Magic numbers

A constant we made earlier

There are a fair amount of numbers used in Physics that really are magic. They are known as physical constants, but the reality is that they might as well be called magic numbers. Now many physicists would not be to happy about using the word magic in the same sentence as physics, but its true.

We have absolutely no idea why these constants have the values they have!

What's more is that there are a fair number of these... Planck's constant, Boltzmann constant, elementary charge (charge on an electron/proton), speed of light, mass of an electron, proton or neutron, the Bohr Magneton, the electron magnetic moment, the atomic mass unit and the fine structure constant (personal favorite of mine) and on it goes.

They define the universe in which we live. We know their values from experiments. Now and again we realize that one is a composite of some of the others, but ultimately we still don't know why they have these values. Here are a few of them...

Atomic mass unit amu	1.66054·10-27 kg
Bohr radius a0	5.29177·10-11 m
Electron radius re	2.81792·10-15 m
Planck constant h	6.6260755·10-34 J·s
Boltzmann constant kB	1.380658·10-23 J/K
Elementary charge e	1.60217733·10-19 C
Avogadro number NA	6.0221367·1023 particles/mol
Speed of light c	2.99792458·108 m/s
Permeability of vacuum μ0	4 π·10-7 T2·m3/J
Permittivity of vacuum ε0	8.854187817·10-12 C2/J·m
Fine structure constant α	1 / 137.0359895
Electron rest mass me	9.1093897·10-31kg
Proton-electron ratios	mp / me = 1836.152701

It turns out that

c² = 1 /(μ₀ε₀)

c is the speed of light
μ₀ is Permeability of vacuum
ε₀ is Permittivity of vacuum

so we can see an example were some of the parameters are related. Most are not. Most appear to be independent. I say independent because we currently don't have a theory linking them all.

More intriguing for me is why they have the values they do. The speed of light, the mass of an electron, the charge on a proton, the fine structure constant all have values that seem somehow arbitrary. Consider Einstein's famous equation

E = m c²

if the value of c (the speed of light) changed then the amount of energy associated with a give mass would be different to what it is now. Imagine if we are looking at this equation the wrong way. Re-arranging we get

 E / m = c² = k

I have just divided each side by m. Imagine the energy of the rest mass of a particle is quantized. By this I mean that an electron has an E of say 10. Now if c had a different value say twice what its value currently is then would this force m to be a quarter of its current value in order to keep E the same? eg

E = 10, c = 1 so m = 10

E / m = c² = 10 / 10 = 1²

but if c doubled  we would have

10 / m = 2² = 4 , so m = 2.5.

So rather than E being governed by m and c, m and c are actually governed by E. So c actually has the value it has because m has the value it has, and vice versa. Similarly there is another equation relating energy and wave frequency

E = h f

where h is Planck's constant, f is the frequency of the radiation and E is the resulting energy. Again, imagine if we have this the wrong way round and h and f are dependent on one another. If this were the case it would be necessary to show why E behaved in the way it did and why it was quantized in such a fashion. This would not be a trivial task!!!

What is the point of this post? Well in a way, there isn't one, other than to show that I think that any theory that explains "everything" needs to explain each of these numbers. If we can do that, then we really are cooking.