[Paper Review] Discriminating properties of compactification in discrete uniform isotropic space-time

Yuri A. Rylov|arXiv (Cornell University)|Sep 15, 2008

Relativity and Gravitational Theory5 references3 citations

TL;DR

This paper proposes that compactification in a discrete, uniform, and isotropic 5D space-time geometry—formalized via a single-valued world function—naturally restricts the possible values of elementary particle electric charge. By modeling the fifth dimension as compactified with period $2L$, the theory derives a geometric upper bound on charge, yielding $|e| \leq e_0$, consistent with experimental observations and resolving a key limitation of conventional Kaluza-Klein theory where charge remains unrestricted.

ABSTRACT

Compactification of the 5-dimensional Kaluza-Klein space-time geometry is considered. The space-time geometry is supposed to be discrete, uniform and isotropic. It is shown, that consideration of the space-time geometry as a physical geometry, i.e. as a geometry described completely by the single-valued world function, leads to a discrimination of some values of the particle charge. At the conventional approach, when the world function becomes to be many-valued after compactification, the value of the elementary particle electric charge remains to be unrestricted, and this fact does not agree with experimental data. It is important, that the discrete geometry is given on the continual set of points. This circumstance makes admissible a compatibility of discreteness with the uniformity and isotropy of the geometry.

Motivation & Objective

• To resolve the inconsistency in Kaluza-Klein theory where compactification leads to a many-valued world function and unrestricted electric charge, contradicting experimental data.
• To demonstrate that discrete, uniform, and isotropic space-time geometry can be consistently described using a single-valued world function, enabling geometric derivation of quantum-like properties.
• To show that compactification in a discrete physical geometry imposes a geometric upper bound on the electric charge of stable elementary particles.
• To establish a geometric foundation for charge quantization without invoking quantum postulates, by linking the elementary charge to the compactification scale and Planck-length scale.

Proposed method

• Formalizing space-time geometry via a single-valued world function $\sigma(P,Q)$, which defines all geometric relations without relying on coordinates or axiomatic systems.
• Modeling the 5D space-time as discrete, uniform, and isotropic, with the fifth dimension compactified into a circle of radius $L/2$.
• Using the world function to define geometric momentum $\pi_5$ and relate it to the physical momentum $p_5$ via $p_5 = bc\pi_5$, with $b$ derived from the geometry.
• Imposing single-valuedness of the wave function $\psi$ in the compactified dimension, which constrains $p_5$ to discrete values $p_5 = \frac{\pi\hbar}{L}s$ for integer $s$.
• Deriving the condition $|s| < \frac{3L^2}{4\pi\lambda_0^2}$ from geometric constraints, where $\lambda_0$ is the elementary length, leading to a bound on charge.
• Relating the maximal charge to the elementary charge $e_0$ via $\varkappa = \frac{e_0 L}{\pi \hbar c}$, showing that $|e| \leq e_0$ when $L \sim \lambda_0$.

Experimental results

Research questions

• RQ1 Can a discrete, uniform, and isotropic space-time geometry consistently describe compactification without introducing a many-valued world function?
• RQ2 Does geometric discreteness in a physical geometry framework lead to quantization of electric charge?
• RQ3 Can the observed upper bound on elementary particle charge ($e_0$) be derived purely from geometric constraints in a 5D Kaluza-Klein model?
• RQ4 How does the use of a single-valued world function in physical geometry enable compatibility between discreteness, isotropy, and uniformity?

Key findings

• The compactification of a 5D discrete, uniform, and isotropic space-time geometry leads to a restriction on the possible values of electric charge, with $|e| \leq e_0$.
• The upper bound on charge arises from the requirement that the wave function remains single-valued in the compactified fifth dimension, which quantizes the momentum $p_5$.
• The maximal charge is achieved when $|s| = 1$, corresponding to the smallest non-zero momentum in the compactified dimension, yielding $|e| = e_0$.
• The bound is geometric: $|s| < \frac{3L^2}{4\pi\lambda_0^2}$, and when $L \sim \lambda_0$, this implies $|s| < 1.91$, so $|s| = 1$ is the only allowed value, giving $|e| \leq e_0$.
• The elementary charge $e_0$ is geometrically related to the compactification scale $L$ and the Planck-scale length $\lambda_0$ via $\varkappa = \frac{e_0 L}{\pi \hbar c}$.
• The result shows that charge quantization can emerge from geometry alone, without invoking quantum mechanics, by using a physical geometry formalism based on a single-valued world function.