Today I’m going to explain why you are not falling through your chair right now using one simple fact, and one object.

The fact is that all electrons are the same as each other, and the object is a structurally critical item of my clothing.

There’s a chance this episode could get a little weird.

By now we’ve established that quantum spin is very weird.

We talked all about that recently - how electrons have spin but aren’t really rotating.

And about how you need to turn an electron around twice - 720 degrees - to get it back to its starting position.

They are, we say, spin-½ - because one normal rotation only gets you halfway around.

That particular weirdness is not just another cute case of quantum mechanics being a bit silly.

The fact that some particles have this property is the entire reason that stuff in our universe has structure, and that matter doesn’t immediately collapse.

It’s the source of the Pauli exclusion principle, and today I’m going to show you exactly why this simple property makes it possible for us to have nice things in our universe.

Particles with spin-½ - or more generally any half-integer spin - 3/2, 5/2, etc.

are called fermions, and include all the particles that we think of as matter - from electrons to quarks to the neutrinos.

The other spin behavior is to have integer spin - spin-1 for example is the much more sensible case of a single 360 degrees rotation back to the starting point.

Integer-spin particles are bosons, and they are the force carrying particles like photons with spin 1 or the Higgs particle with spin 0.

Now it’s possible to stack as many bosons on top of each other as you like.

For example in a laser beam, there’s no limit to the number of photons you can add - all of them in the same quantum state.

But not fermions - no two fermions can share the same quantum state, which is why electrons can’t occupy the same energy states in atoms.

Without this, electrons in multi-electron atoms would all fall into the same lowest energy state - all atoms would be the minimum possible size, and there would be no such thing as chemistry.

No nice things like solids or molecules.

The non-overlap-ability of fermions is called the Pauli exclusion principle.

I’m going to show you why this is the inevitable behavior of groups of particles that have two properties: 1) this weird rotational symmetry, and 2) indistinguishability - which is just the fact that all electrons, for example, are exactly the same - there is no observable change when you swap two electrons.

Combining spin behavior and indistinguishability gives us something called the spin statistics theorem, which sounds complicated but I’m going to derive it with just some basic arithmetic and my belt.

What did you think I was going to say?

Before we get to my belt, let me remind you about spinors.

We talked all about it in the recent episode - but for now just know that it’s just the type of wavefunction that fermions have, and has this property that it returns to its starting state with a 720 degree rotation, not 360.

In that previous episode we saw this amazing animation of an object rotating while being fixed by bands to it's surrounding environment.

Crazily the bands disentangle every 720 degrees.

So a spinor’s rotational weirdness is not necessarily all that “quantum” - it’s a natural function of how it’s connected to the universe.

So allow me to introduce you to the belt trick, first conceived by Paul Dirac himself.

It goes like this: hold the ends of the belt in each hand so the belt is flat.

Let’s think of the belt buckle as a particle - say an electron - and the belt is its connection to whatever - the universe, or to maybe another electron.

Now rotate the electron a full 720 degrees so you have a double twist.

Now I’m going to untwist it without rotating either end of the belt.

Watch.

Now at this point I am letting go of the buckle - but the important thing is that the orientation of the belt ends don’t change with respect to each other.

And there you go, the twists are gone.

So the system under these sorts of rotations is a spinor because a 720 twist is topologically equivalent to no twist - simple translation of the ends transforms between the two states.

On the other hand, if we rotate one end 360 we can’t untangle the belt if we keep the ends fixed.

We can think both ends of the belt as spinor particles like electrons, and in that case we can do another experiment.

What happens if the ends exchange positions?

If we’re careful again not to rotate either end with respect to each other then the belt ends up with a single twist in it - equivalent to a 360 degree rotation of one end.

So it seems that for spinors, a 360 degree rotation is equivalent to the particles switching places.

This is slightly worrying - if we’re using our hands as analogies for electrons then we just demonstrated that it makes a difference if we switch their locations - and that's a problem if electrons are supposed to be indistinguishable which is one of the critical ingredients of the spin statistics theorem.

We’ll come back to how electrons can have this property and yet still be indistinguishable.

Okay, let’s summarize what we’ve learned.

There are objects that require a 720 degree rotation to return to their original configuration in terms of their relationship to other objects.

We call these things spinors.

For these same objects - spinors - a 360 degree rotation is the same thing as swapping places with a second spinor, at least in terms of the relationship between those spinors.

Okay.

How do we connect all of this to actual electrons?

Well electrons don’t really rotate in the classical sense.

They’re quantum objects described by a quantum wavefunction.

A wavefunction is this thing that holds information about the probability of a given property being observed.

For example, the wavefunction representing the position of a particle can look like a sine wave moving through space.

If you have two such wavefunctions overlapping - like two photons in a laser beam, a shift in one of them by half a wave cycle puts the two out of phase with each other.

In that case they actually cancel each other out and you effectively have no photons.

Mathematically, a half-cycle phase shift corresponds to putting a negative sign in front of the wavefunction.

That makes sense right - because adding the two wavefunctions causes them to sum to zero.

A spinor wavefunction of the electron can “wave” through space, but it includes another wavy part.

It has its rotational degree of freedom in which a full wave cycle is a 720 degree rotation.

In that case a 360 degree rotation puts a spinor perfectly out of phase compared to its starting point.

So a 360 rotation introduces a negative sign to the spinor wavefunction.

And that little negative sign is ultimately what drives the difference between fermions and bosons.

OK, so summarizing again: Electrons are spinors and so require a 720 degree rotation to be returned to their initial state.

But a 360 degree rotation shifts their phase by a half cycle and adds a -1 to the wavefunction.

But we also know from the belt trick analogy that swapping two spinors is the same as doing a 360 degree rotation - so that should put a negative sign in front of the combined wavefunction.

And that’s the last piece of the puzzle we need to get to the spin-statistics theorem, and the Pauli exclusion principle.

Now all we need is to do is the math.

Which you should relax about because we’re literally just doing addition and subtraction here.

Let’s think about the quantum state of an electron.

We’ll call it Psi.

Psi gives the distribution of probabilities of some observable - for example, the location of the electron around the atom.

Psi is really the so-called “probability amplitude”.

The actual probability distribution is the square of that.

We can never, ever observe Psi - all we can do is map Psi^2 by making multiple measurements.

The unobservability of Psi is critical to this whole explanation, so remember that.

Now let’s put our electron in an atom - in the ground state.

And add a second electron to the first excited state.

We can think of these two electrons as having a shared wavefunction - a two-particle wavefunction we’ll call Psi(A,B) - which has two electrons A and B, in which one occupies the ground state and one the first excited state.

We’ll see what actually goes into this wavefunction shortly.

But first let’s see how this thing should behave based on the stuff we figured out earlier.

Electrons are fermions, which means that if we swap their locations the wavefunction gets multiplied by -1.

Electron A goes into the first excited state and B goes to the ground state.

So now we have Psi(B,A) and we know that Psi(A,B) = -Psi(B,A) Wavefunctions that change sign like this when it's particle labels are swapped are called anti-symmetric under particle interchange, and those that don’t change sign are, unsurprisingly, called symmetric.

Fermions have antisymmetric wavefunctions, bosons, symmetric.

Remember that we wanted fermions to be indistinguishable from each other.

But it seems like the two-particle wavefunction changes if we swap the particles.

Doesn’t that give us a way to “distinguish” the swap?

Actually no - no observable property is changed by the swap.

Remember, that we only “observe” the square of the wavefunction, and in that square the minus sign goes away.

The square of Psi(A,B) is equal to the square of Psi(B,A) But although we can’t distinguish electron A from electron B through observation of these particles, it turns out that this subtle “unobservable” property has a powerful manifestation in the behaviour of groups of fermions.

To see that, we need to see what this two-particle wavefunction looks like in terms of the individual wavefunctions of our two electrons.

We’ll call the individual electron wavefunctions g and f - if you like for the ground and first excited state of the atom, but this works for any two possible wavefunctions - two possible quantum states - that our electrons could have.

If electron A is in the ground state then its wavefunction is g(A), if it’s in the first excited state it’s f(A), and the same for electron B.

Now let’s say we don’t know which of A or B are in the ground or excited states.

The two-particle wavefunction needs to be a combination of f and g covering all the possibilities - in fact, all possible combinations summed together in what we call a superposition.

There’s one way to do this that captures the anti-symmetric nature that we’re looking for.

It looks like this: Which we can think of as the sum of A being in the ground state, B in the exited, then B in the ground and A in excited.

Each term in a superposition has a coefficient out front - and in this case the coefficient for the second term is minus one.

We’re choosing it because it works.

To prove it, let’s switch the particles and the wavefunction sign should flip.

We want Psi(A,B) to become Psi(B,A).

And Psi(B,A) is just this thing we saw with all the As and Bs switched.

And that is just the negative of the original.

So Psi(B,A) = -Psi(A,B) - swapping electrons flips the sign - so we’ve successfully discovered the wavefunction for a pair of fermions.

Bear with me, we’re getting VERY close now.

I’m now going to show you that we can’t shove both electrons into the same state.

Let’s say we want both electrons to be in the ground state.

The two particle wavefunction would then look like this.

The fs become gs.

Now both components of the superposition are the same, with the exception that the second component has a minus sign - which causes those two components to cancel out.

Essentially, two electrons shoved into the same state end up perfectly out of phase and so destructively interfere.

But you can’t just vanish electrons - so the transition of an electron into an occupied quantum state is impossible.

This statement, together with what we saw from the belt-trick previously about spinors having anti-symmetric wavefunctions, is the pauli exclusion principle.

That is, particles with half integer spin have antisymmetric wavefunctions (the belt trick relation), and particles with anti symmetric wavefunctions cannot have multiple particles in the same quantum state.

The full spin statistics theorem has a lot more to it.

One part of it is a rather more rigorous explanation of why spinors must have anti-symmetric wavefunctions that doesn’t involve pant-retention technology.

It boils down to the fact that you need to use spinors in the Dirac equation - which is the quantum equation of motion for electrons and other spin-½ particles.

That equation by itself doesn’t force you to use symmetric or anti-symmetric wavefunctions - but if you try to use the symmetric wavefunctions of the boson then you get bad-wrong results.

Specifically, the energy level spectrum becomes unbounded from below, meaning you can continually remove energy from the system by lowering the state of a particle forever.

But if you use the correct anti-symmetric wavefunction then everything works just works out great.

So it’s a proof by contradiction: particles described by spinors have to have an anti-symmetric wavefunction and so must obey the Pauli exclusion principle - or you get nonsense.

So there you have it - matter has structure and you don’t fall through your chair because electrons are indistinguishable and they obey a simple, if odd, rotational symmetry.

Ironically, to prove the structural integrity of matter I had to compromise the structural integrity of my pants situation.

Revealing mysteries of the universe sometimes comes at the risk of revealing mysteries.

But that is a risk we’ll always take for you here on space time.