  While a Hamiltonian can be both Hermitian and $PT$ symmetric, it is $PT$
symmetry that is the more general, as it can lead to real energy eigenvalues
even if the Hamiltonian is not Hermitian. We discuss some specific ways in
which $PT$ symmetry goes beyond Hermiticity and is more far reaching than it.
We show that simply by virtue of being the generator of time translations, the
Hamiltonian must always be $PT$ symmetric, regardless of whether or not it
might be Hermitian. We show that the reality of the Euclidean time path
integral is a necessary and sufficient condition for $PT$ symmetry of a quantum
field theory, with Hermiticity only being a sufficient condition. We show that
in order to construct the correct classical action needed for a path integral
quantization one must impose $PT$ 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. With the
spacetime metric being $PT$ even we show that a covariant action must always be
$PT$ symmetric. Unlike Hermiticity, $PT$ symmetry does not need to be
postulated as it is derivable from Poincare invariance. Hermiticity is just a
particular realization of $PT$ symmetry, one in which the eigenspectrum is real
and complete.
