[Paper Review] Reply to Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

Miloslav Znojil|ArXiv.org|Nov 5, 2007

Quantum Mechanics and Non-Hermitian Physics3 references6 citations

TL;DR

This paper defends the unitarity of time-evolution in time-dependent quasi-Hermitian quantum systems by refuting a flawed assumption in a recent comment. It demonstrates that unitarity does not require a time-independent metric operator, clarifying that the standard time-dependent quasi-Hermiticity condition $ H(t)^ op = \Theta(t)H(t)\Theta(t)^{-1} $ suffices, and highlights notation-related ambiguities as the root of confusion.

ABSTRACT

In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B extbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

Motivation & Objective

• To correct a misinterpretation in a recent comment regarding time-dependent quasi-Hermitian Hamiltonians and unitary evolution.
• To demonstrate that unitarity of time evolution is preserved even when the metric operator $ \Theta(t) $ is time-dependent.
• To clarify the role of multiple Hilbert spaces and the proper treatment of dual spaces in quasi-Hermitian quantum mechanics.
• To resolve notational ambiguities that lead to incorrect conclusions, especially concerning the definition of evolution operators and inner products.
• To advocate for the use of double-bracket notation $ \langle\!\langle \cdot | $ to distinguish physical state functionals in $ \mathcal{H}^{(\Theta)} $ from standard duals.

Proposed method

• Re-expresses the time-evolution operator $ u(t) $ via the Hermitian counterpart $ h(t) = \omega(t) H(t) \omega(t)^{-1} $, where $ \omega = \sqrt{\Theta} $, ensuring unitary dynamics.
• Defines the auxiliary operator $ U_R(t) = \omega(t)^{-1} u(t) \omega(0) $, which links the time-evolution in the reference and physical Hilbert spaces.
• Derives the correct time evolution for $ \Theta(t) $ using $ \Theta(t) = U_R(t)^{-1\dagger} \Theta(0) U_R^{-1} $, correcting an incorrect assumption in the comment.
• Identifies and corrects the flawed assumption in the comment that $ i\hbar \partial_t U_R(t) = H(t) U_R(t) $, showing it contradicts the definition of $ U_R(t) $.
• Establishes that the correct condition for time-dependent quasi-Hermiticity is $ H(t)^\dagger = \Theta(t) H(t) \Theta(t)^{-1} $, not the incorrect evolution equation.
• Introduces the distinction between the standard Hilbert space $ \mathcal{H}^{(\text{stand})}_{\text{phys}} $, the reference space $ \mathcal{H}^{(\text{ref})} $, and the physical space $ \mathcal{H}^{(\Theta)} $, with $ \Theta = \Omega^\dagger \Omega $, to clarify dual space structure.

Experimental results

Research questions

• RQ1 Does the time-dependence of the metric operator $ \Theta(t) $ in a quasi-Hermitian Hamiltonian system necessarily violate unitary time evolution?
• RQ2 What is the correct dynamical equation governing the evolution of the metric operator $ \Theta(t) $ in time-dependent quasi-Hermitian systems?
• RQ3 Why does the assumption $ i\hbar \partial_t U_R(t) = H(t) U_R(t) $ lead to incorrect conclusions in the context of time-dependent quasi-Hermitian Hamiltonians?
• RQ4 How do multiple Hilbert space structures—standard, reference, and physical—interact in time-dependent quasi-Hermitian quantum mechanics?
• RQ5 What notation conventions can prevent conceptual and technical errors in the formulation of time-dependent quasi-Hermitian quantum theories?

Key findings

• The unitarity of time evolution in time-dependent quasi-Hermitian systems does not require the metric operator $ \Theta(t) $ to be time-independent.
• The assumption $ i\hbar \partial_t U_R(t) = H(t) U_R(t) $, used in the comment, is incorrect and leads to a contradiction with the definition of $ U_R(t) $.
• The correct time evolution for the metric is given by $ \Theta(t) = U_R(t)^{-1\dagger} \Theta(0) U_R^{-1} $, derived consistently from the definition of $ U_R(t) $.
• The standard time-dependent quasi-Hermiticity condition $ H(t)^\dagger = \Theta(t) H(t) \Theta(t)^{-1} $ remains valid and sufficient for unitary evolution.
• The source of confusion lies in the misuse of notation, particularly the conflation of dual spaces in different Hilbert space representations.
• The authors recommend using double-bracket notation $ \langle\!\langle \cdot | $ for linear functionals in the physical Hilbert space $ \mathcal{H}^{(\Theta)} $ to avoid ambiguity and ensure clarity in time-dependent settings.