By E. Askwith
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28]. 13) we conclude that Vji(pP) V: 1e (p9) = 2: R(j,pP,p9, k)~:~ V~l (p9) VJIe (PP). :~) only depend on the classes Mp and Mq of *morphisms to which pP and pq belong, but are independent of the specific choice of p1' and pqj they are also independent of the choice of the auxiliary cone Ca , (as long as Ca is space-like separated from the localization cones of pP and pq). 14). For proofs see [28). Next, we investigate the properties of the statistics matrices (R#(j,p, q, k )~:~) somewhat systematically.
2 Fusion of intertwiners For p and q in L, we consider *morphisms pP E in space-like cones CP and cq, Mp and pq E Mq localized respectively, which are contained in the interior of a simple region S C M3. The space-like complement of S is assumed to contain a non-empty, space-like auxiliary cone, Ca. 19). In particular, the composition, pP 0 pq, of pP with pq is well defined on the algebra A of quasi-local observables. 2, (PI), (Sect. 28). Let cr be a space-like cone contained in the interior of S, and let pr E Mr be a *morphism of B C• localized in Cr with the property that the representation r of A is unitarily equivalent to the vacuum representation, 1, of pr(A).
6. -=r 2? zl Fig. 6 49 In the situation shown in Fig. 26) p, q. 7). 24) and the remark that R- does not change if f\ Ca is replaced by Ca that But in the situation shown in Fig. ) V;i (pq). 30) But from the calculations in Sect. 67), we infer that where Sj is the spin of representation j. ;- •• ) V:i(pq). 31 ) From Fig. 5 and Fig. 6 we also learn that Vii (PP) = v~j (PP) . ) L I L S R-(i,p,q,k)~ Vjl(pq) V~k(,01') I S S e 21ri('j-"+"-") R-(i,p,q,k)~ Vii (pq) Vlk(pp). 34) (3) where si is the spin of representation j.