  The presence of non-cyclic phases is revealed in the time evolution of mixed
meson systems. Such phases are related to the parameter $z$ describing the
$CPT$ violation; moreover, a non zero phase difference between particle and
antiparticle arises only in presence of $CPT$ symmetry breaking. Thus, a
completely new test for the $CPT$ invariance can be provided by the study of
such phases in mixed mesons. Systems which are particularly interesting for
such an analysis are the $B_{s}^{0}-\bar{B}_{s}^{0}$ and the
$K^{0}-\bar{K}^{0}$ ones. In order to introduce non-cyclic phases, some aspects
of the formalism describing the mixed neutral mesons are analyzed. Since the
effective Hamiltonian of systems like $K^{0}-\bar{K}^{0}$, $B^{0}-\bar{B}^{0}$,
$B_{s}^{0}-\bar{B}_{s}^{0}$, $D^{0}-\bar{D}^{0}$ is non-Hermitian and
non-normal, it is necessary to diagonalize it by utilizing the rules of
non-Hermitian quantum mechanics.
