Spin

One of the variables which characterize the interior degrees of freedom of a quantum particle (or of a quantum field). A non-relativistic particle has spin $s$ ($s = 0, 1/2, 1, 3/2, 2, \ldots$) if its state vector takes values in the representation space of an irreducible unitary representations $D^{(s)}$ of the unitary-unimodular group $\mathrm{SU}(2)$. The dimension of the representation space is $2s+1$. In the relativistic case the spin is defined as the quantum number which characterizes an irreducible representation of the so-called little group, a subgroup of the Poincaré group $P_+^\uparrow$. For a massive particle ($m>0$) the little group is the group $\mathrm{SU}(2)$. For a particle with mass zero the little group is the Euclidean group of the plane. In this case one has, to avoid continuous spin, to restrict to those representations of the little group which are one-dimensional and which are labelled with the quantum number $\lambda$, the so-called helicity. The helicity $\lambda$ takes values $\lambda = 0, 1/2, 1, 3/2, 2, \ldots$. The dimension of the representation space equals one in this case.

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