Eur. Phys. J. Plus (2022) 137: 335
https://doi.org/10.1140/epjp/s13360-022-02567-0

Regular Article

Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

¹ The Czech Academy of Sciences, Nuclear Physics Institute, Hlavní 130, 250 68, Řež, Czech Republic
² Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 50003, Hradec Králové, Czech Republic
³ Institute of System Science, Durban University of Technology, Durban, South Africa

^a znojil@ujf.cas.cz

Received: 3 February 2022
Accepted: 6 March 2022
Published online: 12 March 2022

Abstract

Even the most economical conventional formulation of quantum mechanics called Schrödinger picture (SP) need not be always computationally or conceptually optimal. A Dyson-inspired remedy can sometimes be sought in a transfer of the calculations from the conventional and correct SP Hilbert space $\mathcal{H}$ into a “false but friendlier” Hilbert space $\mathcal{R}$. The difference (i.e., non-equivalence) between $\mathcal{R}$ and $\mathcal{H}$ is given by a judicious simplification of the inner product. Although the Hamiltonian H itself appears non-Hermitian in $\mathcal{R}$, its necessary Hermitization (mediated by a return to $\mathcal{H}$) is usually feasible. If not, another eligible remedy is described in the present paper. Our basic idea lies in the introduction of a “chained” auxiliary manipulation multiplet of inner-product spaces ${\mathcal{R}}_1$, ${\mathcal{R}}_2$, ..., ${\mathcal{R}}_N$. At arbitrary N, the structure and properties of the resulting generalized SP (GSP) formulation of quantum mechanics of unitary systems (characterized by “hiddenly Hermitian” observables) are described and discussed in detail. Nontrivial nature of the related multi-step Dyson-map composition laws and correspondences to the conventional Hermitian models are also revealed and clarified.