Low Dimensional Space-Time and Braid Group Statistics

Abstract

Let us follow the steps of the DHR-analysis for a quantum field theory in 2-dimensional space-time. The results of subsection 2.1 remain unchanged but lemma 2.2.1 and the key lemma 2.2.2 are modified. Consider the set of ordered pairs (𝒪₁, 𝒪₂) with space-like separation. If the dimension of space-time is larger than 2 then this set is connected; we can continuously shift a space-like configuration of two points to any other such configuration without crossing their causal influence zone. In 2-dimensional space-time we have two disconnected components. If 𝒪₁lies to the left of 𝒪₂then 𝒪₁(s) will have to remain on the left of 𝒪₂(s) for any continuous family of pairs 𝒪 _k (s) with space-like separation. In the proof of lemma 2.2.1 we used the possibility of continuously moving the supports of the pair (ϱ₁, ϱ₂) to the supports of the pair (ϱ′₁, ϱ′₂). In 2-dimensional space-time this is only possible if these (ordered) pairs of supports lie in the same connectivity component. The consequence for the key lemma is that ε _ϱ is not completely independent of the choice of the morphisms ϱ₁, ϱ₂ but we may obtain two different operators ε _ϱ , depending on whether in the construction (IV.2.27), (IV.2.28) we choose the support of ϱ₁ to the left of the support of ϱ₂ or to the right. Let us adopt the convention of defining ε _ϱ by the first mentioned choice of the supports. Then one finds that the opposite choice leads to ε⁻¹_ϱ in (IV.2.28). The relation (IV.2.30) is lost. Therefore ε _ϱ does not correspond to a permutation of two elements but generates a braiding of two strands. An illustration is afforded by two strands of hair of a young lady. If they are originally parallel then the position of the loose end points may be interchanged in two inequivalent fashions, rotating by 180° around the center line in a clockwise or in an anticlockwise sense. ε_ϱ corresponds to the one, ε ⁻¹_ϱ to the other operation. Repetition of one of the procedures does not lead back to the original situation but to the beginning of a braid.