1. Abstract We investigate the dynamical consequences of a single scalar charge-phase field, θcharge and show that both electromagnetism and non-relativistic quantum mechanics emerge from its phase geometry and evolution. Starting from a local phase-field Lagrangian, the Euler-Lagrange structure generates a continuity equation together with the associated charge density and current density. Spatial gradients of the charge phase define a propagating spin-1 bond mode which, in the weak-field and long-range limit, gives rise to electromagnetic radiation and reduces to Maxwell electrodynamics. The same charge-phase structure also admits a mechanical representation. The corresponding phase dynamics naturally reproduce the Hamilton-Jacobi equation, establishing a direct connection between charge-phase evolution and dynamical motion. Combining the Hamilton-Jacobi structure with the continuity equation yields a quantum wave description in which the wavefunction emerges from the conserved density and charge phase. Granular corrections associated with the underlying phase structure generate the quantum potential and produce the Schrödinger equation as the effective quantum evolution law. Angular-momentum quantization follows from phase single-valuedness and periodicity, without the introduction of independent quantization postulates. Within this framework, Maxwell electrodynamics, Hamilton-Jacobi mechanics, Schrödinger dynamics, and quantum quantization appear as different dynamical limits of the same charge-phase field. Electromagnetism corresponds to the long-range linear propagation of phase bonds, whereas quantum mechanics reflects the oscillatory and granular dynamics of the underlying phase geometry.
Abstract We investigate the dynamical consequences of a single scalar charge-phase field, θcharge and show that both electromagnetism and non-relativistic quantum mechanics emerge from its phase geometry and evolution. Starting from a local phase-field Lagrangian, the Euler-Lagrange structure generates a continuity equation together with the associated charge density and current density. Spatial gradients of the charge phase define a propagating spin-1 bond mode which, in the weak-field and long-range limit, gives rise to electromagnetic radiation and reduces to Maxwell electrodynamics. The same charge-phase structure also admits a mechanical representation. The corresponding phase dynamics naturally reproduce the Hamilton-Jacobi equation, establishing a direct connection between charge-phase evolution and dynamical motion. Combining the Hamilton-Jacobi structure with the continuity equation yields a quantum wave description in which the wavefunction emerges from the conserved density and charge phase. Granular corrections associated with the underlying phase structure generate the quantum potential and produce the Schrödinger equation as the effective quantum evolution law. Angular-momentum quantization follows from phase single-valuedness and periodicity, without the introduction of independent quantization postulates. Within this framework, Maxwell electrodynamics, Hamilton-Jacobi mechanics, Schrödinger dynamics, and quantum quantization appear as different dynamical limits of the same charge-phase field. Electromagnetism corresponds to the long-range linear propagation of phase bonds, whereas quantum mechanics reflects the oscillatory and granular dynamics of the underlying phase geometry.
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.
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.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
The set of Euler–Lagrange equations that extremelize the action associated with the Lagrangian space-time density of quantum electrodynamics leads to the well-known set of coupled Dirac–Maxwell equations. We compare three alternative Hamiltonian-based descriptions for quantum electrodynamics. We construct a local, a spatially global and a temporally global Hamiltonian and show that the corresponding Hamilton equations of motion are able to reproduce the Dirac–Maxwell equations. While this local Hamiltonian is fully equivalent to quantum electrodynamics, it does not provide any obvious conserved quantities. On the other hand, the two global Hamiltonians can be associated with the temporal and spatial generators of the dynamics and lead to spatially or temporally conserved observables if the fields fulfill certain boundary conditions.
Maxwell’s equations, formulated in 1865, preceded quantum mechanics by six decades. When the logical direction is reversed, however, the electromagnetic field structure can be interpreted as the form required for a quantum theory of charged matter once three standard ingredients are imposed: local gauge invariance, conserved electric current, and a dynamical electromagnetic field. Quantum mechanics supplies the charged-particle state, probability amplitudes, quantum currents, and—through the Dirac equation—a relativistically consistent description of electron charge and spin. Gauge symmetry then organizes the electromagnetic interaction, while current conservation constrains the admissible source structure. The homogeneous Maxwell identities follow naturally from expressing the field through a four-potential, whereas the sourced Maxwell equations require an additional electromagnetic field action or equivalent dynamical assumption. The resulting framework clarifies both the strength and the limit of a “Maxwell from quantum mechanics” program: quantum theory and gauge structure strongly constrain electromagnetism, but Maxwell field dynamics are not obtained from the electron wave equation alone.
This paper extends the preceding Schrödinger-based analysis to the relativistic Dirac framework and examines how gauge covariance, spin magnetism, and quantum-vacuum effects deepen the connection between quantum theory and Maxwell electrodynamics. Local phase invariance of the Dirac equation yields the electromagnetic field tensor, while the conserved Dirac four-current provides a covariant source for Maxwell’s equations. The Pauli limit recovers g=2g=2 and the Bohr magneton, and the Gordon decomposition separates convection and spin-magnetization currents. The Foldy–Wouthuysen expansion further connects relativistic quantum dynamics to spin–orbit and Darwin terms involving the electromagnetic fields. Hydrogen fine-structure and Darwin-shift benchmarks are evaluated quantitatively, while the Dirac velocity operator and Zitterbewegung scale are compared with the magnetic dipole length appearing in the SQC/MSC M1 formulation. The analysis is then extended to the quantum vacuum through the Schwinger critical fields and Euler–Heisenberg effective theory, including vacuum birefringence. The paper distinguishes structural consequences of relativistic quantum mechanics from electromagnetic field dynamics that must still be specified independently.