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 "$\sqrt{hc}$"?


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

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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

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.

  • $\begingroup$ +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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