The physical origin of the de Broglie wavelength remains one of the central interpretive questions in quantum mechanics. This work is developed within the Zero-field framework, a phenomenological model in which space is treated as a continuous medium containing localized regions of inward energy flow toward particle-centered Zero-fields. Within this framework, moving particles generate rotation-induced guiding structures referred to as Frontal Waves. These waves are proposed to accompany particle motion and provide a geometric interpretation of the de Broglie wavelength and wave–particle duality. The proposed Frontal Wave wavelength reproduces the standard de Broglie relation and is interpreted as the physical guiding structure associated with matter-wave behavior. The framework is applied to the Double Slit Experiment and the Mach–Zehnder interferometer, where Frontal Waves propagate through multiple paths and form interference structures that guide particle trajectories toward regions of constructive interference. Connections with Bohmian mechanics are discussed, emphasizing that the model serves as a phenomenological interpretation rather than a modification of the mathematical formalism of quantum mechanics. The Frontal Wave concept provides a geometric picture of matter-wave behavior and offers a unified interpretation of particle guidance, interference phenomena, and the de Broglie wavelength within the broader Zero-field framework.
The physical origin of the de Broglie wavelength remains one of the central interpretive questions in quantum mechanics. This work is developed within the Zero-field framework, a phenomenological model in which space is treated as a continuous medium containing localized regions of inward energy flow toward particle-centered Zero-fields. Within this framework, moving particles generate rotation-induced guiding structures referred to as Frontal Waves. These waves are proposed to accompany particle motion and provide a geometric interpretation of the de Broglie wavelength and wave–particle duality. The proposed Frontal Wave wavelength reproduces the standard de Broglie relation and is interpreted as the physical guiding structure associated with matter-wave behavior. The framework is applied to the Double Slit Experiment and the Mach–Zehnder interferometer, where Frontal Waves propagate through multiple paths and form interference structures that guide particle trajectories toward regions of constructive interference. Connections with Bohmian mechanics are discussed, emphasizing that the model serves as a phenomenological interpretation rather than a modification of the mathematical formalism of quantum mechanics. The Frontal Wave concept provides a geometric picture of matter-wave behavior and offers a unified interpretation of particle guidance, interference phenomena, and the de Broglie wavelength within the broader Zero-field framework.
Within a realist extension of the de Broglie–Bohm framework, this work proposes aphysical interpretation of elementary particle spin as an emergent topological property of anunderlying guiding-wave structure.In previous works, a hypothetical physical wave component associated with photons wasintroduced as a possible experimentally accessible extension of the pilot-wave concept. Thisapproach was subsequently extended to entangled photons through the hypothesis of a com-mon non-separable guiding structure WAB existing in physical three-dimensional space.The present paper develops the internal dynamics of this guiding structure. We proposethat spin does not correspond to the mechanical rotation of a point-like particle, but ratherto a topological excitation of the physical guiding wave itself. The particle is interpreted as alocalized excitation embedded within a structured field W (r, t) possessing an internal phasegeometry.A nonlinear field model is introduced in which stable localized excitations can emerge astopological soliton-like structures. The quantization of spin is associated with a phase wind-ing number characterizing the topology of the guiding field around the particle excitation.The central result of this work is a generalized guiding equation combining the transla-tional guidance induced by the energy flow of the physical wave with an additional topologicalspin current. This equation provides a unified description linking particle motion, internalfield geometry, and spin quantization within a single deterministic framework.The proposed model remains phenomenological and aims at establishing a possible phys-ical ontology for quantum spin rather than replacing the standard mathematical predictionsof quantum mechanics.
This paper proposes a qualitative geometric interpretation of the de Broglie wave within the framework of the “pencil case” ontology — a model constructed from minimal and explicit assumptions about the structure of physical reality. Spacetime is treated as an active, elastic medium called ECV (Expanded Configuration of the Vacuum). Matter is understood as its local densification — CCV (Compact Configuration of the Vacuum). A particle in motion is described as a bubble (Ba) — a spatially extended, dynamically structured region of CCV. Within this framework, the de Broglie wave is not interpreted as a probability wave nor as a separate accompanying entity. Instead, it is understood as the internal pulse of the bubble itself — a periodic process of localization and delocalization occurring within the vacuum structure. The model introduces three energy thresholds: E_Z(1) — threshold for stable bubble formation E_Z(2) — threshold for activation of localization–delocalization oscillations E_Z(3) — threshold beyond which a delocalized bubble state cannot be sustained (associated with the macroscopic regime) In this interpretation, neutrinos are hypothesized to lie in the immediate vicinity of E_Z(2), which may account for their extremely weak oscillatory visibility in interference-type contexts. The framework offers qualitative reinterpretations of interference phenomena, decoherence, delayed-choice experiments, and neutrino detection. It also formulates falsifiable predictions, including: possible material-dependent asymmetry in slit experiments, discontinuous regime transitions in interference patterns, potential long-term geometric “dissolution” of the lightest neutrino into the vacuum. Written in the spirit of conceptual exploration, the paper employs intuitive physical metaphors (sonar, warm rain dissolving into a cold lake, a weakly excited tuning fork) to clarify the underlying geometric picture. The work does not attempt to replace established quantum theory. Rather, it proposes a coherent ontological narrative describing what the de Broglie wave could represent within a geometric vacuum-based framework.
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.
In this work, we extend the 9-dimensional Ultra-Spacetime-Time-Evolution (9D-USTE) framework to explain the origin of quantum double-slit interference from pure geometric field dynamics. Based on the 3+6 dimensional spacetime structure (3D macroscopic spacetime plus a compact 6D internal manifold), a scalar field σ (radion field) governing the volume of the 6D compact internal manifold is shown to exhibit wave-like propagation across the entire spatial region, while the corresponding particle is localized in 3D macroscopic space. We demonstrate that the observed interference pattern is directly identified with the squared gradient of the σ field, whose fluctuations pass through both slits and superimpose. The particle trajectory is guided by the geometric field distribution, yielding a fully deterministic geometric interpretation of quantum duality without wavefunction collapse, nonlocality, or probabilistic postulates. Quantitative estimates of the σ field mass m_σ are provided, and the correspondence between the σ-field wavelength and the de Broglie wavelength is established by fitting existing electron and C₆₀ double-slit experimental data. We further discuss how the compact 6D internal manifold provides a geometric mechanism to address Bell inequality-related nonlocality concerns, distinguishing the 9D-USTE framework from existing Bohmian mechanics and Kaluza-Klein theories. This work advances 9D-USTE as a consistent geometric alternative for unifying classical gravity and quantum phenomena.
Zitterbewegung and de Broglie matter waves are both phenomena associated with the quantum-mechanical properties of the electron, but they are usually discussed in different contexts. Zitterbewegung is understood as a rapid oscillatory motion appearing in the Dirac equation, whereas the de Broglie matter wave is understood as a wave corresponding to the linear momentum of a particle. Some previous studies have attempted to derive the de Broglie matter wave from an internal clock of the electron or from Zitterbewegung. In the present study, however, no causal relation in which the matter wave is generated by Zitterbewegung is assumed. Instead, the two are regarded as different geometrical representations of a common phase degree of freedom. Specifically, Zitterbewegung is interpreted as a phenomenon corresponding to an internal rotational degree of freedom, while the de Broglie matter wave is interpreted as spatial phase progression associated with linear momentum. and are interpreted, respectively, as the rates of phase progression in the temporal and spatial directions. Furthermore, using the relation introduced in previous work, and reinterpreting it as the ratio of the internal rotational velocity to the external propagation velocity, this study shows the possibility of describing internal rotation, translational motion, matter waves, and the fine-structure constant within a common phase geometry.
Abstract The standard formalisms of quantum mechanics and special relativity are extraordinarily successful, yet several foundational features are introduced to students as postulates or mathematical constructs without physical visualisation: why the wavefunction must be complex, what is physically associated with spin, why time dilation occurs in material particles, where the Lorentz-factor scaling of inertia comes from, why energy quantisation follows from a phase-closure condition, and what geometric account can be given of wave-particle duality. This paper presents a geometric picture—already implicit in Schrödinger’s Zitterbewegung and de Broglie’s internal periodicity—that may help students develop physical intuition for each of these features. The electron is represented as executing a phase-coherent helical motion at speed c , with transverse radius a 0 = ℏ / 2 m 0 c and axial pitch equal to the de Broglie wavelength. Within this picture the imaginary unit i admits an intuitive interpretation as the 90 ∘ rotation operator in physical transverse space; quantisation may be visualised as a phase-closure condition with the quantum number n counting helical nodes; time dilation and the Lorentz-factor scaling of inertia both have a transparent geometric reading from a single pythagorean speed constraint; and interference may be understood as a Moiré-type geometric phase effect. The model is fully consistent with standard quantum and relativistic predictions. Its contribution is pedagogical and interpretational: it offers a unified geometric visualisation that may help students develop intuition for these foundational concepts, rather than a derivation of quantum mechanics or special relativity from first principles. The paper is self-contained and addressed to advanced undergraduates and graduate students.