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A Lorentz-Invariant Phase Anchor and a Dirac-Compatible Spinorial Lifting Scaffold

Ping Zhang

Published Mar 18, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.19078680

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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.
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.
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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.
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\psi(\phi)=Ae^{-i\phi/2}ψ(ϕ)=Ae−iϕ/2. This construction reproduces the characteristic two-state structure, spinorial 4π4\pi4π periodicity, and fundamental SU(2)SU(2)SU(2) representation of spin-12\tfrac1221​ 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.
This work develops a phase-based formulation of relativistic wave dynamics, demonstrating that the Klein--Gordon equation emerges naturally from the foundational assumption of intrinsic phase periodicity in material fields. Mapping the phase directly onto the classical action, we postulate that localized excitations possess an invariant rest-frame oscillation governed by a proper frequency ω0ω_0ω0​. This physical condition establishes an operational mass-frequency relation, m=ℏω0/c2m = \hbar ω_0 / c^2m=ℏω0​/c2, without requiring rest energy as an independent, axiomatic input. We show that the Klein--Gordon equation arises as the minimal local, linear, Lorentz-invariant field equation compatible with this internal phase structure. Within this framework, mass acts as an intrinsic frequency scale governing wave propagation, and relativistic kinematics is fully recovered as a structural consequence of phase coherence. This approach provides a unified wave-mechanical interpretation where particle dynamics maps onto the group velocity of dispersive wave packets, offering an intuitive account of free propagation, dispersion, and tunneling across potential barriers.
This paper derives the Dirac equation as the logically forced relativistic completion of the de Broglie wave dynamics established in prior work of this series. Starting from the logical foundation that yields Lorentz/Poincaré invariance, we promote the non-relativistic framework (continuous spacetime, Fourier duality, local plane-wave structures, and probability conservation via statistical energy flux) to the relativistic regime. We show that the requirements of exact local relativistic plane-wave structures (with phase gradients satisfying the mass-shell condition pμpμ=m2c2p^\mu p_\mu = m^2 c^2pμpμ​=m2c2), first-order time evolution (needed for clean probability conservation), and full Lorentz covariance together force the introduction of a specific non-commuting algebraic structure and a multi-component wave function. The Dirac equation emerges as the unique linear differential operator of order at most one in the spacetime derivatives that consistently glues these local structures across smooth backgrounds while preserving a conserved four-current with positive-definite time component. This multi-component structure arises naturally as the minimal algebraic consequence of the constraints, without any separate postulate for intrinsic angular momentum. In exact analogy with the emergence of the finite invariant causal speed in special relativity, the Dirac dynamics is revealed not as an ad-hoc linearization of the Klein--Gordon equation, but as the mathematically inevitable description of indivisible particles in continuous relativistic spacetime. The derivation thereby completes the single-particle sector of the series, demonstrating that core structures of relativistic quantum mechanics flow deductively from a small number of deep, nearly tautological commitments.