Unified presentation of four fundamental inequalities

TL;DR

This paper presents a unified framework for understanding four fundamental physical constants as lower bounds of observable phenomena, enhancing teaching and conceptual clarity about the universe's perceptible limits.

Contribution

It introduces a unified presentation of four fundamental inequalities related to key physical constants, clarifying their roles in defining observable horizons and teaching physics.

Findings

Highlights the role of these constants as perceptual limits.

Connects the constants to critical phenomena and universe perception.

Provides a pedagogical framework for fundamental constants.

Abstract

We suggest an unified presentation to teach fundamental constants to graduate students, by introducing four lower limits to observed phenomena. The reduced Planck constant $\hbar$ is the lowest classically definable action. The inverse of invariant speed, $s$, is the lowest observable slowness. The Planck time, $t_{\rm P}$, is the lowest observable time scale. The Boltzmann constant, $k$, determines the lowest coherent degree of freedom; we recall a Einstein criterion on the fluctuations of small thermal systems and show that it has far-reaching implications, such as demonstrating the relations between critical exponents. Each of these four fundamental limits enters in an inequality, which marks a horizon of the universe we can perceive. This compact presentation can resolve some difficulties encountered when trying to defining the epistemologic status of these constants, and emphasizes their useful role in shaping our intuitive vision of the universe.