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Those of us who are engineers were never fond of the common expression from physicists that

$$ \alpha = \frac{e^2}{\hbar c} $$

implying that the units of the elementary charge are in "$\sqrt{hc}$".

Those of us that are a little more anal (can't spell "analysis" without "anal") about dimension of physical "stuff" and units know that the complete expression for the fine-structure constant is

$$ \alpha = \frac{e^2}{(4 \pi \varepsilon_0) \hbar c} $$

but the seasoned physicists are thinking in terms of electrostatic cgs units where the unit of charge is defined so that the Coulomb constant $k_\mathrm{e} = \frac{1}{4 \pi \varepsilon_0}$ is set to dimensionless 1.

That seemed okay before May 20, 2019 when all of the "variables" in $\varepsilon_0 = \frac{1}{c^2 \mu_0}$ were defined constants and it didn't seem to be a whoop to use a different definition of a dimensionful constant $\varepsilon_0$.

But now $\varepsilon_0$ is a measured universal constant that is expressed with an error specification and is derived from $\alpha$ anyway. Are these HEP physicists gonna continue saying $\alpha = \frac{e^2}{\hbar c}$ or will they have to be more proper with their use of "constants" and dimensionality? Will they continue to say that the units of the elementary charge are "$\sqrt{hc}$"?

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closed as primarily opinion-based by Emilio Pisanty, Jon Custer, Kyle Kanos, GiorgioP, Cosmas Zachos Jul 17 at 23:33

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ HEP theorists (at least all the ones I know) usually just write $e = \sqrt{4 \pi \alpha}$ using natural units rather than electrostatic cgs units. $\endgroup$ – Triatticus Jun 19 at 23:43
  • $\begingroup$ i like that. (but Wikipedia says that these same HEP folks say the unit of charge are $\sqrt{hc}$. maybe that article is crap. I wish that Planck Units were completely rationalized so that $c = \hbar = 4 \pi G = \varepsilon_0 = 1$. I think eventually the HEP guys and cosmologists and TOE guys will eventually come to that. $\endgroup$ – robert bristow-johnson Jun 19 at 23:49
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My prediction is that the 2019 metrological redefinitions will have absolutely no impact on how theoretical physicists use natural units. They will continue to think of $\hbar$ and $c$ as $1$, and of $e$ and $\alpha$ as dimensionless, and there will be nothing improper about doing so. Most will continue to think of SI units as a bizarre historical monstrosity.

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  • $\begingroup$ +1, i'll see if there are other answers before you get a check mark. $\endgroup$ – robert bristow-johnson Jun 20 at 3:30
  • $\begingroup$ but SI is getting better. $\endgroup$ – robert bristow-johnson Jun 20 at 3:31

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