Three alternative model-building strategies using quasi-Hermitian time-dependent observables

Abstract

A $(K+1)-$plet of non-Hermitian and time-dependent operators (say, ${\Lambda }_j(t)$, $j=0,1,\dots ,K$) can be interpreted as the set of observables characterizing a unitary quantum system. What is required is the existence of a self-adjoint and, in general, time-dependent operator (say, $\Theta (t)$ called inner product metric) making the operators quasi-Hermitian, ${\Lambda }_j^{†}(t)\Theta (t)=\Theta (t){\Lambda }_j(t)$. The theory (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the states $\psi (t)$ (realized, via Schrödinger-type equation, by a generator, say, $G(t)$) and of the observables themselves (a different generator (say, $\Sigma (t)(t)$) occurs in the related non-Hermitian Heisenberg-type equation). Every ${\Lambda }_j(t)$ (and, in particular, Hamiltonian $H(t)={\Lambda }_0(t)$) appears isospectral to its hypothetical isospectral and self-adjoint (but, by assumption, prohibitively user-unfriendly) avatar ${\lambda }_j(t)=\Omega (t){\Lambda }_j(t){\Omega }^{-1}(t)$ with ${\Omega }^{†}(t)\Omega (t)=\Theta (t)$. In our paper the key role played by identity $H(t)=G(t)+\Sigma (t)$ is shown to imply that there exist just three alternative meaningful implementations of the NIP approach, viz., ``number one'' (a ``dynamical'' strategy based on the knowledge of $H(t)$), ``number two'' (a ``kinematical'' one, based on the Coriolis force $\Sigma (t)$) and ``number three'' (in the literature, such a construction based on $G(t)$ is most popular but, paradoxically, it is also most complicated).

arXiv comment18 pp

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• Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space