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 $\mathrm{Im}\theta =0$, where $\theta$ 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.

Comments: 9 pages, LATEX, no figues. Misprints corrected and part of the presentation changed, version to be published in Phys. Rev. D