# Anyons: distinguishable or indistinguishable?

1. Apr 6, 2012

### complement

Hey!

I have started reading about anyons in a book called fractional statistics and quantum theory.
In this book and also on the wiki article I have found that anyons are supposedly indistinguishable particles.
However, I went on searching and found info on distinguishable anyons e.g. . I am pretty puzzled now and I hope someone can illuminate me.

This is closely related to:

Thanks!

Last edited by a moderator: Sep 25, 2014
2. Apr 7, 2012

### element4

In three dimensions, exchange between distinguishable particles (say of different spin or mass) are trivial but for indistinguishable ones there are fermions and bosons (forgetting para-statistics for now).

In two dimensions there are non-trivial exchange statistics both for distinguishable and indistinguishable particles. For the indistinguishable ones, there are so-called abelian anyons characterized by a number $0 \leq \theta \leq \pi$, where $\theta = 0$ are bosons while $\theta = \pi$ are fermions. There are also many different types of so-called non-abelian anyons which have much richer properties. Usually the word "anyon-statistics" refer to indistinguishable particles, but people sometimes also use them for the distinguishable ones too.

I have not seen the whole video you just linked to, but the guy talks in a really annoying way.

3. Apr 7, 2012

### Dickfore

In Quantum Mechanics, all particles are indistinguishable, regardless of what statistics they obey.

4. Apr 7, 2012

### Dickfore

Correction, all IDENTICAL particles are indistinguishable. In the video, the distinguishability arises due to the different properties of the particles.

5. Apr 7, 2012

### complement

thanks for the replies!!

@dickfore: sure. if they are different due to their intrisic properties or if they are localizable e.g. if they are fixed.

@element4: what are nonabelian anyons? and what are nontrivial statistics for distinguishable particles in 2 dimensions. do you have any reference for me please?

6. Apr 7, 2012

### element4

Yes, non-identical particles are distinguishable. But I think "complement"'s question is why distinguishable particles have non-trivial exchange statistics while they don't have in 3D and in the book he was reading anyons were considered to be indistinguishable.

The answer is (expressed more technical than my first answer), in 3D the configuration space of $N$ particles is topologically non-trivial only for indistinguishable particles (then the first homotopy group is $S_N$, the permutation group and there are two one-dimensional representations of it, corresponding to fermions and bosons). Thus exchange of non-identical particles is trivial.

While in 2D the situation is different. The configuration space for both distinguishable and indistinguishable particles has non-trivial topology. The first homotopy group for $N$ indistinguishable particles is the so-called Braid group $B_N$, while for distinguishable particles it is the colored Braid group $P_N$ (corresponding to braids where the lines all go back to their original position).

Se also http://en.wikipedia.org/wiki/Braid_theory.

7. Apr 7, 2012

8. Apr 7, 2012

### element4

In two dimensions, multi-particle states (wavefunctions) has to transform as a representation of the Braid group $B_N$. There are infinitely many one-dimensional representations given by an angle $\theta$, such that the wave function changes by a phase $\psi \rightarrow e^{i\theta}\psi$ after exchange of particles.

Non-abelian anyons correspond to higher dimensional representations of the Braid group. Think of it as having a $g$-times degenerate ground state where the wavefunctions are given by $\psi_{\alpha}$, $\alpha = 1,\dots,g$. Braiding particles transforms the wavefunctions as $\psi_{\alpha}\rightarrow U_{\alpha\beta}\psi_{\beta}$, where $U$ is a unitary matrix. Thus exchanging particles in different order, is different since the matrices don't commute (this is just what non-abelian means). Non-abelian anyons are under intense study currently since people are close to realizing them experimentally, and they can potentially be a platform for building (topologically stable) quantum computers. The type of non-abelian anyons people are interested in now days are so-called "Majorana fermions", but beware that they are NOT fermions (and physically not related to Majorana fermions in particle physics). The canonical reference is Rev. Mod. Phys. 80, 1083–1159 (2008) (free arxiv version) and maybe Nature 464, 187-193. For a more easier and popular introduction, see "Computing with Quantum Knots".

As for distinguishable particles, the logic is the same you just need to change the braid group with the colored braid group. But I don't know any reference where the anyon types are classified and analyzed, since people don't find these as interesting now days (it might change in the future, depending on the development of the field).

Last edited: Apr 7, 2012
9. Apr 7, 2012

### element4

Yes, it is what I am talking about. The article only discusses the case of two particles, in that case it is actually possible to draw simple pictures and see intuitively what the difference is between two and three dimensions for distinguishable particles. I can see if I can find a good, simple and intuitive review of the argument.

EDIT: See section 4.2 of these lecture notes, where this is explained in an intuitive way. For distinguishable particles, you need to change the argument slightly and not identify points $q$ and $-q$. This actually makes things a little bit simpler.

Last edited: Apr 7, 2012
10. Apr 7, 2012

### complement

i am curiously waiting for more :D

11. Apr 7, 2012

### complement

do you also have a brief intro to colored braid groups?
i guess this is what im looking for...

12. Apr 7, 2012

### element4

I don't think I know anything simple to read, most references I know are advances math books in knot theory, quantum groups or modular tensor categories. But reading about the usual braid group (the references I gave above), is enough to understand the basics on the colored braid group (sometimes also called pure braid group).

In the usual braid group you allow all sort of braids of the strings. The strings can be thought of trajectories of particles, since all particles are identical, their position can be arbitrary after a braid (then you are back at the starting configuration). But for distinguishable particles, you need to give each string different color since they represent different type of particle. After braiding the particles, you need to make sure the strings of different color end at the same position they started in order to go back to the original configuration. Thus the colored braid group is just the same as the usual braid group, where you throw away the braids which are not consistent with the colors. It's much easier to draw than to write write in words.

13. Apr 7, 2012

### complement

thanks!
i still would be happy if you could give me a reference. i know about knot theory and all that a fair bit to hopefully grasp it :)

14. Apr 7, 2012

### element4

You can take a look at the book "Braid Groups" by Kassel and Turaev, what I call colored braid group they call pure braid group. I think there is a section in the beginning where they talk about a certain configuration space and prove that its fundamental group is the pure braid group, its essentially the configuration space of distinguishable particles. "Quantum Groups" by Kassel also contain some discussion of the pure braid group I in the middle of the book I think, but in the somewhat more abstract setting of Braided Monoidal Categories (these mathematical objects are actually very fundamental for anyons).