The wave-particle duality is one of the most intriguing properties of quantum objects. The duality is rather pervasive in the quantum world in the sense that not only classical particles sometimes behave like waves, but also classical waves, e.g., light, may behave as point particles. In this article, motivated by the wave-particle duality, I develop a deterministic time-crystal Lorentz-covariant mechanical model for an electron with spin. In the proposed time-crystal model an electron is formed by two components: a particle-type component that transports the electric charge, and a wave component that moves at the speed of light and whirls around the massive component. Interestingly, the motion of the particle-component is completely ruled by the trajectory of the wave-component, somewhat analogous to the pilot-wave theory of de Broglie-Bohm. The dynamics of the time-crystal electron is controlled by a generalized least action principle. The model predicts that the electron stationary states have a constant spin angular momentum, predicts the spin vector precession in a magnetic field and gives a possible explanation for the physical origin of the anomalous magnetic moment. Remarkably, the developed model has nonlocal features that prevent the divergence of the self-field interactions. The classical theory of the electron is recovered as an effective theory valid on a coarse time scale. The reported results suggest that time-crystal models may be used to describe some features of the quantum world that are inaccessible to the standard classical theory.
The wave-particle duality is one of the most intriguing properties of quantum objects. The duality is rather pervasive in the quantum world in the sense that not only classical particles sometimes behave like waves, but also classical waves, e.g., light, may behave as point particles. In this article, motivated by the wave-particle duality, I develop a deterministic time-crystal Lorentz-covariant mechanical model for an electron with spin. In the proposed time-crystal model an electron is formed by two components: a particle-type component that transports the electric charge, and a wave component that moves at the speed of light and whirls around the massive component. Interestingly, the motion of the particle-component is completely ruled by the trajectory of the wave-component, somewhat analogous to the pilot-wave theory of de Broglie-Bohm. The dynamics of the time-crystal electron is controlled by a generalized least action principle. The model predicts that the electron stationary states have a constant spin angular momentum, predicts the spin vector precession in a magnetic field and gives a possible explanation for the physical origin of the anomalous magnetic moment. Remarkably, the developed model has nonlocal features that prevent the divergence of the self-field interactions. The classical theory of the electron is recovered as an effective theory valid on a coarse time scale. The reported results suggest that time-crystal models may be used to describe some features of the quantum world that are inaccessible to the standard classical theory.
Time crystals are exotic phases of matter characterized by a broken time-translational symmetry, such that the ground state of the system evolves in time in a periodic fashion. Even though the time-crystal concept was introduced relatively recently, a related proposal can be traced back to Louis de Broglie. In his thesis, de Broglie conjectured that elementary particles may have some sort of internal clock that rules their behavior in the microscopic world. Here, I revisit de Broglie's idea and demonstrate that a special extreme case of classical mechanics yields in a natural way time-crystal type dynamics. Remarkably, it is found that time-crystal particles are characterized by a spin 4-vector that has a purely kinematic origin and is determined by the binormal of the velocity trajectory in the Bloch sphere. The dynamics of time-crystal particles is ruled by a generalized least action principle, such that the particle dynamically probes the nearby space and moves on average towards the direction that minimizes the action. I apply the theory to the case of charged particles and find that it predicts spin precession and that it provides a simple and intuitive picture for the origin of the anomalous magnetic moment of the electron. The classical formalism is recovered as an "effective theory" valid on a coarse time scale.
A theory employing the vortex shape of the electron was presented to resolve the enigma of the wave-particle duality. Conventions such as “particle” and “wave” were used to describe the behavior of quantum objects such as electrons. A superfluid vacuum formed the base to describe the basic vortex structure and properties of the electron, whereas various formulations derived from hydrodynamic laws described the electron vortex circumference, radius, angular velocity and angular frequency, angular momentum (spin) and magnetic momentum. A vortex electron fully explained the associations between momentum and wave, and hydrodynamic laws were essential in deriving the energy and angular frequency of the electron. In general, an electron traveling in space possesses internal and external motions. To derive the angular frequency of its internal motion, the Compton wavelength was used to represent the length of one cycle of the internal motion that is equal to the circumference of the electron vortex. The angular frequency of the electron vortex was calculated to obtain the same value according to Planck’s theory. A traveling vortex electron has internal and external motions that create a three-dimensional helix trajectory. The magnitude of the instantaneous velocity of the electron is the resultant of its internal and external velocities, being equal to the internal velocity reduced by the Lorentz factor (whose essence is presented in a detailed formulation). The wavelength of the helix trajectory represents the distance traveled by a particle along its axis during one period of revolution around the axis, resulting in the same de Broglie wavelength that corresponds to the helix pitch of the helix. Mathematical formulations were presented to demonstrate the relation between the energy of the vortex and its angular frequency and de Broglie’s wavelength; furthermore, Compton’s and de Broglie’s wavelengths were also differentiated.
The spherical model of electron cannot explain Bohr magneton, de Broglie wave, etc. To explain these properties physically, we propose that spin, one core concept of quantum theory, is a reality, and introduce this alternative solely based on experimental findings, Planck’s, Einstein’s, and Dirac’s insights, and follows classical laws, that an electron is an electrical wave with a literal 21 spin (Dirac), i.e., spins two circles in one cycle, both angular momentum are ℏ. Its wavelength is determined by E=hνm0=λm0hc=m0c2 (Plank and Einstein), λ m 0 is the Compton wavelength. As E⃗ ⊥ c⃗ for a transverse wave, it has a net electricity due to the first spin, f(Eq)=21E0(1−cos(2ω0t)), half of its cycle, i.e., the minimal meaningful unit as F = E q , is related with its charge, q = E q ϵ 0 . Moreover, an electron has a ring shape due to the second spin, 2 π R 0 = λ m 0 , λ m 0 is the electron-impact cross section. As ℏ=λm0h2πλm0=m0cR0, it has a quantized classical angular momentum, with a speed of c . The Bohr magneton, μB=2m0qℏ, g -factor is 2. Critically, the full Dirac’s equation of an electron with a constant velocity is mc=ℏ⊥+p⊥+ℏ//+p//, and its electromagnetic properties due to charge are f(Eq)=f(Eq)cc2−v′2+f(Eq)cv′=Enet//+Eq⊥, v′//ℏ⊥ or v′//ℏ//, and v′//Enet. Assuming heavy charged particles have equivalent though shall not identical mechanisms, mathematically this model is equivalent to Special Relativity, and supports ALL its formulas, i.e., v < c , E = m c 2 , direction-insensitive m = γ m 0 and t=γt0; Moreover, the net electricity Enet// and Bohr magneton is orientation-quantized, direction-sensitive and follows Lorentz transformation, leading to Lorentz transformation of length contraction; while following Relativity, magnetism can be defined by f(B)=f(Eq)c2v′, which is not circularly symmetric and self-evidently does not follow Lorentz transformation. The temporospatial averaged magnetic intensity of a current is relativistic, Bˉ=∑f(Eq)c2Δv=2πϵ0c2I=2πμ0I, and μ 0 ϵ 0 c 2 = 1, supporting Maxwell’s equations. Further implications include chirality therefore magnetic spatial quantization, de Broglie wave, etc. The spherical model should be just an observation due to orientation quantization of Enet. We conclude this model of electron may provide one step further to the ultimate Quantum Theory in a nonrelativistic spacetime.
This paper presents a novel theory of wave mechanics derived from a re-examination of the foundations of relativity. By expanding the traditional linear transformation ansatz to include a vector-valued temporal component, we demonstrate that Einstein's postulates, coupled with the principle of single-valued time, are sufficient to predict wave-like, quantizable behavior in particle motion. This approach naturally leads to a Clifford algebra framework and reveals the necessity of a dynamic, higher-dimensional temporal space. We derive Lorentz transformations and demonstrate the origin of the spacetime metric's signature. Furthermore, we obtain relations analogous to De Broglie's and show compatibility with Planck's distribution law. Notably, our theory provides a unified framework for spin-0 and spin-1/2 particles and offers a complete, normalized solution for a particle at rest, resolving a long-standing challenge in quantum mechanics. The inherent coupling of the Einstein Field Equations with the derived wave equation suggests potential implications for cosmology, including alternative explanations for dark matter and MOND. This work establishes a foundation for a new, consistent, and intuitive understanding of the quantum world, with experimentally testable predictions that distinguish it from conventional quantum mechanics.
This paper presents a novel theory of wave mechanics derived from a re-examination of the foundations of relativity. By expanding the traditional linear transformation ansatz to include a vector-valued temporal component, we demonstrate that Einstein's postulates, coupled with the principle of single-valued time, are sufficient to predict wave-like, quantizable behavior in particle motion. This approach naturally leads to a Clifford algebra framework and reveals the necessity of a dynamic, higher-dimensional temporal space. We derive Lorentz transformations and demonstrate the origin of the spacetime metric's signature. Furthermore, we obtain relations analogous to De Broglie's and show compatibility with Planck's distribution law. Notably, our theory provides a unified framework for spin-0 and spin-1/2 particles and offers a complete, normalized solution for a particle at rest, resolving a long-standing challenge in quantum mechanics. The inherent coupling of the Einstein Field Equations with the derived wave equation suggests potential implications for cosmology, including alternative explanations for dark matter and MOND. This work establishes a foundation for a new, consistent, and intuitive understanding of the quantum world, with experimentally testable predictions that distinguish it from conventional quantum mechanics.
This paper presents a novel theory of wave mechanics derived from a re-examination of the foundations of relativity. By expanding the traditional linear transformation ansatz to include a vector-valued temporal component, we demonstrate that Einstein's postulates, coupled with the principle of single-valued time, are sufficient to predict wave-like, quantizable behavior in particle motion. This approach naturally leads to a Clifford algebra framework and reveals the necessity of a dynamic, higher-dimensional temporal space. We derive Lorentz transformations and demonstrate the origin of the spacetime metric's signature. Furthermore, we obtain relations analogous to De Broglie's and show compatibility with Planck's distribution law. Notably, our theory provides a unified framework for spin-0 and spin-1/2 particles and offers a complete, normalized solution for a particle at rest, resolving a long-standing challenge in quantum mechanics. The inherent coupling of the Einstein Field Equations with the derived wave equation suggests potential implications for cosmology, including alternative explanations for dark matter and MOND. This work establishes a foundation for a new, consistent, and intuitive understanding of the quantum world, with experimentally testable predictions that distinguish it from conventional quantum mechanics.