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Ok, this question looks a bit ridiculous at the outset. However, I was thinking, and I couldn't actually come up with a reason why there shouldn't exist a representation of the Lorentz group that was, say, spin-$1/4$. These wouldn't be spinors, and I don't exactly know what they would look like, but I can't find any reason to prevent them from existing.

Therefore, are such theories forbidden? If so, why? If not, can any such theory ever be physically relevant?

Qmechanic
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2 Answers2

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The Lorentz group, SO(1,3) has a universal cover that is a double cover, Spin(1,3).

This is the origin of spin-1/2.

This suggests that had we a spacetime which had a rotation group $G$ whose universal cover $UG$ was a quadruple cover then we ought to find that this theory has spin-1/4 particles.

Mozibur Ullah
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  • Out of curiosity, what would such a spacetime look like (with a quadruple cover)? –  May 03 '18 at 15:02
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    @riemanntensor: I've no idea! It would be toy theory at best in the sense of formal physics to see what such a world would look like. It might be worth a thesis if someone wanted to pursue it systematically ;). – Mozibur Ullah May 03 '18 at 15:43
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    A quadruple cover of a group A by another group B is possible iff the Lie-manifold of the group A is 4-fold connected or, in other words, the fundamental homotopy group of this manifold is discrete with cardinality 4. – DanielC May 03 '18 at 16:26
  • @DanielC: I was wondering about what Lie groups admit quadruple covers. Thanks for filling me in. Is this a specialisation of n-connectedness to the case n equals 4? – Mozibur Ullah May 03 '18 at 16:38
  • Yes, that is correct. – DanielC May 03 '18 at 16:51
  • Note also: Poincaré admits continuous-spin representations for massless particles, for any number of spacetime diemensions. But these representations are dynamically pathological, so they are rarely considered. – AccidentalFourierTransform May 03 '18 at 17:56
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In 2D you can have anyons, which can have any spin.

See https://en.m.wikipedia.org/wiki/Anyon

my2cts
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