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) 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) employs the real Clifford algebra 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 ω2>μ2, this manifold supports elliptical complementary motion with frequency ωeff=ω2−μ2, 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.
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) and the half-phase relation χ2=Υ. In the fundamental representation,S^a=2bσa,where b is the CPG action scale; with the empirical identification b=ℏ, the standard spin-1/2 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 dependencesin2ϑsin2(2vωML),while their dimensional prefactors are different. When the rapid propagation phase is unresolved, the common angular dependence reduces to sin2ϑ. 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.
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 ω0. This physical condition establishes an operational mass-frequency relation, m=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.
We develop a geometric framework in which spatial coordinates are minimally extended through coupled internal degrees of freedom associated with intrinsic change. Within this complexified transport geometry, phase and quantization emerge naturally from closure conditions on admissible configurations. Direct differentiation of the resulting geometric state representation yields a complex wave equation whose stationary real sector reproduces the conventional Schrödinger structure, while the effective potential arises intrinsically from the internal geometry itself. A complementary variational formulation independently recovers the same stationary structure through extremization of intrinsic geometric strain. Relational temporal evolution is obtained through progression across admissible closure cycles, yielding the conventional time-dependent Schrödinger form without introducing an external fundamental time parameter. The present treatment is restricted to a reduced one-dimensional transport sector, with multidimensional and relativistic extensions left for future work.
This work resolves a foundational tension in Dirac's formulation of relativistic quantum mechanics: momentum appears in two distinct but undifferentiated roles—as a dispersion-locked parameter generating unitary evolution in the relativistic wave equation, and as a free phase variable subject to symplectic flow in the constraint formalism. We demonstrate that this duality reflects an intrinsic Lorentzian structure in energy-momentum space, characterized by the interval dΣ² = dE²/c² − dp². This geometry divides naturally into three sectors: symmetric (dispersion-locked), null (massless, E/|p| = c), and antisymmetric (free-momentum) regimes, each governed by different geometric constraints. Dispersion emerges as the global integral of an exact action one-form, while the (E,p) symplectic structure arises locally as the exterior derivative of the action's inexact form θ_E = E dp. Energy quantization appears as a topological stability condition requiring closed spectral loops to satisfy ∮ E dp = nh, defining "proto-eigenstates" whose Fourier encodings yield standard Hilbert-space eigenstates. We recover both the Bohr spectrum and Dirac fine-structure energies from spectral phase closure, revealing α = e²/(ℏc) as a geometric action-normalization ratio between Maxwell U(1) curvature and Lorentz spinor action—not a coupling parameter from QED perturbation theory. The framework synthesizes unitary flow and symplectic uncertainty into a single bilinear Lorentzian structure, showing quantum states as objective in their symmetric unitary existence but path-dependent in their antisymmetric observed manifestation. This geometric reconstruction bridges Dirac's wave mechanics and constraint formalism through native (E,p) phase space, offering a foundation for quantum theory that treats energy-momentum as primary coordinates rather than conjugate variables derived from spacetime position.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.