We show that the de Broglie (Prince) equation λ p = h naturally produces two conjugate momenta—a group momentum p_group = mv_n and a phase momentum p_phase = mc^2/v—whose product equals the square of the photon momentum:p_phase ×p_group = p^2(ϕ).This reformulation eliminates the superluminal phase velocity problem that has plagued wave mechanics since 1924. We first derive the constancy of Planck’s constant from the classical work equation and the Leibniz product rule, requiring no quantum postulates. The photon boundary at v = c fixes H = h and eliminates the integration constant, rendering the assumption v = 0 unnecessary. Velocity quantization v_n = c (n−1)/(n+1) emerges from first principles, producing discrete momentum levels where pn = mvn recovers classical mechanics at each level. The imaginary unit 'i' emerges algebraically when the Prince equation forbids rest. By expressing the wave’s phase structure as a momentum rather than a velocity, the physical ontological reality of matter waves becomes manifest without any quantity exceeding c.
We show that the de Broglie (Prince) equation λ p = h naturally produces two conjugate momenta—a group momentum p_group = mv_n and a phase momentum p_phase = mc^2/v—whose product equals the square of the photon momentum:p_phase ×p_group = p^2(ϕ).This reformulation eliminates the superluminal phase velocity problem that has plagued wave mechanics since 1924. We first derive the constancy of Planck’s constant from the classical work equation and the Leibniz product rule, requiring no quantum postulates. The photon boundary at v = c fixes H = h and eliminates the integration constant, rendering the assumption v = 0 unnecessary. Velocity quantization v_n = c (n−1)/(n+1) emerges from first principles, producing discrete momentum levels where pn = mvn recovers classical mechanics at each level. The imaginary unit 'i' emerges algebraically when the Prince equation forbids rest. By expressing the wave’s phase structure as a momentum rather than a velocity, the physical ontological reality of matter waves becomes manifest without any quantity exceeding c.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
We present a complete, self-contained wave mechanics derived entirely from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as an ontological spatial compaction field, and by anchoring the waveon the Euler Complex Plane (ECP)—a two-faced, rotating, compact topological surface—we derive a nonlinear wave equation with a coefficient 8π^2, a soliton identity 1 = 1, a logarithmic spiral geodesic, and a phase-velocity relation v(ϕ) v = c^2.The framework yields quantized energy levels, spin, the selection rule Δn = ±1, and the principle of least action as a geometric consequence of the ECP’s topology. Planck’s constant h is shown to be a scale-invariant geometric constant, and thewave is proven to be ontically real, self-localizing, and non-dispersive. We also map the geometry to clinical neurology, showing that the brain is a natural standing Prince field.
We derive a complete wave equation framework from de Broglie’s fundamental relation λ = h/p, expressed in differential form as δp = hδψ where ψ = 1/λ represents spatial compaction. Without invoking Hamiltonian formalism or particleconcepts, we obtain a general nonlinear, anisotropic wave equation that reduces to both Maxwell’s electromagnetic equations and d’Alembert’s universal wave equation in appropriate limits. We further develop the multiplicative superpositionprinciple for combining multiple Prince waves, showing that ψ_n+1 = (Product of ψ_i)^1/n is the unique formulation maintaining dimensional consistency and physical unity. Remarkably, Planck’s constant h cancels from all final wave equations, revealing the scale-independent nature of wave propagation. All results emerge from the single principle δp = hδψ.
We derive a complete wave equation framework from de Broglie’s fundamenta relation λ = h/p, expressed in differential form as δp = hδψ where ψ = 1/λ represents spatial compaction. Without invoking Hamiltonian formalism or particle concepts, we obtain a general nonlinear, anisotropic wave equation that reduces to both Maxwell’s electromagnetic equations and d’Alembert’s universal wave equation in appropriate limits. We further develop the multiplicative superposition principle for combining multiple Prince waves, showing that ψ_n+1 = (PIψ_i)1/n is the unique formulation maintaining dimensional consistency and physical unity. Remarkably, Planck’s constant h cancels from all final wave equations, revealing the scale-independent nature of wave propagation. All results emerge from the single principle δp = hδψ.
We present the complete, ontically grounded wave mechanics derived from the de Broglie equation λp = h, interpreted literally as a real, physical wave. By treating ψ = 1/λ as a spatial compaction field and anchoring it on the Euler ComplexPlane (ECP)—a two-faced, rotating, compact topological surface—we derive the nonlinear wave equation, the soliton identity 1 = 1, the logarithmic spiral geodesic, the dual velocities and frequencies, the eigen-energy spectrum, and the photon asthe uncompacted reference state. We show that the Maxwell and d’Alembert equations are linear, uncompacted limits of the Prince nonlinear wave equation. The framework resolves the wave-particle duality, quantizes time and energy, and vindicatesde Broglie’s law of phase harmony as the foundational principle of quantum mechanics.
From a combination of Maxwell’s electromagnetism with Planck’s law and the de Broglie hypothesis, we arrive at quantized photonic wave groups whose constant phase velocity is equal to the speed of light c = ω/k and to their group velocity dω/dk. When we include special relativity expressed in simplest units, we find that, for particulate matter, the square of rest mass , i.e., angular frequency squared minus wave vector squared. This equation separates into a conservative part and a uniform responsive part. A wave function is derived in manifold rank 4, and from it are derived uncertainties and internal motion. The function solves four anomalies in quantum physics: the point particle with prescribed uncertainties; spooky action at a distance; time dependence that is consistent with the uncertainties; and resonant reduction of the wave packet by localization during measurement. A comparison between contradictory mathematical and physical theories leads to similar empirical conclusions because probability amplitudes express hidden variables. The comparison supplies orthodox postulates that are compared to physical principles that formalize the difference. The method is verified by dual harmonics found in quantized quasi-Bloch waves, where the quantum is physical; not axiomatic.