We revisit the helical decomposition model of a massive relativistic particle and correct the fun damental defect of fixing the three-dimensional total speed to the speed of light. By formulating the helical structure on four-velocity space with proper time as the parameter, the four-velocity satisfies the mass-shell identity U · U = −c 2 identically, while the observable three-speed v = √ vz 2 + v⊥ 2 remains strictly below c. Mapping the internal degree of freedom onto a two-level quantum state whose Bloch-sphere parameter space is the direction of the unit three-velocity n = v/v, we derive a candidate U(1) geometric phase γ geo = 2π(1 − cos α) = 2π ( 1 − v z v ) , where cos α = v z /v is the polar angle of the velocity direction. This phase depends only on the direction of motion and is distinguishable in principle from the dynamical phase, the standard Berry phase, and the Sagnac effect. For v⊥/v = 0.1, we obtain γ geo ∼ 0.031 rad, corresponding to a visible-light interferometric shift of about 2.5 nm, which lies at the edge of current precision. We also discuss the speculative promotion of this kinematic connection to an intrinsic geometric
We revisit the helical decomposition model of a massive relativistic particle and correct the fun damental defect of fixing the three-dimensional total speed to the speed of light. By formulating the helical structure on four-velocity space with proper time as the parameter, the four-velocity satisfies the mass-shell identity U · U = −c 2 identically, while the observable three-speed v = √ vz 2 + v⊥ 2 remains strictly below c. Mapping the internal degree of freedom onto a two-level quantum state whose Bloch-sphere parameter space is the direction of the unit three-velocity n = v/v, we derive a candidate U(1) geometric phase γ geo = 2π(1 − cos α) = 2π ( 1 − v z v ) , where cos α = v z /v is the polar angle of the velocity direction. This phase depends only on the direction of motion and is distinguishable in principle from the dynamical phase, the standard Berry phase, and the Sagnac effect. For v⊥/v = 0.1, we obtain γ geo ∼ 0.031 rad, corresponding to a visible-light interferometric shift of about 2.5 nm, which lies at the edge of current precision. We also discuss the speculative promotion of this kinematic connection to an intrinsic geometric
This paper summarizes the current state of the helical worldline model developed across eleven previous publications. The model posits that all particles follow helical paths through spacetime with invariant speed v_path = √[v² + (c/γ)²] = c, where v is spatial velocity and c/γ represents motion through time. This single principle unifies special relativity, de Broglie matter waves (λ = h/p as spatial projection of the tilted helix), the Compton effect (λ_f - λ_i = (h/m₀c)(1-cos θ) from helical phase matching), and the photoelectric effect (work function φ derived from helical compression by crystal lattices). Spin ħ/2 emerges from the 4π symmetry of the helix, and Bohr quantization follows from helical standing wave conditions in atoms. The model makes testable predictions including tunneling time ≈ ħ/(2m₀c²) and work function scaling φ ∝ 1/L² with crystal structure. The author presents the current mathematical framework, summarizes confirmed connections to established physics, identifies open questions, and outlines further study needed to extend the model toward a full unification.
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.
Author Note: This deposit represents an early exploratory working paper in the development of the Quanta Kinetic framework. The core relativistic kinematics and dynamics have since been formally isolated and axiomatically derived in the peer-reviewed manuscript preprint here: 10.5281/zenodo.23567446. This paper formalizes momentum not as an independent dynamic vector, but as the active geometric reallocation of a system's invariant four-velocity budget away from internal temporal phase progression and into spatial propagation. Within the invariant four-velocity constraint v² + 𝕋² = c² g₀₀ (where 𝕋 ≡ c(dτ/dt) = c/γ defines the chronokinetic rate), acquiring spatial velocity directly throttles internal temporal phase processing. By substituting γ = c / 𝕋 into standard relativistic momentum (p = γ m₀ v), this work derives the Invariant Momentum Product: p · 𝕋 = m₀ c v This identity forms an exact mathematical symmetry with Einstein's energy product (E · 𝕋 = m₀ c³). Taking the ratio of both products directly recovers the foundational relativistic relation without ad-hoc postulates: (p · 𝕋) / (E · 𝕋) = (m₀ c v) / (m₀ c³) ⇒ p = E v / c² Key theoretical and operational developments include: Relativistic Quantum Operator Mechanics: Projecting the Klein-Gordon mass-shell relation (ʲ - p̂² c² = m₀² c⁴) onto the physical Hilbert space isolates the spatial velocity operator (v̂ ≡ c² p̂ Ê⁻¹, mapping to the Foldy-Wouthuysen mean-position operator) and the chronokinetic phase operator (𝕋̂ ≡ m₀ c³ Ê⁻¹). Evaluating expectation values across dispersed quantum wavepackets establishes the variance-bounded conservation identity: ⟨v̂⟩² + ⟨𝕋̂⟩² + (σ_v² + σ_𝕋²) = c² proving that spatial momentum and internal de Broglie phase evolution (ω₀ = m₀ c² / ℏ) are locked in an orthogonal, zero-sum competition across the invariant speed ceiling. Thermodynamic Coupling & Multi-Tier Velocity: Expanding spatial allocation into three orthogonal tiers—⟨v²⟩ ≡ v_bulk² + ⟨v_thermal²⟩ + σ_v² (bulk translation, thermal jitter, and quantum dispersion)—demonstrates that coherent macroscopic acceleration and incoherent microscopic heating draw additively from the exact same temporal budget. Collision Dynamics as Clock Re-Balancing: Classical and relativistic momentum exchange during physical collisions is reinterpreted as a dynamic re-balancing of local clock rates across interacting bodies. As Particle A transfers velocity to Particle B, Particle A sheds spatial velocity and recovers its proper-time existence rate (𝕋_A' > 𝕋_A), while Particle B acquires spatial velocity and sacrifices temporal passage (𝕋_B' < 𝕋_B), preserving the total four-velocity norm of the closed system: (v_A')² + (𝕋_A')² + (v_B')² + (𝕋_B')² = 2c²
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.
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.
The nonlocal correlation mechanism of quantum entanglement and the underlying dynamics of barrier penetration in quantum tunneling constitute long-standing foundational explanatory gaps in standard quantum mechanics. While the standard theory can accurately describe the phenomena and provide probabilistic outcomes, it cannot offer a geometric-topological origin at the spacetime level. Based on the foundational axioms of three-dimensional vector light-speed helical spacetime, this paper establishes a self-consistent geometric explanatory framework: microscopic quantum effects are attributed to emergent behaviors arising from the helical structure of spacetime, topological bonding, and spacetime curvature resonance. The contributions of this paper include: 1. Establishing the core axioms of three-dimensional helical spacetime and a three-level spacetime normalization equation, constructing a low-energy emergence mechanism for four-dimensional Minkowski spacetime, and resolving the self-consistency conflict between the old and new definitions of spacetime; 2. Deriving the topological throat structure for two-particle entanglement based on the principle of topological bonding stability, providing the spacetime geometric origin of nonlocal correlations in entanglement and a finite propagation time delay; 3. Building an eigenfrequency model of spacetime distortion in the potential barrier, formulating the geometric resonance condition for quantum tunneling, and deriving a discrete resonance tunneling prediction distinct from the WKB approximation; 14. Independently deriving the mass-energy equivalence relation, the Lorentz transformation, and the Bohr radius from the model’s axioms, thereby verifying the low-energy self-consistency of the frame work; 5. Presenting a complete first-order perturbation integral derivation for the hydrogen 1S-2S transition, yielding a quantifiable and verifiable prediction for a spectral shift; 6. Explicitly listing the mathematical limitations of the model, its boundary of applicability, and falsifiable experimental protocols. This framework offers a geometric alternative explanatory system distinct from standard quan tum mechanics and quantum field theory. All quantitative results and predictions are preserved for long-term academic scrutiny. Keywords: vector light speed; helical spacetime; topological throat; spacetime topological bonding; geometric resonant tunneling; quantum entanglement; hydrogen atomic spectral shift