Violation and Arrows of Time : Evolution of a Neutral K or B Meson from an Incoherent to a Coherent State

We study the evolution of a neutral K meson prepared as an incoherent equal mixture of K and K̄. Denoting the density matrix by ρ(t) = 1 2 N(t) [1+ ~ ζ(t) · ~σ], the norm of the state N(t) is found to decrease monotonically from one to zero, while the magnitude of the Stokes vector |~ ζ(t)| increases monotonically from zero to one. This property qualifies these observables as arrows of time. Requiring monotonic behaviour of N(t) for arbitrary values of γL, γS and ∆m yields a bound on the CP-violating overlap δ = 〈KL |KS 〉, which is similar to, but weaker than, the known unitarity bound. A similar requirement on |~ ζ(t)| yields a new bound, δ < 1 2 (