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

TL;DR

This paper argues that $PT$ symmetry is a more fundamental and general property of Hamiltonians than Hermiticity, showing it is always present due to Poincaré invariance and essential for consistent quantum field theory and path integral formulations.

Contribution

It demonstrates that $PT$ symmetry is inherently linked to the Hamiltonian as a generator of time translations and is always present, unlike Hermiticity which is a special case.

Findings

Hamiltonian must always be $PT$ symmetric due to Poincaré invariance.

$PT$ symmetry is necessary and sufficient for the reality of Euclidean path integrals.

Classical actions in path integral quantization must be $PT$ symmetric.

Abstract

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.