# Anyon Demystified

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Every quantum physicist knows that all particles are either bosons or fermions. And the standard textbook arguments that this is so do not depend on the number of dimensions.

On the other hand, you may have heard that in 2 dimensions particles can be anyons, which can have any statistics interpolating between bosons and fermions. And not only in theory, but even in reality. But how that can be compatible with the fact that all particles are either bosons or fermions? Where is the catch?

This, of course, is discussed in many papers (and a few books) devoted to anyons. But my intention is not to present a summary of the standard literature. I want to explain it in my own way which, I believe, demystifies anyons in a way that cannot be found explicitly in the existing literature. I will do it in a conceptual non-technical way with a minimal number of explicitly written equations. Nevertheless, the things I will say can be viewed as a reinterpretation of more elaborated equations that can easily be found in the standard literature. In this sense, my explanation is meant to be complementary to the existing literature.

Consider a 2-particle wave function ##\psi({\bf x}_1,{\bf x}_2)##. The claim that it is either bosonic or fermionic means that it is either symmetric or antisymmetric, i.e.
$$\psi({\bf x}_1,{\bf x}_2)=\pm \psi({\bf x}_2,{\bf x}_1) ….. (1)$$
But suppose that the wave function satisfies a Schrodinger equation with a potential ##V({\bf x}_1,{\bf x}_2)##, which has a property of being asymmetric
$$V({\bf x}_1,{\bf x}_2) \neq V({\bf x}_2,{\bf x}_1) .$$
In general, with an asymmetric potential, the solutions of the Schrodinger equation will not satisfy (1). And yet, no physical principle forbids such asymmetric potentials. It looks as if it is very easy to violate the principle that wave function must be either bosonic or fermionic.

But that is not really so. The principle that wave function must be symmetric or antisymmetric refers only to identical particles, i.e. particles that cannot be distinguished. On the other hand, if the potential between the particles is not symmetric, then the particles are not identical, i.e. they can be distinguished. In that case, (1) does not apply.

Now assume that the asymmetric potential takes a very special form, so that the wave function of two non-identical bosons or fermions takes the form
$$\psi({\bf x}_1,{\bf x}_2)=e^{i\alpha} \psi({\bf x}_2,{\bf x}_1)$$
where ##\alpha## is an arbitrary real number. This is the anyon. And there is nothing strange about it, it is simply a consequence of the special interaction between two non-identical particles. The effect of interaction is to simulate an exotic statistics (exotic exchange factor ##e^{i\alpha}##), while the “intrinsic” statistics of particles (i.e. statistics in the absence of exotic interaction) is either bosonic or fermionic.

The only non-trivial question is, does such interaction exists? It turns out (the details of which can be found in standard literature) that mathematically such an interaction exists, provided that the particles live in 2 dimensions and that the potential is not really a scalar potential ##V({\bf x}_1,{\bf x}_2)## but a vector potential ##{\bf A}({\bf x}_1,{\bf x}_2)##. And physically, that is in the real world, such interaction does not exist for elementary particles such as electrons, but only for certain quasi-particles in condensed matter physics. These are the main conceptual ideas of anyons, while the rest are technical details that can be found in standard literature.

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51 replies
1. A. Neumaier says:

What's the basis for you saying this?

The literature on the subject. Read some of the links given here: http://www.physicsoverflow.org/32114/

Well then I'm free to make up any particle thats true in some dimension.

You are free to make up anything. The question is whether Nature will be described by your make-up.

People had studied low-dimensional quantum physics and their peculiarities for theoretical reasons, long before it was found that there are many interesting systems in Nature described by them.

2. A. Neumaier says:

OK that caliber of paper is above my skill level.

3. Demystifier says:

4. Demystifier says:

But the Lorentz group is based on 4-dimensional Minkowski space if nothing is said.

Yes, but you can repeat their proof step by step in any number of dimensions, including 2+1. Indeed, as I already said, there is no consistent local Lorentz invariant QFT in 2+1 dimensions with intrinsic anyon statistics.

5. Demystifier says:

Hmm. John Baez in an old article claims that the spin-statistics theorem only applies for 4 or more spacetime dimensions:

http://math.ucr.edu/home/baez/braids/node2.html

"Now for the catch: the spin-statistics theorem only holds for spacetimes of dimension 4 and up."

See my post above. So the spin-statistics theorem, with all the explicit assumptions of the theorem, is valid even in 2+1.

6. stevendaryl says:

See my post above. So the spin-statistics theorem, with all the explicit assumptions of the theorem, is valid even in 2+1. In 2+1, anyons are OK as non-relativistic (first-quantized) QM, but not as  relativistic local (second-quantized) QFT.

I'm still trying to reconcile your claim that the theorem is valid in 2+1 dimensions with John Baez' claim that it is only valid in 4 (and higher) dimensions. Are you saying that there is a general theorem that doesn't assume Lorentz invariance, which is valid in 4 or more dimensions, and then a specialization that does assume Lorentz invariance, which is valid in any number of dimensions?

7. stevendaryl says:

I'm still trying to reconcile your claim that the theorem is valid in 2+1 dimensions with John Baez' claim that it is only valid in 4 (and higher) dimensions. Are you saying that there is a general theorem that doesn't assume Lorentz invariance, which is valid in 4 or more dimensions, and then a specialization that does assume Lorentz invariance, which is valid in any number of dimensions?

Or are you saying that Baez is just wrong about the loophole in 2+1 dimensions?

8. stevendaryl says:

Or are you saying that Baez is just wrong about the loophole in 2+1 dimensions?

The argument I heard, which I didn't really understand, is that if you consider paths in configuration space, the path where two identical particles exchange places is continuously deformable to a path were one of the particles rotates through 360 degrees and nothing else happens. So somehow there must be a relationship between particle exchange and rotations. But the deformation requires at least 3 dimensions, so there is a 2-D loophole. Something like that.