# Understanding of the Fermion, Boson difference

1. Oct 14, 2009

Ok my understanding of the Fermion, Boson difference is this:

Identical Particles carry a representation of the permutation group. Since we have not found any para statistics, this representation must be one dimensional. And there are only two one dimensional irreducible representations of the permutation group: the identity and sign(P). Depending on the representation we are dealing with Bosons or Fermions.

In 3d we can permute particles by moving them in space, and if the space is simply connected there can be no "orientation" in these swaps.

In 2d this is different. And now I don't really know how to make the connection. There is the claim, that one has to replace the permutation group with its simply connected extension aka Atkin's braid group, allowing for more statistics and there is the claim that this helps in understanding the fractional quantum Hall effect.

But I don't understand how confining an electron is supposed to allow it to change its statistics. The world is still three dimensional. Confinement cannot change this.
- So what are we talking about here? Electron excitations?
- Can the braid group statistics even be constructed from particles that don't adhere to them individually?
- And if it can what makes people think that the solid state environment will jump through what looks like very elaborate hoops to make anyon statistics work?

2. Oct 14, 2009

### xepma

Re: Anyons

In the fractional quantum Hall effect the electrons are effectively confined to move in two spatial dimensions. This means that whatever excitations comes out forth out of the collective behaviour of the electrons is also confined to these two dimensions. It's these excitations which can be viewed as quasiparticles living in a 2+1 dimensional world.

Your correct that the true world is still three dimensional. But the anyons in the fractional quantum Hall effect are described using an effective field theory. This effective field theory has a minimal length scale. As long as we do not approach this limit, we can employ the effective description. At some point the effective theory will break down, but then we lose whole the meaning of a quasiparticle anyway.

In the end, the funky statistics of the anyons is a reflection of the very complicated collective behaviour of the electrons. From some point of view you could also say that you are not even talking about statistics of the anyons, but rather a Berry curvature associated with the position of the quasiparticles. Braiding two quasiparticles means we are 'simply' adiabatically changing the wavefunction of the electrons instead. And it's this adiabatic change which induces a Berry phase (which is precisely the phase you expect if you just look at the statistics of the anyons).

3. Oct 15, 2009

Re: Anyons

Can the effective field theory be derived, or is it deus ex machina?

4. Oct 16, 2009

### xepma

Re: Anyons

Not from first principles. But this is the case for most strongly correlated electron systems.

But the effective field theory picture reproduces a lot of the experimental and numerical stuff, so do not write it off so fast. It's actually been quite succesful. Conductivity, fractional charge of the excitations, energy spectrum, etc, all come out quite nicely.

But you are pinpointing one gap: the statistics of the anyons has not been observed. It is sort of the thing everyone in the field is waiting for.

5. Oct 17, 2009

Re: Anyons