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>=\(M) over bar (0)>, where M-0 is a neutral flavored 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,F-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><---->\F-2>. Also implied is another relation involving the coefficients p(a,b), q(a,b), which can be interpreted as Im theta=0, where theta is the rephasing-invariant parameter describing CPT violation in M-0(M) over bar (0) mixing for Khalfin's choice of \F-1,F-2>. The states \F-1,F-2> of our theorem need not form a particle-antiparticle pair, nor even be restricted to particle physics.