$PT$-symmetric quantum mechanics is a Hermitian quantum mechanics

TL;DR

The paper demonstrates that $PT$-symmetric quantum mechanics can be understood as a form of $J$-Hermitian quantum mechanics, establishing conditions for unitary evolution and discussing implications for different state spaces.

Contribution

It shows that $PT$-symmetric quantum mechanics is equivalent to $J$-Hermitian quantum mechanics and clarifies conditions for unitarity in different state space frameworks.

Findings

$PT$-symmetric quantum mechanics is a form of $J$-Hermitian quantum mechanics.

Time evolution is unitary in the Krein space if the Hamiltonian is $J$-Hermitian.

Unitarity issues arise when considering evolution in the Hilbert space instead of the Krein space.

Abstract

The author discusses a different kind of Hermitian quantum mechanics, called $J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian (or equivalently $PT$-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.