Search Results (76)

The appearance of a negative mass term in the classical, non-relativistic Klein–Gordon equation deduced from mechanical interactions describes a repulsive interaction. In the case of a traveling wave, this results in an increase in amplitude and a decrease in the wave propagation velocity. Since this leads to dissipation, it is a symmetry-breaking phenomenon. After the repulsive interaction is eliminated, the system evolves towards the original state. Given that the interactions within the system are conservative, it would be assumed that even the original state is restored. The analysis to be presented shows that a wave with a lower angular frequency than the original one is transformed back to a slightly larger amplitude. This description is a suitable model of the axion effect, during which an electromagnetic wave interacts with a repulsive field and becomes of a continuously lower frequency. Full article

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

We establish a rigorous kinetic-theoretical framework to analyze causal propagation in thermal transport phenomena within relativistic ideal fluids, building a more rigorous framework based on the kinetic theory of gases. Specifically, we provide a refined derivation of the wave equation governing thermal and density fluctuations, clarifying its hyperbolic nature and the associated characteristic propagation speeds. The analysis confirms that thermal fluctuations in a simple non-degenerate relativistic fluid satisfy a causal wave equation in the Euler regime, and it recovers the classical expression for the speed of sound in the non-relativistic limit. This work offers enhanced mathematical and physical insights, reinforcing the validity of the hyperbolic description and suggesting a foundation for future studies in dissipative relativistic hydrodynamics. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article

We investigate the non-relativistic scattering of a plane wave by a vertical segment formulating the problem in terms of the Lippmann–Schwinger equation in two spatial dimensions. Adjusting the coupling strength function we show how to implement the scattering by a system of multiple slits and by a Cantor set. We present detailed calculations of the scattered wave function for the line segment, as well as for the single, double, and multiple slits. We define reflection and transmission functions that are position-dependent in a defined region. From these results, we obtain the probability densities and differential and total cross-sections for these problems. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

We address the scientific “time” concept in the context of more general relaxation processes toward the Wärmetod of thermodynamic equilibrium. More specifically, we sketch a construction of a conceptual ladder of chemical reaction steps that can rigorously bridge a description from the microscopic domain of molecular quantum chemistry to the macroscopic materials domain of Gibbsian thermodynamics. This conceptual reformulation follows the pioneering work of Kenichi Fukui (Nobel 1981) in rigorously formulating the intrinsic reaction coordinate (IRC) pathway for controlled description of non-equilibrium passages between reactant and product equilibrium states of an overall material transformation. Elementary chemical reaction steps are thereby identified as the logical building-blocks of an integrated mathematical framework that seamlessly spans the gulf between classical (pre-1925) and quantal (post-1925) scientific conceptions and encompasses both static and dynamic aspects of material change. All modern chemical reaction rate studies build on the apparent infallibility of quantum-chemical solutions of Schrödinger’s wave equation and its Dirac-type relativistic corrections. This infallibility may now be properly accepted as an added“inductive law” of Gibbsian chemical thermodynamics, the only component of 19th-century physics that passed intact through the revolutionary quantum upheavals of 1925. Full article

The origin and acceleration of high-energy particles, constituting cosmic rays, is likely to remain an important topic in modern astrophysics. Among the two categories galactic and solar cosmic rays, the latter are much less investigated. The primary source of solar cosmic ray particles are impulsive explosions of the magnetized plasma, known as solar flares and coronal mass ejections. These particles, however, are characterized by relatively low energies compared to their galactic counterparts. In this work, we explore the resonance wave–wave (RWW) interaction between the polarized electromagnetic radiation emitted by the solar active regions and the quantum waves associated with high-energy, relativistic electrons generated during solar flares. Mathematically, the RWW interaction problem boils down to analyzing a Klein–Gordon Equation (spinless electrons) embedded in the electromagnetic field. We find that RWW could accelerate the relativistic electrons to enormous energies even comparable to energies in the galactic cosmic rays. Full article

Kaon condensation in hyperon-mixed matter [(Y+K) phase], which may be realized in neutron stars, is discussed on the basis of chiral symmetry. With the use of the effective chiral Lagrangian for kaon–baryon and kaon–kaon interactions; coupled with the relativistic mean field theory and universal three-baryon repulsive interaction, we clarify the effects of the s-wave kaon–baryon scalar interaction simulated by the kaon–baryon sigma terms and vector interaction (Tomozawa–Weinberg term) on kaon properties in hyperon-mixed matter, the onset density of kaon condensation, and the equation of state with the (Y+K) phase. In particular, the quark condensates in the (Y+K) phase are obtained, and their relevance to chiral symmetry restoration is discussed. Full article

We start with the 50-component relativistic matrix equation for a hypothetical spin-2 particle in the presence of external electromagnetic fields. This equation is hypothesized to describe a particle with an anomalous magnetic moment. The complete wave function consists of a two-rank symmetric tensor and a three-rank tensor that is symmetric in two indices. We apply the general method for performing the nonrelativistic approximation, which is based on the structure of the $50\times 50$ matrix ${\Gamma }^0$ of the main equation. Using the 7th-order minimal equation for the matrix ${\Gamma }^0$, we introduce three projective operators. These operators permit us to decompose the complete wave function into the sum of three parts: one large part and two smaller parts in the nonrelativistic approximation. We have found five independent large variables and 45 small ones. To simplify the task, by eliminating the variables related to the 3-rank tensor, we have derived a relativistic system of second-order equations for the 10 components related to the symmetric tensor. We then take into account the decomposition of these 10 variables into linear combinations of large and small ones. In accordance with the general method, we separate the rest energy in the wave function and specify the orders of smallness for different terms in the arising equations. Further, after performing the necessary calculations, we derive a system of five linked equations for the five large variables. This system is presented in matrix form, which has a nonrelativistic structure, where the term representing additional interaction with the external magnetic field through three spin projections is included. The multiplier before this interaction contains the basic magnetic moment and an additional term due to the anomalous magnetic moment. The latter characteristic is treated as a free parameter within the hypothesis. Full article

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory. Full article

A simple method of non-relativistic variational calculations of the electronic structure of a two-electron atom/ion, primarily near the nucleus, is proposed. The method as a whole consists of a standard solution of a generalized matrix eigenvalue equation, all matrix elements of which are reduced to a numerical calculation of one-dimensional integrals. The distinctive features of the method are as follows: The use of the hyperspherical coordinate system. The inclusion of logarithms of the hyperspherical radius R in the basis functions, similar to the Fock expansion. Using a special basis function including the leading angular Fock coefficients to provide the correct behavior of the wave function near the nucleus. The main numerical parameters characterizing the properties of the helium atom and a number of helium-like ions near the nucleus are calculated and presented in tables. Among others, the specific coefficients, $a_{21}$, of the Fock expansion, which can only be calculated using a wave function with the correct behavior near the nucleus, are presented in table and graphs. Full article

Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the $1s$ ground state and $2s$ excited state of the one-electron systems ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}^{+}$, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic $2p_{3/2}$ orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure. Full article

Recent discoveries of massive galaxies existing in the early universe, as well as apparent anomalies in Ω_m and H₀ at high redshift, have raised sharp new concerns for the ΛCDM model of cosmology. Here, we address these problems by using new solutions for the Einstein field equations of relativistic compact objects originally found by Ni. Applied to the universe, the new solutions imply that the universe’s mass is relatively concentrated in a thick outer shell. The interior space would not have a flat, Minkowski metric, but rather a repulsive gravitational field centered on the origin. This field would induce a gravitational redshift in light waves moving inward from the cosmic shell and a corresponding blueshift in waves approaching the shell. Assuming the Milky Way lies near the origin, within the KBC Void, this redshift would make H₀ appear to diminish at high redshifts and could thus relieve the Hubble tension. The Ni redshift could also reduce or eliminate the requirement for dark energy in the ΛCDM model. The relative dimness of distant objects would instead arise because the Ni redshift makes them appear closer to us than they really are. To account for the CMB temperature–redshift relation and for the absence of a systematic blueshift in stars closer to the origin than the Milky Way, it is proposed that the Ni redshift and blueshift involve exchanges of photon energy with a photonic spacetime. These exchanges in turn form the basis for a cosmic CMB cycle, which gives rise to gravity and an Einsteinian cosmological constant, Λ. Black holes are suggested to have analogous Ni structures and gravity/Λ cycles. Full article

The relativistic positive-energy wave equation proposed by P. Dirac in 1971 is an old but largely forgotten subject. The purpose of this note is to speculate that particles described by this equation (called here Dirac particles) are natural candidates for the dark matter. The reasoning is based on a fact that the internal structure of such particles simply prohibits their interaction with electromagnetic fields (at least with the minimal coupling) which is exactly what is required for dark matter. Dirac particles have quite unusual properties. In particular, they are transformed by an infinite-dimensional representation of the homogeneous Lorentz group, which clearly distinguishes them from all known elementary particles described by finite-dimensional representations and hints to a physics beyond the Standard Model. To clarify the topic, a brief review of the main features of the above-mentioned Dirac equation is given. Full article