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.
Categories: physics.gen-ph hep-ph