Irreducible spin representation

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

In the ordered angular momentum eigenstate basis $(|1,1⟩,|1,0⟩,|1,-1⟩)$, with positive raising coefficients,To derive them, $[J_3,J_{+}]=\hslash J_{+}$ restricts $J_{+}$ to its two adjacent superdiagonal entries. Write these real entries as $A,B\ge 0$. Since $J_{-}=J_{+}^{†}$, the relation $[J_{+},J_{-}]=2\hslash J_3$ reads $\mathrm{diag}(A^2,B^2-A^2,-B^2)=2{\hslash }^2\mathrm{diag}(1,0,-1)$, forcing $A=B=\sqrt{2}\hslash$. Adding and subtracting the ladder operators gives the displayed matrices. Direct multiplication verifies the angular momentum commutation relations and $J^2=2{\hslash }^{2}I$.

For active rotations with $U=e^{-i\varphi J_3}{e}^{-i\theta J_2}{e}^{-i\psi J_3}$, the irreducible spin representation matrix is $D_{m{m}^{\prime }}^{(j)}=e^{-im\varphi }{d}_{m{m}^{\prime }}^{(j)}(\theta )e^{-i{m}^{\prime }\psi }$, where $d_{m{m}^{\prime }}^{(j)}=⟨j,m|e^{-i\theta J_2}|j,m^{\prime }⟩$. Its spin-one-half middle matrix is $\left(\begin{array}{cc}\cos (\theta /2)& -\sin (\theta /2)\\ \sin (\theta /2)& \cos (\theta /2)\end{array}\right)$. A $2\pi$ rotation acts by $(-1{)}^{2j}$; only integer-spin representations descend to the special orthogonal group in three dimensions.

The rotation matrix about the $y$ axis in a spin-$j$ angular-momentum representation. For $j=1$, $d_{-1,+1}^1=(1-\cos \theta )/2$; its square gives the polar dependence of a transition between those spin projections.

For Hermitian $J_3$ and adjoint angular momentum ladder operators $J_{+}^{†}=J_{-}$ on a positive-definite inner-product space, a highest-weight vector of weight $j$ obeys $J_{+}|j,j⟩=0$. The commutation relations give $J_{+}(J_{-}{)}^{\ell }|j,j⟩=\ell (2j-\ell +1)(J_{-}{)}^{\ell -1}|j,j⟩$. Taking inner products forces $2j$ to be a nonnegative integer: otherwise the first negative factor would give a negative squared norm. The norm first vanishes at $\ell =2j+1$, so the generated unitary representation has dimension $2j+1$. Without positivity an abstract highest-weight module can remain infinite-dimensional even at an integral highest weight.

In the ordered spin-one basis $(m=1,0,-1)$, the second angular momentum operator satisfies $J_2^3=J_2$. Therefore its matrix exponential reduces to $I-i\sin \theta J_2+(\cos \theta -1)J_2^2$. A half-turn exchanges the two outer weights and negates the middle weight. Applied to the pion isospin triplet, this fixes the rotation factor in G parity.