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

Abstract

A $(K+1)-$plet of non-Hermitian and time-dependent operators (say, $\Lambda_j(t)$, $j=0,1,\ldots,K$) 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, $\Lambda_j^\dagger(t)\Theta(t)=\Theta(t)\Lambda_j(t)$. The theory (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the states $\psi(t)$ (realized, via Schr\"{o}dinger-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 $\Omega^\dagger(t)\Omega(t)=\Theta(t)$. 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).