Observations on the fine structure constant - Part 1

A while back I did an introductory post on the fine structure constant. More recently I did a post on what was not the fine structure constant. In this post I am going to take another look at this delightful mystery. This is probably going to end up 2 or even 3 posts by the looks of things. Let's get on.

The fine structure is described on wikipedia, it has a number of physical interpretations, for example, the square of the ratio of the elementary charge to the Planck charge. Despite this interpretation and the others listed on the wiki page we still don't know what it is all about. You would think that 8 different descriptions we would be able to figure out what it is. This is not the case. We just can't figure it.

What is apparent to me though is that once we do understand this number we will have a far greater understanding of the universe we inhabit. The fact that it pops up so often in so many places. This thing is a big deal and anyone who managers to explain it will really have made a major break through in physics.

I have been looking at the Planck units in recent months and have found out some great stuff, but once again I have drawn a total blank when it comes to the fine structure constant. Above we mentioned that it can be represented by

$\alpha = $ $ \frac {e^2} {q_p^2}$    ... (1)

$\alpha$ - fine structure constant
e - elementary charge
$q_p$ - Planck charge

This is an amazing result, after all the Planck charge is given by

$q_p = \sqrt {4 \pi \epsilon_0 \hbar c}$    ...(2)

$hbar$ - Reduced Planck constant
$1 / 4 \pi \epsilon_0$ - Coulomb's constant
c - speed of light

Why do we get the result in (1), why isn't it 1? $\pi$ - 3.141592653? natural log - 2.718281828? the golden ratio - 1.61803398875? or even 42?

It isn't any of these, it's $7.297352569   10^{-3}  \approx 1 / 137.03599917$

137.03599917! - seriously what is that? Some strange solution to a Kepler triangle, some bizarre integral? Even the root looks no better

$\sqrt {7.297352569   10^{-3}} = 0.085424543 \approx 1 / 11.7062376$

Well what do we know?

1) The mass of an electron divided by the Planck mass does not seem to have any obvious relationship to the fine structure constant. This is also true for the classic radius of an electron and the Planck length.

2) It is dimensionless. So it is not energy or momentum, acceleration, velocity, charge or temperature, it is a number.

3) It is a ratio. In equation (1) above it is the ratio of two charge values. Whenever it is a ratio the two values have to be of the same type, eg charge, time, mass, momentum, energy etc so that the dimensions cancel.

That said, it could be the ratio of a potential energy and a kinetic energy, both are energy. It could also be the ratio of energy density and pressure, both of these have the same dimensions.

4) It isn't 1, that may sound obvious but think about it for a minute. The Planck charge and the elementary charge are remarkably close, they are pretty much within an order of magnitude.

5) You cannot actually get the Planck charge. The Planck charge is NOT an integer number of the elementary charge. Given our current understanding of charge, you cannot get isolated charges that are not integer values of the elementary charge. Therefore, it is not possible to isolate an amount of charge that is equal to the Planck charge.

May need to rethink this for Fractional Quantum Hall Effect.


Planck charge

Lets take a look at equation (2), the definition of the Planck charge. if contains 4 parts,

$\pi$ - 3.141592653 - if it contains $\pi$ then it may have something to do with circles and waves
$\epsilon_0    -     8.854178817    10^{-12} F.m^{-1}$
$\hbar    -    1.05457172     10^{-34}  Js$
$c - 299 792 458    m  s^{-1}$

It does NOT contain anything relating to Gravity and big G. If we divide by the Planck mass it becomes

$\frac {q_p} {m_p} = \sqrt {4 \pi \epsilon_0 G}$    ...(3)

The charge to mass ratio does NOT contain $\hbar$ or c. It does contain big G. $\epsilon_0$ is present in both equations.

The Planck charge ratio, just like the Coulomb to Gravity ratio

$\frac {F_c} {F_g} = \frac {e^2} {m^2}  \frac {1} {4 \pi \epsilon_0 G}  $   ...(4)

which is just

$\frac {F_c} {F_g} = \frac {e^2} {m^2}  \frac {m_p^2} {q_p^2}  = \frac {\alpha} {\alpha_G}$   ...(5)

where

$\alpha_G = $ $\frac {m^2} {m_p^2}$     ....(6) 

Alternatively if we replace the elementary charge with the Planck charge we get

$F_c = $  $ \frac {q_p^2} {4 \pi \epsilon_0 r^2} $    ... (7)

which becomes

$F_c = $ $\frac {4 \pi \epsilon_0 \hbar c} {4 \pi \epsilon_0 r^2} = \frac {\hbar c} {r^2}$ ...(8) 

In equation (8) we have a form of Coulomb's law that does not involve charge! What does that mean?

Ok, I think I'll leave this post here. Will continue shortly.

I wrote this while listening to this.