The natural units

$$G = c = \hbar = k_{B} = 1$$

set fundamental constants of gravity, relativity, quantum physics, and statistical physics to simple numbers.

However, surprisingly, the natural units imply that charge has the dimensions of length$^{-1}$.

How does this follow?

For reference, see the text in [3] of page 6 of the paper.

  • $\begingroup$ In the Standard Model, all three of the coupling constants for the U(1), SU(2) and SU(3) groups are dimensionless. $\endgroup$ – ohwilleke Jun 2 '17 at 1:18

The note uses Planck units to set the dimension of charge equal to the dimension of mass.

However, it does not state explicitly that this is true in Planck units. It only mentions that

$$G = c = \hbar = k_{B} = 1.$$

In Planck units, we also have

$$\frac{1}{4\pi\epsilon_{0}} = 1.$$

The dimension of $1/4\pi\epsilon_{0}$ is $L^{3}MT^{-2}Q^{-2}$. Therefore, we find that $Q = M$ in Planck units.

  • $\begingroup$ well, in Planck units, every physical quantity is dimensionless. not just $Q$ and $M$. $\endgroup$ – robert bristow-johnson Sep 13 '18 at 23:16

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