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Does fractional charge imply fractional statistics (e.g. anyons)?

If not, are there some relations between them?

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Not sure about the condensed matter context, but in general the answer is NO. For instance, quarks have fractional charge but are regular fermions.

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The answer is 'no' in condensed matter also. As anyons are neither bosons nor fermions, they can follow some statistics other than BE or FD, but it has nothing to do with fractional charge. just-learning has already given you a perfect example.

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Would suggest, if fractional charges were in Nature.

Fractional charges are the result of error of James Chadwick in 1934. He identified (with no real proof) that neutron is particle similar to proton.

If you trust that positive proton and neutral neutron as similar particles, you have to trust in fractional charges.

But analysis shows that alternative is than as neutron as well atomic nuclei consist of protons and negative pions. In that case fractional charges are not needed to explain whole picture.

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