On the Orthogonality of Independently Propagating States as Occurring in the Lee-Oehme-Yang Theory

Abstract

We generalize a theorem by Khalfin, originally derived for the states $| F_1 > = | M^0>$, $| F_2 > = |\bar M^0>$, where $M^0$ is a neutral flavoured meson (e.g., $K^0$ or $B_d^0$), by assuming CPT invariance. Dispensing with CPT invariance and allowing for an arbitrary pair of orthogonal states $| F_{1,2}>$, we show that any linear combinations $| P_a > = p_a | F_1 > + q_a | F_2>$ and $| P_b > = p_b | F_1 > - q_b | F_2>$, if postulated to be independently propagating in time, as in the Lee--Oehme--Yang Theory, must be mutually orthogonal. This implies a reciprocity relation: equality of the probabilities of the transitions $| F_1 > \leftrightarrow | F_2>$. Also implied is another relation involving the coefficients $p_{a,b}$, $q_{a,b}$, which can be interpreted as $\mathrm{Im} θ= 0$, where $θ$ is the rephasing-invariant parameter describing CPT violation in $M^0 \bar M^0$ mixing for Khalfin's choice of $| F_{1,2}>$. The states $| F_{1,2}>$ of \emph{our} theorem need not form a particle-antiparticle pair, nor even be restricted to particle physics.