Coupling Constants

I revisited Constants that Resolve to Unity recently and I updated the table there, replicated below.

For the most part these were minor changes, correcting the symbols for current and temperature, fixing the column width, making the subscript notation consistent and so on.  But there was a more significant change, which was a mention of the ratio between the square of elementary charge e and Planck charge that is expressed by fine structure constant α.

I had been thinking about doing something similar with another value, the gravitational coupling constant α_G, but this is a bit of an odd constant as it’s often poorly defined.  I kept niggling at it, trying to get a good explanation of what all these coupling constants are, including the fine structure constant.  I noticed that there is a mathematical similarly between α and α_G, beyond them both using alpha as their notation.  Namely, that α=e²/q_Pl² and α_G=m²/m_Pl² (where m is the mass of whatever you are using as the basis of the gravitational coupling constant, which you can clarify by using a subscript to both the m term and after the G in the α_G term, but they often don’t).

The clearest explanation for these coupling constants that I found was here, but there is an error in it and it is not completely clear, so I will have a hack at it in a way that totally avoids the introduction of that error.

Thanks to Professor Watkins, who has since retired.

---

I will start with gravitational coupling since this is the most intuitive and then go back to charge coupling.

We know that a body with mass m₁ that is a distance r away from a second body with mass m₂ will experience a force of gravity F_G given by:

F_G=Gm₁m₂/r²

We could think of m₁ and m₂ being accumulations of unspecified numbers n₁ and n₂ of neutrons that each have mass m_n such that m₁=n₁m_n and m₂=n₂m_n.  This means that the force F_G varies by the number of neutrons involved and the distance and is mediated by a constant value (dubbed a “force constant” by Watkins, with the choice to use φ to denote it being mine).  So:

φ_G=G.m_n²

And:

F_G=Gm₁m₂/r²=φ_G.(n₁n₂/r²)

Watkins then introduces the term ħc and shows, numerically, that:

φ_G/ħc=α_G

where α_G is specifically the gravitational coupling constant for a neutron, so to be more accurate, after rearranging:

α_Gn=φ_Gn/ħc=m_n².(G/ħc)

Note that the value of Planck mass is m_Pl=√(ħc/G).  Therefore, G/ħc=1/m_Pl² and so:

α_Gn=m_n²/m_Pl²

Watkins does not explain the introduction of the term ħc, so I suggest a slightly different approach.

The term φ_G, or rather φ_Gn, can be broken down further into two terms, one that carries the units and one that is dimensionless, noting that the dimensionless term m_Pl²/m_Pl²=1:

φ_Gn=G.m_n².m_Pl²/m_Pl²=(m_n²/m_Pl²).(Gm_Pl²)

The ratio of the two masses is, of course, dimensionless and can therefore be assigned as α_Gn.  We know that m_Pl=√(ħc/G) so the second term becomes:

Gm_Pl²=G.(√(ħc/G))²= G.ħc/G=ħc

So we arrive at φ_Gn expressed as two terms (the second being the ħc that Watkins introduces without explanation):

φ_Gn=α_Gn.ħc

or, rearranged:

α_Gn=φ_Gn/ħc [=m_n².(G/ħc)]

---

Note that the derivation above was done in terms of a neutron mass m_n but there was no stage at which the value of m_n needed to be specified.  Therefore, the exact same logic applies irrespective of what mass we use, so we could say:

α_Gx=m_x²/m_Pl²

where x can refer to any mass; that of a neutron, electron, proton, Planck mass and any other random mass you want.  Filling in the first four, we arrive at:

 

The error that Watkins introduced was with the value of α_Gn.  For some reason, Watkins arrives at 3.76915×10^-39 as the ratio between 1.871855×10^-64 and 3.1616×10^-26, which is clearly wrong (1.871855/3.1616=0.5920594, with the small variation in magnitude being due to a difference in our values for G).  This error was irrelevant to the logic of the derivation.

---

The same sort of process as above can be used to derive an electromagnetic coupling constant via the formula:

F=q₁q₂/4πε₀r²

Using the same logic, thinking of q₁ and q₂ as unspecified numbers n₁ and n₂ of units of elementary charge e, we see that q₁=n₁e and q₂=n₂e.  There is another force constant φ_E implied here (E for electromagnetic):

φ_E=e²/4πε₀

Watkins just introduces ħc, so that we get:

α_E=φ_E/ħc=e²/(4πε₀.ħc)

And, because q_Pl=√(ħc.4πε₀):

α_E=e²/q_Pl²

The electromagnetic coupling constant associated with the elementary charge isn’t normally called that, but is rather referred to as the fine structure constant, and the subscript E is not used.  So:

α=e²/q_Pl²

As with the gravitational case, the φ_E term can be broken down, this time using q_Pl²/q_Pl²=1:

φ_E=e²/4πε₀.q_Pl²/q_Pl²=(e²/q_Pl²).(q_Pl²/(4πε₀))

The ratio of the two charges is dimensionless and can be assigned as α.  Noting that q_Pl=√(ħc.4πε₀), the second term becomes:

q_Pl²/(4πε₀)=ħc.4πε₀/(4πε₀)=ħc

So:

φ_E= α.ħc

Or, rearranged:

α_E=φ_E/ħc [=e²/(4πε₀.ħc)]

---

Note also that k_e=1/4πε₀, so:

α_E=e²/(4πε₀.ħc)=e²/(ħc/k_e)

---

Similarly with the gravitational coupling constant derived above, a particular charge was selected with the electromagnetic coupling constant (or fine structure constant) relating specifically to the elementary charge.  Other charges are available, such as the charge on up and down quarks, +2e/3 and -e/3 respectively, or the Planck charge.

Therefore, as per the logic above, we could say that:

α_Ex=q_x²/q_Pl²

where x can refer to any charge.  This gives us, for the options above:

 

---

Note that I have included two terms α_GPl and α_EPl.  These could be referred to as the “natural coupling constants” (gravitational and electromagnetic respectively).  For the purposes of adding to the suite of constants that resolve to unity, this means I now have two more.  Doing so demotes the fine structure constant to what it actually is, just the square of the ratio between the charge selected to represent (so, the elementary charge) and the natural unit (the Planck charge).  The same applies to whatever variant of the gravitational coupling constant you would like to talk about.  The natural value is unity, and any other value is merely the square of the ratio of the mass selected and the Planck mass.

Of course, it would no longer be possible, if anyone actually does this, to say that the fine structure constant tells the universe what the relationship between the elementary charge and the Planck charge is.  But it doesn’t, the fine structure constant is merely something that falls out of the fact that the elementary charge has a specific value.

I remain totally unconvinced that the fine structure constant tells us anything more than the ratio between the elementary charge and the Planck charge.  The physical interpretations given at Wikipedia all seem to fall out of this ratio.  (I plan to write something on this in a later post.)

The same logic applies to the various options for the gravitational coupling constant.

To ram the point home further, we could establish the gravitational coupling constant in terms of standard duck masses (m_duck), because the equation is, very simply:

α_Gduck=m_duck²/m_Pl²

And the same if we use the standard charge on a post-launch duck (q_duck), in which case the duck centred fine structure constant becomes:

α_duck=q_duck²/q_Pl²

By careful selection of the values of m_duck or q_duck, a sufficiently motivated researcher could come up with startling numbers the coupling constants – much more exciting one than ~1/137.  Want it to equal pi to ten significant figures, just make q_duck=3.3233509704×10^-18C.

The immediate response is (or should be) that this duck centred fine structure constant (α_duck=π) doesn’t mean anything.  It certainly wouldn’t tell us why the standard charge on a post-launch duck (q_duck) had that value, if it did.  The value of the α_duck would simply be a different way of saying that q_duck has the value that it has (q_duck= q_Pl.√α_duck) – a simple truism.

Similarly, none of the variants of the gravitational coupling constant nor the fine structure constant are really telling us anything useful in themselves.

---

There are (at least) two other coupling constants (associated with weak and strong interaction), but I don’t really want to get into them.  I don’t think they would relate to Planck units, having done a quick check.  Perhaps someone else with a better knowledge of such things could have a good think about it and let me know.

It is possible that they tell us something, but it is equally possible that these coupling constants are similar to the gravitational and electromagnetic coupling constants in at least that sense and tell us nothing new at all.  My brief and patently inadequate check indicated that while the wording “coupling constant” is used for all four, and the meaning is comparable between the weak and strong interactions, they are not actually the same sort of thing as we talking about with respect to electromagnetic and gravitational force coupling.

That said, it is interesting to note that the strong force is usually thought of having a relative strength of 1.  Maybe you might thing, ah, but that's different.  Note however, that while the relative strength is listed at the link above as being in the order of 10^-3, it's more often listed as being in the order of 10^-2 and frequently more specifically as 1/137 (or more precisely, α).  Sadly, however, if you dig a bit deeper, you will find that the strong force is only approximated as 1, and sometimes has a value closer to 14.5 (or the inverse thereof, the explanation I saw was a bit vague on that point), and - in terms of a nucleus - is only the residual force left over from holding the constituent quarks of a proton or neutron together. 

---

Finally, here is the updated table of constants that resolve to unity (noting that I have removed the charge to structure ratio as irrelevant, added some more formulas for clarity and also changed to a font that makes the difference between a large “i” and a small “L” a little more distinct):

And, for ease of reference, a clean version with the greyed-out items removed:

All unity, as far as the eye can see.  Truly beautiful.