Completing the International System of units with $c=\hbar=\mu_\circ=\epsilon_\circ=k_B=N_A=1$

Abstract

A drawback of the new SI is that by fixing the value of the elementary charge $e$, the vacuum magnetic permeability $\mu_\circ$ and impedance $Z_\circ=\mu_\circ c$ are no longer fixed, but get written proportionately to $\alpha$. All electrical units get dependent on $\alpha$ (and might even, conceivably, vary with time). This may be cured by embedding the SI in a new framework in which the "fundamental constants of nature" are fixed and equal to 1, i.e. $c=\hbar=\mu_\circ=\epsilon_\circ=Z_\circ= k_B=N_A=1$. The metre, joule, and kilogram get identified as 1 m = (1/$c_\circ$) s = (1/299 792 458) s, 1 J = $(1/\hbar_\circ)$ s$^{-1}= (2\pi /6.626\;070\;15) \times 10^{34}\ \rm s^{-1}$ and 1 kg = $(c_\circ^2/\hbar_\circ)\ \rm s^{-1}= 0.852 ... \times\ 10^{51}\ s^{-1}$. Fixing $\mu_\circ= \mu_{\circ\circ}$ N/A$^2=1 $ provides 1 A = $\sqrt{\mu_{\circ\circ} \rm N} =\!\sqrt{\mu_{\circ\circ} c_\circ/\hbar_\circ}\ \rm s^{-1}$ and 1 C = $\sqrt{\mu_{\circ\circ} c_\circ/\hbar_\circ} = 1.890 ...\times\ 10^{18} $, with $e = 1.602 ... \times\ 10^{-19}\ \rm C$ also equal to $ \sqrt{4\pi\alpha} = 0.3028 ... $. All SI units can be defined in terms of the second, with the coulomb, ohm and weber dimensionless, and the mole identified as the very large Avogadro number.