A New Dual System For The Fundamental Units, including and going beyond the newly revised SI

Abstract

We propose a new system for the fundamental units, which includes and goes beyond the present redefinition of the SI, by choosing also $c=\hbar=1$. By fixing $c=c_\circ $m/s = 1, $\hbar=\hbar_\circ $ Js = 1 and $\underline{\mu_\circ}=\mu_\circ $N/A$^2$ = 1, it allows us to define the metre, the joule, and the ampere as equal to (1/299 792 458) s, $(1/\hbar_\circ = .948 ... \times \ 10^{34})\ {\rm s}^{-1}$ and $\sqrt{\mu_\circ \rm N}= \sqrt{\mu_\circ c_\circ / \hbar_\circ}\ {\rm s}^{-1}= 1.890...\times 10^{18}\ {\rm s}^{-1}$. It presents at the same time the advantages and elegance of a system with $\hbar = c = \underline{\mu_\circ} = \underline{\epsilon_\circ } = k = N_A = 1\,$, where the vacuum magnetic permeability, electric permittivity, and impedance are all equal to 1. All units are rescaled from the natural ones and proportional to the s, s$^{-1}$, s$^{-2}$, ... or just 1, as for the coulomb, ohm and weber, now dimensionless. The coulomb is equal to $\sqrt{\mu_\circ c_\circ / \hbar_\circ}= 1.890... \times 10^{18}$, and the elementary charge to $e=1.602...\times 10^{-19} {\rm C} = \sqrt{4\pi\alpha}=.3028... $ . The ohm is equal to $1/\mu_\circ c_\circ$ so that the impedance of the vacuum is $Z_\circ = 376.730... \Omega =1$. The volt is $ 1/ \sqrt{{\mu_\circ c_\circ \hbar_\circ}}\ {\rm s}^{-1} = 5.017... \times\ 10^{15}\ {\rm s}^{-1}$, and the tesla $c_\circ $V/m = $ \sqrt{{c_\circ^3}/{\mu_\circ\hbar_\circ}}\ {\rm s}^{-2} = 4.509... \times 10^{32}\ {\rm s}^{-2}$. The weber is $ 1/ \sqrt{{\mu_\circ c_\circ \hbar_\circ}} = 5.017... \times\ 10^{15}$. $\ K_J =483\,597. \ $... GHz/V $= e/\pi = $ .09639..., and $R_K = 25\,812.\ ...\ \Omega =1/2\alpha\simeq 68.518$. One can also fix $e$ = 1.602 176 634 $\times 10^{-19}$ C, at the price of adjusting the coulomb and all electrical units with $\mu_\circ=4\pi\times 10^{-7}\eta^2$ where $\eta^2, \propto \alpha$, is very close to 1.