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What you should really be thinking of is a group and representations. In the massive case, the little group is $SU(2)$ whose representations are labelled by a half-integer $j$ and has dimension $2j+1$. In the massless case, the little group os $U(1)$ whose representations are labelled by a half-integer $h$ and have dimension $1$. However, we want our Hilbert ...


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