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

TL;DR

This paper proposes a new framework for the SI units where fundamental constants are fixed to 1, simplifying unit definitions and linking electrical units directly to the fine-structure constant.

Contribution

It introduces a reformulation of SI units fixing key constants to 1, unifying units and reducing dependence on variable constants like alpha.

Findings

All SI units can be expressed in terms of the second.

Electrical units become dimensionless or directly related to alpha.

The mole is identified with Avogadro's number.

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.

Equations

$$\left\{\,\begin{array}[]{c}1\,\rm C=\eta\ C_{\rm old},\vspace{2mm}\\ 1\,\rm A=\eta\ A_{\rm old}\,,\end{array}\right.\ \ \hbox{and thus }\ \ \left\{\,\begin{array}[]{ccl}1\,\rm V\!&=&\rm\eta^{-1}\ \ V_{\rm old}\,,\vspace{1mm}\\ 1\,\rm Wb\!&=&\rm\eta^{-1}\ Wb_{\rm\,old}\,,\vspace{1mm}\\ 1\,\rm T\!&=&\rm\eta^{-1}\ \ T_{\rm old}\,,\end{array}\right.\ \ \ \left\{\ \begin{array}[]{cccc}1\,\rm F\!&=&\!\eta^{2}&\!\!\rm F_{\rm old}\,,\vspace{1mm}\\ 1\,\rm H\!&=&\!\eta^{-2}&\!\!\rm H_{\rm old}\,,\vspace{1mm}\\ 1\,\rm\Omega\!&=&\!\eta^{-2}&\!\!\Omega_{\rm old}\,,\end{array}\right.$$

$${\mu }_{\circ }=4\pi \times 10^{-7}N/{\mathrm{A}}^2\to {\mu }_{\circ }={\mu }_{\circ \circ }N/{\mathrm{A}}^2=4\pi \times 10^{-7}{\eta }^{2}N/{\mathrm{A}}^2.$$

$${ϵ}_{\circ }={ϵ}_{\circ \circ }{\mathrm{C}}^2/(\mathrm{J}\mathrm{m})=\frac{1}{4\pi \times 10^{-7}{c}_{\circ }^2}{\eta }^{-2}{\mathrm{C}}^2/(\mathrm{J}\mathrm{m})=8.854187817620...\times 10^{-12}{\eta }^{-2}F/m.$$

$$\alpha =\frac{e^2}{4\pi {ϵ}_{\circ }\hslash c}=\frac{{\mu }_{\circ }c{e}^2}{4\pi \hslash }=\frac{10^{-7}{c}_{\circ }{e}_{\circ }^2}{{\hslash }_{\circ }}{\eta }^2={\alpha }_{\circ }{\eta }^2=\frac{{\eta }^2}{137.035999158713...}=\frac{1}{137.035999084(21)}.$$

$${\mu }_{\circ \circ }=4\pi \times 10^{-7}{\eta }^2=4\pi \times 10^{-7}\alpha /{\alpha }_{\circ }=12.5663706212(19)\times 10^{-7}.$$

$${\eta }^2=1+2.0(2.3)\times 10^{-10}.$$

$$\begin{array}[]{c}\displaystyle\eta^{2}\,=\,\frac{\alpha\,}{\alpha_{\circ}}\,=\,\frac{\mu_{\circ}}{4\pi\!\times\!10^{-7}\ \rm N/A^{2}}\,=\,1+5.4\,(1.5)\times 10^{-10}\,.\vspace{-1.5mm}\\ \end{array}$$

$$\begin{array}[]{ccl}Z_{\circ}=(4\pi\times 10^{-7}\,\eta^{2}\rm\ N/A^{2})\times(299\,792\,458\ m/s)=\mu_{\circ\circ}c_{\circ}\ \Omega\!&=&\!376.730\;313\;461\;770\,...\ \eta^{2}\ \Omega\vspace{2mm}\\ \!&=&\!376.730\;313\;665\,(57)\,...\ \Omega\,.\end{array}$$

$$\frac{\dot{\mathrm{C}}}{\mathrm{C}}=\frac{\dot{\mathrm{A}}}{\mathrm{A}}=-\frac{\dot{\mathrm{V}}}{\mathrm{V}}=-\frac{\dot{\mathrm{T}}}{\mathrm{T}}=-\frac{1}{2}\frac{\dot{\mathrm{\Omega }}}{\mathrm{\Omega }}=...=\frac{\dot{\eta }}{\eta }=\frac{1}{2}\frac{\dot{\alpha }}{\alpha }$$

$$c=c_{\circ }m/s=299792458m/s=1⟺1\mathrm{m}\equiv (1/c_{\circ })\mathrm{s}=(1/299792458)\mathrm{s}.$$

$$\hslash ={\hslash }_{\circ }\mathrm{J}\mathrm{s}=(6.62607015\times 10^{-34}/2\pi )\mathrm{J}\mathrm{s}=1⟺1\mathrm{J}\equiv (1/{\mathrm{\hslash }}_{\circ }){\mathrm{s}}^{-1}=0.948252156246...\times 10^{34}{s}^{-1}.$$

$$\left\{\ \begin{array}[]{ccccccc}{\rm 1\,N}\!&=&\!\rm 1\ J/m\!&=&\!{c_{\circ}}/{\hbar_{\circ}}\ \rm s^{-2}\!&=&\!2.842\;788\;447\;250\,...\times 10^{42}\ \,\hbox{s}^{-2}\,,\vspace{2mm}\\ {\rm 1\,kg}\!&=&\!1\ \rm J\,s^{2}\!/m^{2}\!&=&\!{c_{\circ}^{2}}/{\hbar_{\circ}}\ \rm s^{-2}\!&=&\!0.852\;246\;536\;175\,...\times 10^{51}\ \,\hbox{s}^{-1}\,.\end{array}\right.$$

$$\mu_{\circ}=\mu_{\circ\circ}\ \rm N/A^{2}=4\pi\!\times\!10^{-7}\ \eta^{2}\ \rm N/A^{2}\,=1\ \ \ \Longleftrightarrow\ \ \ 1\,A=\sqrt{\mu_{\circ\circ}\,\rm N}=\sqrt{4\pi\!\times\!10^{-7}\,\rm N}\ \,\eta\,.\ \ \$$

$${ϵ}_{\circ }={ϵ}_{\circ \circ }{\mathrm{C}}^2/(\mathrm{J}\mathrm{m})=\frac{1}{4\pi \times 10^{-7}{c}_{\circ }^2}{\eta }^{-2}{\mathrm{C}}^2/(\mathrm{J}\mathrm{m})=1⟺1\mathrm{C}=\sqrt{\mathrm{J}\mathrm{m}/{ϵ}_{\circ \circ }}=\sqrt{4\pi \times 10^{-7}{c}_{\circ }^2\mathrm{J}\mathrm{m}}\eta ,$$

$$Z_{\circ }=\sqrt{{\mu }_{\circ }/{ϵ}_{\circ }}={\mu }_{\circ }c=4\pi \times 10^{-7}{c}_{\circ }{\eta }^2\Omega =376.730313461770...{\eta }^2\Omega =1.$$

$$1\Omega =1/{\mu }_{\circ \circ }{c}_{\circ }=1/(376.730...\times {\eta }^2),$$

$$\left\{\,\begin{array}[]{cccclcl}\rm 1\,A\!&=&\!\rm\sqrt{\mu_{\circ\circ}\,N}\!&=&\!\sqrt{\mu_{\circ\circ}c_{\circ}/\hbar_{\circ}}\ \rm s^{-1}\!&=&\!1.890\;067\;014\;853\,...\times 10^{18}\ \eta\ \rm s^{-1},\vspace{2mm}\\ \rm 1\,C\!&=&\!\rm 1\,A\,s\!&=&\!\sqrt{\mu_{\circ\circ}c_{\circ}/\hbar_{\circ}}\!&=&\!1.890\;067\;014\;853\,...\times 10^{18}\ \eta\ =\,1.890\;067\;015\;36\,(14)\,...\times 10^{18}.\end{array}\right.$$

$$e=\sqrt{{\mu }_{\circ \circ }{c}_{\circ }{e}_{\circ }^2/{\hslash }_{\circ }}=\sqrt{4\pi \alpha }=\sqrt{4\pi {\alpha }_{\circ }}\eta =0.302822120789...\eta =0.302822120871(23).$$

$$1\mathrm{Wb}=1\mathrm{C}\times 1\mathrm{\Omega }=1/\sqrt{{\mu }_{\circ \circ }{c}_{\circ }{\hslash }_{\circ }}=5.017029284119...\times 10^{15}{\eta }^{-1}=5.01702928276(38)...\times 10^{15},$$

$${\Phi }_{\circ }=\frac{h}{2e}=\frac{\pi {\hslash }_{\circ }}{e_{\circ }}\mathrm{Wb}=2.067833848461...\times 10^{-15}\mathrm{Wb}=\frac{\pi }{\mathrm{e}}=\sqrt{\frac{\pi }{4\alpha }}\simeq 10.374382972...{\eta }^{-1},$$

$$\!\left\{\!\begin{array}[]{cccccccccccc}\displaystyle K_{J}=\frac{1}{\Phi_{\circ}}\!&=&\!\displaystyle\frac{e_{\circ}}{\pi\hbar_{\circ}}&\!\rm Hz/V\!\!&=&\!\!483\;597.\,848\;416\;983\,...&\!{\rm GHz/V}\!\!&=&\!\displaystyle\frac{e}{\pi}=\sqrt{\frac{4\alpha}{\pi}}\!&=&\!.0963\;912\;748\;023\;449\,...\ \eta\,,\\[11.38109pt] \displaystyle R_{K}=\,\frac{h}{e^{2}}\!\!&=&\,\displaystyle\frac{h_{\circ}}{e_{\circ}^{2}}&\!\!\!\Omega&=&25\,812.\,807\;459\;304\,...&\Omega&=&\displaystyle\!\displaystyle\!\!\frac{1}{2\alpha}\!&=&68.517\;999\;579\;356\,...\ \eta^{-2}.\end{array}\right.$$

$$\begin{array}[]{ccl}\displaystyle R_{K}=\frac{h}{e^{2}}\!&=&\!\displaystyle\mu_{\circ}c\ \,\frac{1}{2}\ \,\frac{4\pi\epsilon_{\circ}\hbar c}{e^{2}}=\,\frac{Z_{\circ}}{2\alpha}\,=\,\frac{1}{2}\ [Z_{\circ}\!=376,730\,...\,\eta^{2}\,\Omega=1]\ \eta^{-2}\left[\,\frac{1}{\alpha_{\circ}}=137.035\,999\,...\,\right]=\frac{1}{2\alpha}\vspace{2mm}\\ &=&25\,812.\,807\,459\,304\,...\ \Omega\ .\vspace{-4mm}\\ \end{array}$$

$$\left\{\ \begin{array}[]{ccccccrc}\rm 1\,V&=&\rm 1\,J/C=1\ Wb\ s^{-1}&=&\displaystyle{1}/{\sqrt{\mu_{\circ\circ}c_{\circ}\hbar_{\circ}}}\ \,{\rm s}^{-1}&=&5.017\;029\;284\;119\,...\times 10^{15}&\!\eta^{-1}\ \,{\rm s}^{-1}\,,\vspace{2mm}\\ 1\,\hbox{V/m}&=&\,1\,{\rm N/C}\ \!&=&\!\displaystyle\sqrt{{c_{\circ}}/{\mu_{\circ\circ}\hbar_{\circ}}}\ \ {\rm s}^{-2}\!&=&\displaystyle 1.504\;067\;540\;944\,...\times 10^{24}&\!\eta^{-1}\ {\rm s}^{-2}\,,\vspace{2mm}\\ 1\,\hbox{T}&=&1\,{\rm N/(A\,m)}\!&=&\!\sqrt{c_{\circ}^{3}/\mu_{\circ\circ}\hbar_{\circ}}\ \ {\rm s}^{-2}\!&=&\displaystyle 4.509\;081\;050\;976\,...\times 10^{32}&\!\eta^{-1}\ {\rm s}^{-2}\,,\\[5.69054pt] 1\ \hbox{F}&=&\displaystyle 1\,{\rm C/V\,=\,1\,s/\Omega}&=&\mu_{\circ\circ}c_{\circ}\ \ {\rm s}\!&=&\!\ \ 376.730\;313\;461\;770\,...&\!\!\!\!\!\eta^{2}\ \ {\rm s}\,,\\[5.69054pt] 1\ \hbox{H}&=&1\ \hbox{J/A}^{2}=\,1\ \Omega\,{\rm s}&=&\displaystyle({1}/{\mu_{\circ\circ}c_{\circ}})\ \ {\rm s}\!&=&\!\displaystyle\!\!1/376.730\;313\;461\;770\,...&\!\!\!\!\!\eta^{-2}\ \,{\rm s}\,.\end{array}\right.$$

$$K_J^2{R}_K=4/h=2/\pi =(4/h_{\circ })(\Omega /{\mathrm{V}}^2{\mathrm{s}}^2)=4/(h_{\circ }\mathrm{J}\mathrm{s})=.603676071856860...\times 10^{34}(\mathrm{J}\mathrm{s}{)}^{-1}.$$

$$\left\{\ \begin{array}[]{ccccl}{1\,\rm K}\!&=&\!k_{B\circ}\,{\rm J}=(k_{B\circ}/\hbar_{\circ})\ {\rm s}^{-1}\!&=&1.\,380\;649\times 10^{-23}\ \rm J=1.309\;203\;391\;269\,...\times 10^{11}\ s^{-1}\,,\vspace{2mm}\\ {\rm 1\,mol}\!&=&\!\ \ \ \ \ \ N_{A\circ}\!&=&6.022\;140\;76\times 10^{23}\,.\end{array}\right.$$

$$\begin{array}[]{ccl}\displaystyle 1\,e{\rm V}=e_{\circ}\,{\rm J}=\frac{e_{\circ}}{\hbar_{\circ}}\ \rm s^{-1}\!&=&\!\rm 1.602\,176\,634\times 10^{-19}\ J=1.519\,267\,447\,878\,...\,\times 10^{15}\ s^{-1}\vspace{0mm}\\ &=&\!(e_{\circ}/k_{B\circ})\ \rm K=11\,604.518\,121\,550\,...\ K\,,\end{array}$$

Full text

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 = $(1/\hbar_{\circ})$ $\rm s^{-1}=(2\pi/6.626\,070\,15)\times 10^{34}\ \rm s^{-1}$ and 1 kg = $(c_{\circ}^{2}/\hbar_{\circ})\ \rm s^{-1}=0.852\,...\times\,10^{51}\ \rm s^{-1}$.

Fixing $\mu_{\circ}\!=\mu_{\circ\circ}\rm\,N/A^{2}=1$ 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.

I Having to cope with an unfixed $\mu_{\circ}$ ,  in the new SI

The International System of units is based on six fundamental units, leaving aside the candela si .  It has just been revised so that the kilogram, the ampere, the kelvin and the mole get redefined in terms of fixed numerical values of the Planck constant $h$, the elementary charge $e$, the Boltzmann constant $k_{B}$, and the Avogadro constant $N_{A}$ mills ; mills2 ; rev ; rev2 ; cr . All basic units are now defined from reproducible universal phenomena, rather than from unique material artefacts, as were the metre and kilogram stored for years at the Bureau International des Poids et Mesures (BIPM) m ; kg .

But, by deciding to fix the elementary charge to be, exactly, $e=1.602\,176\;634\times 10^{-19}$ C, the coulomb, and thus the ampere, have to be slightly adjusted accordingly. With $e=e_{\circ}\,\rm C$ replacing its earlier expression as $e=e_{\rm old}\,\rm C_{\rm old}$, one has pfcr

[TABLE]

with $e_{\circ}=\eta^{-1}\,e_{\circ\,\rm old}$. The numerical value $e_{\circ}$ of the elementary charge is chosen very close to its former best determination so that $\eta$ is very close to 1, up to a few $10^{-10}$.

The ampere was previously defined as Aold, that constant current which, if maintained in two straight parallel conductors of infinite length, 1 m apart in vacuum, would produce between them a force of $\,2\times 10^{-7}$ N per metre of length amp . The ampere getting multiplied by $\eta$, the new force per metre of length will now be $2\!\times\!10^{-7}\ \eta^{2}$ N/m. If we want to keep the usual expression of Ampère’s force law, $F/L=\mu_{\circ}I^{2}/\,2\pi r$, we must abandon the fixed expression for the vacuum magnetic permeability $\mu_{\circ}$, through the replacement

[TABLE]

The vacuum electric permittivity $\epsilon_{\circ}=1/\mu_{\circ}c^{2}$ becomes also unfixed,

[TABLE]

$\mu_{\circ\circ}$ and $\epsilon_{\circ\circ}$ are dimensionless coefficients related by $\epsilon_{\circ\circ}\mu_{\circ\circ}=1/c_{\circ}^{2}\,$, free to adjust to the new definitions of the ampere and coulomb.

As a result the sizes of electrical units now depend on $\alpha$. They are no longer rigidly tied to the mechanical ones. The connection is made softer, with a freely-floating $\mu_{\circ\circ}$ as a parameter, so that $\mu_{\circ}$ may now qualify as a new “fundamental constant of nature”, on the same ground as $c$ and $\hbar$. The fine structure constant gets expressed as

[TABLE]

The vacuum magnetic permeability $\mu_{\circ}$, measured in N/A2 or H/m, is proportional to $\alpha$ and depends on its future measurements, through

[TABLE]

The 2017 adjustment codata1 ; codata2 , taken with the exact values of the fixed constants ampmise , corresponds to

[TABLE]

But with the new 2018 recommended values alpha of $\alpha$ and $\mu_{\circ}$ as expressed in (4,5) we now obtain a slightly higher value

[TABLE]

This already shows (at $\approx 3.6\ \sigma$) a small discontinuity in the definition of the new electrical units, as compared to the older ones. It provides back from (3) $\epsilon_{\circ}=8.854\;187\;8128\,(13)\times 10^{-12}$ F/m, as in eps .

Similarly the vacuum impedance $Z_{\circ}=\!\sqrt{\mu_{\circ}/\epsilon_{\circ}}=\mu_{\circ}c$, previously fixed, now has a slightly higher value

[TABLE]

Its value in ohms depends on future measurements of $\alpha$, even if $\,Z_{\circ}$ is supposed to be a characteristic of the vacuum. Fortunately, the ohm itself is now proportional to $\eta^{-2}$ as seen in (1), so that $Z_{\circ}=376.730\,...\,\eta^{2}\,\Omega\,$ is independent of $\eta$, and $\alpha$ pfcr . This indetermination of the values of the vacuum impedance (in $\Omega$), magnetic permeability $\mu_{\circ}$ (in H/m), and electric permittivity (in F/m) is a new feature introduced in the SI on its last 2019 revision, decoupling electrical units from mechanical ones. It may be viewed, conceptually, as an unfortunate feature of the new SI, resulting from the desire to fix the value of $e$ so that $K_{J}=2e/h$ and $R_{K}=h/e^{2}$ be numerically fixed in GHz/V and $\Omega$, respectively.

The electrical units could even depend on time, in the new SI,  should $\alpha$ depend on time, as occasionally considered. They would no longer be time-invariant (in principle a basic requirement for fundamental units) but could vary, according to

[TABLE]

(with $\rm 1\,C\!\times\!1\,V=1\,J$ and $\rm 1\,A\!\times\!1\,m\!\times\!1\,T=1\,N\,$ independent of $\eta$ and $\alpha$), even if such variations are constrained to be extremely small (and practically insignificant). This is the price to pay for fixing $e$ and leaving $\mu_{\circ}$ and $Z_{\circ}=\mu_{\circ}c$ unfixed. Fortunately, as we shall see, we can still embed the new SI within a larger framework, bringing in all the advantages of a system in which $c=\hbar=\mu_{\circ}=$ $\epsilon_{\circ}=1\,$.

II Fixing * c* = $\hbar$ = $\mu_{\circ}$ = $\epsilon_{\circ}$ = 1 ,  within the new SI

In contrast with the Système International d’unités, revised since May 20th, 2019, theoreticians often like to consider an “ideal” choice of $c=\hbar=1$, extended to electromagnetism with $\,\mu_{\circ}\!=\epsilon_{\circ}\!=1$.  But this is usually considered as a utopic system, as opposed to the generality, practicality and universality

of the SI. Still, the shall show how the new SI, as officially redefined, can be included within a larger framework in which $c=1$ as suggested by relativity, and $\hbar=1$ as suggested by quantum mechanics. $\hbar=1$ is implicitly adopted, in fact, when we say that the electron is a spin-1/2 particle.

Then we can again directly view the vacuum magnetic permeability $\mu_{\circ}$, electric permittivity $\epsilon_{\circ}$ and impedance $Z_{\circ}=\sqrt{\mu_{\circ}/\epsilon_{\circ}}$ as fixed quantities, as for the speed of light $c=1/\sqrt{\mu_{\circ}\epsilon_{\circ}}$ and reduced Planck’s constant $\hbar$. All of them may be regarded as “fundamental constants of nature”.

We may then, in addition, fix them to 1, which provides expressions of the metre in terms of the second (from $c$),

of the joule in terms of the s*-1* (from $\hbar$), of the ampere in terms of $\sqrt{\rm N}$ (from $\mu_{\circ}$), etc..

The theory of relativity already allowed for defining the metre from the second, by fixing the value of $c$ to be, exactly, $299\;792\;458\ \rm m/s\,$ c . If we impose, in addition, $c=1$ pfcr , we can consider the second as being also a unit of length, and identify the metre as a fixed fraction of the second:

[TABLE]

The definition of the kilogram has just changed, from the historical “grand $\cal K$” stored at BIPM kg to a new one based on quantum physics rev ; rev2 ; cr . This is done by fixing the value of Planck’s constant to $h=6.626\;070\;15\times 10^{-34}\ \rm J\,s\,.$ If we now impose, in addition,  $\hbar=h/2\pi=1$, we can consider the second*-1* as being, also, a fundamental unit of energy, and identify the joule as a fixed number of  s*-1*:

[TABLE]

The newton and the kilogram can then be identified as

[TABLE]

But can we go further in our plan to include the new SI within a system where $c=\hbar=\mu_{\circ}=\epsilon_{\circ}=1$ ? This may seem difficult, especially as $\mu_{\circ}$ is no longer exactly fixed but given by $\mu_{\circ}=4\pi\times 10^{-7}\,\eta^{2}\ \rm N/A^{2}$, its value being now dependent on $\alpha$. Irrespectively of this, we can impose $\mu_{\circ}=1$, which provides the ampere as proportional to a square root of the newton:

[TABLE]

With $c=1$, asking for $\mu_{\circ}=1$ or $\epsilon_{\circ}=1/\mu_{\circ}c^{2}=1$ is equivalent, with

[TABLE]

verifying from (13,14) that  1 C = $c_{\circ}$ m A = 1 A s. This is also equivalent to choosing in (8)

[TABLE]

The new SI gets embedded within a larger framework, so that $\mu_{\circ},\ \epsilon_{\circ}$ and $Z_{\circ}=\sqrt{\mu_{\circ}/\epsilon_{\circ}}=\mu_{\circ}c$  are all fixed and equal to 1 . The ohm is a dimensionless unit,

[TABLE]

as required for $Z_{\circ}=\mu_{\circ\circ}\,c_{\circ}\ \Omega\,$ in (8,15) to be independent of $\alpha$, and equal to 1, the natural unit of impedance. The ampere and the coulomb are obtained from (13,14) as

[TABLE]

With $e=e_{\circ}\,\rm C=1.602\;176\;634\times 10^{-19}\,\rm C$ and $\alpha=e^{2}/4\pi$ we recover

[TABLE]

The weber is also dimensionless, with

[TABLE]

reflecting that 1 V = 1 A $\times\,1\,\Omega$. The flux quantum, $h/2e=\pi/e$, has a fixed value when expressed in Wb,

[TABLE]

but is proportional to $\eta^{-1}$ i.e. to $1/\sqrt{\alpha}$, as for the weber. The Josephson and von Klitzing constants, numerically fixed in Wb and GHz/V in the new SI ampmise , are

[TABLE]

To illustrate better how $R_{K}$ involves the vacuum impedance $Z_{\circ}\!=\mu_{\circ}c=1$ and $\alpha=\eta^{2}\,\alpha_{\circ}$, we can write

[TABLE]

The remaining electrical units may be expressed in terms of the second as pfcr

[TABLE]

$\eta=1+2.7\,(.8)\times 10^{-10}$, very close to 1 as seen in (6,7), originates from fixing $e$ in view of fixing $K_{J}$ and $R_{K}$, resulting in electric units no longer rigidly tied to the mechanical ones. This could have been avoided by conserving the earlier definitions of the electrical units, with $\eta=1$ and $\mu_{\circ}=4\pi\!\times\!10^{-7}\ \rm N/A^{2}$. $K_{J}$ and $R_{K}$ would then have been left unfixed. Still they are related, independently of $e$, by

[TABLE]

Finally we may also fix $k_{B}=k_{B\circ}\ \rm J/K=1$, $N_{A}=N_{A\circ}\ \rm mol^{-1}=1$, which define the kelvin and the mole as

[TABLE]

The resulting expression of the Kelvin is in agreement with

[TABLE]

and the mole gets identified with the very large Avogadro number, $N_{A\circ}$.

III conclusion

The new SI brought a huge progress by providing a new definition of the kilogram based on quantum physics. Still by fixing the value of $e$, it requires to adjust the coulomb and ampere, changing the vacuum magnetic permeability $\mu_{\circ}$ and impedance $Z_{\circ}=\mu_{\circ}c$ into unfixed parameters, to be measured experimentally. All electrical units become dependent on $\alpha$, and might even vary (very slightly) with time.

Still the new SI can be embedded within a larger framework, with $\hbar=c=\mu_{\circ}=\epsilon_{\circ}=k_{B}=N_{A}=1$, the vacuum magnetic permeability, electric permittivity and impedance being fixed and equal to 1. All units, metre, joule, kilogram, ampere, … and kelvin get fixed and defined in terms of the second, with the ohm, coulomb and weber dimensionless, $e=1.602\,...\times 10^{-19}\ \rm C=\sqrt{4\pi\alpha}=0.3028\,...$, and the mole equal to the Avogadro number.

Références

• (1)
• (2)

SI brochure, The International System of Units (SI) (8th edition,updated in 2014), https://www.bipm.org/utils/common/pdf/si_brochure_8.pdf

• (3)

I. Mills, P. Mohr, T. Quinn, B. Taylor and E. Williams, Redefinition of the kilogram: a decision whose time has come, Metrologia 42 (2005) 71, https://iopscience.iop.org/article/10.1088/0026-1394/42/2/001/meta

• (4)

I. Mills, P. Mohr, T. Quinn, B. Taylor and E. Williams, Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1, Metrologia 43 (2006) 227, https://stacks.iop.org/Met/43/227

• (5)

On the revision of the International System of Units (SI), Resolution 1 of the 26th CGPM (2018), https://www.bipm.org/en/CGPM/db/26/1/ ;

• (6)

SI Brochure: The International System of Units (SI) (9th edition, 2019), https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf ;

• (7)

The new International System of Units, Compte Rendus Physique (2019), to appear

• (8)

Definition of the metre by the international Prototype, Resolution of the 7th CGPM (1927), https://www.bipm.org/en/CGPM/db/7/1/

• (9)

(Former) International Prototype of the Kilogram, https://www.bipm.org/en/bipm/mass/ipk/

• (10)

P. Fayet, A new dual system for the fundamental units, Comptes Rendus Physique (2019), http://authors.elsevier.com/sd/article/S163107051930026X

• (11)

Definition of electric units, CIPM (1946), https://www.bipm.org/en/CIPM/db/1946/2/

• (12)

P. Mohr, D. Newell, B. Taylor and E. Tiesinga, Data and analysis for the CODATA 2017 special fundamental constants adjustment, Metrologia 55 (2018) n∘1, 125, https://stacks.iop.org/Met/55/125

• (13)

D. Newell et al., The CODATA 2017 values of $h$, $e$, $k$, and $N_{A}$ for the Revision of the SI, Metrologia 55 (2018) L13, https://stacks.iop.org/Met/55/L13

• (14)

Mise en pratique for the definition of the ampere and other electric units in the SI, https://www.bipm.org/utils/en/pdf/si-mep/SI-App2-ampere.pdf

• (15)

CODATA recommended 2018 values of the Fundamental Physical Constants, https://physics.nist.gov/cgi-bin/cuu/Value?alphinv, https://physics.nist.gov/cgi-bin/cuu/Value?mu0

• (16)

Vacuum electric permittivity, https://physics.nist.gov/cgi-bin/cuu/Value?ep0

• (17)

Definition of the metre, Resolution 1 of the 17th CGPM (1983), https://www.bipm.org/en/CGPM/db/17/1/  and Comptes Rendus des séances de la 17 ème CGPM, https://www.bipm.org/utils/common/pdf/CGPM/CGPM17.pdf , p. 98