I "Natural" systems of units

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This is a question arising from the discussion in another thread.

It's my understanding that among systems of natural units, the Planck units system sets not only "all three of those to 1", but also does the same with the Coulomb and Boltzmann constants, leaving the elementary charge not similarly normalized, because trying to normalize that too, would introduce an inconsistency.

That's part of my impression from the Wikipedia Natural units article; however, @PeterDonis said you can't set ##c = 1, \hbar = 1, G = 1## -- that no more than 2 of those can be set to 1 without inconsistency.

So far, it's been my experience that when he flatly disagrees with me about something, he's right and I'm wrong, but in this instance I'm still in doubt:

The cited article says:

Thus, we cannot set all of ##k_e, e, ħ,## and ##c## to 1, we can normalize at most three of this set to 1.​
[In that statement, ##e## refers to the elementary charge.]

and later,

Planck units are defined by​

##c = ħ = G = k_e = k_B = 1##,​

where ##c## is the speed of light, ##ħ## is the reduced Planck constant, ##G## is the gravitational constant, ##k_e## is the Coulomb constant, and ##k_B## is the Boltzmann constant.​

That seems to me to show that in the Plank units system of natural units, not only are ##c, \hbar,## and ##G## being set to 1, but so are ##k_e## and ##k_B##, while ##e## is not.

But @PeterDonis posted while I was editing, so I'll read his reply now, which I'm sure will be insightful.
 
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You can't set all three of ##c##, ##G##, and ##\hbar## to ##1##, at least not if the ##1## is supposed to be dimensionless. The best you can do is set ##c = 1## and pick either ##G = 1## or ##\hbar = 1## (or something similar), but not both.

the Planck units system
This system doesn't make all three of ##c##, ##G##, and ##\hbar## equal to a dimensionless number ##1##; for the reasons explained above, that can't be done. What it does is, basically, set ##c = 1## dimensionless and ##\hbar = 1## dimensionless; this sets the units of length and time to be the same (Planck length/time), and the units of mass/energy to be the same (Planck mass/energy). You can see from the first equation for the Planck energy ##E_P## on the Wikipedia page for Planck units [1] that with these definitions, the units of mass/energy are the inverse of the units for length/time (because ##\hbar## is dimensionless and ##E_P = \hbar / t_P##); note that this equation also shows how the units of mass and energy are the same if we set ##c = 1## so ##l_P = t_P##.

Then the system defines a "Planck force" ##F_P## and uses that to define ##G## and the Coulomb constant, and uses the Planck energy to define the Boltzmann constant and Planck temperature. Setting those three constants numerically equal to ##1## just fixes the units of force, charge, and temperature to the "Planck" values. But this does not make any of those three constants dimensionless; we can work out what units they must have from the equations:

Since the units of force are inverse length squared (because force = mass x acceleration and each of those has units of inverse length), the equation ##F_P = G m_P^2 / l_P^2## tells us that ##G## must have units of length squared, and the equation ##F_P = q_P^2 / 4 \pi \varepsilon_0 l_P^2## tells us that the Coulomb constant ##4 \pi \varepsilon_0## has units of charge squared.

Since the units of energy are inverse length, the equation ##E_P = k_B T_P## tells us that Boltzmann's constant has units of inverse length times inverse temperature. (Note that if we choose to measure temperature in energy units, which is common, then ##k_B## will in fact be dimensionless, but this is nothing specific to Planck units.)

[1] https://en.wikipedia.org/wiki/Planck_units
 
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As a further note, Planck units as described in my previous post are an example of typical quantum field theory units; in QFT it's most convenient to set ##c = \hbar = 1## and go from there. In General Relativity, however, it's more convenient to use "geometric units", in which ##c = G = 1##. In these units, mass/energy and length/time have the same units, and ##\hbar## has units of mass/length squared (though ##\hbar## is basically not used in GR since GR is a classical theory). It is also often convenient to use units of length for charge as well; this actually allows the Coulomb constant to be dimensionless since force turns out to be dimensionless in these units (since acceleration still has units of inverse length, so mass x acceleration is dimensionless). And temperature is usually measured in energy units, so Boltzmann's constant is dimensionless as well in these units.
 
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Thanks for that explanation. I'll mull it over for a time.
 

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