All particles belong to one of 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.

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's also roughly reflective of the  way these particle types were discovered in the math 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 and then retreated.

Because the particles are  indistinguishable it’s not   possible to tell the difference between  particles passing by each other to swap   places versus an extremely 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.

And 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 line 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 that wavefunction to give zero probability on the other side of this boundary and   to reflect the wavefunction at the boundary.

The types of wavefunction that satisfy these conditions   get reflected perfectly by the boundary, but with  one important change: they pick 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's electromagnetic field multiplied by e to the power of i times the phase shift.

From now   on though we’ll define 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 fermion pair.

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.

And 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!

So let’s add some dimensions and see if anyons  still exist in those universes.

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 with respect to each other.

We call this center of mass coordinates, and  it 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 seperation, and 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 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 in 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?

Well, they can't really.

But the only valid answer is 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.

So 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 is 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 for electrons.

Anyons could also be useful in quantum computing.

Quantum computers 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 these.

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.