[Paper Review] Tubular geometry construction as a reason for new revision of the space-time conception

Yuri A. Rylov|ArXiv.org|Apr 5, 2005

Quantum Mechanics and Applications10 references15 citations

TL;DR

This paper proposes tubular geometry (T-geometry) as a generalization of Euclidean geometry based on the world function σ = ρ²/2, introducing nondegeneracy and finite divisibility of spacetime. By replacing the Euclidean world function with a generalized σ, T-geometry naturally explains quantum phenomena—such as stochastic particle motion and mass geometrization—without postulating quantum principles, offering a deterministic, geometric foundation for microcosmic physics.

ABSTRACT

The tubular geometry (T-geometry) is a generalization of the proper Euclidean geometry, founded on the property of sigma-immanence. The proper Euclidean geometry can be described completely in terms of the world function $σ=ρ^{2}/2$, where $ρ$ is the distance. This property is called the sigma-immanence. Supposing that any physical geometry is sigma-immanent, one obtains the T-geometry $\mathcal{G}$, replacing the Euclidean world function $σ_{E}$ by means of $σ$ in the sigma-immanent presentation of the Euclidean geometry. One obtains the T-geometry $\mathcal{G}$, described by the world function $σ$. This method of the geometry construction is very simple and effective. T-geometry has a new geometric property: nondegeneracy of geometry. The class of homogeneous isotropic T-geometries is described by a form of a function of one parameter. Using T-geometry as the space-time geometry one can construct the deterministic space-time geometries with primordially stochastic motion of free particles and geometrized particle mass. Such a space-time geometry defined properly (with quantum constant as an attribute of geometry) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion (without a use of quantum principles). Geometrization of the particle mass appears to be connected with the restricted divisibility of the straight line segments. The statement, that the problem of the elementary particle mass spectrum is rather a problem of geometry, than that of dynamics, is a corollary of the particle mass geometrization.

Motivation & Objective

• To develop a new physical geometry (T-geometry) based on the world function σ, generalizing Euclidean geometry.
• To address the limitations of Minkowski spacetime in describing intrinsic stochasticity in microcosmic particle motion.
• To propose that quantum phenomena arise not from intrinsic indeterminism but from geometric properties of spacetime.
• To show that particle mass and the mass spectrum can be geometrized as consequences of spacetime's finite divisibility.
• To argue that the need for field quantization arises from outdated assumptions, not geometric reality.

Proposed method

• Construct T-geometry by replacing the Euclidean world function σ_E with a general world function σ, preserving σ-immanence as the foundational principle.
• Define homogeneous isotropic T-geometries via a distortion function D(σ_M), which encodes the thickness and shape of timelike geodesics.
• Use the world function σ to fully reconstruct geometric objects (points, lines, planes) without relying on coordinates or metric tensors.
• Introduce nondegeneracy as a new geometric property: spacetime segments cannot be infinitely subdivided, leading to intrinsic discreteness.
• Model free particle motion as stochastically determined in non-Minkowski T-geometries, with stochasticity arising from geometric structure, not dynamics.
• Geometrize particle mass by linking it to the restricted divisibility of straight-line segments in T-geometry.

Experimental results

Research questions

• RQ1 Can quantum phenomena such as stochastic particle motion be explained geometrically without invoking quantum principles?
• RQ2 Is the mass spectrum of elementary particles a geometric property of spacetime rather than a dynamical one?
• RQ3 Does the finite divisibility of spacetime segments naturally lead to a geometric origin of mass and quantum constants?
• RQ4 Can the electromagnetic and gravitational fields be understood as metrical fields without requiring their quantization?
• RQ5 What is the role of the distortion function D(σ_M) in defining a class of deterministic, non-Minkowski spacetimes that reproduce quantum behavior?

Key findings

• T-geometry is a complete, σ-immanent generalization of Euclidean geometry, fully reconstructible from a single world function σ.
• The class of homogeneous isotropic T-geometries is parameterized by a distortion function D(σ_M), which controls the thickness and shape of timelike geodesics.
• In T-geometries with D(σ_M) ≠ 0, free particle motion is primordially stochastic, explaining quantum-like behavior without postulating indeterminism.
• Particle mass is geometrized as a consequence of the finite divisibility of spacetime segments, not a dynamical property.
• The quantum constant emerges as a geometric attribute of spacetime, not a fundamental constant of nature.
• The electromagnetic and gravitational fields are metrical fields; their apparent quantization may be an artifact of assuming Minkowski spacetime, not a physical necessity.