This work proposes a minimal geometric framework connecting three characteristic structures associated with relativistic electrons: the Compton-scale internal periodicity, the de Broglie longitudinal modulation, and the spinorial double-cover structure of spin-1/2. Starting from a Lorentz-unfolded phase associated with rest-frame Compton frequency, a cylindrical kinematic model is used to recover the de Broglie wavelength as a geometric phase pitch under Lorentz transformation. The construction is then extended to a minimal spinorial setting via a Hopf-fibration structure, providing a natural geometric completion that links internal phase periodicity with SU(2) double-cover behavior. The framework is purely kinematic and does not assume a specific dynamical model of the electron, but highlights a structural correspondence between Lorentz transformations, phase unfolding, and Hopf-type topology. Recent exact solutions of the Dirac equation with Hopfion-like structure (Białynicki-Birula, 2019) are noted as a complementary realization of related geometric features, suggesting a broader underlying Lorentz–Hopf structure in relativistic electron states.
This work develops a minimal relativistic phase framework that connects internal periodicity, de Broglie modulation, and a Dirac-compatible branch structure within a single kinematic construction. Starting from a Lorentz-invariant internal phase anchor, the analysis shows that the de Broglie wave can be interpreted as the observer-frame modulation of an underlying proper-time periodicity, rather than as an independent ontological wave imposed from outside. This yields a compact geometric picture in which matter-wave phase arises from relativistic unfolding of an intrinsic internal clock. To extend this structure beyond the scalar phase level, a minimal half-angle lifting is introduced through a phase variable ΘΘ, defined only at the level required to construct the branch scaffold. No strong physical interpretation of ΘΘ is imposed in this paper. Within this restricted setting, the lifted phase naturally generates a four-branch organization that is compatible with the formal sign structure of the Dirac plane-wave sectors. The purpose of this construction is not to claim a full derivation of the Dirac equation, nor a strict one-to-one identification of all branches, but to establish a compact relativistic phase scaffold that is structurally consistent with Dirac-type splitting. The main contribution of this paper is therefore a kinematic bridge: from Lorentz-unfolded internal periodicity, to de Broglie phase modulation, to a minimal Dirac-compatible branch architecture. This provides a controlled intermediate layer between scalar relativistic phase and spinorial structure, while leaving detailed physical interpretation, strict branch mapping, and full dynamical completion to subsequent work. This paper is intended as a structural foundation for later developments in the Vacuum Fluid Theory (VFT) program, especially those concerning the emergence of spinorial sectors, relativistic morphology, and the relation between internal phase organization and observable quantum branches.
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.
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 structuresaimed at clarifying their physical content.
We present a geometric interpretation of intrinsic spin in which a point-like particle possesses an internal rotational degree of freedom associated with its proper-time evolution. Rather than identifying spin with the mechanical rotation of an extended body, the proposed framework introduces an internal phase variable parameterizing a compact manifold along the particle's worldline. The resulting internal clock evolves at twice the de Broglie mass-energy frequency, allowing the rest-frame wavefunction to be written as
ψ(ϕ)=Ae−iϕ/2. This construction reproduces the characteristic two-state structure, spinorial
4π periodicity, and fundamental
SU(2) representation of spin-
21 particles. Because the internal phase acts as an evolution parameter rather than a canonical operator, the framework avoids the dimensionality and algebraic obstructions associated with quantizing a spatial angular coordinate. The construction extends consistently to the standard Dirac formalism, preserving the established structure of relativistic quantum mechanics and quantum field theory while providing a geometric interpretation of the origin of intrinsic spin.
We present a three-level organization of a bilocal relational theory inwhich structures usually associated with quantum mechanics arise atlogically distinct stages. The first level is purely geometric. Starting from the bilocalconfiguration space
M×M and its canonical cotangent bundle,the relative sector admits a radial--angular decomposition. For
N=3,the angular phase space is
T∗S2. Its non-singular sector isnaturally organized as
R+×SO(3) and therefore possesses the canonical doublecover
R+×SU(2)≃C2∖{0}. A classical two-component
SU(2)-spinorial carrier thus appearsbefore any Lorentzian dynamics or quantum postulate; its identificationwith a Lorentz-covariant Weyl spinor is not assumed at this stage.Reduction at fixed internalCasimir yields the Hopf fibration with
∣c1∣=1. For
N=3, defining
pμ=(pt,P) and
μint=κ1+iκ2rewrites the full constraint as
pμpμ+∣μint∣2=0.Its Clifford factorization then admits an exact chiral first-order formwith a two-dimensional complex kernel. At this stage the constructionremains classical: the momenta are phase-space variables and the kernelspinor is not yet a quantum wave function.The third level promotes this geometric and dynamical structure toquantum spinor dynamics. Geometry supplies an integer topological classbut not a universal scale of action; the identification
I=nℏ/2 therefore requires the quantum scale
ℏ.Together with the operator rule
pμ↦−iℏDμ, the classical chiral factorization becomesa quantum Dirac-type equation. If the internal complex contributionadmits a sector or limit in which it reduces to a real constant
m, the standard massive Dirac equation
(iℏγμDμ−m)ΨD=0is recovered. The construction therefore separates three ingredients that arenormally introduced together: spinorial and topological statestructure, Lorentzian dynamics, and quantum operator dynamics. We donot derive quantum mechanics from classical geometry. Rather, we showthat the spinorial carrier and the algebraic architecture of the Diracequation can arise before quantization, while the quantum scale andoperator realization remain independent inputs. This identifiesprecisely which elements of quantum spinor dynamics are geometric,which depend on the Lorentzian dynamical postulate, and which remaingenuinely quantum assumptions.