Upper bounds on the highest phonon frequency and superconducting temperature from fundamental physical constants

Abstract Fundamental physical constants govern key effects in high-energy particle physics and astrophysics, including the stability of particles, nuclear reactions, formation and evolution of stars, synthesis of heavy nuclei and emergence of stable molecular structures. Here, we show that fundamental constants also set an upper bound for the frequency of phonons in condensed matter phases, or how rapidly an atom can vibrate. This bound is in agreement with \textit{ab initio} simulations of atomic hydrogen and high-temperature hydride superconductors, and implies an upper limit to the superconducting transition temperature $T_c$ in condensed matter. Fundamental constants set this limit to the order of 10$^2-10^3$ K. This range is consistent with our calculations of $T_c$ from optimal Eliashberg functions. As a corollary, we observe that the very existence of the current research of finding $T_{\mathrm{c}}$ at and above $300$ K is due to the observed values of fundamental constants. We finally discuss how fundamental constants affect the observability and operation of other effects and phenomena including phase transitions.