Search Results (147)

We derive the geodesic equation for point particles propagating in Moyal-type noncommutative spacetimes using a field-theoretic approach based on the quasi-classical limit of the noncommutative Klein–Gordon equation. Starting from a twisted-geometric construction of the covariant Laplace–Beltrami operator, we obtain the noncommutative Hamilton–Jacobi equation and show that all noncommutative effects are absorbed into an effective, position-dependent mass function $M\left(x\right)$ appearing in an otherwise standard relativistic dispersion relation. The corresponding particle dynamics then acquires an additional term in the geodesic equation that takes the form of a fixed external force $F_{\mathrm{NC}}^{\mu }=-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\nu }{M}^2\left(x\right)$, sourced entirely by the quantum nature of spacetime. We compute this effective mass perturbatively up to fourth order in the noncommutativity parameter for a general metric, proving that all odd-order corrections vanish identically. For the specific case of an $(r-\theta )$ twist applied to spherically symmetric backgrounds, we obtain explicit expressions demonstrating that the leading correction to geodesic motion appears at order ${\Theta }^2$ and is proportional to the probe particle’s mass, while massless particles remain unaffected. Full article

In this article, we present a methodological and systematic approach to solving the Klein–Gordon–Fock equation using the separation of variables method, with particular emphasis on its formulation in light-front coordinates. Although the plane-wave solution is well known in relativistic quantum mechanics, the explicit procedure leading to this solution is not always developed in detail, especially when the equation is written in light-front variables. We first revisit the Klein–Gordon–Fock equation for a free particle in Minkowski spacetime, showing how the usual separation between temporal and spatial variables leads to the expected plane-wave form. This treatment is used as a reference for the corresponding analysis in light-front coordinates. We then rewrite the equation in light-front coordinates, adopting ${\alpha }_{LF}=\sqrt{2}$, and apply the separation of variables method to the coordinates $x^{+}$, $x^{-}$, and $x^{\perp }$. In this formulation, $x^{+}$ and $x^{-}$ appear coupled through the mixed derivative term ${\partial }_{+}{\partial }_{-}$, with the separation process requiring an additional decoupling step involving an inverse relation and a nonzero constant $\lambda$. We show that an appropriate choice of this constant, together with a suitable choice of the superposition coefficients, allows the separated solution to recover the plane-wave structure obtained from the covariant transformation of the scalar product $p_{\mu }{x}^{\mu }$. Thus, the results clarify the consistency between the direct coordinate-transformation approach and the explicit solution of the differential equation in light-front coordinates, while also highlighting the usefulness of separation of variables as a methodological tool in the study of relativistic wave equations. Full article

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (${\beta }_{\mathrm{NM}}=1/4$) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders $\alpha \ne \beta$, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory. Full article

The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group $SU\left(2\right)\otimes U\left(1\right)$ using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates. Full article

While superdeterministic and fractal spacetime models offer compelling alternative perspectives on quantum foundations, the simulation and validation of effective wave dynamics in such non-differentiable, deterministic settings remain computationally and theoretically challenging. To address this, a framework built around the Fractional Nonlinear Klein–Gordon Equation (FNKGE), defined through the spectral fractional Laplacian, was developed. This equation was solved and benchmarked through a comparative study between Physics-Informed Neural Networks (PINNs) with Fourier features and Physics-Informed Graph Neural Networks (PI-GNNs). Additionally, detection patterns were simulated via deterministic agents, and theoretical links between fractal geometry, computational irreducibility, and deviations from statistical independence were formalized. Regarding the computational evaluation, superior accuracy was achieved by the PI-GNNs, yielding a mean relative error of $0.5\%$ ($\overline{ϵ}=0.005$), alongside faster convergence and a more well-conditioned Hessian spectrum compared to PINNs. Crucially, a continuous power-law decay ($S\left(k_y\right)\sim k_y^{-1.8}$) was revealed by the spectral analysis of the simulated detection patterns, confirming the emergence of classical optical caustics rather than discrete quantum-interference peaks. Furthermore, a modified dispersion relation that accurately predicts linear instability regimes was derived, and specific boundary artifacts in non-periodic domains were identified. Taken together, the FNKGE is validated by these results as a viable effective model for fractal wave phenomenology and as a robust benchmark for physics-informed learning architectures. Full article

We investigate the effects of Lorentz symmetry violation (LSV) on a scalar particle via a non-minimal coupling in the Klein–Gordon equation within the charge–parity–time CPT-odd gauge sector. Through an analytical approach, we derive bound-state solutions for two distinct anisotropic backgrounds: time-like and space-like. In the time-like case, the LSV induces an effective centrifugal potential, modifying the angular momentum spectrum. When a hard-wall confining potential is included, discrete energy levels emerge, explicitly dependent on the LSV parameters. In the space-like scenario, the particle becomes confined by a Coulomb-type potential induced by the LSV, leading to a quantized energy spectrum that reduces to the free-particle limit when the LSV parameters vanish. Our results illustrate how spacetime anisotropies, encoded in a background vector field, can significantly alter the quantum dynamics of scalar particles in the presence of a magnetic field. Full article

This paper deals with several equations of mathematical physics written in explicit form with their solutions. In Theorem 1, an oblique derivative problem for the string equation is studied. More precisely, the initial-boundary value problem for the string equation is investigated. The corresponding vector field on the boundary is non-vanishing and does not have a characteristic direction, but can be tangential to some part of the boundary, and it is allowed to change sign. A classical solution exists with suitable compatibility conditions at the corner points. The picture changes significantly in the case of the wave equation with several (say two: 2D) space variables in a circular cylinder. The initial-boundary value problem turns out to be underdetermined with an infinite-dimensional kernel if the boundary vector field is orthogonal to the time axis. By prescribing extra conditions on the generatrices of the cylinder where the vector field is tangential to the cylinder, we obtain a unique classical solution. In Theorem 2, we consider the Cauchy problem in the interior of the parabola of the Lorentzian-type eikonal equation and find its unique classical solution in $\{0\le x_2\le 1/2\}\cap \{x_2\ge {\displaystyle \frac{x_1^2}{2}}\}$. Propagation of singularities for the D and 3 D hyperbolic (Klein–Gordon) equations in ${\mathbf{R}}^4$, ${\mathbf{R}}^8$ is studied in Theorem 3. In the double characteristic points, the wave front propagates either along the surface of the characteristic cone, or in the solid cone starting from $(t_0,x_0)$. Full article

In cosmological scenarios where the Peccei–Quinn symmetry is broken after inflation, small-scale axion field inhomogeneities can undergo gravitational collapse, leading to the formation of bound structures. The dynamics of these systems are commonly described using cosmological perturbation theory applied to the Einstein–Klein–Gordon equations. In the non-relativistic regime, this description reduces to the Gross–Pitaevskii–Poisson or Schrödinger–Poisson equations, depending on whether axion self-interactions are included. In this work, we extend the axion’s relativistic action by introducing a non-minimal scalar-curvature coupling of the form $\xi R{\varphi }^4$, which effectively induces a gravitationally mediated pairwise interaction. By performing a perturbative expansion and subsequently taking the non-relativistic limit, we derive a modified set of evolution equations governing the early stages of axion structure formation. Full article

This paper investigates the conservation of mass and energy in the space-fractional Klein-Gordon-Schrödinger system with fractional Laplacian operators. Firstly, the invariant energy quadratization method is applied to transform the original system into an equivalent form. For spatial discretization, Fourier spectral methods are employed, yielding a semi-discrete scheme. Subsequently, an invariant energy quadratization Runge-Kutta approach is used for temporal discretization, resulting in a fully discrete scheme. Owing to its diagonally implicit structure, the proposed scheme is both highly accurate and efficient while preserving mass and energy exactly. Numerical experiments are conducted to verify the accuracy and conservation properties of the method. Full article

In this work, we investigate massive charged scalar perturbations in the background of three-dimensional dilaton black holes with a cosmological constant. We demonstrate that the wave equations governing the dynamics of these perturbations are exactly solvable, with the radial part expressible in terms of confluent Heun functions. The quasibound state frequencies are computed analytically, and we examine their dependence on the scalar field’s mass and charge, as well as on the black hole’s mass and electric charge. Our analysis also underscores the crucial role played by the cosmological constant in shaping the behavior of these perturbations. This specific black hole metric arises as a solution to the low-energy effective action of string theory in $2+1$ dimensions, and it holds potential for experimental realization in analog gravity systems due to the similarity between its surface gravity and that of acoustic analogs. Moreover, the analytic tractability of this system offers a valuable testing ground for exploring aspects of black hole spectroscopy, stability, and quantum field theory in curved spacetime. The exact solvability facilitates deeper insights into the interplay between geometry and matter fields in lower-dimensional gravity, where quantum gravitational effects can be more pronounced. Such studies not only enrich our understanding of dilaton gravity and its string-theoretic implications but also pave the way for potential applications in simulating black hole phenomena in laboratory settings using analog models. Full article

We develop a fully gauge-invariant analysis of gravitational-wave polarizations in metric $f\left(R\right)$ gravity with a particular focus on the modified Starobinsky model $f\left(R\right)=R+\alpha R^2-2\mathsf{\Lambda }$, whose constant-curvature solution $R_d=4\mathsf{\Lambda }$ provides a natural de Sitter background for both early- and late-time cosmology. Linearizing the field equations around this background, we derive the Klein–Gordon equation for the curvature perturbation $\delta R$ and show that the scalar propagating mode acquires a mass $m_{\psi }^2=1/\left(6\alpha \right)$, highlighting how the same scalar degree of freedom governs inflationary dynamics at high curvature and the propagation of gravitational waves in the current accelerating Universe. Using the scalar–vector–tensor decomposition and a decomposition of the perturbed Ricci tensor, we obtain a set of fully gauge-invariant propagation equations that isolate the contributions of the scalar, vector, and tensor modes in the presence of matter. We find that the tensor sector retains the two transverse–traceless polarizations of General Relativity, while the scalar sector contains an additional massive scalar propagating degree of freedom, which manifests through breathing and longitudinal tidal responses depending on the wave regime and detector frame. Through the geodesic deviation equation—computed both in a local Minkowski patch and in fully covariant de Sitter form—we independently recover the same polarization content and identify its tidal signatures. The resulting framework connects the extra scalar polarization to cosmological observables: the massive scalar propagating mode sets the range of the fifth force, influences the time evolution of gravitational potentials, and affects the propagation and dispersion of gravitational waves on cosmological scales. This provides a unified, gauge-invariant link between gravitational-wave phenomenology and the cosmological implications of metric $f\left(R\right)$ gravity. Full article

This study presents an efficient numerical scheme for solving the generalized nonlinear time-fractional Klein–Gordon equation. The Caputo time-fractional derivative is discretized using a conventional finite-difference approach, while the spatial domain is approximated with uniform hyperbolic polynomial B-splines. These discretizations are coupled through the $\theta$-weighted scheme. The uniform hyperbolic polynomial B-spline framework extends classical spline theory by incorporating hyperbolic functions, thereby enhancing flexibility and smoothness in curve and surface representations—features particularly useful for problems exhibiting hyperbolic characteristics. A rigorous stability and convergence analysis of the proposed method is provided. The effectiveness of the scheme is further validated through numerical experiments on benchmark problems. The results demonstrate up to two orders of magnitude improvement in $L_{\infty }$ error norms compared to prior spline methods. This substantial accuracy enhancement highlights the robustness and efficiency of the proposed approach for fractional partial differential equations. Full article

The canonical quantization of a field theory for spin-1/2 massive bosons that satisfy the Klein–Gordon equation is presented. The breakdown of the usual spin–statistics connection is due to the redefinition of the dual field, rendering the theory pseudo-Hermitian. The normal-ordered Hamiltonian is bounded from below with real eigenvalues, and the theory is consistent with microcausality and invariant under parity, charge conjugation and time reversal. Full article

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using Maple. Subsequently, classical Lie symmetry analysis is employed to derive the infinitesimal generators of the equation. A Lagrangian formulation is constructed for these generators, from which similarity variables are systematically obtained. This framework enables a complete similarity reduction, transforming the complex nonlinear partial differential equation into a more tractable ordinary differential equation. To solve this reduced ordinary differential equation and to obtain a spectrum of soliton solutions, we implement the new generalized exponential differential rational function method. This advanced technique utilizes a rational trial function based on the $ith$ derivatives of exponentials, generating a diverse spectrum of closed-form soliton solutions through strategic choices of arbitrary constants. The novelty of this approach provides a unified framework for handling higher-order nonlinearities, yielding solutions such as multi-peakons and lump solitons, which are vividly characterized using Mathematica-generated 3D, 2D, and contour plots. These findings provide significant insights into nonlinear wave dynamics with potential applications in quantum field theory, nonlinear optics, plasma physics, etc. 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