Three-Hilbert-Space Formulation of Quantum Mechanics

TL;DR

This paper extends the two-Hilbert-space formulation of quantum mechanics to a three-Hilbert-space framework, clarifying conceptual issues and improving the mathematical formalism, especially in time-dependent scenarios.

Contribution

It introduces a three-Hilbert-space reformulation of quantum mechanics to address ambiguities in the two-Hilbert-space approach and refines the notation and understanding of time evolution.

Findings

Identifies a weakness in the two-Hilbert-space formalism.

Proposes a three-Hilbert-space reformulation.

Clarifies the concept of covariance in time-dependent quantum systems.

Abstract

In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as ${\cal H}^{\rm (auxiliary)}$ and ${\cal H}^{\rm (standard)}$) we spot a weak point of the 2HS formalism which lies in the double role played by ${\cal H}^{\rm (auxiliary)}$. As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the same theory. As a byproduct of our analysis of the formalism we offer an amendment of the Dirac's bra-ket notation and we also show how its use clarifies the concept of covariance in time-dependent cases. Via an elementary example we finally explain why in certain quantum systems the generator $H_{\rm (gen)}$ of the time-evolution of the wave functions may differ from their Hamiltonian $H$.