Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2301.12898

The Role of Fractional Dimension in Study Physics: A Two-Channel Representation with Geometric Memory

Ali Dorostkar
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this study, we explore the field of physics through the lens of fractional dimensionality. We propose that space is not confined to integer dimensions alone, but can also be understood as a superposition of spaces that exist between these integer dimensions. The concept of fractional dimensional space arises from the idea that the space between integer dimensions is filled, which occurs through the application of a fractional derivative operator (the local part) that rotates the integer dimension to encompass all spaces between two integers. It examines how fractional dimensional frameworks can enhance our understanding of classical mechanics, particularly regarding the duality of memory versus no-memory behavior, or local versus non-local dynamics. In the lens of fractional dimension, motion in classical physics can be analyzed through two distinct solutions. When the fractional dimensional trajectory takes values of 1 or 2 corresponding to the first and second integer derivatives, it represents linear and accelerated systems, respectively, yielding trivial solutions via differentiation. However, if the fractional dimensional trajectory evolves as a linear function of time in fractional dimensional space or follows a nonlinear path, surprisingly it results in a non-trivial solution for linear and accelerated systems, respectively. This approach offers a broader framework for describing motion, extending to memory (non-local) effect beyond traditional local integer-order differentiation . Moreover, we propose that the coupling of space and time, commonly referred to as space-time, is better understood as space-dimension-time within this framework, where the dimension serves as an interconnecting platform.

Explore related subjects

physics.gen-ph

Keep this discovery

BibTeXRIS

Ali Dorostkar. 2023-01-06. The Role of Fractional Dimension in Study Physics: A Two-Channel Representation with Geometric Memory. https://arxiv.org/abs/2301.12898

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories

We present a joint observational analysis confronting viable $f(R)$ modified gravity theories, specifically the Hu \& Sawicki and Starobinsky models, with background and large-scale structure (LSS) data. Utilizing Monte-Carlo Markov chain (MCMC) sampling across datasets including baryon acoustic oscillations (BAO), type Ia supernovae (SNeIa), cosmic microwave background (CMB) distance priors, and linear growth measurements ($f\sigma_8$, $f$, $\sigma_8$), we place tight constraints on the model parameters governing deviations from General Relativity. For the full dataset combination, we obtain $\log_{10} b_\mathrm{HS} = -6.325_{-1.138}^{+1.216}$ for the Hu \& Sawicki model and $b_\mathrm{S} = (0.8\pm61.0)\times10^{-4}$ for the Starobinsky model. Model comparison based on the Akaike Information Criterion indicates that these $f(R)$ extensions are statistically favored over flat $\Lambda\text{CDM}$ ($|\Delta\text{AIC}| \ge 3.99$) for the combined data. However, when considering the Bayesian Information Criterion, the evidence for support is significantly reduced. Furthermore, we construct two-dimensional phase-space diagrams in the $(\mu, \gamma)$ and $(\mu, \Sigma)$ planes across several redshifts, establishing a novel diagnostic null-test allowing us to probe for deviations from $\Lambda\text{CDM}$, corresponding to the fixed point $(1,1)$ in both planes, using LSS observables. Should future weak-lensing and galaxy surveys provide data points with $\mu-1<0$ and $\gamma-1>1$ or $\Sigma-1 < 0 $, then the aforementioned models could be directly ruled out.

physics.gen-ph↗
arXiv2026

Bound states in the continuum of gravitational waves

Bound states in the continuum (BICs) are ubiquitous wave phenomena, but have not yet been demonstrated for gravitational waves (GWs). Here, in-plane periodic perturbations, exponentially localized at the plane $z = 0$, are shown to lead to distributional surface energy tensors at this plane and to be regular vacuum solutions ($T_{\mu \nu} = 0$) of the linearized Einstein field equations outside of it. These are achieved by explicitly calculating the Ricci tensor components and the Ricci scalar from the metric perturbation tensor. To fulfill each vacuum solution ($R_{{\sigma \nu}_{(+, \times)}} = 0$ and $R_{{}_{(+, \times)}} = 0$), different surface polariton-like dispersions are required. These bound perturbations decay exponentially to a flat metric ($h_{{BIC}_{(+,\times)}} \propto e^{- k_z |z|}$), and each localized metric has a correspondence to a different planar GW polarization ($+,\times$). The Lorenz gauge-fulfilling solutions exist at the $\Gamma$ point in momentum space, dwelling within the continuum of wavevectors of propagating GWs. The strains in the unit cell of the periodic perturbations have opposite parities relative to the corresponding planar GWs under a $C_2$ in-plane rotation, making them incompatible by symmetry with their propagating counterpart, indicating localization via symmetry protection.

physics.gen-ph↗
arXiv2026

Sector-Resolved Bayesian Model Averaging for DESI-Era Cosmology

We present a quotient-space Bayesian formulation for DESI-era anomaly interpretation. Given a pattern-labeled catalog with map \(i\mapsto \Act(i)\), the induced posterior \(p(\Act\mid D)\), sector inclusion probabilities \(P_\alpha\), co-activation probabilities \(P_{\alpha\beta}\), and grouped Bayes factors \(B_\alpha(D)\) are exact summaries over predeclared physical activation events. Pairwise comparisons such as \(\lcdm\) versus \(\wacdm\) remain ordinary Bayes-factor tests between specified families; the quotient construction addresses the coarser question of which physical sector carries posterior support when different sectors are represented by unequal numbers of catalog elements. We derive a sector-resolved DESI-CMB-SN likelihood specification for late-time background, early-time ruler, supernova calibration, perturbation, and gravitational-wave propagation sectors. The construction includes an Alcock-Paczynski/isotropic-scale BAO decomposition, a pure-ruler projection, analytic marginalization of low-rank supernova calibration modes, Fisher-normalized sector priors, inactive-sector leakage tests, log-evidence uncertainty propagation, and prior/sector-partition diagnostics. The result is a quantitative procedure for reporting model-comparison support at the level of physically interpretable sectors.

physics.gen-ph↗