Wavefunction's unconventional statistics manifested. In three dimensions, exchanging identical particles has a simple effect on a wavefunction: no change for bosons, multiplication by −1 for fermions. In two dimensions, things are more complicated. Consider the two ways to switch identical particles "A" and "B" shown in the figure. Because the clockwise and counterclockwise switches can't be continuously deformed into each other, 2D exchange doesn't just swap coordinates; it also involves a topological component. When many particles are involved, the topological issues are correspondingly more complex, and exchange operations might not commute. In that case the particles are said to have non-abelian (that is, noncommuting) anyon statistics. Non-abelian anyons are more than a mathematical curiosity: Condensed-matter physicists have plausibly argued that the quasiparticles that participate in the so-called ν = 5⁄2 fractional quantum Hall state are objects of that type (see the article by Sankar Das Sarma, Michael Freedman, and...

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