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Renowned Austrian physicist Wolfgang Pauli, for whom the Pauli exclusion principle is
named, once said: “When I die my first question to the Devil will be: What is the meaning
of the fine structure constant?”
American physicist, Richard Feynman described it as “a magic number that comes to us with
no understanding.”
This constant represented by the Greek letter alpha, appears on the surface to be nothing
special.
It’s just a dimensionless number, so no matter what units you use, it will always
have the same value - approximately 1/137.
Yet, it’s precise value could not be more important.
Some studies show that if it was different by just 4%, you and I would not exist because stars could not make
carbon and oxygen, elements essential for life.
It’s value being so precise is one of the reasons that some believe nature must be fine tuned
for life.
This constant seems to turn up everywhere in physics, like the quantum theory governing
the electromagnetic force.
Yet its value cannot currently be predicted directly from any theory.
It’s one of the fundamental constants we simply have to measure experimentally in nature.
It’s one of the parameters that is inserted by hand into the Standard Model of particle
physics, our best theory of nature, to make the theory work.
Some numerologists have even suggested a mystical significance to the number 137 because it
seems to show up in other places in nature as well.
What is this mysterious constant?
Why does it have the value that it does? And what does it represent?
The complex physics behind this constant, simplified to the best of my ability, is coming up right now!
First what is the fine structure constant?
Well, the equation for alpha looks like this:
At one time it was believed to be exactly 1/137, but advancing technology has allowed
us to measure this much more precisely.
It’s equal to the following: And it is now one of the most precisely measured constants.
The predicted and measured value correspond with a standard uncertainty of only
0.23 parts per billion.
There are no units attached to this number because it is dimensionless.
Dimensionless units are interesting because they are independent of any arbitrary measuring
units we might use, such as meters or seconds, or grams.
It’s like pi.
It’s a kind of ratio.
If an alien race was advanced enough to figure out the same science that we have, they would
come up with exactly the same number for pi and for the fine structure constant, when
converted to log in base 10.
We use base 10 because we have 10 fingers and toes.
Aliens may use base 2 or 8 or 12 or whatever.
10 is arbitrary.
Numbers and equations are nice, but now the question is, what is the physical interpretation
of the formula that leads to this mystery number?
If we examine the formula closely, there are a few different ways we can write it out and think of what
this number represents.
First, if we rewrite the formula in a slightly different way, we can get a ratio that looks
like this: This represents the ratio of two energies
– the energy needed to overcome the electrostatic repulsion of two electrons that are a distance
d apart, and the energy of a photon with the wavelength lambda λ.
Another way to rearrange this formula results in a ratio that may be even easier to understand:
This essentially represents the ratio of the speed of an electron in a classical orbit
to the speed of light.
In other words, the speed of an electron orbiting the classical model of an atom is about 1/137th
the speed of light. Or the electron travels at 1/137th the maximum speed allowed in nature.
Keep in mind that this understanding would be in the Bohr model of the atom which modeled
it as a negative point charge spinning around a positively charged nucleus.
Our understanding of the model of an atom today is different.
With the advent of the Schrodinger equation and a deeper knowledge of quantum mechanics,
our modern model of the atom shows the electron being a negatively charged probability cloud
around a fuzzy positively charged nucleus.
How is the number useful in modern physics?
The number is directly related to the strength of the electromagnetic force.
In other words it represents the strength of the interaction between two charged particles.
So for example the higher the value, the greater the strength of attraction between an electron
and a proton, and equivalently, the greater the strength of repulsion between two of the
same charges, such as two electrons.
You might say, well, this sounds a lot like Coulomb’s constant in Coulomb’s law which
relates what the force between two charges, separated by a distance r, would be.
Well, you would be spot on because one way to think of the fine structure constant is like coulomb's constant expressed in
dimensionless units.
This constant is found everywhere because in our macro world, the two fundamental forces
we directly experience most are gravity and electromagnetism.
You know how important gravity is, but electromagnetism is everywhere too,
for example in the form of light which are electromagnetic waves,
your cell phone electronics,
and chemistry since it’s the interaction of negatively charged electrons and atomic
nuclei in atoms and molecules.
Thus the chemistry of life is determined by electromagnetism.
Since the fine structure constant characterizes the strength of the electromagnetic force,
it’s critical for life.
One way to visualize the fine structure constant is by using Feynman diagrams.
This diagram comes from the math of the quantum theory describing the electromagnetic force
called Quantum ElectroDynamics or QED.
The arrows represent two charged particles and the wavy line represents photons.
So in this diagram, what we are seeing is two electrons moving near each other, exchanging
a photon, and then being repelled away from each other.
QED shows that it’s the exchange of photons that causes repulsion or attraction.
I have a full video on QED if you want to know the details of this.
The vertex is the point where the charged particles release or absorb a photon.
And it’s this vertex where alpha, or the fine structure constant, plays a crucial role
because it is the coupling constant that tells us the exact strength of this interaction.
Alpha is related to the probability that an electron will emit or absorb a photon.
We have known the constant for more than 100 years.
It was German theoretical physicist, Arnold Sommerfeld who introduced it in 1916 when
he was expanding the Bohr model of the atom.
In 1912, Niels Bohr came up with a model of the atom, where the electrons orbited around
the atomic nucleus like planets orbiting the Sun.
But unlike planets, Bohr proposed that electrons can only occupy certain energy levels proportional
to Planck’s constant, which he called "atomic orbitals."
In other words, the energies of electrons in these orbitals were quantized.
And when electrons changed from a higher orbital to a lower orbital, they released energy in
the form of photons.
And they absorbed energy when they moved from lower to higher orbitals.
His original equation looked like this: The original Bohr equation from 1913 simply states
that the energy depends on the quantum number “n," which is simply the orbital that the electron is sitting in.
This model solved several problems in classical physics, and contributed to our current understanding
of quantum mechanics.
But it failed to precisely reproduce experimental results of the light emitted by atoms.
The energy levels appeared to split into two, whereas Bohr's model only predicted one.
Those additional levels were very close to each other.
But they indicated that Bohr’s model was incomplete.
It was too simplistic.
It did not take into account the spin of electrons or relativistic effects.
It only predicted the coarse structure of atoms – based on the main orbitals.
Sommerfeld was able to show that there is finer structure to the atom, that it has suborbitals.
Sommerfeld modeled the hydrogen atom as Bohr did, but incorporated a factor that was the
ratio of the electron's velocity to the speed of light.
He incorporated the fine structure constant, which more accurately described the observed
results of the energy emission of the atom.
Sommerfeld’s modified version of Bohr’s equation looks like this:
This equation takes Bohr’s original equation and adds an addition factor – the stuff
to the right of the plus sign.
You’ll notice that alpha, the fine structure constant, shows up in the additional factor.
This factor has to be added by hand after making measurements to find out what it is.
This was a brilliant modification by Sommerfeld that gave us a deeper understanding.
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So the next question is, how is the fine structure constant measured?
Alpha can be measured experimentally at cyclotron accelerators like Fermilab in Illinois.
This is done by accelerating an electron in a magnetic field, and measuring its magnetic
moment.
In very general terms, the electron acts like a spinning bar magnet.
The magnetic moment is related to the strength and direction of the magnetic field created
by this electron.
The alpha value can then be figured out from this measurement.
Why is this number 1/137 so significant?
Well, for one thing it’s a small number.
This means that electromagnetism is relatively weak at least compared to the strong nuclear
force.
The consequence of this is that electrons orbit on average, at a substantial distance
away from the proton.
This can be understood explicitly from the Bohr radius, which is the radius of a hydrogen
atom.
Where a naught is the radius, h-bar is the reduced plank constant, m_e is the electron
mass, c is the speed of light and alpha is the finestructure constant.
We see that because the radius of the atom depends on the inverse of alpha, then a smaller
alpha gives us a large radius.
For example if the nucleus of an atom was the size of a golf ball, the size of the electron
probability cloud would extend to 2.5 kilometers.
This distance allows electrons to be relatively available for exchange with other atoms, so
that chemistry can take place.
And thus life is possible.
On the other hand, this number is not too large, because otherwise atoms would not form
in the first place.
So, in our universe it is relatively balanced, not too low, not too high.
You might say the fine structure constant is in the Goldilocks zone.
In 1957, English astronomer Fred Hoyle and others worked out the process by which stars
produce heavy elements such as carbon, nitrogen and oxygen.
They found that the abundance of carbon in the universe could be explained only if the
fine structure constant had a value that made the nuclei of helium atoms more likely to
fuse to produce carbon nuclei than otherwise.
This higher probability of fusion makes it possible for all these heavier atoms like
carbon to exist.
And the existence of these elements is the reason life exists on earth.
Scientists calculated that if this constant was different by about 4%,
we may not have had these heavier elements because either Helium would not fuse or stars
would burn out too quickly due to too much fusion.
What then determines the value of α?
Well, scientists are still trying to figure this out since there is no theory that predicts
it.
Some believe that the value of alpha was set at the moment of the big bang due to the initial
conditions from quantum fluctuations.
Some think that there are tiny hidden dimensions as predicted for example in string theory,
and that it’s the geometry of these dimensions that fixes the value the fine structure along
with other fundamental constants as well.
Most scientists believe there must be some way to calculate this value.
We just haven’t figured it out yet.
Why do some people assign it mystical meaning?
Mainly because the number 137 seems to appear elsewhere too.
For example some say that it can’t be coincidence a that the age of the universe is 13.7 billion
years.
This is a stretch because current best estimates say the number is actually about 13.8 billion
years.
The number shows up in Jewish mysticism, as well as the Christian bible.
I think Richard Feynman said it best, “It’s one of the greatest damn mysteries of physics,
you might say the hand of God wrote that number, but we don’t know how he pushed his pencil.”
Is alpha really a constant though?
The answer is no.
Alpha actually changes as a function of the energy conditions under which you perform
your experiments..
This is what the theory of QED shows - that the coupling, that is alpha, changes depending
on the energy of the system.
It is indeed very close 1/137 at about zero Kelvin which is roughly the temperature of the universe.
Actually it’s a little over 2 Kelvin.
In the grand scheme of things, even room temperature or about 300 Kelvin, is also very low energy.
But at very high temperature like 10^15 Kelvin, such as what was present near the time of
the big bang, it would not be the same as it is today.
It would have been around 1/127 or larger,
but after a few minutes as temperatures and energies reduced, it would have reached 1/137
as today.
So this means that alpha was higher in the early history of the universe.
So why do we then call it a constant, if it actually isn’t?
The answer is simple, for most practical purposes we use it to describe systems that are so
cold, that this temperature dependence doesn’t affect its value.
So when we say that alpha is 1/137, we mean that this is this value at low temperatures,
and in that limit it is constant.
This however doesn’t tell us why it is 1/137.
As with the other parameters of the standard model, we don’t know why they have the values
they do.
You could invoke the anthropic principle because we do know that universe would probably look
very different if the value of alpha was significantly different from what it is right now.
But until we have a more complete theory of nature like a grand unified theory, the enigma
of α remains.
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