English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 eigenvalueseven if the Hamiltonian is not Hermitian. We discuss some specific ways inwhich $PT$ symmetry goes beyond Hermiticity and is more far reaching than it.We show that simply by virtue of being the generator of time translations, theHamiltonian must always be $PT$ symmetric, regardless of whether or not itmight be Hermitian. We show that the reality of the Euclidean time pathintegral is a necessary and sufficient condition for $PT$ symmetry of a quantumfield theory, with Hermiticity only being a sufficient condition. We show thatin order to construct the correct classical action needed for a path integralquantization one must impose $PT$ symmetry on each classical path, arequirement that has no counterpart in any Hermiticity condition sinceHermiticity of a Hamiltonian is only definable after the quantization has beenperformed and the quantum Hilbert space has been constructed. With thespacetime metric being $PT$ even we show that a covariant action must always be$PT$ symmetric. Unlike Hermiticity, $PT$ symmetry does not need to bepostulated as it is derivable from Poincare invariance. Hermiticity is just aparticular realization of $PT$ symmetry, one in which the eigenspectrum is realand complete.

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