Isospin multiplet ($|I,M⟩$)

An isospin multiplet is an irreducible finite-dimensional SU(2) representation with labels $I,M$, where $I\ge 0$, $2I$ is an integer and $M=I,I-1,\dots ,-I$. With Hermitian generators, the lowering coefficient is $\sqrt{(I+M)(I-M+1)}$. The quadratic Casimir operator has value $I(I+1)$, fixing all ladder norms and the termination at $M=-I$.

An isospin-invariant decay amplitude is an intertwining map from the initial isospin multiplet into the final tensor product. The Clebsch-Gordan decomposition for SU2 contains each allowed final isospin once. Commutation with the third generator and the quadratic Casimir operator places the image in the matching coupled state. Commutation with the ladder operators makes its scalar coefficient independent of the magnetic component. Orthogonality of the Clebsch-Gordan coefficients then gives a summed squared amplitude $|a{|}^2$. A physical decay rate additionally includes phase space and spin sums, which can be absorbed into a common reduced coefficient when exact isospin symmetry makes them identical between channels.

With positive ladder coefficients, the second-axis half-turn is $e^{-i\pi I_2}|I,M⟩=(-1{)}^{I-M}|I,-M⟩$. Conjugation sends $I_{\pm }$ to $-I_{\mp }$, making consecutive phases differ by a minus sign. The highest-state rotation formula, expanded through its last nonzero lowering term, fixes the highest-state phase to one.