Identical Particles - The Homotopy Class

The Homotopy Class

To understand why we have the statistics that we do for particles, we first have to note that particles are point localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact (at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower), then we are dealing with the space /{coincident points}, the subspace with coincident points removed. (x, y) describes the configuration with particle I at x and particle II at y. (y, x) describes the interchanged configuration. With identical particles, the state described by (x, y) ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by (y, x). Let's look at the homotopy class of continuous paths from (x, y) to (y, x). If M is Rd where d ≥ 3, then this homotopy class only has one element. If M is R2, then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc., a clockwise interchange by half a turn, etc.). In particular, a counterclockwise interchange by half a turn is NOT homotopic to a clockwise interchange by half a turn. Lastly, if M is R, then this homotopy class is empty. Obviously, if M is not isomorphic to Rd, we can have more complicated homotopy classes...

What does this all mean?

Let's first look at the case d ≥ 3. The universal covering space of /{coincident points}, which is none other than /{coincident points} itself, only has two points which are physically indistinguishable from (x, y), namely (x, y) itself and (y, x). So, the only permissible interchange is to swap both particles. Performing this interchange twice gives us (x, y) back again. If this interchange results in a multiplication by +1, then we have Bose statistics and if this interchange results in a multiplication by −1, we have Fermi statistics.

Now how about R2? The universal covering space of /{coincident points} has infinitely many points that are physically indistinguishable from (x, y). This is described by the infinite cyclic group generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not lead us back to the original state. So, such an interchange can generically result in a multiplication by exp(iθ) (its absolute value is 1 because of unitarity...). This is called anyonic statistics. In fact, even with two DISTINGUISHABLE particles, even though (x, y) is now physically distinguishable from (y, x), if we go over to the universal covering space, we still end up with infinitely many points which are physically indistinguishable from the original point and the interchanges are generated by a counterclockwise rotation by one full turn which results in a multiplication by exp(iφ). This phase factor here is called the mutual statistics.

As for R, even if particle I and particle II are identical, we can always distinguish between them by the labels "the particle on the left" and "the particle on the right". There is no interchange symmetry here and such particles are called plektons.

The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.