Anyons

To understand anyons, it is useful to recall first that ordinary particles are of two sorts, bosons and fermions. These differ in several respects, beginning with the action of spatial rotations on the corresponding Hilbert spaces. For particles in ordinary 3dimensional space, the group SO (3) of Euclidean rotations of that space acts on the states of the particle. (More precisely, the group of all Euclidean motions acts, but we abstract from the particle’s position and consider only its orientation in space; thus we ignore translations and consider only the group of rotations.) Because the vector representing a state is defined only up to a phase factor, the action of the rotation group is not a representation in the usual sense but a projective representation. This means that each rotation g of physical 3-dimensional space is represented by a unitary operator p(g) on the Hilbert space, but this p(g) is unique only up to a phase factor. It is customary to make some arbitrary choice of these phase factors, so that we can speak unambiguously of p(g). The arbitrariness of the choice is, however, reflected in the fact that p(gh) and p(g)p(h) need not be equal but can differ by a phase factor. Furthermore, p and p' are considered equivalent representations if they differ only by these arbitrary phase factors. It is reasonable to ask, in this connection, why the operators p (g) need to be unitary or even linear, rather than only linear up to phase factors. The reason is that, unlike absolute phases, relative phases are relevant in superpositions, so physical symmetries must preserve them.

It turns out that any projective representation p of SO(3) is given by a genuine unitary representation p of the universal covering group of SO(3), namely SU (2)

(see for example [1] and [10]). That is, if p : SU(2) ^ SO(3) denotes the 2-to- 1 projection map, we have p о p equivalent to p. More concretely, it means that there are two sorts of projective representations of SO(3), up to equivalence. One sort is the ordinary unitary representations of SO(3); the other is given by unitary representations of SU (2) that send the non-trivial element -1 of the kernel of p to the operator -1. (Throughout this paper, we use I, sometimes with subscripts, to denote identity transformations, functions, morphisms, etc.) The first sort of representation corresponds to bosons, whose state vectors (not merely their states) are unchanged when rotated gradually through a full revolution. The second sort corresponds to fermions, where a rotation through 2n changes the state vector by a sign.

A second distinction between bosons and fermions, even more important for our purposes, is the behavior of systems of several identical particles. Because the particles are identical, any permutation of the particles leaves the state unchanged and therefore changes the state vector by at most a phase factor. As a result, we have a one-dimensional projective representation of the symmetric group. Again, it turns out that there are just two possibilities (both of which are actual unitary representations of the symmetric group). Either all permutations leave the state vectors unchanged, or the even permutations leave the state vectors unchanged while the odd permutations reverse the vectors’ signs.

A deep theorem of relativistic quantum field theory, the spin-statistics theorem, says that these two behaviors of multi-particle states under permutations exactly match the two behaviors of single-particle states under rotations. Interchanging two identical bosons leaves the state vector of the pair unchanged; interchanging two identical fermions reverses the sign of the state vector.

The preceding discussion of bosons and fermions depends crucially on the fact that the particles are in ordinary 3-dimensional space. If particles were confined to a 2-dimensional space, more possibilities would arise.

Specifically, the rotation group in two dimensions, SO(2), has more sorts of projective representations than SO(3) does; the reason is ultimately that the universal covering group of the circle group SO (2) is the additive group of real numbers, and the covering projection is not 2-to-1 but ro-to-1. The result is that a gradual rotation of a particle through 2n can multiply its state vector by an arbitrary phase factor, not just ±1. The possibility of getting any phase here led to the name anyon.

Reducing the dimensionality of space from 3 to 2 also affects the possibilities for permuting identical particles. For simplicity, consider the case where there are just two particles, and we interchange them. We can perform the interchange gradually, in the plane, by rotating the 2-particle system counterclockwise by n around the midpoint between the particles. Alternatively, we can achieve the same interchange by a clockwise rotation. In 3-dimensional space, these two options are equivalent in the sense that they can be gradually deformed into each other, by rotating the plane of the particles’ motion about the line through the particles’ initial positions. In 2-dimensional space, there is no such deformation without making the particles collide. Winding one particle around the other any number of times, we get infinitely many ways to achieve one and the same permutation. With more than two particles, there are more complicated ways to achieve the same permutation by moving the particles around in the plane. As a result, in place of (projective) representations of symmetric groups, we have representations of braid groups. For example, in the case of two particles, in place of the group of two possible permutations of the particles, we have the group of all integers, with integer n representing a counterclockwise rotation by nn (and negative n representing clockwise rotations).

The preceding discussion was oversimplified in that (among other things) when moving particles around each other, we ignored any rotation that the individual particles might have undergone during the motion. A more accurate presentation would need to suitably combine the braid and rotation groups.

 
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