For a generic particle with spin s, we would need 2 s + 1 such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices: ∑ ∑ m = − − j j a m ∗ ∗ b m = ∑ ∑ m = − − j j ( ∑ ∑ n = − − j j U n m a n ) ∗ ∗ ( ∑ ∑ k = − − j j U k m b k ) , ∑ ∑ n = − − j j ∑ ∑ k = − − j j U n p ∗ ∗ U k q = δ δ p q .