetic propagation momentum is therefore
Π=∇S−qA,and the associated local propagation velocity is
v=mΠ. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term
Q provides an additional modification of the propagation structure. The same equations can be written in a second-order form,
mdtdv=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.
This paper proposes that gravity arises from the constructive interference of de Broglie matter waves in a density-dependent refractive medium. The argument proceeds in five steps: (1) greater mass implies greater density; (2) by Snell's law, greater density implies a higher refractive index; (3) a higher refractive index implies a shorter wavelength; (4) when matter waves propagate outward from an object in all directions, the wavelength is shorter on the side facing the larger mass; (5) constructive interference is stronger and more densely packed where the wavelength is shorter, so the object is preferentially redistributed toward the side of greater mass. This shows gravity to be a wave-mechanical phenomenon: matter is not pulled toward mass by some force; it is redistributed toward mass through the constructive interference of its own matter wave. Newton and Einstein described the trajectory and the geometric structure of gravitational motion; this paper proposes the physical mechanism by which that motion occurs. This paper makes three propositions explicit, with carefully distinguished epistemic status: (A) the mass distribution determines the effective refractive medium through which matter waves propagate — an experimentally established fact, confirmed independently by the COW experiment (1975) and by semiconductor heterojunction experiments confirming Snell's law for matter waves at density boundaries; (B) the displacement of the constructed-interference maximum is the physical mechanism of actual gravitational motion — the paper's sole new physical identification, going beyond Born's rule as standardly stated; it is consistent with but not compelled by the COW data, and constitutes the primary target of experimental falsification (Section 6.6d); (C) the COW fringe shift is a direct measurement of the gravitational mechanism — a logical corollary that follows necessarily from (A) and (B), not an independent assumption. Section 6.6d translates these propositions into five differentiating experimental predictions, ordered by experimental accessibility, including an immediately executable reanalysis of existing COW data and a novel slit-material density experiment that provides a clean differential test between the present framework and standard wave optics.
Abstract The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
The Lorentz transformation (LT) is explained by changes occurring in the wave characteristics of matter as it changes inertial frame. This explanation is akin to that favoured by Lorentz, but informed by later insights, due primarily to de Broglie, regarding the underlying unity of matter and radiation. To show the nature of these changes, a massive particle is modelled as a standing wave in three dimensions. As the particle moves, the standing wave becomes a travelling wave having two factors. One is a carrier wave displaying the dilated frequency and contracted ellipsoidal form described by the LT, while the other (identified as the de Broglie wave) is a modulation defining the dephasing of the carrier wave (and thus the failure of simultaneity) in the direction of travel. The superluminality of the de Broglie wave is thus explained, as are several other mysterious features of the optical behaviour of matter, including the physical meaning of the Schrödinger equation and the relevance to scattering processes of the de Broglie wave vector. Consideration is given to what this Lorentzian approach to relativity might mean for the possible existence of a preferred frame and the origin of the observed Minkowski metric.