  Currently there is much interest in Hamiltonians that are not Hermitian but
instead possess an antilinear $PT$ symmetry, since such Hamiltonians can still
lead to the time-independent evolution of scalar products, and can still have
an entirely real energy spectrum. However, such theories can also admit of
energy spectra in which energies come in complex conjugate pairs, and can even
admit of Hamiltonians that cannot be diagonalized at all. Hermiticity is just a
particular realization of $PT$ symmetry, with $PT$ symmetry being the more
general. These $PT$ theories are themselves part of an even broader class of
theories, theories that can be characterized by possessing some general
antilinear symmetry, as that requirement alone is a both necessary and
sufficient condition for the time-independent evolution of scalar products,
with all the different realizations of the $PT$ symmetry program then being
obtained. Use of complex Lorentz invariance allows us to show that the
antilinear symmetry is uniquely specified to be $CPT$, with the $CPT$ theorem
thus being extended to the non-Hermitian case. For theories that are separately
charge conjugation invariant, the results of the $PT$-symmetry program then
follow. We show that in order to construct the correct classical action needed
for a path integral quantization one must impose $CPT$ symmetry on each
classical path, a requirement that has no counterpart in any Hermiticity
condition since Hermiticity of a Hamiltonian is only definable after the
quantization has been performed and the quantum Hilbert space has been
constructed. We show that whether or not a $CPT$-invariant Hamiltonian is
Hermitian is a property of the solutions to the theory and not of the
Hamiltonian itself. Thus Hermiticity never needs to be postulated at all.
