Another Fine Mess You Have Got Yourself Into, Luke Barnes

In his New Atlantic article, The Fine Tuning of Nature's Laws, Luke Barnes provided this image:

Below it was explanatory text:

“What if we tweaked just two of the fundamental constants? This figure shows what the universe would look like if the strength of the strong nuclear force (which holds atoms together) and the value of the fine-structure constant (which represents the strength of the electromagnetic force between elementary particles) were higher or lower than they are in this universe. The small, white sliver represents where life can use all the complexity of chemistry and the energy of stars. Within that region, the small “x” marks the spot where those constants are set in our own universe.”

While he doesn’t specify clearly, I think he is making an error here.  Unfortunately he doesn’t talk much about charge (both mentions are in reference to what electromagnetic force is not about charge per se), but it seems like he might be suggesting that it could be possible to change the fine-structure constant without changing the elementary charge.  I have a vague recollection of having even read it, make it less of a suggestion and more of an explicit statement, but for the life of me I can no longer find it.

Even if he is not making such a claim, he is still putting the cart before the horse.  Electromagnetic force between elementary charges is going to be proportional to the magnitude of the elementary charge, full stop.  It has nothing to do with the value of the fine structure constant which is merely a representation of the ratio between the elementary charge and the Planck charge (α_(e)=e²/q_Pl², where the (e) subscript highlights that the calculation of the electromagnetic coupling constant [also known as the fine structure constant] is calculated on the basis of the elementary charge).  He should be talking about the value of the elementary charge perhaps, and not the fine structure constant.

Sure, if a hypothetical elementary charge were z times that of the elementary charge (e_hyp=ze), then (at the same separation, r) the repulsive electromagnetic force between two protons would be z² times as strong – and the attractive force between an electron and a proton would also be z² times as strong.

For two protons, it seems that the maximum value of e_hyp may well be the Planck charge, noting that it was calculated in SI World and Planck World that the strong force is more than sufficient to hold two Planck charges together at a distance of femtometre.  This means that there is another way to look at the fine structure constant – that is as a representation of how finely tuned the strong force is not.  It is quite bit stronger than it needs to be (perhaps in the order of a thousand times).  Alternatively, if there is a natural limit to the possible charge on subatomic particles to the Planck charge, then it is possible that the elementary charge could have any non-zero magnitude below that of the Planck charge, giving the fine structure constant any non-zero value below unity.

Consider then an electron and a proton bound in a hydrogen atom.  The electron can be thought of as being prevented from spiralling into the nucleus by the balance of forces (there is also an argument from the basis of the kinetic/potential energy balance but note that this argument, as presented, uses a leap that is not explained and is thus not accounted for adequately – see also the Bohr model which co-incidentally points towards the nature of the leap). 

For hydrogen, (note the subscript used here is to emphasise that we are talking about an electron):

m_ev_e²/r_e=e²/4πε₀r_e²

m_ev_e²=e²/4πε₀r_e

Note that 2πr_e=λ_e so

m_ev_e²=2π.e²/4πε₀λ_e

e²/4πε₀= m_eλ_ev_e²/2π

but note that p_e=m_ev_e and λ_e=h/p_e, so m_eλ_e=h/v_e and so

e²/4πε₀=(hv_e/2π)=ħv_e

Thus

v_e= e²/4πε₀ħ

Note that this lines up with the value calculated here for a hydrogen atom where the principal quantum number n=1.

Now consider that α_(e)=e²/4πε₀ħc and q_Pl=√(4πε₀ħc), we have

v_e=(e²/q_Pl²).c=α_(e).c

It follows, therefore that

v_hyp=α_hyp.c

This clearly places a natural limit on the charge of an electron such that 0<α_hyp<1, meaning that 0<e_hyp<q_Pl (assuming non-zero mass, otherwise “≤” might apply at the upper end).  This is a form of mathematical confirmation of the intuition obtained from consideration of the case of two protons.

Note the immutability of this equation.  If you change the magnitude of the elementary charge, you change the magnitude of the electromagnetic coupling constant and therefore you change the speed of the electron.

Looking back at an earlier equation and multiplying through by ħc/ħc

m_ev_e²= ħc.e²/4πε₀ħcr_e

Recalling that α_(e)=e²/4πε₀ħc and that v_e=α_(e).c

m_e(α_(e).c)²= ħc.α_(e)/r_e

m_e.α_(e).c = ħ/r_e

Rearranging for r_e

r_e=ħ/(m_e.α_(e).c)

Compare this with the Bohr radius which is given by

a₀=4πε₀ħ²/m_ee²

Multiplied through by c/c and noting that α_(e)=e²/4πε₀ħc

a₀= ħ.4πε₀ħc/e²/(m_e.c)

a₀=ħ.1/α_(e)/(m_e.c)= ħ/(α_(e).m_e.c)=r_e

By extension, we see that

r_hyp=ħ/(m_hyp.α_hyp.c)

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The implication here is that there is some flexibility in the related values for our hypothetical electron, α_hyp, m_hyp and r_hyp.  We have already established that 0<α_hyp<1.  Looking at the extremes:

If α_hyp→0, and m_hyp has any value, then

r_hyp→∞

If α_hyp→1, and m_hyp=m_Pl=√(ħc/G), then

r_hyp→ħ/(√(ħc/G).c)=√(ħG/c³)=l_Pl

If α_hyp→1, and m_hyp>m_Pl=√(ħc/G), then

r_hyp<l_Pl

If α_hyp→1, and m_hyp→0, then

r_hyp→∞

In other words, the orbital radius of the electron could (depending on the choices for the other two values) be anything greater than the Planck length.  The mass of the electron has a soft limit at the Planck mass, but could have higher values if the fine structure constant were sufficiently low.  Note that we would eventually run into problems with low values of the fine structure due constant gravitational attraction swamping the electromagnetic repulsion meaning that there is another soft limit hidden in there.  Also, there is a limit due to the size of the nucleus, meaning that r_hyp would have to be significantly above the femtometre scale. The magnitude of the fine structure constant is limited to between 0 and 1, as explained above.

In reality, the only possible driver of any fine tuning here, if anything, is the value of the strong coupling constant, but this only affects particles in the nucleus, and it is more than sufficient to bind two particles with a unit of Planck charge each as close as a femtometre to each other.  The tuning, such that it is, is with respect to the separation at which the force is maximised – but even then, this is about 10⁴ tighter than the electron orbit.

It appears, therefore, that the much vaunted “fine tuning” is, in fact, pretty damn coarse.

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I don’t know enough about the (residual) strong force to work out if there are any natural limitations to its strength.  But the valid magnitude for the fine structure constant is certainly limited to between zero and unity, so Barnes’ image should at least look like this:

Since nothing happens in the region above where the strong constant has a strength of 1, we could safely ignore that space – which Barnes sort of does with his squeezing 10-∞ in a region that is about half that of 1-10.  Just keep in mind that this is arguably a hypercorrection:

Barnes’ scale is strangely pseudo-logarithmic, centred on unity with more than one half of it taken up by values, on both axes, between 0.1 and 10.  At first glance (especially prior to correction) it seems that the line above the “carbon-impossible” region might have a shape that is merely an artifact of his selection of scale.  But with the cut-down version, we can see clearly that this isn’t the case, the point [0.1,01] sits on the line, but neither [0.01,0.01] nor [1,1] do.

While there is very limited data, I suspect that what Barnes did was use a combination of logarithms and square roots of the offset from unity to construct his scale.  I don’t know why he did that.  If he’d not used such a scale, he could still have made his point, perhaps even more strongly.  If he didn’t use his strange logarithm scale, he could have had something more like this:

Note that I’ve just used his scale and plotted the intersections, I’ve not tried to reproduce the curves.  I am not commenting on his claims in the coloured sections per se, but I have added my caveat to try to prevent the image being misused.  Remember, you cannot change the fine-structure constant without affecting the elementary charge – by definition.

Why didn’t Barnes present his case more like this?

I suspect that the problem is that it would appear to make his case too strongly and that would have attracted closer, unwanted sceptical scrutiny.  There is good reason to label apologists as “liars for Jesus”, Barnes among them.

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Interestingly, in a later paper, Barnes does not mention the fine-structure constant at all (although he does use the symbol α without saying what it means, it is used in a claim about the mass of a proton).

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.