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    Degeneracy Implies Non-abelian Statistics

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    A non-abelian anyon can only occur in the presence of ground state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension >1>1 that the resulting representations of all nn-strand braid groups BnB_n are overall phases, even though the ground state manifolds for nn such anyons in the plane are in general Hilbert spaces of dimensions >1>1. We observe that degeneracy is all that is needed: for an anyon with quantum dimension >1>1 the non-abelian statistics cannot all be overall phases on the degeneracy ground state manifold. Therefore, degeneracy implies non-abelian statistics, which justifies defining a non-abelian anyon as one with quantum dimension >1>1. Since non-abelian statistics presumes degeneracy, degeneracy is more fundamental than non-abelian statistics.Comment: State the main result as a theorem and add several clarification
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