This work proposes a geometric representation of quantum phase fields motivated by the long-standing tension between localized particle cores and dispersive wave propagation in quantum mechanics. Following the original insight of Louis de Broglie's phase harmony, the quantum phase is reinterpreted as a helical structure surrounding a localized core. The construction is developed using an inverse methodology: instead of deriving the structure dynamically from first principles, a candidate geometry is postulated and tested for internal consistency with the kinematic constraints of quantum mechanics. At the matching boundary defined by the Compton radius Rc = ħ/(mc), the phase field acquires a quantized azimuthal winding, forming a helical structure that connects the localized core with the extended navigation wave. Within this framework, the de Broglie wavelength λ = h/p emerges naturally as the geometric pitch of the helical phase surface. The analysis demonstrates that this helical configuration satisfies three essential requirements for a double-solution structure:topological phase quantization, velocity matching between wave and particle, and the correct decomposition of rest and kinetic energy. The work is presented as a proof-of-existence for a geometrically consistent phase structure that could support a localized vortex-like particle guided by a continuous phase field. It forms part of a broader research program exploring the mechanical and topological origins of quantum phenomena.
This work proposes a geometric representation of quantum phase fields motivated by the long-standing tension between localized particle cores and dispersive wave propagation in quantum mechanics. Following the original insight of Louis de Broglie's phase harmony, the quantum phase is reinterpreted as a helical structure surrounding a localized core. The construction is developed using an inverse methodology: instead of deriving the structure dynamically from first principles, a candidate geometry is postulated and tested for internal consistency with the kinematic constraints of quantum mechanics. At the matching boundary defined by the Compton radius Rc = ħ/(mc), the phase field acquires a quantized azimuthal winding, forming a helical structure that connects the localized core with the extended navigation wave. Within this framework, the de Broglie wavelength λ = h/p emerges naturally as the geometric pitch of the helical phase surface. The analysis demonstrates that this helical configuration satisfies three essential requirements for a double-solution structure:topological phase quantization, velocity matching between wave and particle, and the correct decomposition of rest and kinetic energy. The work is presented as a proof-of-existence for a geometrically consistent phase structure that could support a localized vortex-like particle guided by a continuous phase field. It forms part of a broader research program exploring the mechanical and topological origins of quantum phenomena.
This work revisits the kinematic foundations of wave–particle duality by re-examining the logic originally proposed by Louis de Broglie within the framework of Special Relativity developed by Albert Einstein. Starting from the assumption that a massive particle possesses an intrinsic periodic degree of freedom associated with its rest energy, the study explores how Lorentz covariance acts on such periodicity. By representing the internal phase on a transverse cylindrical manifold, it is shown that the relativity of simultaneity naturally converts temporal periodicity into a spatial helical phase structure when observed from a moving frame. Within this geometric representation, the longitudinal phase gradient is fixed by relativistic kinematics, leading directly to the relation kz=p/ℏkz=p/ℏ which reproduces the well-known de Broglie relation. In this framework, particle momentum appears as the torsional density of the helical phase field, while the transverse radius forms a Lorentz-invariant geometric seat for the intrinsic structure. The analysis further shows that this transverse scale naturally aligns with the characteristic length Rc=ℏ/mcRc=ℏ/mc previously identified as a structural scale in the author's earlier work on helical phase-field models. The resulting picture suggests a geometric convergence in which the de Broglie wavelength and the characteristic particle radius emerge as the longitudinal and transverse signatures of a single relativistic phase structure. This work is purely kinematic and does not attempt to resolve the detailed dynamical origin of the internal energy partition. The dynamical mechanism responsible for the transverse stabilization and the associated energy coefficients will be addressed in subsequent studies.
A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structures aimed at clarifying their physical content.
A geometric interpretation of electron dynamics is presentedbased on phase-coherent helical motion constrained by relativistic kinematics.Building on de Broglie’s concept of internal periodicity and theHamilton–Jacobi formulation of mechanics, the electron is modeled asan intrinsic light-speed helical trajectory whose axial pitch correspondsto the de Broglie wavelength and whose transverse scale is set by thereduced Compton length. Within this framework, spin-1/2 symmetry,magnetic moment, rest energy, and bound-state quantization emerge asinterconnected consequences of phase closure and geometric constraints,rather than as independent postulates. The formal operator structure ofquantum mechanics is recovered as a statistical description of ensemblesof phase-coherent helical trajectories, with momentum and energy operatorsarising as generators of spatial and temporal phase evolution. The frameworkprovides a geometric visualization of interference, tunneling, and quantumtransitions, while remaining fully consistent with the standard probabilisticpredictions of quantum mechanics and the Lorentz invariance of specialrelativity. The helical phase interpretation is intended as a conceptualand pedagogical complement to orthodox formalisms, offering physicalintuition for otherwise abstract quantum structures without modifying theirmathematical foundations. Thus, this work does not propose a modificationof quantum theory, but offers a geometric interpretation of its formal structuresaimed at clarifying their physical content.
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.
Abstract In this work, we investigate the emergence of quantum structures within a purely geometric field framework based on a unified helical configuration. Building on the geometric foundations and limiting recovery principles established in the preceding papers of this series, we demonstrate that quantization arises as a consequence of topological and spectral constraints imposed on the underlying helical field, rather than being introduced as a fundamental postulate. We show that discrete spectral sectors naturally follow from global and local topological conditions on the field configuration space, leading to an intrinsic form of quantization without invoking canonical commutation relations or operator axioms. Effective quantum operators are constructed directly from geometric functionals of the helical field, yielding non-commutative structures as emergent features of constrained geometry. Within this setting, Planck’s constant appears as an effective parameter characterizing the spectral density of admissible configurations, rather than as a fundamental universal constant. The framework consistently recovers standard quantum kinematics in appropriate limiting regimes while remaining fully compatible with the classical and relativistic limits previously derived. This approach provides a coherent geometric interpretation of quantum discreteness, operator structure, and effective constants, thereby offering a unified perspective in which classical, relativistic, and quantum descriptions arise from a single geometric field without assuming independent foundational layers.
This work develops a relational helical formulation of quantum mechanics in which the complex structure of the wavefunction acquires a concrete geometric interpretation. The imaginary unit i is identified as the generator of helical transport across a foliation of discrete hypersurfaces, transforming quantum phase into an angular coordinate and amplitude into a radial coordinate on a logarithmic spiral. Within this framework, standard quantum mechanics emerges as the projection of a deeper helical symmetry, revealing that its complex structure already encodes an intrinsic relativistic geometry. Building on this foundation, the construction incorporates a spiral mass equation, a modified ER-analogue metric, and a hypersurface-discrete spacetime, yielding a relational Hilbert- bundle structure governed by a least-action principle defined between layers. The resulting dual gravitational system, combining geometric (spiral) and energetic (mass/curvature) components, introduces canonical variables such as the spiral pitch a(λ) and the helical connection A(λ,x), each admitting well-defined conjugate momenta. These variables satisfy canonical commutation relations and form the operator algebra required for a quantum-gravity-like dynamics. The string-length postulate, identifying hypersurface circumferences as fundamental geometric quanta, further discretizes the underlying geometry. Together, these elements constitute a coherent proto-quantum-gravity framework in which the conceptual tension between quantum mechanics and relativity is dissolved, and the quantization of geometric degrees of freedom arises organically from the relational helical structure.