Advancing the case for $PT$ Symmetry -- the Hamiltonian is always $PT$ Symmetric

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.

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