Editing Indistinguishable particles (section)

== Homotopy class ==
{{See also|Homotopy|Braid statistics}}

To understand why particle statistics work the way that they do, note first that particles are point-localized excitations and that particles that are spacelike separated do not interact. In a flat {{mvar|d}}-dimensional space {{mvar|M}}, at any given time, the configuration of two identical particles can be specified as an element of {{math|''M'' × ''M''}}. If there is no overlap between the particles, so that they do not interact directly, then their locations must belong to the space {{math|[''M'' × ''M''] \ {coincident points},}} the subspace with coincident points removed. The element {{math|(''x'', ''y'')}} describes the configuration with particle I at {{mvar|x}} and particle II at {{mvar|y}}, while {{math|(''y'', ''x'')}} describes the interchanged configuration. With identical particles, the state described by {{math|(''x'', ''y'')}} ought to be indistinguishable from the state described by {{math|(''y'', ''x'')}}. Now consider the [[homotopy class]] of continuous paths from {{math|(''x'', ''y'')}} to {{math|(''y'', ''x'')}}, within the space {{math|[''M'' × ''M''] \ {coincident points} }}. If {{mvar|M}} is {{tmath|\mathbb R^d}} where {{math|''d'' ≥ 3}}, then this homotopy class only has one element. If {{mvar|M}} is {{tmath|\mathbb R^2}}, 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 {{mvar|M}} is {{tmath|\mathbb R}}, then this homotopy class is empty.

Suppose first that {{math|''d'' ≥ 3}}. The [[universal covering space]] of {{math|[''M'' × ''M''] &setminus; {{mset|coincident points}}}}, which is none other than {{math|[''M'' × ''M''] &setminus; {{mset|coincident points}}}} itself, only has two points which are physically indistinguishable from {{math|(''x'', ''y'')}}, namely {{math|(''x'', ''y'')}} itself and {{math|(''y'', ''x'')}}. So, the only permissible interchange is to swap both particles. This interchange is an [[involution (mathematics)|involution]], so its only effect is to multiply the phase by a square root of 1. If the root is +1, then the points have Bose statistics, and if the root is –1, the points have Fermi statistics.

In the case <math>M = \mathbb R^2,</math> the universal covering space of {{math|[''M'' × ''M''] &setminus; {{mset|coincident points}}}} has infinitely many points that are physically indistinguishable from {{math|(''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 recover the original state; so such an interchange can generically result in a multiplication by {{math|exp(''iθ'')}} for any real {{mvar|θ}} (by [[unitarity]], the absolute value of the multiplication must be 1). This is called [[anyon]]ic statistics. In fact, even with two ''distinguishable'' particles, even though {{math|(''x'', ''y'')}} is now physically distinguishable from {{math|(''y'', ''x'')}}, the universal covering space still contains infinitely many points which are physically indistinguishable from the original point, now generated by a counterclockwise rotation by one full turn. This generator, then, results in a multiplication by {{math|exp(''iφ'')}}. This phase factor here is called the [[mutual statistics]].

Finally, in the case <math>M = \mathbb R,</math> the space {{math|[''M'' × ''M''] &setminus; {{mset|coincident points}}}} is not connected, so even if particle I and particle II are identical, they can still be distinguished via labels such as "the particle on the left" and "the particle on the right". There is no interchange symmetry here.