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Complementary Phase Geometry: Lagrangian Dynamics and Deterministic Wave-Like Trajectories

Evren Belenlioğlu

Published Sep 21, 2026• 0 citations
Publisher

Abstract

Complementary Phase Geometry (CPG) employs the real Clifford algebra $\mathrm{Cl}(3,4)$ and a distinct kinetic quadratic form of signature $(1,6)$. Building on its Hamiltonian foundations, this work develops the Lagrangian dynamics of coupled observable and complementary degrees of freedom within a prescribed unit-lapse transport model. The coupled Euler--Lagrange equations and canonical correspondence are derived for massive particles. An exact representation of the reduced nonlinear dynamics establishes preservation of the timelike domain and identifies a dynamically invariant zero-momentum manifold. For $\omega^2>\mu^2$, this manifold supports elliptical complementary motion with frequency $\omega_{\rm eff}=\sqrt{\omega^2-\mu^2}$, producing deterministic observable modulation. The propagating zero mode is recovered within the Lagrangian formulation as a single deterministic wave-like trajectory combining secular drift, bounded harmonic modulation, and bounded transport. An effective aperture construction based on a Hamilton--Jacobi boundary action-span approximation, uniform phase weighting, and assumed linear composition reproduces the standard Fraunhofer functional forms for single-, double-, and multiple-slit configurations. The microscopic particle--boundary interaction, boundary-selected outgoing trajectories, and quantitative connection to detector statistics remain unresolved.

DOI
10.5281/zenodo.22874409

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Complementary Phase Geometry (CPG) is developed as a deterministic canonical model in which the material phase of a particle is represented through additional intrinsic geometric degrees of freedom. The construction extends Minkowski spacetime by three real complementary coordinates and employs the real Clifford algebra Cl(3,4), while complex structure enters through the Complementary Generator Principle acting on infinitesimal dynamical quantities. From the invariant transported velocity norm, the particle action, Lagrangian, Hamiltonian, canonical symplectic structure, first-order evolution equations, and the complete 21-dimensional bivector generator structure are derived. A constitutive Action-Phase Principle relates the material phase to Hamilton's principal function through a universal action scale b, yielding canonical phase-gradient relations and distinct local and material phase frequencies. In the transport-free sector, exact first-order differential relations are obtained, while a Klein-Gordon-type second-order representation follows under a compatibility condition on the local canonical gradients. Its non-relativistic slow-envelope limit yields an extended Schr\"odinger-type equation. With the restriction to theobservable sector, the standard free Klein-Gordon and Schrödinger forms are recovered mathematically. A reduced Hamiltonian model further predicts phase-dependent oscillatory trajectory and flight-time modulations.
Complementary Phase Geometry (CPG) contains an intrinsic three-dimensional complementary rotational sector. Here this structure is examined as a geometric carrier of spin and polarization. A normalized complementary rotation bivector defines an oriented plane whose dual axis is identified with polarization. Its spinorial lift yields the standard double-cover structure Spin(3)≃SU(2)Spin(3)\simeq SU(2)Spin(3)≃SU(2) and the half-phase relation χ2=Υ\chi^2=\Upsilonχ2=Υ. In the fundamental representation,S^a=b2σa, \hat S_a=\frac{b}{2}\sigma_a,S^a​=2b​σa​,where bbb is the CPG action scale; with the empirical identification b=ℏb=\hbarb=ℏ, the standard spin-1/21/21/2 algebra, projections, and Casimir are recovered. Combining the polarization--plane identification with the pre-existing mixed observable--complementary dynamics of CPG yields a further conditional prediction. Both the free-flight timing variance and the transverse-position variance contain the common dimensionless dependencesin⁡2ϑ sin⁡2 ⁣(ωML2v), \sin^2\vartheta\, \sin^2\!\left( \frac{\omega_{\rm M}L}{2v} \right),sin2ϑsin2(2vωM​L​),while their dimensional prefactors are different. When the rapid propagation phase is unresolved, the common angular dependence reduces to sin⁡2ϑ\sin^2\varthetasin2ϑ. The absolute magnitudes remain controlled by presently undetermined complementary and mixed-coupling parameters. Phenomenological magnetic coupling reproduces standard Stern--Gerlach eigenchannels and analyzer-overlap weights. Magnetic moments, microscopic branch selection, and Born-rule statistics remain open.
A field-theoretic framework is developed in which a real scalar phase field serves as the fundamental object, and spacetime geometry, wave dynamics, and probabilistic structure emergefrom constraints on its gradients. The theory is defined by a scalar functional on momentumspace, leading to nonlinear equations of motion and an induced effective metric governing propagation. Under Lorentz invariance, the functional reduces to a single-variable dependence, andadditional assumptions on observability motivate a projective structure that constrains its form.In the weak-gradient regime, the theory reproduces linear relativistic wave equations andadmits a complex representation consistent with standard quantum wave behavior. Beyondthis regime, nonlinear effects lead to gradient steepening and caustic formation, resulting inmulti-branch phase structure. A detector model based on nonlinear coupling to the phase fieldis introduced, providing a dynamical mechanism for branch selection. A conserved current associated with phase transport yields a density that approximates the Born rule under controlledconditions.The framework unifies aspects of Hamilton–Jacobi theory, nonlinear scalar field dynamics, and emergent geometry, while providing a structured pathway from phase dynamics towave-like behavior and measurement-like phenomena. Possible phenomenological consequencesinclude background-dependent propagation, nonlinear deviations from superposition, and smallcorrections to standard probabilistic predictions.
This paper shows how classical transport dynamics arise naturally from the same variational dissipative–conservative framework that also supports quantum-analogue behaviour. In the transport-dominated regime, the oscillatory field reduces to Hamilton–Jacobi dynamics with well-defined trajectories and effective inertia, while coupling to auxiliary dissipative degrees of freedom suppresses interference and stabilises classical motion. No stochastic noise, no collapse postulates, and no hidden variables are introduced. Quantum, classical, and decoherent behaviour appear as distinct dynamical regimes of a single deterministic system, controlled by the balance between phase transport and amplitude curvature. Classical mechanics is recovered not as a foundational axiom, but as an effective description with a clear domain of validity.
Wave particle duality remains a central interpretational challenge in quantum theory. In this work, we develop a trajectory-based description of photon dynamics formulated in an extended complex space within the relativistic quantum Hamilton Jacobi framework. In this approach, photon motion is represented by complex trajectories whose real projections describe propagation, while imaginary components encode oscillatory structure. We show that momentum eigenstates correspond to straight line trajectories with uniform propagation at the speed of light, whereas superposition states lead to nontrivial quantum potentials and oscillatory motion in the complex plane. Extending the analysis to complex two dimensional space reveals richer dynamical behavior, including propagating wave like trajectories and standing wave like patterns in real projections. Energy momentum consistency is verified through an internal coherence analysis based on projected standing wave wavelengths. Rather than introducing new dynamical laws or additional physical dimensions, the complex space is employed as an interpretational framework that renders wave like and particle like aspects as complementary projections of a single underlying motion. The results suggest a unified geometric perspective on wave particle duality, while remaining fully compatible with standard quantum mechanics.
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,\Psi(\mathbf r,t)=R(\mathbf r,t)e^{iS(\mathbf r,t)/\hbar},Ψ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂R2∂t+∇⋅[R2∇S−qAm]=0,\frac{\partial R^2}{\partial t}+\nabla\cdot\left[R^2\frac{\nabla S-q\mathbf A}{m}\right]=0,∂t∂R2​+∇⋅[R2m∇S−qA​]=0,∂S∂t+(∇S−qA)22m+qϕ+U+Q=0,\frac{\partial S}{\partial t}+\frac{(\nabla S-q\mathbf A)^2}{2m}+q\phi+U+Q=0,∂t∂S​+2m(∇S−qA)2​+qϕ+U+Q=0,whereQ=−ℏ22m∇2RR.Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}.Q=−2mℏ2​R∇2R​. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,\boldsymbol{\Pi}=\nabla S-q\mathbf A,Π=∇S−qA,and the associated local propagation velocity isv=Πm.\mathbf v=\frac{\boldsymbol{\Pi}}{m}.v=mΠ​. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term QQQ provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdvdt=q(E+v×B)−∇U−∇Q,m\frac{d\mathbf v}{dt}=q(\mathbf E+\mathbf v\times\mathbf B)-\nabla U-\nabla Q,mdtdv​=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.