  The author discusses a different kind of Hermitian quantum mechanics, called
$J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics
is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the
Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian
(or equivalently $PT$-symmetric). An issue with unitarity comes up when time
evolution is considered in the Hilbert space of states rather than in the Krein
space of states. The author offers possible scenarios with this issue.
