Gravity, Fine-Structure Constant and Natural Units -- some Thoughts based on Dimensional Analysis --

TL;DR

This paper explores the relationships between fundamental physical constants and dimensions through dimensional analysis, highlighting the unique status of gravity and the implications for natural units and the fine-structure constant.

Contribution

It presents a simple dimensional analysis approach linking fundamental constants to natural units and explains the form of the fine-structure constant.

Findings

The form of the fine-structure constant follows from dimensional analysis.

Gravity remains a distinct, non-quantized area of physics.

Different unit systems can be derived from fundamental constants.

Abstract

Here we discuss direct links of the number of fundamental dimensions to the fundamental natural constants using simple arguments of dimensional analysis \corr{based on Maxwell's dimensions length (L), time (T) and mass (M) as well as the constants $G$, $c$, $\hbar$ and $e$}. We find that the \corr{form} of the fine-structure constant is a direct consequence of this connection. Additionally, our approach emphasises that gravity is a quite distinct area of physics which is not yet successfully quantised, i.e. not yet combined with quantum mechanics. We also discuss different unit systems based on dimensional analysis and natural constants.