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Given the fusion rule (for anyons) "A x B = 1 + C" it is possible for the anyons A and B to fuse to the vacuum "1" or to fuse to the anyon C.

What determines what will happen if they fuse? Is the outcome probabalistic?

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Morally speaking, the fusion rules tell you that if you take particle A and bring it close to Particle B they will interact and fuse together. The resulting state will be a super-position of no particles -- corresponding to the identity -- and the C particle. If we denote the quantum dimensions of particle $X$ by $d_X$, then the resulting state will be $|\psi \rangle = \sqrt{\frac{1}{d_{A}d_B}} |vac \rangle + \sqrt{\frac{d_C}{d_A d_B}} |C\rangle$.

(note that the fusion rules tell you that $d_A d_B = 1+ d_C$ so the state is indeed normalized).

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  • $\begingroup$ How do you know the relative phase is +1? $\endgroup$ – ZeroTheHero May 3 at 18:35

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