Unified Propagation of Massive Particles: Phase Dynamics, Gauge Structure, and Generalized Refraction

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 function\[Ψ( r,t)=R( r,t)eiS( r,t)/,\]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,\[∂ R^2/∂ t+∇·[R^2∇ S-q A/m]=0,\]\[∂ S/∂ t+(∇ S-q A)^2/2m+qφ+U+Q=0,\]where\[Q=-^2/2m∇^2R/R.\] The gauge-invariant kinetic propagation momentum is therefore\[Π=∇ S-q A,\]and the associated local propagation velocity is\[ v=Π/m.\] We call this coupled amplitude--phase representation the 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: 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 \(Q\) provides an additional modification of the propagation structure. The same equations can be written in a second-order form,\[md v/dt=q( E+ v× B)-∇ U-∇ Q,\]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.