This paper introduces a field-theoretic framework for wave–particle duality in which matter is described by a complex scalar field ψ carrying a conserved U(1) chargeIn the particle phase, the field condenses into a compact non-topological soliton (a Q-ball) with effective radius L ≈ r₀. In the wave phase, the same energy delocalizes in space and the field equation approaches the linear Klein–Gordon regime
This paper introduces a field-theoretic framework for wave–particle duality in which matter is described by a complex scalar field ψ carrying a conserved U(1) chargeIn the particle phase, the field condenses into a compact non-topological soliton (a Q-ball) with effective radius L ≈ r₀. In the wave phase, the same energy delocalizes in space and the field equation approaches the linear Klein–Gordon regime
Abstract Two of the most deeply entrenched postulates of quantum mechanics—complementarity and wave function collapse—have been regarded as irreducible principles since their formulation in 1928 and 1932, respectively. This paper provides a first-principles resolution of both postulates through a single physical mechanism. A particle is a Q-ball—a stable localized non-topological soliton of a complex scalar field under a fixed U(1) conserved charge constraint, stabilized by the joint action of Derrick's theorem and Noether's theorem. A wave is the Yukawa tail of the Q-ball—the unique decaying solution of the massive scalar field radial equation under natural boundary conditions. Complementarity is replaced by the core-tail unified entity. Collapse is replaced by the physical process of conserved charge transfer: measurement is the gradient-flow-driven transfer of conserved charge from the Q-ball tail space to the detector local state, with the determinism of single detection events guaranteed by the exclusivity mechanism arising from the integer quantization of the conserved charge. Conserved charge transfer requires a finite time τ_c—a core prediction confirmed by recent data across five independent experimental domains: molecular charge transfer (CF₃I⁺, 1.46±0.41 fs), hydrated ion charge delocalization (20–hundreds of attoseconds), superconducting qubit quantum jumps (∼4 μs), entanglement formation (232 as), and solid-state interfacial charge transfer (300±200 as). Under the same postulate chain, the Q-ball is the unique stable field configuration capable of producing wave-particle duality. The two most fundamental puzzles of quantum mechanics since its inception—complementarity and collapse—are simultaneously resolved by a single physical process within this mechanism. ORCID: 0009-0007-1999-0293 Email: 1352533302@qq.com phone:+86 18278275520(china)
Why does matter possess wave-like properties? Is wave-particle duality an inherent dual nature of matter, or merely a projection of a deeper underlying physical structure under two distinct observational perspectives? This paper presents a rigorous first-principles derivation of matter wave dynamics within the axiomatic framework of Time Field Theory (TFT). The entire derivation relies solely on two core axioms: local intrinsic total velocity conservation and metric self-consistency (flux conservation), without introducing any extra postulates. Following the logical sequence of ontological foundation, phase derivation, wave generation, hierarchical classification, physical correspondence, extreme-case verification, and unified spectrum construction, this paper first establishes the core ontological premise that matter corresponds to localized condensed states of the time field. It then derives the geometric structure of the existence phase angle on the unit circle of the complex plane from the velocity conservation axiom, revealing the fundamental origin of intrinsic material phases. On this basis, the de Broglie relations are strictly derived, eliminating the dualistic contradiction between particle and wave characteristics. This work further distinguishes two distinct wave regimes: intrinsic waves originating from phase evolution within condensed states, and apparent waves arising from the global spatial translation of field distributions, which are respectively mapped onto electromagnetic waves and gravitational waves as two classes of physical reality. Using black hole horizons as an extreme physical testbed, we demonstrate that intrinsic wave phenomena vanish completely at time-freezing surfaces while apparent wave behaviors remain intact, providing a unified explanation for the observational fact that black holes emit no light yet produce gravitational waves. We also clarify the ontological origin of black hole darkness: light is not trapped by gravity, but the physical conditions required for photon generation cease to hold at time-freezing surfaces. Finally, we construct a continuous spectrum of wave behaviors ranging from the vacuum ground state to black hole horizons. This paper proves that wave-like behavior is not a binary ”present/absent” attribute of matter, but a continuous function of the condensation strength of the time field. Quantum mechanics and classical mechanics do not describe two separate physical worlds; instead, they represent two limiting regimes of a single universal time field under different condensation levels.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ = h/p[cite: 15, 225]. By treating ψ = 1/λ as an ontological spatial compaction field [cite: 15, 226], we eliminate arbitrary te xtbook global coordinate grids and unphysical Hamiltonian scaffolding[cite: 16, 225]. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp = hδψ) along a local, physics-determined 1D radial axis[cite: 19, 163], all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase coherent non-linear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structural identity (1 = 1) when evaluated against the Euler Complex Plane (ECP) baseline wave, flawlessly vindicating de Broglie’s original vision of a localized wave-particle soliton.
One of the foundational pillars of modern physics—wave-particle duality and the absoluteness of the speed of light (c)—lies at the core of the ontological tensions between quantum mechanics and relativity. This study rejects the assumption that the photon is an abstract, zero-dimensional mathematical object, proposing instead a two-component mechanical structure consisting of a micro-"core" at the center—exhibiting microscopic fluctuations (less than a billionth fraction) at low energy and evolving into a spatial macro-superposition (form and position uncertainty) at high energy—surrounded by a dynamic "energy envelope" carrying the core's frequency-dependent vibration kinetics. This model provides a mechanical foundation demonstrating how low-frequency optical events allow fields to pass through one another (wave/ghost behavior) due to minimal core kinetic smearing and sub-nanoscale fluctuations, while high-frequency gamma conditions drive core interactions and Breit-Wheeler matter condensation through increased vibration intensity and macro-superposition (effective cross-section growth). Furthermore, it is discussed that the measured speed of light is not an abstract spacetime curvature, but a local terminal speed shaped by the quantum background.
Abstract Wave-particle duality is the fundamental phenomenon of particles and fields in quantum mechanics. In the past, the trajectory-like (particle-like) behaviour and wave-like behaviour has been considered separately. In this article, a superimposed model is derived to characterise wave-particle duality of classical particles. The superimposed model reflects an invariant mathematical structure (analogous variables and differential equations) from classical mechanics, classical field theories and quantum mechanics. Its analytical solution carries trajectory-like property (phase-independent) and wave-like property (phase-dependent) of particles that is consistent with to Schrodinger’s picture. Subsequently, the presented model is applied to model duality of classical waves in electromagnetism, acoustics and elasticity. The analysis implies the existence of quantum effects of classical waves at macroscopic scale. It predicts quantum picture on energy and momentum exchange between classical particles and waves. As seen in the model, wave-particle duality reflects inherent and indispensable characteristics of classical objects.