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 e ϱ is not completely independent of the choice of the morphisms ϱ1, ϱ2 but we may obtain two different operators e 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 e ϱ by the first mentioned choice of the supports. Then one finds that the opposite choice leads to e−1ϱ in (IV.2.28). The relation (IV.2.30) is lost. Therefore e 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. eϱ corresponds to the one, e −1ϱ 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.