where 1 n is the n -dimensional unit matrix and J ( n ) = ( J 1 ( n ) , J 2 ( n ) , J 3 ( n ) ) are the (2 n + 1) -dimensional irreducible representations of s o ( 3 ) ≅ ≅ s u ( 2 ) also termed spin matrices or angular momentum matrices. These are explicitly given as ( J 1 ( j ) ) a ′ a = 1 2 ( ( j − − a ) ( j + a + 1 ) δ δ a ′ , a + 1 + ( j + a ) ( j − − a + 1 ) δ δ a ′ , a − − 1 ) ( J 2 ( j ) ) a ′ a = 1 2 i ( ( j − − a ) ( j + a + 1 ) δ δ a ′ , a + 1 − − ( j + a ) ( j − − a + 1 ) δ δ a ′ , a − − 1 ) ( J 3 ( j ) ) a ′ a = a δ δ a ′ , a where δ denotes the Kronecker delta. In components, with − m ≤ a, a′ ≤ m, − n ≤ b, b′ ≤ n, the representations are given by ( π π ( m , n ) ( J i ) ) a ′ b ′ , a b = δ δ b ′ b ( J i ( m ) ) a ′ a + δ δ a ′ a ( J i ( n ) ) b ′ b ( π π ( m , n ) ( K i ) ) a ′ b ′ , a b = − − i ( δ δ b ′ b ( J i ( m ) ) a ′ a − − δ δ a ′ a ( J i ( n ) ) b ′ b )