Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics

TL;DR

The paper addresses the interpretational challenge of non-Hermitian Hamiltonians with real spectra in quasi-Hermitian quantum mechanics by employing Dyson maps to generate Hermitian twins and consistent inner-product metrics. It develops a three-tier classification of Dyson maps, analyzes both forward and inverse constructions, and provides explicit examples (including a two-dimensional model and a fermionic four-level system) to illustrate how different choices of the Dyson map and unitary transformations affect the Hermitian avatar and observable avatars. The work clarifies how probabilistic interpretations emerge and how multiple observables can be made compatible within a unified framework, highlighting the role of the metric $\Theta$ and the parameters $K$ and $\mathcal{U}$ in shaping physical interpretations. Overall, it offers a concrete, extensible scheme for constructing physically meaningful QHQM models with flexible yet well-defined probabilistic content, advancing both foundational understanding and potential applications.

Abstract

In quasi-Hermitian quantum mechanics (QHQM) of unitary systems, an optimal, calculation-friendly form of Hamiltonian is generally non-Hermitian, $H \neq H^\dagger$. This makes its physical interpretation ambiguous. Without altering $H$, this ambiguity is resolved by specifying a nontrivial inner-product metric $Θ$ in Hilbert space. Here, we focus on an alternative strategy: transforming $H$ into a Hermitian form via the Dyson map $Ω: H \to \mathfrak{h}$. This construction of the Hermitian isospectral twin $\mathfrak{h}$ of $H$ does not only restore the conventional correspondence principle between quantum and classical physics, but it also provides a framework for the exhaustive classification of all admissible probabilistic interpretations of quantum systems in QHQM framework.