# 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. rubi says:

The proof of spin-statistics in Streater & Wightman depends heavily on 4 dimensions through the theory of complexified Lorentz transformations and the analytic continuation of the Wightman functions to the extended tube. It's just that Streater & Wightman prove these facts in earlier chapters and the proof of spin-statistics just silently uses them.

2. Demystifier says:

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.

There are two ways to define quantum statistics. One is in terms of particle exchange. The other is in terms of algebra of field operators. The former is a definition for QM, where the fundamental degrees are particles. The later is a definition for QFT, where the fundamental degrees are fields. The two definitions are closely related, but not fully equivalent.

The argument above is a valid argument in QM. But it is not a valid argument in QFT. QM and QFT are different theories. Even if they are not so different in the case of bosons and fermions, they are very different in the case of anyons. The standard spin-statistics theorem is a theorem about relativistic QFT, not a theorem about QM. We have a consistent QM theory of anyons in 2+1, but not a consistent local relativistic QFT of anyons in 2+1. The standard spin-statistics theorem and the anyon theory cannot be directly compared because they talk about different objects; one is talking about fields and the other about particles.

It seems that Baez (and apparently many others) failed to realize that two different ways of defining statistics cannot be directly compared. The fact that anyons as particles can live in 2+1 QM does not contradict the other fact that anyons as fields can not live in 2+1 local relativistic QFT.

3. Demystifier says:

The proof of spin-statistics in Streater & Wightman depends heavily on 4 dimensions through the theory of complexified Lorentz transformations and the analytic continuation of the Wightman functions to the extended tube. It's just that Streater & Wightman prove these facts in earlier chapters and the proof of spin-statistics just silently uses them.

You may be right about that, but the original Pauli version of spin-statistics theorem does not depend on it.

4. stevendaryl says:

There are two ways to define quantum statistics. One is in terms of particle exchange. The other is in terms of algebra of field operators. The former is a definition for QM, where the fundamental degrees are particles. The later is a definition for QFT, where the fundamental degrees are fields. The two definitions are closely related, but not fully equivalent.

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

5. Demystifier says:

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

I guess I could say something about that too, but not in this thread.

6. rubi says:

You may be right about that, but the original Pauli version of spin-statistics theorem does not depend on it.

Well, Pauli's proof also requires 4 dimensions, since it also makes use of the ##SU(2)## angular momentum theory.

This is a topic for another thread, but I am intrigued by the possibility of a pure-particle formulation of relativistic quantum mechanics that is equivalent to QFT. It would at first seem impossible, because of particle creation, but there are at least two hand-wavy approaches to getting around that: (1) the Dirac sea idea, and (2) viewing particle creation in terms of a particle that can travel back and forth in time. I don't know whether there has been any serious effort to make either of these work rigorously.

There is a nice theorem by Currie, Jordan & Sudarshan that says that (Hamiltonian) relativistic particle theories must be non-interacting, both classically and quantum mechanically.

7. A. Neumaier says:

but you can repeat their proof step by step in any number of dimensions, including 2+1.

but not in 1+1. And it applies only in the relativistic case. The observable anyons are nonrelativistic.

8. A. Neumaier says:

There is a nice theorem by Currie, Jordan & Sudarshan that says that (Hamiltonian) relativistic particle theories must be non-interacting, both classically and quantum mechanically.

C, J & S only handle the classical case.

9. rubi says:

C, J & S only handle the classical case.

That's not true. They only make use of the bracket relations, which are the same in the classical and the quantum formalism. They even emphasize this explicitely in their paper.