Braided Statistics and Worldlines

Why two dimensions changes the question

Exchange two identical particles in three dimensions and the paths by which you did it are all deformable into one another; only the endpoint matters, and the state can pick up at most a sign — bosons or fermions. In two dimensions paths that wind around one another cannot be deformed into paths that do not. The relevant group is the braid group rather than the permutation group, and the state may acquire an arbitrary phase, or, if the ground state is degenerate, may be rotated within that degenerate space by a matrix.

Those are anyons: abelian if the exchange multiplies by a phase, non-abelian if it applies a matrix that does not commute with other exchanges. In the non-abelian case the worldlines' braid — the actual history of crossings in spacetime — is the computation. This is the standard against which every looser use of 'braid' in this volume is measured.

Where the evidence stands

Anyonic statistics in the fractional quantum Hall regime have been observed in interferometry and in collision experiments, and abelian anyonic phases are now reasonably well established. Non-abelian braiding is the harder claim; results in quantum-Hall systems and in engineered superconducting devices are suggestive, contested, and in some cases have been withdrawn. A chapter written in 2026 should say partial and recent rather than demonstrated.