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Spin-1/2: An Internal Rotational Degree of Freedom in Proper Time

Hou Y. Yau

Published Jul 7, 2026• 0 citations
Publisher

Abstract

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 $\psi(\phi)=Ae^{-i\phi/2}$. This construction reproduces the characteristic two-state structure, spinorial $4\pi$ periodicity, and fundamental $SU(2)$ representation of spin-$\tfrac12$ 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.

DOI
10.5281/zenodo.21251341

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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.
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
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.
Spin is one of the fundamental properties of elementary particles and plays a central role in quantum theory and the Standard Model of particle physics. Despite its fundamental role and extensive experimental verification, its physical origin remains unclear, and spin is generally described within the standard quantum formalism without an explicit geometrical interpretation. This work investigates the possibility that spin may be understood as a manifestation of an underlying geometrical dynamics within the framework of the Theory of Absolute (ToA). The analysis is based on a toroidal model involving two coupled periodic motions. It is shown that a simple coupling between the angular coordinates of the toroidal trajectory naturally reproduces several characteristic properties of spin-1/2 systems, including two spin orientations, the characteristic spin-1/2 behavior, and the requirement of a 4𝜋 rotation for complete restoration of the dynamical state. The model further leads naturally to the relation S=L/2, from which the experimentally observed value g=2 follows. The analysis is further extended by incorporating the local temporal dynamics of ToA. Applying the theory's relation between velocity and local time to the proposed internal motion leads to a natural limitation of admissible internal velocities. As the internal velocity approaches the speed of light, the rate of local time decreases toward zero, providing a self-consistent dynamical constraint on the internal motion. When applied to the electron as an illustrative spin-1/2 system, the model also reproduces the characteristic scales of the Bohr magneton and the reduced Compton wavelength. The results suggest that the Theory of Absolute provides a coherent framework for investigating a realistic geometrical interpretation of spin. Although the proposed model does not constitute a complete physical theory of spin, it demonstrates that several experimentally observed spin properties can emerge naturally from a simple geometrical structure combined with local temporal dynamics.
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.