Unitary Symmetry and Weak Interactions. II. SU(3) Transformation Properties

It is shown that if weak interactions are $T-L$ invariant and satisfy the $\Delta T=\frac{1}{2}$ rule, their SU(3) transformation properties are severely limited. The nonleptonic Hamiltonian must belong to an octet, and the leptonic Hamiltonian is restricted to an octet and decuplet. To prove this result, the matrix elements of certain spintype operators are expressed in terms of the Casimir operators for SU(3). $T-L$ invariance and the $\Delta T=\frac{1}{2}$ rule impose constraints upon the eigenvalues of these Casimir operators and hence limit weak interactions in the manner described above. The converse of this result is not true: for example, a Hamiltonian belonging to an octet is not necessarily $T-L$ invariant.