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arXiv:2301.06128 · Physics Letters A

Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation

Miloslav Znojil

Published Jan 23, 2023• 3 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2023.128655

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In a consistent quantum theory known as “non-Hermitian interaction picture” (NIP), the standard quantum Coriolis operator Σ(t) emerges whenever the observables of a unitary system are given in their quasi-Hermitian and non-stationary rather than “usual” representations. With Σ(t) needed, in NIP, in both the Schrödinger-like and Heisenberg-like dynamical evolution equations we show that another, amended and potentially simplified theory can be based on an auxiliary N−term factorization of the Dyson’s Hermitization map Ω(t). The knowledge of this factorization is shown to lead to a multiplet of alternative eligible Coriolis forces Σn(t) with n=0,1,…,N. The related formulae for the measurable predictions constitute a new formalism refered to as “factorization-based non-Hermitian interaction picture” (FNIP). The conventional NIP formalism (where N=1) becomes complemented by an (N−1)-plet of its innovative “hybrid” alternatives. Some of the respective ad hoc adaptations of observables may result in an optimal representation of quantum dynamics.
Abstract The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (i.e., non-stationary) aspects that are characteristic of the coupled cluster method (CCM) techniques that nowadays form one of the most versatile and most accurate of all available formulations of quantum many-body theory. In so doing we expose and exploit multiple parallels between the NHIP and the CCM in its time-dependent versions.
In the framework of quantum mechanics using quasi-Hermitian operators the standard unitary evolution of a non-stationary but still closed quantum system is only properly described in the non-Hermitian interaction picture (NIP). In this formulation of the theory both the states and the observables vary with time. A few aspects of implementation of this picture are illustrated via the "wrong-sign" quartic oscillators. It is shown that in contrast to the widespread belief, both of the related Schrödinger-equation generators G(t) and the Heisenberg-equation generators Σ(t) are just auxiliary concepts. Their spectra are phenomenologically irrelevant and, in general, complex. It is argued that only the sum H(t)=G(t)+Σ(t) of the latter operators retains the standard physical meaning of the instantaneous energy of the unitary quantum system in question.