There has been a lot of fuss about the fine-structure
constant (α),
perhaps because it’s a specifically odd value, at very very close to
1/137. And 137 is an odd number, both in
that it’s not even and also in that it’s a prime. And it’s a special prime, being a Pythagorean
prime because 88*88+105*105 = 137*137, and the square root of 137 is the
hypotenuse of a triangle with integer legs (4*4+11*11=137). 1/137 has a palindromic period number.

The value of the fine-structure constant is not, however,
precisely 1/137. It’s closer to 137.036,
which is not as sexy.

This doesn’t stop some people from getting excited about,
including our fine-tuning friends – for example Luke
Barnes. The reason for this
(they argue) is that if the fine-structure constant were even slightly
different then stars would either fail to produce oxygen (which I think we can
all agree is important) or fusion could not occur at all – with a margin of
about 4% either way.

The thing that’s a bit odd is that the discussion is all
about this fine-structure constant, and yet the value of the elementary charge
seems never to be mentioned.

What, you may ask, does the elementary charge have to do
with the fine-structure constant? If so,
that means you didn’t follow the

*Wikipedia link*regarding what the fine-structure constant is, because the very first two sentences are:
In

*physics*, the**fine-structure constant**, also known as**Sommerfeld's constant**, commonly denoted*α*(the*Greek letter alpha*), is a*dimensionless physical constant*characterizing the strength of the*electromagnetic interaction*between elementary charged particles. It is related to the*elementary charge**e*, which characterizes the strength of the coupling of an elementary charged particle with the*electromagnetic field*, by the formula (4π*ε*_{0}).*ħcα*=*e*^{2}.
So the fine-structure constant is proportional to the square
of the elementary charge, because

*ε*_{0},_{ }*ħ*and*c*are all constants (and 4 and π are also constant – note that I added the brackets above, they aren’t there on the Wikipedia site). What I find interesting is that 4π*ε*_{0},_{ }*ħ*and*c*are not only constant but, in Planck units, they all resolve to 1. Note also that*ħ*is the reduced Planck constant, the Planck constant divided by 2π. We could call 4π*ε*_{0}“increased permittivity of free space” or the “increased electric constant”.
This might seem to be a little bit of a cheat, but it should
be noted that

*µ*_{0}has as similar but inverse relationship to Planck units, in that*µ*_{0}/4π (“reduced permeability of free space” or the “reduced magnetic constant”) resolves to 1 in Planck units, so that not only does c^{2}=1/*µ*_{0}*ε*_{0}but that relationship remains the same when*µ*_{0}and*ε*_{0}are replaced with their increased and reduced versions respectively. Note also that the fine-structure constant can be expressed in terms of permeability, by the formula (*ε*_{0}/4π).*ħcα*=*e*^{2}. And these two constants frequently appear with a 4π in the appropriate place, almost they are begging someone to normalise them in a similar way to how the Planck constant it normalised.
Normalisation removes the mystery of why, when all the other
fundamental constants seem to resolve to 1 at the Planck scale, these two don’t. They do when normalised. What remains outstanding however is the fine
structure constant. It’s a dimensionless
value, so how could we possibly resolve it down to 1?

The answer is hiding in those equations - (4π

*ε*_{0}).*ħcα*= (*µ*_{0}/4π).*ħcα*=*e*^{2}. Or, once reorganised -*α*=*e*^{2}*/*(4π*ε*_{0}).*ħc*= (*e/**√(*(4π*ε*_{0}).*ħc*))^{2}. So does*√(*(4π*ε*_{0}).*ħc*) have any meaning that we should be aware of? You bet it does – it’s the*Planck charge*, or the charge on the surface of a sphere that is one*Planck length*in diameter and has a potential energy of one*Planck energy*.
So, put another way:

*α*=*e*^{2}*/q*, the fine-structure constant is effectively an expression of the ratio of the elementary charge (_{pl}^{2}*e*) to the Planck charge (*q*), in much the same way as the_{pl}*gravitational coupling constant*is effectively an expression of the ratio of the*rest mass of an electron*(*m*) to the_{e}*Planck mass*(*m*), or_{pl}*α*=_{G}*m*_{e}^{ 2}*/m*. (If you look up “electromagnetic coupling constant”, you’ll be redirected to the fine-structure constant.)_{pl}^{2}
If you read about the

*gravitational coupling constant*, you will note that there “is some arbitrariness in the choice of which particle’s mass to use”. It appears less arbitrary to select the*elementary charge*when considering the electromagnetic coupling constant (ie the fine-structure constant), but it is still a**arbitrary. There is a smaller charge that could be selected, that associated with quarks, which could be as low as***little**e*/3 (positive or negative depending on the type of quark).
Before I take the next step, I have to point out that while the
gravitational and electromagnetic coupling constants (as commonly understood)
are effectively an expression of the ratio between the relevant characteristic
of an electron to the relevant Planck unit, this isn’t the

**of these coupling constants. They are both defined as “a constant characterizing the attraction between a given pair of***meaning**elementary particles*”, electromagnetic attraction in the case of the fine-structure/electromagnetic coupling constant and gravitational attraction in the case of the gravitational coupling constant. There is also a definition based on the interaction of these elementary particles with the related field.
We could naturalise both of these constants by considering
instead “the attraction between a pair of Planck particles” or the interaction
of Planck particles with the relevant field, considering them to have both Planck
charge and Planck mass. When we do, the
values both resolve to 1.

Another way of saying this is the fact that the coupling
constants don’t have a value of 1 is merely because the electron mass and
charge are both smaller than the Planck equivalents (the mass is much smaller, but
the gravitational coupling constant is also much smaller than the fine-structure
constant). When people are talking about
the range in which the fine-structure constant could be varied without
affecting life in this universe (by preventing stars from doing what they need
to do to create the basic building blocks of life as we know it), they are
really talking about how much higher or lower the charge on the electrons and
protons can be. It’s actually a bit odd
that fine-tuners don’t do this because when they say that the fine-structure
constant can only vary by as much as 4% before we run into trouble, this is equivalent
to saying that the charge on an electron or proton can only vary by as much as
2%. If there

**fine-tuning here, then there’s actually twice as much fine-tuning (on this single measure) as the fine-tuners are claiming.***is*
Either way, it’s a bit unreasonable to point at the fine-structure
constant as an example of fine-tuning in and of itself. If the fine-tuners want to claim any
fine-tuning here, they need to point to the elementary charge (and, if they can
establish a correspondingly apocalyptic argument for gravity, the electron rest
mass). However, if they can explain why
elementary charge is odd in some way or could be something else than it is,
they are welcome to try. There doesn’t
seem to be anyone else looking into that and when people ask awkward questions
there’s a lot of “

*we just don’t know*”. And, so far, the fine tuners appear to steered clear of the elementary charge.