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 $c=\hslash =1$. By fixing $c=c_{\circ }$m/s = 1, $\hslash ={\hslash }_{\circ }$ 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, $(1/{\hslash }_{\circ }=.948...\times 10^{34}){\mathrm{s}}^{-1}$ and $\sqrt{{\mu }_{\circ }\mathrm{N}}=\sqrt{{\mu }_{\circ }{c}_{\circ }/{\hslash }_{\circ }}{\mathrm{s}}^{-1}=\mathrm{1.890...}\times 10^{18}{\mathrm{s}}^{-1}$. It presents at the same time the advantages and elegance of a system with $\hslash =c=\underline{{\mu }_{\circ }}=\underline{{ϵ}_{\circ }}=k=N_A=1$, 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 $\sqrt{{\mu }_{\circ }{c}_{\circ }/{\hslash }_{\circ }}=\mathrm{1.890...}\times 10^{18}$, and the elementary charge to $e=\mathrm{1.602...}\times 10^{-19}\mathrm{C}=\sqrt{4\pi \alpha }=.3028...$ . 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 $1/\sqrt{{\mu }_{\circ }{c}_{\circ }{\hslash }_{\circ }}{\mathrm{s}}^{-1}=\mathrm{5.017...}\times 10^{15}{\mathrm{s}}^{-1}$, and the tesla $c_{\circ }$V/m = $\sqrt{c_{\circ }^3/{\mu }_{\circ }{\hslash }_{\circ }}{\mathrm{s}}^{-2}=\mathrm{4.509...}\times 10^{32}{\mathrm{s}}^{-2}$. The weber is $1/\sqrt{{\mu }_{\circ }{c}_{\circ }{\hslash }_{\circ }}=\mathrm{5.017...}\times 10^{15}$. $K_J=483597.$... GHz/V $=e/\pi =$ .09639..., and $R_K=25812....\Omega =1/2\alpha \simeq 68.518$. 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.

Comments: 26 pages, 2 tables