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.