Maxwell Electrodynamics and Schrödinger Wave Mechanics from a Charge Phase Field

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.