In this article, we consider a classical charged particle in a one-dimensional classical linear oscillator potential interacting with classical electromagnetic zero-point radiation. We show that the oscillator has a stable ground state and unstable resonant excited states which are analogous to those of the quantum harmonic oscillator. We develop six themes. (1) Classical electromagnetic zero-point radiation is Lorentz invariant and requires that any mechanical system which comes to equilibrium in it is at least approximately Lorentz invariant. Very few potentials meet this approximately-Lorentz-invariant criterion. (2) Classical relativistic waves travel at finite speed $c,$ which requires that the interaction of a point charge $e$ located at $\mathbf{r}\left( t\right) $ be treated using the full electric field $\mathbf{E}\left[\mathbf{r}\left( t\right) ,t\right] $ , not simply in the dipole approximation as $\mathbf{E}\left[ 0,t\right] $ . (3) The mechanical linear oscillator has $SO\left( 2\right) $ symmetry associated with the time behavior of the angle variable $\phi\left(t\right) = \omega_{0}t+\phi_{0}$ , and leads to representations associated with integer indices for the action variable $J$ . (4) In the large-mass approximation, the ground state is stable in classical zero-point radiation, even when higher radiation multipoles are included in the analysis. (5) For the resonant excited states, the predominant dipole radiation emitted by the charged oscillator is balanced by the energy gained from higher frequencies of classical electromagnetic zero-point radiation corresponding to odd multiples of $\omega_{0}$ . (6) The Bohr frequency condition $\Delta\mathcal{E = \hbar \omega}_{0}$ appears naturally in a classical context. The presence of the Lorentz-invariant zero-point radiation seems completely hidden in providing the energy balance for the oscillating charged particle . The present analysis puts a classical electromagnetic understanding under the old quantum picture of electrons in classical orbits, but requires resonance of the mechanical system in classical electromagnetic zero-point radiation.
Ideally, the characterization of non-classical states requires measurements on timescales much shorter than the system’s decoherence time. However, in quantum cavity electrodynamics, photon loss rates are often comparable to measurement rates, which degrades the fidelity of the readout. In this work, we propose an active protocol to accelerate the phase estimation of squeezed vacuum states trapped in a lossy cavity. Instead of relying on passive leakage or linear coherent driving, we excite the system with a time-optimized degenerate parametric drive. This adapted two-photon pump actively amplifies the relevant quantum fluctuations, rapidly projecting the phase information of the squeezed state onto the integrated homodyne photocurrent of the output field before significant decoherence occurs. Because a squeezed vacuum carries no phase information in its first moments, this quadratic control is the minimal resource able to imprint the squeezing angle onto the detected signal. Using a numerical optimal control approach, we derive pumping envelopes that maximize the ability to distinguish between orthogonal squeezed states, achieving approximately a 6.02-fold increase in signal contrast compared to constant excitation schemes. We demonstrate that this active extraction protocol produces a strictly monotonic calibration curve, enabling rapid and unambiguous identification of the squeezed phase angle. Furthermore, robustness analysis confirms that the phase estimate remains reliable even in the presence of intense Gaussian detection noise, providing a practical framework for rapid state verification in open-system quantum technologies.
We propose a positive approximation scheme for source terms arising in local truncated $M$ -fractional source problems. Here, the term ‘truncated $M$ -fractional’ refers to a local weighted derivative: for differentiable functions, it reduces to $Kx^{1-\nu}\mathrm{d}/\mathrm{d}x$ . Thus, the benchmark considered is a scalar local initial-value problem rather than a nonlocal evolution equation with memory. The scheme combines Baskakov–Schurer–Stancu basis functions with a normalized local average induced by the truncated $M$ -fractional integral. This construction preserves positivity, yields explicitly computable moments, and incorporates a fractional-type local averaging parameter into the operator nodes. The resulting explicit source approximation can be inserted directly into local fractional-type initial-value problems without solving an additional nonlinear algebraic system. It is particularly useful when source, density, heat-input, or wave-amplitude profiles must remain nonnegative during numerical approximation. For the resulting operators, we compute the first and second algebraic moments and the second central moment, establish weighted Korovkin convergence on the half-line, and derive direct estimates using the ordinary modulus of continuity and Lipschitz classes. We also apply the operators to an inhomogeneous problem in which the truncated $M$ -fractional derivative acts as a local weighted derivative. The induced approximate solutions are shown to converge uniformly to the exact solution on compact intervals. A dimensionless thermal-source benchmark supports the theoretical findings: for the polynomial source considered, the grid-based maximum error decreases from $0.428002$ at $n = 5$ to $0.042014$ at $n = 80$ . Additional benchmark and sensitivity studies clarify the roles of $\alpha$ , $\rho$ , and $\vartheta$ . The results provide a transparent bridge between Baskakov-type positive approximation and local truncated $M$ -fractional source models, without portraying the scalar benchmark as a full heat-conduction or nonlocal memory model.
Phase transitions play a central role in statistical physics and are well understood in systems at thermal equilibrium. However, their nonequilibrium counterparts remain less explored because of the lack of a unified theoretical framework. In this study, we examine the potential of deep learning, specially convolutional neural networks (CNN), to detect nonequilibrium phase transitions (NEPT) in a 2D Ising model-like square lattice ferromagnetic system under effective parameter $h$ . Without this parameter ( $h = 0$ ), we generate equilibrium spin configurations using the standard Metropolis algorithm as usual. With $h\ne 0$ , we generate nonequilibrium spin configurations using an effective Glauber update rule which breaks detailed balance. While the former is used to train CNN, the later is not subsequently described by the usual Boltzmann-Gibbs distribution rather used to test the model. The study shows that the trained CNN can distinguish between ordered and disordered nonequilibrium steady states and accurately predict the critical temperature of the system undergoing NEPT. Finite size scaling analysis of the model’s output (perdition) reproduces critical exponents that agree with theoretical expectations, indicating that this method provides a strong, data driven approach for examining nonequilibrium critical phenomena.
We formulate matter scale field as a relativistic framework in which a homogeneous scalar field controls the evolution of the physical metric and material standards through one common scale. Cosmological redshift is interpreted as a comparison between propagated light and material standards whose scale has evolved between emission and detection. The action is written on a single physical metric $g_{\mu\nu}$ : gravity, electromagnetism, the Standard Model, clocks and rulers, emission and detection are all defined on this metric. Consequently, the local Standard-Model description, local Lorentz invariance, universal matter coupling and the operational meaning of dimensionless constants are preserved. For a homogeneous solution, we write $g_{\mu\nu}(t) = b(\phi(t))^2g^0_{\mu\nu}$ , where $b(\phi) \gt 0$ is the scalar-controlled metric scale and $g^0_{\mu\nu}\equiv g_{\mu\nu}(t_0)$ is the physical metric at the present epoch, with $b(\phi(t_0)) = 1$ . When the same action is expressed relative to the present value, source-free Maxwell propagation contains no explicit factor $b(\phi)$ , whereas the coefficients setting the Higgs vacuum scale, generated particle masses and atomic transition frequencies acquire the corresponding powers of $b(\phi)$ . Particle masses and atomic frequencies therefore scale with $b(\phi)$ , while material lengths scale as $b(\phi)^{-1}$ . A photon emitted at $t_\textrm{em}$ has a present-calibrated frequency proportional to $b_\textrm{em}$ , which is conserved during source-free propagation. At observation, it is compared with the present material standard proportional to $b_0$ , giving $1+z = b_0/b_\textrm{em}$ . Light-curve time dilation and the Friedmann–Lemaitre–Robertson–Walker distance relations follow from the physical metric. We derive the homogeneous kinematics, physical-metric action and Standard-Model scale organization behind this mechanism. For a canonical scalar, a log-linear field-to-scale map yields a model-specific expansion and distance–redshift relation directly testable against $\Lambda$ CDM and other background cosmologies. Cosmological perturbations, structure growth, gravitational lensing and CMB anisotropies are decisive tests of the framework.
