Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

TL;DR

The paper critiques a recent reply on time-dependent quasi-Hermitian Hamiltonians, highlighting that the derived evolution operator leads to a non-observable effective Hamiltonian when the metric is time-dependent.

Contribution

It clarifies that the time-evolution operator used in the discussion results in a non-observable Hamiltonian, supporting previous conclusions about unobservable Hamiltonians in such systems.

Findings

Time-evolution operator leads to non-observable Hamiltonian when metric is time-dependent

Effective Hamiltonian differs from original Hamiltonian under certain conditions

Supports the view that unitary evolution can be generated by unobservable Hamiltonians

Abstract

I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schr\"odinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.