4. (12 pts) The Ko and Ko mesons are both neutral particles with zero baryon and lepton number, but with opposite strangeness. Let 11 <0) denote a state containing a single Ko at rest (and nothing else), and likewise let 11 <0) denote a single Ko at rest. (a) In a hypothetical world without weak interactions, these particles would be absolutely stable and degenerate in mass. Within the two-dimensional space of states spanned by 11 <0) and 11 <0), the Hamiltonian would have the diagonal form H = l), where A = mycoc2. Justify these assertions. What symmetry implies the equality of mass of these particles? (b) Weak interactions add terms to the Hamiltonian which do not respect all the symmetries of strong (and electromagnetic) interactions. Within this two-dimensional space of states, the effect of weak interactions is to add an odd-diagonal term, so that H = (A B with B real. What are the eigenstates of this Hamiltonian? What are the corresponding rest-energies? (c) Given this perturbed Hamiltonian, what is the time-evolution of the state 11 <0)? In other words, if = 11 <0), what is IW(t))? (d) The phases of the states 11 <0) and 1K 0) may be chosen so that charge conjugation interchanges the two states, CIKO) = 11 <0) and CIKO) = 11 <0). What does a CP trans- formation do to these states? Show that the eigenstates of the perturbed Hamiltonian are also eigenstates of CP.