Can a Particle Be Neither Matter Nor Force? (Script) (Patreon)

All particles belong to two large groups: fermions like protons and electrons make everything we consider "matter", while bosons like photons and gluons transmit the fundamental forces. And that about covers the universe: matter moving through space and time under the action of forces.  But what if we could create particles in between these two possibilities. Physics says these neither matter nor force anyons can exist, and they may have some pretty incredible uses. 

[Intro]

We live in a conveniently simple universe. Two particle types—bosons and fermions. But what if there are actually infinitely many particle types? To see how this might be true, let's first do a quick recap on fermions and bosons. The most obvious behavioral characteristic of fermions is that they can never be in the same state as another fermion. If two electrons are spinning in the same direction and you bring them close together there will be a sort of force that repels them. 

This is known as degeneracy pressure or just  "quantum pressure" and it's the reason you don't fall through the floor. Bosons on the other hand have no problem being in the same state as another boson, they can pile up, and this is the reason light beams can pass through each other and that lasers are possible. But why can bosons share a state while fermions can't? Well, we explained this before in terms of the relationship between particle spin and the exchange symmetry of the wave function. That’s a great video to watch after this one, by the way. The short version is that swapping identical fermions introduces a phase shift to the combined wave function, putting the swapped wave function perfectly out of phase with the original

That means if both fermions are in the same state, then a superposition of swapped and unswapped wave functions cancel out, causing the wave function to vanish. Quantum mechanics forbids this, so instead a type of force appears to arise preventing fermions from overlapping. That’s our quantum pressure.

On the other hand, swapping a pair of bosons leaves the combined wave function unchanged, so you can have superpositions of pairs in the same state without any problems. 

The final part of that frankly awesome video was that we can connect the spin of the particle with this exchange symmetry—but you’ll have to watch for that, because I don’t want to have to take my belt off again. This connection to spin comes from bringing Einstein’s relativity into the picture and it was discovered by Wolfgang Pauli in 1940

Fermions had anti-symmetric exchange symmetry and half-integer spin, like ½, 3/2, 5/2, etc., while bosons have symmetric exchange symmetry and integer spin—so 0, 1, 2, etc. This story about what happens when you have particles swap places is the standard explanation of the difference between bosons and fermions and for the cause of quantum pressure. It also roughly reflects the way these particle types were discovered. 

But there’s a bit of a problem with this explanation. The whole idea of swapping particles is pretty ambiguous. We showed moving two particles together and through each other. But there’s a fundamental quantum limit to how precisely we can track the positions of the electrons, so what if the particles come extremely close but then retreat. Because the particles are indistinguishable it’s not possible to tell the difference between particles passing by each other to swap places versus an extreme close approach and then a retreat. In fact it’s not reasonable to even assign labels that remain fixed to an individual electron.

This is exactly why in 1977 Jon Magne Leinaas and Jan Myrheim set out to find a new way to think about fermions and bosons that took seriously the fact that naming electrons is silly. They succeeded, and in the process discovered a brand new type of particle—anyons. Let's go back to our example of two electrons moving towards and through each other. We’ll make it easy by pretending that this is a one-dimensional universe—the line the electrons move on is the only space that exists. That means we can describe their positions with one variable each. 

Every point in this plane represents a combination of locations for the two electrons. But we just said that we shouldn’t go around labeling electrons. In this space it’s clear how to avoid that. For example, this point here has electron 1 at position 1 and electron 2 at position 2. But that’s the same physical situation as here, where electron 2 is at position 1 and electron 1 is at 2 . In fact, this whole upper half of the plane is an exact duplicate of the lower half. So let’s delete the upper half from our space. This is what we call the configuration space for our two electrons, every point represents one possible configuration of our system. Maybe the electrons begin here, but then they both move in different ways to each other. As they move the configuration changes, tracing a line through configuration space.

But what happens if this reaches the diagonal border of this space? This represents the particles getting so close to each other we lose track of which is which. They could have continued on, passing by each other, which would look like this if we hadn’t eliminated that region. Or they could have both been deflected, like this. As far as our indistinguishable electrons are concerned these two paths have to be thought of as the same thing. That means this trajectory is the same as this trajectory so we only keep the part that’s inside our configuration space. 

So it seems this boundary has a very particular effect on the motion of our two electron system—it causes it to bounce back. Because we’re trying to do quantum mechanics here, we need to think in terms of the wave function, not particles. So what happens to the wave function at the boundary?

The wavefunction in quantum mechanics represents the distribution of probabilities for quantum properties—in this case the positions of electrons. As the name implies, the wavefunction is wavey. It’s a spread-out thing in which the value at each point is influenced by the values at neighboring points, and the whole thing moves around over time according to the Schrodinger equation

We can learn a lot about the two-particle wavefunction by requiring it to give zero probability on the other side of this boundary and to reflect at the boundary. The types of wavefunction that satisfy these conditions get reflected perfectly by the boundary, but with one important change: it picks up a phase shift.

Let’s look at a more physical analogy. When light bounces off glass or the surface of water its phase is shifted by a half of a wavecycle—that’s a phase shift of pi radians, 180 degrees. We can write this as the photon EM field multiplied by e to the power of i times the phase shift. From now on we’ll be defining the phase shift as the number half wavecycles that the wave is shifted—so e to the i pi times eta, where the Greek letter eta is that number of half cycles. So we’d say that light gets a phase shift of one when reflected off a mirror. One times pi is a half wavecycle shift. 

The wave function of our pair of electrons also gets a phase shift when it bounces off this abstract boundary in configuration space. That phase shift is also  one for the electron pair, and for any fermions. In quantum mechanics, a phase shift is applied to the wavefunction by multiplying by e to the i times pi times the shift. E to the i pi times one is Euler’s identity—it’s equal to -1.

So the combined two-electron wavefunction gets a minus sign on reflection. This is exactly the same “anti-symmetric” quality that we talked about in the previous video—swapping electrons flips the sign of the wavefunction. But this new way of looking at it is arguably better because it explicitly accounts for the fact that we don’t know which electron is which after their close approach.

This also works for bosons. In their case, the phase shift at the boundary is 0, and e to the i times 0 is 1. That means the boson pair bounces in configuration space with no change. That’s equivalent to our particles being symmetric on swapping the particles. 

OK, so we’ve recovered the basic behavior of bosons and fermions without pretending we can tell one particle from another. But something new and a bit worrying comes out of this argument. We know what the phase shift has to be for bosons and fermions, but are these the only phase shifts allowed? 

Back to the example of light—it’s possible to build a mirror that causes literally any phase shift you want in the reflected light. So could there be wavefunctions that experience a phase shift that’s not 0 or pi at the configuration space boundary? Actually yeah, in fact it’s possible to come up with wavefunctions that have any phase shift

Any at all. And a particle with a phase shift of anything but 0 or pi is called an anyon. For bosons and fermions, the phase shift or symmetry on exchange is related to their spin. The same would be true of anyons. That means anyons should be able to have any spin, not just half or full integer spins.

So if this is supposedly a more robust method for understanding particle statistics, and it predicts any conceivable spin, why does this universe only seem to have bosons and fermions with their very limited allowable spins? Actually, this prediction we just made for the existence of anyons is probably right - they should exist in the situation I just described—a one dimensional universe! 

Let’s add some dimensions and see if anyons still exist. If we have 2 electrons moving in 2 dimensions we would normally need a 4-dimensional configuration space—that’s 2-D for each electron. But we have a trick to get around this. We only care about the locations of the electrons relative to each other, so imagine our axes move around with the electrons. All we need is the distance between the electrons and their orientation.

We call this center of mass coordinates, and this gives us a 2-D configuration space for our 2 electrons in 2 spatial dimensions. This is what it looks like for the electrons to move around in real space and this new config space. Distance between them gets smaller and the configuration moves towards zero, also the position of the center of mass rises.

Once again, the indistinguishability of the particles means some of this new configuration space is redundant. Whenever they pass through zero separation we don’t know whether they’ve passed by each other or deflected back the way they came.

This has the effect of eliminating half the plane, because we can never know which particle is to left or to the right. Now when the configuration would cross the boundary it has to go to an equivalent configuration that is allowed. In one dimension that was just the reflection and that's why it bounced but in two dimensions any configuration is equivalent to a configuration that is swapped AND rotated by 180 degrees so that's what we have to do, and this results in a sort of pacman-like motion

This has the effect of eliminating half the plane, because we can never tell which particle is to the left or to the right but oddly enough when the configuration reaches this border it doesn't bounce back, but rather it "pacmans" to the other side.

In this case to be able to describe continuous non-pac-man-like motion we do a little arts and crafts—we fold the remaining half of the config space and join the edges.  Now the same motion in this space looks like moving around the cone. If our 2-electron system makes one circuit around the cone it means the electrons must have passed the Pac-Man boundary, so they passed very close to each other so that we lost track of which is which. This is the equivalent of bouncing off the boundary in the previous case. It also means the 2-particle wavefunction picks up a phase shift if it does this. If our particles are anyons, that phase shift could be anything.

If we want to reverse the phase shift we just have the system retrace its steps. If the phase shift for a clockwise motion in config space is some value eta, then the shift for an anti clockwise shift is minus that—minus eta. That has to be the case because rewinding things should subtract the phase shift that was added. For this 2-D case that phase shift can be anything we want—therefore 2-D anyons should still exist.

OK, so what about a universe like our own with 3 spatial dimensions? If we have 2 particles moving in 3 dimensions we need a 6-D configuration space to represent it. Even with tricks like center of mass coordinates we still have a 4-D space, which isn’t going to be easy to represent on your 2-D screen. What we can do is restrict our particles to a 2-D plane in that 3-D space. If we do that, the configuration space looks just like the previous one where the actual universe was 2-D

But there’s a difference in the case of the 2-D plane embedded in a 3-D universe compared to an actual 2-D universe. See, in a 2-D universe you can’t turn around to see the plane of motion from the other direction, but in a 3-D one you can. And believe it or not, it’s your ability to do that that ultimately makes anyons impossible. 

What do I mean by that? Remember the config space of the 2-D universe—the cone. Motion around the cone introduces a phase shift and motion in the opposite direction introduces the negative of the same shift. In the 2-D slice of our 3-D universe that should be exactly the same. Pass two indistinguishable particles through the same point and they pick up one phase shift, pass them backwards and they should pick up the negative to return to their initial state. In configuration space, the 2-D slice of our 3-D universe should give opposite phase shifts for clockwise versus counterclockwise motion.

But now we’re in 3-D space we can do something we couldn’t in 2-D space. We can rotate our point of view by turning around in the new dimension. Think of this like looking at a transparent clock from the front and then looking at it from the back. Clockwise will look like counterclockwsie even though the watch itself is moving in the exact same way

If we change our reference frame by rotating it 180 degrees the motion looks different—as though reflected in a mirror. Now if we construct the configuration space with respect to our new perspective, we see that motions are reversed. Clockwise becomes counterclockwise and vice versa

But this time we reversed the apparent particle motion by changing our frame of reference, not by moving the particles differently. So really there should be no difference to the phase shift compared to when we were looking from the other direction. Why should the quantum nature of electrons care about which way we’re facing? After all, two observers could be facing different directions and should observe the same phase shift. On the other hand, if we really did cause those particles to reverse their path we’d have counterclockwise motion on configuration space and we WOULD expect a negative phase shift. So it seems we have a contradiction. How can this clockwise or counterclockwise motion both give opposite sign phase shifts and the same sign phase shifts?

This can only happen if the ultimate effect on the wavefunction is the same because then there’s no way to distinguish why the rotation direction flipped.

Remember that the effect of a phase shift on the wavefunction is to multiply the wavefunction by e to the power of i times pi times the shift. For the most powers, e to the i pi eta is not equal to e to the minus i pi eta. Unless eta is an integer. Then the effect of a positive or negative shift on the wavefunction is identical.  Any even integer—0, 2, 4, or -2, -4, etc leads to multiplying the wavefunction by 1–these are bosons. Every odd integer—1, 3, 5, -1, -3, etc—multiplies the wavefunction by -1–those are our fermions.

Long story short: in a 3-D universe, the particle exchange symmetry has to work the same no matter the orientation of the system or the observer. That allows only phase shifts that are multiples of pi, which in turn translates to only bosons or fermions depending on whether it’s an even or odd multiple. Any other shift is disallowed because it breaks the universe. Which means anyons can’t exist in any universe with 3 or more dimensions. But in a 1- or 2-D universe there’s no way to view the particle motion from a different orientation in a way that reflects clockwise to counterclockwise. Therefore non-integer shifts don’t break anything and anyons are possible. So, what’s the use of all of this? Well, besides being a beautiful explanation of the origin of bosons and fermions, perhaps it does allow us to make anyons in this universe after all. 

Just last year a team in France figured out that if you stick two very specific alloys together the electrons in between them are sorta trapped and they can only move through the 2D surface where the metals they touch. Then by carefully applying magnetic fields they can give these particles more or less spin than what they should normally have, and as a result they behave like anyons.

This team even managed to move the anyons in specific directions and make them collide with each other, and by analyzing these collisions they were able to prove their anyons had a spin of ⅓ instead of the usual ½.  Anyons could also be useful in quantum computers. These work by controlling the flow of probability between two possibilities, but with anyons probability can flow in more directions, allowing quantum computers to do more complex algorithms, or even storing quantum information in ways that are very resistant to noise. We know all of this is possible, but we are still years away from a quantum computer based on anyons. And there are many other potential applications besides. 

Nature intended to give us only fermions and bosons to work with, but by looking very closely at how probability behaves in lower dimensions we were able to find a treasure trove of infinitely many particles with strange properties. They cannot exist without our help, but we’re now learning how to promote these strange lower dimensional entities up into our 3+1-dimensional spacetime.