Return to Constants that Resolve to Unity

When I finished The Conservatory – Notes on the Universe, I went away and began to worry that I should remove the last couple of sentences.  Part of my concern, mulling over it, was due to me leaving out the t_Pl term to not only emphasise that it’s the age of the universe in units of Planck time, but also to both make it work if you try to use different units for the age of the universe and ensure that the temporal term is cancelled out.

The other concern was that perhaps I was being strong in my implication that Planck units are the fundamental units.  I know that there are people out there who will argue differently.  So, I did what I normally do in a situation like this and fiddled with a spreadsheet, and I am now prepared to double down.  Planck units are the fundamental units.  My logic is that, using these terms, everything fundamental resolves to unity, even (argued in the next post) the coupling constants.

I tried to find a different set of natural units that might replace Planck.  First I assumed that we’d need something smaller, so I posited “Centiplanck” units that are all 1/100^th of Planck units.

Recall the table at Constants that Resolve to Unity:

I modified this to reflect Centiplanck units:

With this change to the units, the reduced Planck constant no longer resolves to unity. 

Since it is one of the least quoted derived Planck value, and in that sense least important, I put the Planck charge back to unity:

This change simply makes things worse as now three fundamental physical constants no longer resolve to unity (because raised permittivity and Coulomb's constant are just the inverse of each other, I don't distinguish between them).

So I tried again by returning mass to its normal Planck value:

This is yet worse, with four fundamental physical constants no longer resolving to unity.

I was finally ready to take the nuclear option, to see what happens when I break the link between temporal and spatial units, knowing that by doing so I would lose the speed of light as a unit that resolves to unity:

This merely shifts the deckchairs around because while one fundamental physical constant now resolves to unity, we only get that by losing the speed of light.  The other option is worse:

None of fundamental physical constants resolve to unity.

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From the analysis above, it is obvious that there is only one natural unit scheme which allows all fundamental physical constants to resolve to unity.  Therefore, I do not believe that there can be more fundamental natural units – and it would be very strange indeed if Planck units were not woven into the very fabric of the universe*.

That is not to say that there is no value in other “natural units” to make calculations in your field of expertise easier, merely that none of the constants require any further explanation for their value when that value, across the board, when expressed in a fundamental natural unit scheme, is unity.

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* I was waxing poetic here, there is no fabric of the universe per se, and certainly no weaver.  Hopefully the reader understands what I was getting at when I wrote that.

The Conservatory - Notes on the Universe

In late February 2024, Sabine Hossenfelder put out a video on energy conservation.  The link there is to the point at 2:43 where she talks about the effect of space expanding on energy.

The key point, for the purpose of this discussion, is that a photon ends up having less energy after the space it is in expands.  That got me thinking and, as a consequence, I want to go through a thought experiment.

Imagine that you are the god of physics, starting off with nothing.  You want to create something, but you have an energy budget of precisely zero (in part because, of course, you don’t exist).

Say then that you create, over a period Δt, a circle of radius r=x.  Effectively, what you have done is stretch, out of nothing in this case, a curve to a length of l=2πx, the circumference of a circle with radius r=x.  We can think of this length having potential energy, because all else being equal, it would want to go the ground state (being nothing).  Note that potential energy can be thought of as negative.

The other way to balance the energy budget is for there to be energy associated with the circumference of the circle, positive energy that is of equal magnitude to the potential energy.

Now imagine that the energy that balances out the negative energy due to the expansion is expressed as if it were carried by a single photon. As Sabine mentioned, the energy of a photon is inversely proportional to its wavelength, E=hc/λ.  The lowest possible energy, therefore, relates to the longest possible wavelength associated with the circumference l=2πx.

It might seem intuitive to say that that wavelength is λ=2πx, but note that the distance between the nodes (null points) of a wave is actually half a wavelength:

So, the longest wavelength associated with any segment is double the length of that segment or, in this case, λ=4πx.  We will call this longest associated wavelength the “fundamental wavelength” from here on.  Note that it is equivalent to the circumference of a circle with a radius of 2x.

The minimum energy carried by our hypothetical photon would, therefore, be E=hc/4πx=ħc/2x.  For ease of reference, I am going to call this hypothetical photon a “carrier photon” in reference to that fact that it is “carrying” the mass-energy.  Being hypothetical, it should not be thought of as a real photon.

Note that the energy of a photon is proportional to its angular frequency E=ħω, so in this case, ω=c/2x.  Angular frequency is a measure of the number of wavelengths in a given time and, in this case, can be thought of at the number of wavelengths made possible during the period Δt.  Looking at the fundamental, we can see that the minimum is one half of a wavelength per Δt.

Let us now give x a value.  As a god of physics, you are not only non-existent, but also very lazy, so your effort at creation is the minimum possible, meaning that the radius of your circle is as small as possible.  As per my last post (Why I Like Planck), I will assume that this is one unit of Planck length, x=l_Pl=√(ħG/c³), making the fundamental wavelength λ=4πx=4π.√(ħG/c³.  This means that the energy of the carrier photon would be:

E=ħc/(2.√(ħG/c³))=√(ħc⁵/G)/2=E_Pl/2

where E_Pl is one unit of Planck energy.

The related angular frequency is inversely related wavelength, such that ħω=c/λ, or ω=c/ħλ.  If the wavelength λ=4πx=4π.√(ħG/c³), then the angular frequency is ω=c/2x=c/(2.√(ħG/c³))=√(c⁵/ħG)/2.  Given that t_Pl=√(ħG/c⁵), the angular frequency of the carrier photon is one half of the inverse of one unit of Planck time.  Remember that we are considering angular frequency, in this case, to be the number of wavelengths in the period Δt. You made one half of a wavelength, which means Δt is one unit of Planck time.

The only way to increase the angular frequency, associated with a circle with a radius of one unit of Planck length, and thus increase the energy of the carrier photon, would be to decrease the value of Δt, but its value is already the minimum possible.

This means that energy involved, at one half of one unit of Planck energy, is both the maximum and the minimum possible.

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Right, now we have to go back to the notion of potential energy.  By expanding a circle out of nothingness to a radius of one unit of Planck length, you (as the god of physics) generated negative energy.  Since the only energy that a carrier photon on the circumference of the circle can have is one half of one unit of Planck energy, we have three options:

The carrier photon’s energy exactly matches the potential energy of the expansion, leading to a balanced energy budget of zero.

The carrier photon’s energy is greater than the potential energy of the expansion, making the circle explode.

The carrier photon’s energy is less than the potential energy of the expansion, leading to the collapse of the circle.

The final two options aren’t very interesting, partly because they aren't reflective of a flat universe such as the one we live in.  So, let us assume that the amount of potential energy generated by expanding a circle such that its circumference increases by 2πl_Pl (that is, the radius increases by one unit of Planck length) is E_Pl/2.  If you expand the circle again, over a period of one unit of Planck time, by another unit of Planck length, then the total energy of the carrier photon becomes a full unit of Plank energy, E_Pl.  And so on.

This is, of course, reminiscent of the FUGE model so the question becomes – what about in our universe?

Our universe has been expanding for 13.787 billion years, give or take.  This corresponds to 4.351×10¹⁷ seconds and 8.070×10⁶⁰ units of Planck time.  We want to know how many wavelengths are being brought forward each unit of Planck time (right now).  Given that one additional half of a wavelength has been created per unit of Planck time of expansion, that cumulative number is now 4.035×10⁶⁰ per unit of Planck time.

Converting that into SI units, this is equivalent to a ridiculously high angular frequency of 7.486×10¹⁰³Hz.

A carrier photon with this angular frequency has an energy of 7.895×10⁶⁹J which corresponds to 8.784×10⁵²kg.  Which is in the ballpark of estimates of the mass of the observable universe.

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So, what am I saying?  Basically, I am saying that energy is conserved.  The whole reason that there is much more energy in the universe today than there was back 13.787 billion years ago is that we are in a situation with 7.895×10⁶⁹J of potential energy that is being balanced precisely by the energy introduced by expansion.

I am not saying that we live on the circumference of a circle, nor the two-dimensional surface of a black hole in a hologram universe, merely that the mathematics seems to work out – if you think of the conservation of energy in this slightly bizarre way.

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What might not be immediately obvious here is that the above is a very complicated way to arrive at the conclusion that the total amount of mass-energy in the universe is E_TOTAL=ꬱ(t).E_Pl/t_Pl/2, where ꬱ(t) is the age of the universe in units of Planck time (t_Pl) and E_Pl/2 is the amount of energy added to the universe every unit of Planck time, a conclusion I arrived at by considering the critical density of the universe (see here).  Note that energy is not related to time per se, but rather to the speed of light.  Therefore, in a different scheme of natural units, this relationship would not stand unless the natural unit of mass varied in inverse proportion to the variation in the natural unit of time.  These variations would impact in turn on the natural units of electrical potential and current and would affect the resolution of the reduced Planck constant to unity.

In summary, the fact that E_TOTAL=ꬱ(t).E_Pl/t_Pl/2 seems to be telling us that Planck units are fundamental in some sense.

Why I Like Planck

Any poor soul who has read all of my stuff, probably an imaginary person, or at the most a bot of some kind, will know that I like Planck units.  I’ve even been accused of assigning more importance to Planck units that one should.

To support an upcoming post, I want to explain why I like Planck.

Key to the FUGE model, making up fully half of the initialism, is granular expansion.  The notion is that the universe is granular at a very small scale.  The question then is precisely what scale most accurately reflects the granularity of the universe.

I prefer the Planck units as natural units for this task, since all but one of the fundamental physical constants resolve to unity when using them.

There is an exception, being the elementary charge, e.  However, it should be noted that the value to which the elementary charge resolves using Planck units is such that α=e², where α is the fine structure constant.  This falls out of the equation α=e²/4πε₀ħc, noting the resolution to unity of the three terms: the reduced Planck constant, ħ; the speed of light, c; and the “raised permittivity”, 4πε₀.  Note also that I consider that q_Pl=4πε₀ħc, because it is more meaningful than the other option (because as a consequence α=e²/ q_Pl²).

Note that α is a dimensionless constant which cannot therefore resolve to unity.  As far as I can tell, the only fundamental physical constant that would change in a universe which had a different value of α would be the elementary charge.

That all said, it is possible that there is another set of natural units that underlie the granularity of the universe.  If so however, each fundamental physical constant when expressed in terms of those natural units would make some sort of sense.  The problem with using an alternative is that the Planck units already make maximal sense of the fundamental physical constants.  Any deviation from them merely adds problems.

There are currently five alternative schemes:

• Stoney units,
• Schrödinger units,
• Atomic units,
• (Atomic) natural units, and
• Strong units

The sixth apparent alternative, “geometrized units”, is really just a subset of any other alternative scheme in which c and G resolve to unity.

The first two, Stoney units and Schrödinger units, are better than the schemes used in atomic physics because in both cases the related energy units are equal to Planck energy.  This might not be immediately apparent for the Schrödinger units, but it must be noted that in that scheme, the speed of light is not 1, but rather 1/α.  These schemes are not suitable for representing the granularity of the universe they both include the fine structure constant α in the definition of dimensional units, which is not dimensional and thus cannot be resolved to unity.  It is introduced via the use of the elementary charge as the basic unit of charge.  The reduction of the natural unit of charge to unity deletes any meaning from the value of α which is, on the other hand, obvious with the Planck scheme (that is, as mentioned above, α=e²/q_Pl²).

The schemes used in atomic physics (atomic units and the unhelpfully named “natural units”) both use the electron mass as a basis and is therefore not suitable for representing the granularity of the universe.  The same applies to strong units used in nuclear physics, which has proton mass as a basis.

If there is a better scheme, I certainly cannot think of one.  If there is any action that takes place in a period of less than one unit of Planck time, I cannot think of one.  If there is any fundamental particle that is shorter than one unit of Planck length or has a wavelength shorter than one unit of Planck length, I cannot think of one.

For that reason, when I think of the shortest possible time or distance, I always think Planck.

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Note that, buried in the next post, is another reason to consider Planck units to be eminently suitable.  It’s just a little difficult to explain.

Constants that Resolve to Unity

Note that all the non-grey items in the table below resolve to unity or are unity by definition, if one uses the Planck units.

The two grey items are included because they are linked to a constant that isn't formally recognised, as far as I know, which I have dubbed the "charge to structure ratio" for ease of reference.

Note that I also made up the terms “reduced permeability” and “raised permittivity” as I have not noticed any similar usage anywhere.  Perhaps the terms are already used, or something similar and, if so, again I’d appreciate knowing.

Normalising permeability and permittivity to unity is certainly not a new concept.  I use the term “resolve” because I am not really doing anything to these values above, other than expressing them in Planck units.  I consider the step of dividing or multiplying by 4π to be a “normalisation”.  This normalisation is applied to my selection of the Planck charge, such that q_Pl²=4πε₀ħc.  Note also that the term 4πε₀ appears in the expression for Coulomb's constant, k_e=1/4πε₀.  In other words, Coulomb's constant is merely the inverse of raised permittivity.  For that reason I have placed them together and generally consider them to be the same constant.

Wikipedia's page on Planck units, at time of writing, indicates that the selection of Planck charge as either q_Pl=√(4πε₀ħc) or q_Pl=√(ε₀ħc) is at the author's choice.  I disagree for two reasons.

First, Planck charge can be written in terms of the Boltzmann constant, noting again that k_e=1/4πε₀:

q_Pl=√(ħc/k_e)

Compare this with the equation for Planck mass:

m_Pl=√(ħc/G)

Second, it can be seen clearly that when q_Pl=√(4πε₀ħc) is selected, the expression for the fine structure constant becomes α=e²/q_Pl², indicating that the value is merely a representation of the ratio between the elementary charge and the Planck charge.  (See also Why I Like Planck, Constants that Resolve to Unity and Coupling Constants.)

Expansion on Uncertainty

In Is the Universe Getting More Massive?  (Flatness, not Fatness), I concluded that (if the universe is and has always been flat then) the mass of the universe is increasing at a rate of M / r = c²/2G.  I should have put deltas in there, ie:

ΔM / Δr = c²/2G

I went on to write:

This implies, to me, that if the universe is and has always been flat, then the mass of universe is increasing by one unit of Planck mass every two units of Planck time.  (Note that I reached the same conclusion in Is the Universe (in) a Black Hole? but I expressed it in terms of energy.)

In the linked article I wrote (where ꬱ is the age of the universe):

Of interest is the fact that, with the assumption that the universe is a black hole and that it is expanding at the speed of light, we can recall the equation for the Schwarzschild radius and get this result a little more easily:

r_s = ꬱ.c = 2GM/c²  =>  M = ꬱ.c³/2G

This last equation is for mass of the universe now but the implication is that for a given period of time Δt, ΔM = Δt.c³/2G, or

ΔM / Δt = c³/2G

From which we can conclude that Δr / Δt = c, but all this is saying is that the universe is expanding at c, which we already know.

We can go further though, using this relationship, noting that l_pl / t_pl = c and thinking of incremental changes (increments of Planck length and Planck time):

ΔM / Δr = c²/2G

=> ΔM.c . Δr = c³/2G . Δr² (multiplying through by c. Δr²)

=> ΔM.c . Δr = c³/2G . ℏG/c³ (noting that Δr² = l_pl²)

=> Δp . Δx = ℏ/2

This is the lower limit of Heisenberg’s Uncertainty principle (Δp . Δx ≥ ℏ/2).  Alternatively:

ΔM / Δr = c²/2G

=> ΔM.c² . Δr/c = c³/2G . Δr²

=> ΔE . Δt = ℏ/2

Which is the lower limit of an alternate expression of Heisenberg’s Uncertainty principle (ΔE . Δt ≥ ℏ/2). 

How then should this be interpreted?  The way I understand it is that if we consider the tiniest meaningful increment of time, by which I mean one Planck time, then we are being told by the Heisenberg Uncertainty Principle that the minimum change in energy must be half a Planck energy.  Now this might be the wrong way around, since the expansion and the flatness of the universe point to the lower limit of the Heisenberg Uncertainty Principle, so it could be that this principle is merely pointing to an emergent feature of “flat expansion”.  Or it could just be another big fat coincidence.

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Interestingly, if you find your way to the vacuum energy page at Wikipedia, you will find that there is an “unsolved problem in physics” note:

Why does the zero-point energy of the vacuum not cause a large cosmological constant? What cancels it out?

The cosmological constant is the energy density of space and in Is the Universe Getting More Massive?  (Flatness, not Fatness) I concluded that that the density of mass-energy of the universe which is not baryonic or dark matter is about 6x10^-10 J/m³, which is precisely what is measured.  In my model, this “unsolved problem in physics” is not a problem.

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I should point out that when I was looking for more information on “Planck atoms”, a term that I think was used in The Story of Loop Quantum Gravity - From the Big Bounce to Black Holes (as mentioned in Another Teeny Tiny Struggle), I chanced upon some documents by José Garrigues-Baixauli.  It was when I was perusing those that it struck me that my one Planck energy per two Planck time result was reminiscent of the Heisenberg Uncertainly Principle equation.  I am not in a position to agree with everything that José has written there, but I do notice some parallels in that he has arrived at a couple of similar ideas from a different direction.

This image is particularly evocative considering the contents of Spherical Layers, the image that followed and the rather opaque follow up in The Messiness of Layered Spheres (I promise that it made sense to me even before the clarifying edit that I have just performed, but I was inside my head at the time of writing so I had an advantage).

Half-Integer Spin and the Free Space Constants

Most of the time that you see the Planck constant used, it is used in terms of angular frequency (ω), so it’s not so much the Planck constant as the reduced Planck constant (ħ).  This is because it’s referring to an entire cycle of 180 degrees or two radians or 2π.

Most of the time you see the permittivity of free space or permeability of free space use (also known as the electric constant and magnetic constant respectively), you see that 4π is involved.  It’s as though we could talk about the reduced magnetic constant (µ₀) to remove that 4π term.  For example:

µ₀=2α.h/(e².c)=4π.α.ħ/(e².c) <=> µ₀-bar=α.ħ/(e².c)

Similarly, we could have a naturalised version of the electric constant (ε₀), which also almost always has a 4π involved:

ε₀=e²/(2α.h.c)=e²/(4πα. ħ.c) <=> ε₀-bar=e²/(α. ħ.c)

As discussed in What is the Planck Constant? these both resolve to unity in Planck units.  Note also, as discussed in Fine-Structured but not Fine-Tuned, α= e²/q_pl², meaning that µ₀ and ε₀ can be expressed in terms of ħ (unity in Planck units), c (unity in Planck units) and q_pl (unity in Planck units), ie:

µ₀-bar=ħ/(q_pl².c)=(ħ/q_pl²)/c=1

and

ε₀-bar=q_p ²/(ħ.c)=(q_p ²/(ħ)/c=1

This is quite useful, basically everything (at the Planck scale) resolves to unity and the only reason we have odd numbers is because of our units of length, time, mass, charge and temperature – or because of our arbitrary choice of reference mass (for α_G) and slightly less arbitrary choice of reference charge (for α).

The question arises though, why 4π?  The 2π for the reduced Planck constant, ħ, makes sense because of the angular frequency, because it’s referring to a full rotation through 360 degrees, or 2π radians, but how could 4π make sense?  Of course, I’ve given the game away in the title of this post.

A photon has what you could call a “normal” spin.  After a spin of 360 degrees it is identical to how it started.  Sub-atomic particles however, like electrons, have a quantum level half-integer spin, or spin-1/2 – they need a spin of 720 degrees to arrive back at an identical state from which they started out, or 4π radians.

This suggests that the electric and magnetic constants might be linked to a characteristic peculiar to quarks and leptons, ie half-integer spin.

What is the Planck Constant?

Recently, the Planck constant was mentioned in the news due to an agreement to change the definition of the kilogram.  Rather than returning to the reference kilogram mass, the kilogram is now to be determined based on the value of the Planck constant, which is now a defined value in the same way as the speed of light is a defined value.

The question that is a little difficult to find the answer to in all the new reports is “what precisely is the Planck constant?”  Of course you could go and ask Google, but when you’re reading an article, you sort of want all your answers in one place.

I did see one article which tried to address the question, with the following:

The Planck constant is the amount of energy released in light when electrons in atoms jump around from one energy level to another, explained physicist Tim Bedding of Sydney University.

Well, yes, sort of.  I am pretty sure that this is a journalist error rather than a physicist error.

The Planck constant (h) can be used to work out the energy of a photon, when we know its wavelength or its frequency: E=h.f=h/λ.  But that’s not all.  The Planck constant is often used as part of a sort of exchange rate, allowing you to convert from everyday units like seconds, metres and kilograms into quantum level units: Planck units like Planck time, Planck length and Planck mass.

A particularly useful thing about using natural units, like Planck units, is that fundamental constants resolve down to unity, that is, they equal 1.  For example, using Planck units:

•         Speed of light, c = 1
•         Gravitational constant, G=1
•         Coulomb constant, k_e=1
•         Boltzmann constant, k_B=1

As it is, the Planck constant itself is not a fundamental constant that resolves to unity, but the reduced Planck constant is and does – meaning that the value of the Planck constant in Planck units is 2π, so the reduced Planck constant is ħ=h/2π.  There are two other key constants, the permittivity and permeability of free space, or the electric constant and the magnetic constant, ε₀ and μ₀, which don’t quite resolve down to 1 either, but these resolve down to ε₀=1/4π and μ₀=4π, so you could have a raised permittivity and a reduced permeability of ε₀-bar=1 and μ₀-bar=1.

In an earlier article, I wrote about how it is possible to resolve even the fine-structure constant and the gravitational coupling constant (which are both dimensionless) to 1.

None of this would be possible without the Planck constant, so the added role of defining the kilogram should not be too much for it to bear.

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I thought I might add here that for many purposes in physics what matters more is angular frequency rather than just frequency per se.  That is to say its not just how many whatevers per unit of time, but how many rotations per unit of time.  A full rotation is 2π radians so the relationship between vanilla frequency (f) and angular frequency (ω) is ω=2π.f which means that there is another equation for the energy of a photon, E=(h/2π).ω=ħ.ω.  There is an argument that the unreduced Planck constant is only used for historical reasons and that the reduced Planck constant is that one that we should be using primarily.

I'll go into this a bit more in a later article.

Fine-Structured but not Fine-Tuned

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 (4πε₀).ħcα = e².

So the fine-structure constant is proportional to the square of the elementary charge, because ε₀, 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 4πε₀, 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 4πε₀ “raised permittivity of free space” or the “raised electric constant”.

This might seem to be a little bit of a cheat, but it should be noted that µ₀ has as similar but inverse relationship to Planck units, in that µ₀/4π (“reduced permeability of free space” or the “reduced magnetic constant”) resolves to 1 in Planck units, so that not only does c²=1/ µ₀ε₀ but that relationship remains the same when µ₀ and ε₀ 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 (ε₀/4π).ħcα = e².  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 is normalised.

Normalisation removes the mystery of why, when all the other fundamental constants seem to resolve to 1 at the Planck scale, these two (µ₀ and ε₀) 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 - (4πε₀).ħcα = (4π/µ₀).ħα/c = e².  Or, once reorganised - α = e²/(4πε₀).ħc = (e/√((4πε₀).ħc))².  So does √((4πε₀).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 = e²/q_pl², the fine-structure constant is effectively an expression of the ratio of the elementary charge (e) to the Planck charge (q_pl), in much the same way as the gravitational coupling constant is effectively an expression of the ratio of the rest mass of an electron (m_e) to the Planck mass (m_pl), or α_Ge = m_e ²/m_pl².  (If you look up “electromagnetic coupling constant”, you’ll be redirected to the fine-structure constant.)

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 little arbitrary.  There is a smaller charge that could be selected, that associated with quarks, which could be as low as 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 meaning of these coupling constants.  They are both defined as “a constant characterizing the attraction between a given pair of 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 is fine-tuning here, then there’s actually twice as much fine-tuning (on this single measure) as the fine-tuners are claiming.

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.  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 have steered clear of the elementary charge.