Search Results (27)

We used Clifford algebra $C{l}_{2,0}$ to find the 2D orbit of a hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency $\omega$, which is turned on at time $t=0$ via a unit step switch. Using a coordinate system co-rotating with the electron’s unperturbed circular orbit at angular frequency ${\omega }_0$, we derived a complex differential equation that is similar to but different from that of the Lorentz oscillator equation for light–atom interaction. We solved the homogeneous and particular differential equation and showed that the position of the electron is a linear combination of five exponential Fourier terms or orbital wave functions with frequencies 0, ${\omega }_0$, $2{\omega }_0$, $(2{\omega }_0-\omega )$, and $\omega$, whose coefficients depend on the light-to-atom frequency ratio $\omega /{\omega }_0$ and light-to-atom force magnitude ratio $A/r_0$. We showed that the electron orbits are approximately Keplerian at light-to-atom frequency ratio $\alpha =\omega /{\omega }_0=\{0,1,2\}$, with the orbits becoming discontinuous and divergent as $\alpha \to 1^{\pm }$, but continuous and non-divergent at $\alpha =\{0,2\}$. These Keplerian orbits are approximated by the sum of the zeroth, first, and second harmonics of the electron’s unperturbed orbital wave function ${\widehat{\psi }}_0=e^{\widehat{\mathsf{ı}}{\omega }_{0}t}$, corresponding to the eccentric, deferent, and epicycle in the Copernican construction of planetary orbits. Full article

This work places the invariant $d{s}^2$ at the center of the gravitational interaction, interpreting it not as a purely geometric object but as the differential of proper time, endowed with direct physical meaning. Starting from the extension of Fermat’s principle to massive particles—namely, the requirement that freely falling bodies follow trajectories that extremize proper time, which for timelike motion corresponds to a local maximum—and invoking the universality of Galilean free fall, we derive the form of $d{s}^2$ in a static gravitational field. Lorentz invariance then provides the natural framework to extend this result to systems involving moving matter. The invariant derived through this procedure matches the weak-field limit of General Relativity formulated in the harmonic gauge. Within this linearized regime, we show that the structure of the theory already contains the seeds of its nonlinear completion: any dynamically consistent extension to strong gravitational fields necessarily involves the Ricci tensor. From this viewpoint, Einstein’s field equations appear not as a postulated geometric law but as the unique covariant closure required to ensure energy–momentum conservation and the self-consistency of the gravitational interaction. Full article

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically. Full article

The historical and conceptual foundations of General Relativity are revisited, putting the main focus on the physical meaning of the invariant $d{s}^2$, the Equivalence Principle, and the precise interpretation of spacetime geometry. It is argued that Albert Einstein initially sought a dynamical formulation in which $d{s}^2$ encoded the gravitational effects, without invoking curvature as a physical entity. The now more familiar geometrical interpretation—identifying gravitation with spacetime curvature—gradually emerged through his collaboration with Marcel Grossmann and the adoption of the Ricci tensor in 1915. Anyhow, in his 1920 Leiden lecture, Einstein explicitly reinterpreted spacetime geometry as the state of a physical medium—an “ether” endowed with metrical properties but devoid of mechanical substance—thereby actually rejecting geometry as an independent ontological reality. Building upon this mature view, gravitation is reconstructed from the Weak Equivalence Principle, understood as the exact compensation between inertial and gravitational forces acting on a body under a uniform gravitational field. From this fundamental principle, together with an extension of Fermat’s Principle to massive objects, the invariant $d{s}^2$ is obtained, first in the static case, where the gravitational potential modifies the flow of proper time. Then, by applying the Lorentz transformation to this static invariant, its general form is derived for the case of matter in motion. The resulting invariant reproduces the relativistic form of Newton’s second law in proper time and coincides with the weak-field limit of General Relativity in the harmonic gauge. This approach restores the operational meaning of Einstein’s theory: spacetime geometry represents dynamical relations between physical measurements, rather than the substance of spacetime itself. By deriving the gravitational modification of the invariant directly from the Weak Equivalence Principle, Fermat Principle and Lorentz invariance, this formulation clarifies the physical origin of the metric structure and resolves long-standing conceptual issues—such as the recurrent hole argument—while recovering all the empirical successes of General Relativity within a coherent and sound Machian framework. Full article

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. Note that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix. Full article

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression $e=h\nu$, de Broglie’s equation for momentum $p=h/\lambda$, and Einstein’s special relativity energy, where h is the Planck constant, $\nu$ and $\lambda$ are the frequency and wavelength, respectively, of an associated wave having a wave speed $w=\nu \lambda$. The author has extended these relations to a family that is characterised by a second fundamental constant $h^{\prime }$ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant $\kappa =h^{\prime }/h$ such that $\kappa =0$ corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation. Full article

Miller’s rule originated as an empirical relation between the nonlinear and linear optical coefficients of materials. It is now accepted as a useful tool for guiding experiments and computational materials discovery, but its theoretical foundation had long been limited to a derivation for the classical Lorentz model with a weak anharmonic perturbation. Recently, we developed a mathematical framework which enabled us to prove that Miller’s rule is equally valid for quantum anharmonic oscillators, despite different dynamics due to zero-point fluctuations and further quantum-mechanical effects. However, our previous derivation applied only to one-dimensional oscillators and to the special case of second- and third-harmonic generation in a monochromatic electric field. Here we extend the proof to three-dimensional quantum anharmonic oscillators and also treat all orders of the nonlinear response to an arbitrary multi-frequency field. This makes the results applicable to a much larger range of physical systems and nonlinear optical processes. The obtained generalized Miller formulae rigorously express all tensor elements of the frequency-dependent nonlinear susceptibilities in terms of the linear susceptibility and thus allow a computationally inexpensive quantitative prediction of arbitrary parametric frequency-mixing processes from a small initial dataset. Full article

We develop novel extensions in the theory of weighted Lorentz spaces. In particular, we generalize classical results by introducing variable-exponent Lorentz spaces, establish sharp constants and quantitative bounds for maximal operators, and extend the framework to encompass fractional maximal operators. Moreover, we analyze endpoint cases through the study of oscillation operators and reveal new connections with weighted Hardy spaces. These results provide a unifying approach that not only refines existing inequalities but also opens new avenues in harmonic analysis and partial differential equations. Full article

This paper elucidates the Lorentz group, a fundamental subgroup of the Poincaré group. The orbits and little groups associated with the Lorentz group are described in detail, along with their corresponding properties. The Poincaré group is presented. Another fundamental aspect of the Poincaré group is Wigner’s little groups obtained from this group. An in-depth discussion on the cases of both massive and massless relativistic particles within the context of little groups is given. Our examination extends to the properties of various special groups associated with the Poincaré group. Applications of these groups are elaborated by physical examples taken from high-energy physics and optics from both classical and quantum domains. Specifically, covariant harmonic oscillators including entangled states, proton form factors, and the parton picture as proposed by Feynman are discussed. In this context, laser cavities and shear states are also addressed. We lay out the underlying mathematics that connects these apparently disparate realms of physics. Full article

This study identified time-varying harmonic characteristics in a high-density plasma (HDP) chemical vapor deposition (CVD) chamber by depositing low-k oxide (SiOF). The characteristics of harmonics are caused by the nonlinear Lorentz force and the nonlinear nature of the sheath. In this study, a noninvasive directional coupler was used to collect harmonic power in the forward and reverse directions, which were low frequency (LF) and high bias radio frequency (RF). The intensity of the 2nd and 3rd harmonics responded to the LF power, pressure, and gas flow rate introduced for plasma generation. Meanwhile, the intensity of the 6th harmonic responded to the oxygen fraction in the transition step. The intensity of the 7th (forward) and 10th (in reverse) harmonic of the bias RF power depended on the underlying layers (silicon rich oxide (SRO) and undoped silicate glass (USG)) and the deposition of the SiOF layer. In particular, the 10th (reverse) harmonic of the bias RF power was identified using electrodynamics in a double capacitor model of the plasma sheath and the deposited dielectric material. The plasma-induced electronic charging effect on the deposited film resulted in the time-varying characteristic of the 10th harmonic (in reverse) of the bias RF power. The wafer-to-wafer consistency and stability of the time-varying characteristic were investigated. The findings of this study can be applied to in situ diagnosis of SiOF thin film deposition and optimization of the deposition process. Full article

Here we consider theoretically an exciton-like dipole formed by a magnetic monopole and a magnetic antimonopole. This type of quasiparticles may be formed in a magnetic counterpart of a one dimensional semiconductor crystal. We use the familiar Lorentz driven damped harmonic oscillator model to find the eigenmodes of magnetic monopole dipoles strongly coupled to light. The proposed model allows predicting optical signatures of magnetic monopole excitons in crystals. Full article

Despite the extensive research on nanocomposites, a fundamental theory on the interface region is still difficult to achieve. In the present paper, we chose epoxy resin and nano-SiO₂, nano-SiC, nano-ZnO to prepare three kinds of nanocomposites. The dispersion characteristics at the terahertz range and dielectric permittivity at 1 Hz of epoxy resin-based nanocomposites were investigated. The reduction of the permittivity of nanocomposites at a slight filler concentration was absent at the terahertz range. The measurement results at 1 Hz show that the interaction between nano-SiO₂, nano-SiC particles and epoxy resin was strong with the modification of the silane coupling agent. However, the modification of nano-ZnO particles was invalid. The Lorentz harmonic oscillator model was employed to fit the dispersion characteristics. The relevance between the damping constant and the dielectric permittivity at low frequency was established, indicating that the increase in the damping coefficient results from the restriction of the molecular chain motion by the interfacial region. The present results in this paper reveal a bright prospect of terahertz time-domain spectroscopy in establishing the theory of nanocomposite dielectric. Full article

We studied the effects of the Lorentz invariance violation on the rotation of neutron stars (NSs) in the minimal gravitational Standard-Model Extension framework, and calculated the quadrupole radiation generated by them. Aiming at testing Lorentz invariance with observations of continuous gravitational waves (GWs) from rotating NSs in the future, we compared the GW spectra of a rotating ellipsoidal NS under Lorentz-violating gravity with those of a Lorentz-invariant one. The former were found to possess frequency components higher than the second harmonic, which does not happen for the latter, indicating those higher frequency components to be potential signatures of Lorentz violation in continuous GW spectra of rotating NSs. Full article

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”. Full article

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^{*}$-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space. Full article