arXiv subjects

Muhammad Farchani Rosyid

Publications and source records attributed to Muhammad Farchani Rosyid.

4 recordsLinked to original sources

arXiv2026

Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds

We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $Σ$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem. To resolve the spectral degeneracy that obscures "momentum space" in such settings, we require the degenerate sector to be resolved by a local, symmetry-adapted maximal Abelian commuting set (a fiberwise MASA), constructed from geometric differential operators, most notably from Killing data when such symmetries are available. We provide a constructive algorithm for generating these commuting operators and show that the resulting momentum label spaces $\mathcal{F}$ (discrete, continuous, or mixed) reflect geometric symmetry constraints. We introduce a dual classification: (i) by MASA completeness and Stackel separability, and (ii) by the topology of $\mathcal{F}$. Finally, we distinguish unitary changes induced by true isometries (which preserve the GFT structure) from changes of coordinate-adapted degeneracy resolution/separation schemes, which may induce inequivalent $k$-space labelings (e.g. Cartesian vs spherical constructions in $\mathbb{R}^{3}$) while remaining unitarily equivalent on $\mathcal{L}^{2}\left[Σ\right]$. This symmetry-adapted harmonic analysis is intended as a foundation for curved-space mode decompositions; dynamical applications are developed in the subsequent work.

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arXiv2025

The Ontic Necessity of the Quantum Wavefunction: Why Epistemic Views Struggle with the Uncertainty Principle

The ontological status of the quantum wavefunction remains one of the most debated questions in quantum theory. While epistemic interpretations regard the wavefunction as a reflection of our knowledge or beliefs, ontic interpretations treat it as a real physical object. In this paper, we argue that epistemic approaches struggle to explain the universality and precision of the uncertainty principle, a core feature of quantum mechanics. By contrast, treating the wave-function as ontic allows a consistent and natural derivation of quantum uncertainty from the mathematical structure of Hilbert space. We examine key interpretations on both sides and highlight why the epistemic view falls short in addressing constraints that appear to be intrinsic to nature.

quant-ph↗

arXiv2021

On the Stochastic Flows on $(m+n+1)$-Dimensional Exotic Spheres

Stochastic flows of Stratonovich stochastic differential equations on exotic spheres have been studied. The consequences of the choice of exotic differential structure on stochastic processes taking place on the topological space $S^{m+n+1}$ as state space of the processes have been investigated. More precisely, we have investigated the properties of stochastic processes where the state spaces of the stochastic processes under consideration are $({m+n+1})$-dimensional differentiable manifolds which are homeomorphic but not necessarily diffeomorphic to standard ${(m+n+1)}$-dimensional sphere. The differentiable manifolds have been constructed from disjoint union $\mathbb{R}^{m+1}\times S^{n}\sqcup S^m\times \mathbb{R}^{n+1}$ by identifying every pair of its points using a map $u :\mathbb{R}^{m+1}\times S^n\rightarrow S^m\times \mathbb{R}^{n+1}$ which is constructed from a diffeomorphism $h_1\times h_2:S^m\times S^n\rightarrow S^m\times S^n$. The diffeomorphisms $h_1$ and $h_2$, therefore, can be regarded as the carriers of the "exoticism" of the constructed manifolds. For all of the above purposes, homeomorphisms $h$ from the above-constructed manifolds onto the standard sphere explicitly in term of the diffeomorphisms $h_1$ and $h_2$ have been constructed. Using the homeomorphisms $h$ and all their associated maps derived from them and expressed in terms of $h_1$ and $h_2$ as well as their derivatives, we construct the stochastic processes or flows on the above-constructed manifolds corresponding to stochastics processes on the standard sphere $S^{m+n+1}_s$. The stochastic processes yielded from the above construction on the constructed manifolds can be regarded as the same stochastic processes on $S^{m+n+1}_s$ but described in exotic differential structures on $S^{m+n+1}$.

math-ph↗

arXiv2021

On the Stochastic Processes on $7$-Dimensional Spheres

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with the so-called ${\bullet}$-action of $S^3$, whereas the Gromoll-Meyer exotic sphere $Σ^7_{GM}$ as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with respect to the so-called ${\star}$-action of $S^3$. The Stratonovich stochastic differential equation which describes a continuous-time stochastic process on the standard sphere is constructed and studied. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism $h: S^7_s\rightarrow Σ^7_{GM}$. The corresponding Fokker-Planck equation and entropy rate in the Stratonovich approach is also investigated.

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