Composite Quantum Coriolis Forces

Abstract

In a consistent quantum theory known as “non-Hermitian interaction picture” (NIP), the standard quantum Coriolis operator $\mathsf{\Sigma }\left(t\right)$ emerges whenever the observables of a unitary system are given in their quasi-Hermitian and non-stationary rather than “usual” representations. With $\mathsf{\Sigma }\left(t\right)$ 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 $\mathsf{\Omega }\left(t\right)$. The knowledge of this factorization is shown to lead to a multiplet of alternative eligible Coriolis forces ${\mathsf{\Sigma }}_n\left(t\right)$ with $n=0,1,\dots ,N$. 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.

1. Introduction

In this paper, a thoroughly innovated formulation of quantum mechanics (herein referred to as “factorization-based non-Hermitian interaction picture”, FNIP) will be proposed and described. In contrast to the majority of the more dated literature offering a conventional presentation of quantum theory (cf., e.g., [1]), the operators of observables will be assumed non-Hermitian and non-stationary. In this context, reviews [2,3,4,5,6] or, at least, Section 2 and Section 3 below may be recommended for introductory reading.

The dominant inspiration of our present innovative methodical proposal may be seen in the existence of certain exactly solvable unitary quantum models in which the evolution is controlled by a stationary non-Hermitian Hamiltonian $H\ne H\left(t\right)$ (for an example see Appendix A below) and, more importantly, in the existence of the exactly solvable unitary quantum models in which the dynamics is represented by a set of certain manifestly time-dependent, non-stationary non-Hermitian operators of observables denoted, say, as ${\mathsf{\Lambda }}_0\left(t\right)$, ${\mathsf{\Lambda }}_1\left(t\right)$, etc., (see, e.g., review [4] for more details).

Our interest in the latter and, admittedly, important but still slightly exotic subclass of the solvable non-Hermitian quantum models was motivated, first of all, by the related mathematics. Indeed, in both the stationary and non-stationary scenarios one must, after all, re-Hermitize the observables in terms of a respective stationary Dyson map $\mathsf{\Omega }$ [2,5,7] or of its time-dependent analogue $\mathsf{\Omega }\left(t\right)$ [4,8,9]. Such a Hermitization has the form

$${\mathsf{\Lambda }}_k\left(t\right)=\mathsf{\Omega }\left(t\right){\lambda }_k\left(t\right){\mathsf{\Omega }}^{-1}\left(t\right)\mathrm{with}{\lambda }_k\left(t\right)={\lambda }_k^{†}\left(t\right),k=0,1,\dots$$

(1)

of a suitable explicit (or, at least, possible) transformation of all of the operators of observables. It is unavoidable for purposes of consistency [2]; however, in applications it represents one of the decisive technical challenges. In this context, our attention has been attracted by a more or less purely empirical observation that in virtually all of the above-mentioned solvable models, the authors managed to circumvent the technical obstacles via an auxiliary Dyson-map-factorization assumption

$$\mathsf{\Omega }\left(t\right)={\mathsf{\Omega }}_N\left(t\right){\mathsf{\Omega }}_{N-1}\left(t\right)\dots {\mathsf{\Omega }}_1\left(t\right).$$

(2)

A more in-depth study of this coincidence was performed in [10,11]. For the sake of simplicity, we decided to keep the Dyson map stationary, $\mathsf{\Omega }\left(t\right)=\mathsf{\Omega }\left(0\right)=\mathsf{\Omega }$, i.e., we decided to work just in the simpler formulation of quantum mechanics called “non-Hermitian Schrödinger picture” (NSP). The decision proved productive because we revealed that after the factorization (2) at a nontrivial $N\ge 2$, the two conventional and well-known forms of the NSP theory (cf. [5]) become complemented by a fairly rich menu of its $N-1$ alternatives called “hybrid NSP” (HNSP). Via a few schematic toy-model examples we also demonstrated, in [10,11], that the availability of such a menu opens, for a given stationary input Hamiltonian H, the possibility of the choice of an “optimal” version of the HNSP-based strategy of the quantum model-building.

The stationarity constraints $H\ne H\left(t\right)$ and $\mathsf{\Omega }\ne \mathsf{\Omega }\left(t\right)$ have subsequently been removed in [12]. There, we employed $N=2$, and we managed to find, describe and illustrate there the specific merits of a unique “hybrid” $N=2$ reformulation of quantum mechanics in non-Hermitian interaction picture. Paradoxically, one of the weaknesses of the latter result lied in the process of its derivation because it seemed to indicate that all of its $N>2$ generalizations would be trivial. Later on, we imagined that this was not the case. On the contrary, we found that the transition from the choice of $N=2$ to the general $N>2$ scenarios proved truly remarkable from both mathematical and phenomenological points of view. Furthermore, this also represents the main message to be delivered by our present paper.

The presentation of our results will be preceded by an introductory review of the state of art in Section 2 and Section 3. The first part of the new, $N>2$ amendment of the non-stationary, NIP theory will be then, in Section 4, devoted to the construction of the general composite Coriolis forces and to the explanation of their role in the Schrödinger-like NIP evolution equations.

In Section 5, this result will be complemented by the derivation of the $N>2$ form of the Heisenberg-like equations controlling the characteristic NIP time-dependence of the operators representing the observables. Moreover, an outline of dynamics in the statistical quantum mechanics will be provided, with the state of the system being merely specified as a mixture of pure states (i.e., by the so-called density matrix).

In general, the proposed FNIP formalism is supposed to be useful, efficient and applicable to any unitary quantum system in which there appears the possibility of a simplification of its dynamical evolution equations. This is to be realized by means of a suitable factorization (2) of the overall non-stationary Dyson map $\mathsf{\Omega }\left(t\right)$ entering the simplification ansatz (cf. Equation (5) below). An explicit illustrative example of such an eligible system will be provided and described in Section 6. We will show there that for the specific non-stationary non-Hermitian anharmonic-oscillator model of Fring and Tenney [13], the fundamental factorization (2) can even be realized non-numerically and in several alternative arrangements. To provide the climax of our paper, an extensive discussion is added in Section 7, and our message is finally summarized in Section 8.

2. Classical Coriolis Force and Its Quantum Analogue

The phenomenon known as the “Coriolis force” [14,15] emerges, in classical mechanics, due to the non-inertial choice of the frame of reference. A quantum analogue of such a force (i.e., the main subject of our present interest) enters the fray when one considers the Schrödinger equation

$$\mathrm{i}\frac{d}{dt}|\psi \left(t\right)\succ =\mathfrak{h}\left(t\right)|\psi \left(t\right)\succ$$

(3)

in a conventional Hilbert space of states (say, $\mathcal{L}$) in which the Hamiltonian is assumed self-adjoint,

$$\mathfrak{h}\left(t\right)={\mathfrak{h}}^{†}\left(t\right).$$

(4)

Such a constraint is equivalent to the unitarity of the evolution [16] so that, in some sense, it specifies a quantum analogue of the classical choice of the inertial reference frame.

A quantum analogue of transition to the non-inertial frame may be interpreted as mimicking a gravitational force (see a deeper development of this idea in [17]). In the language of mathematics the parallelism can be encountered when one preconditions the ket-vector element of $\mathcal{L}$ using the Dyson-inspired map,

$$\left|\psi \right(t)\succ =\mathsf{\Omega }(t\left)\right|\psi \left(t\right)\rangle .$$

(5)

One may expect the emergence of a quantum analogue of the Coriolis force in the various realistic applications of preconditioning (5) (see, e.g., [18]). Indeed, the invertible non-stationary operator $\mathsf{\Omega }\left(t\right)$ may simulate important features of dynamics, provided only that the transformations $\mathsf{\Omega }\left(t\right)$ are sufficiently flexible.

In a way recommended and tested by Dyson [7], such operators may even be non-unitary yielding nontrivial product

$${\mathsf{\Omega }}^{†}\left(t\right)\mathsf{\Omega }\left(t\right)=\mathsf{\Theta }\left(t\right).$$

(6)

Even when $\mathsf{\Theta }\left(t\right)\ne I$ (i.e., when the preconditioning is non-unitary), the symbol $\left|\psi \right(t)\rangle$ may still be treated as a useful representative of the state of the quantum system [2]. In general, it can be interpreted as a ket-vector element of an auxiliary, “fictious”, manifestly unphysical but mathematically friendlier Hilbert space (say, $\mathcal{F}$) [4]. Subsequently, the insertion of ansatz (5) converts the textbook Schrödinger Equation (3) living in $\mathcal{L}$ into its equivalent version in $\mathcal{F}$,

$$\mathrm{i}\frac{d}{dt}|\psi \left(t\right)\rangle =G\left(t\right)|\psi \left(t\right)\rangle ,G\left(t\right)=H\left(t\right)-\mathsf{\Sigma }\left(t\right),H\left(t\right)={\mathsf{\Omega }}^{-1}\left(t\right)\mathfrak{h}\left(t\right)\mathsf{\Omega }\left(t\right)$$

(7)

where

$$\mathsf{\Sigma }\left(t\right)=\frac{\mathrm{i}}{\mathsf{\Omega }\left(t\right)}\dot{\mathsf{\Omega }}\left(t\right),\dot{\mathsf{\Omega }}\left(t\right)=\frac{d}{dt}\mathsf{\Omega }\left(t\right).$$

(8)

The latter operator $\mathsf{\Sigma }\left(t\right)$ defined in terms of the known non-stationary Dyson map $\mathsf{\Omega }\left(t\right)$ is precisely the quantum analogue of the classical Coriolis force.

Equation (7) plays the role of a starting point of a reformulation of quantum mechanics in non-Hermitian interaction picture (NIP) as described in paper [8] and reviews [6,9,19,20,21]. In comparison, the older, stationary non-Hermitian Schrödinger picture (NSP, cf. reviews [2,3,5]) is much simpler to apply. In particular, in NSP one finds that the Coriolis forces are absent, $\mathsf{\Sigma }\left(t\right)=0$. As a consequence, the NSP generator $G\left(t\right)$ in (7) coincides then with the hiddenly Hermitian NSP Hamiltonian $H\left(t\right)$.

Certainly, the non-stationary approach is technically much more challenging. One is forced to search, more intensively, for the simplifications of the NIP formalism. In this sense, the main purpose of our present paper can briefly be formulated as a transfer of some of the most recent technical amendments of the stationary NSP quantum theory to the non-stationary NIP context.

In the early stages of such a project, naturally, we were well aware of the fact that in the non-stationary NIP framework the emergence of the Coriolis forces would indeed lead to a number of new technical challenges. This was one of the reasons why we did not dare to move, in the subsequent non-stationary framework of Ref. [12], beyond the first non-trivial non-stationary $N=2$ setup. Nevertheless, the appeal and urgency of some of the related realistic applications of the NIP approach (say, in the coupled-cluster context of review [20]) forced us to return to the description of the extension of the NIP analysis of paper [12] to the general case in which one would be able to use the more general ansatz (2) with $N>2$.

3. Composite Dyson Maps (Stationary Case)

Although the abstract formulation of the stationary version of the quasi-Hermitian quantum mechanics (QHQM, [2]) is now more than thirty-years old, the widely accepted opinion is that “it is generally very difficult to implement” (cf. p. 1216 in [5]). Still, after a number of technical amendments [3,5,6,22] the related model-building strategy became enormously popular [6]. One of the reasons may be seen in the extremely helpful assumption of the stationarity of the Dyson map $\mathsf{\Omega }\left(t\right)=\mathsf{\Omega }\left(0\right)=\mathsf{\Omega }$ as well as of the related operator $\mathsf{\Theta }\left(t\right)=\mathsf{\Theta }\left(0\right)=\mathsf{\Theta }$ called the Hilbert-space metric. In some models, indeed, this made the practical applicability of the QHQM approach (mentioned in the second line of

Table 1. Three alternative realizations of unitarity in stationary NSP framework.

Quantum Theory	Hamiltonian	Metric	Condition
conventional, [1]	$\mathfrak{h}$	I	${\mathfrak{h}}^{†}=\mathfrak{h}$
Dyson-inspired, [2]	$H={\mathsf{\Omega }}^{-1}\mathfrak{h}\mathsf{\Omega }$	$\mathsf{\Theta }={\mathsf{\Omega }}^{†}\mathsf{\Omega }$	$H^{†}\mathsf{\Theta }=\mathsf{\Theta }H$
hybrid [10], $\mathsf{\Omega }={\mathsf{\Omega }}_2{\mathsf{\Omega }}_1$	$H_1={\mathsf{\Omega }}_{1}H{\mathsf{\Omega }}_1^{-1}$	${\mathsf{\Theta }}_2={\mathsf{\Omega }}_2^{†}{\mathsf{\Omega }}_2$	$H_1^{†}{\mathsf{\Theta }}_2={\mathsf{\Theta }}_2{H}_1$

) comparable with the use of the conventional quantum theory using Hermitian Hamiltonians and mentioned in the first line of Table 1.

In the stationary QHQM alias NSP framework the unitary evolution of the quantum system in question is in fact represented, simultaneously, in the two non-equivalent Hilbert spaces, viz., in the unphysical space $\mathcal{F}={\mathcal{H}}_{mathematical}$ (with the conventional trivial identity-operator metric) and in its “physical-interpretation” partner (say, $\mathcal{H}$ or ${\mathcal{H}}_{physical}$, differing from $\mathcal{F}$ just by an amendment of the inner product, $I\to \mathsf{\Theta }$). From a purely pragmatic perspective, such an enhancement of the flexibility of the formalism (and, presumably, a significant simplification of the Hamiltonian) just transferred the potential mathematical difficulties (if any) from the conventional self-adjoint operator $\mathfrak{h}$ to the new nontrivial self-adjoint operator $\mathsf{\Theta }$.

In a brief letter [10], we proposed, therefore, another, “hybrid” theoretical option (cf. the third line of Table 1). A more detailed exposition of the idea was then provided in [11]. We obtained a reformulation of the unitary theory called hybrid non-Hermitian Schrödinger picture (HNSP).

A formal mathematical guide to the latter amendment of the theory has been found in the assumption of an $N-$term factorization of the Dyson map, i.e., in a time-independent special case of Equation (2),

$$\mathsf{\Omega }={\mathsf{\Omega }}_N{\mathsf{\Omega }}_{N-1}\dots {\mathsf{\Omega }}_1.$$

(9)

In Ref. [10], we outlined only the main consequences of the special, $N=2$ realization of such a factorization. A more consequent implementation of the idea was then performed in [11]. An extension of applicability of the $N=2$ ansatz $\mathsf{\Omega }={\mathsf{\Omega }}_2{\mathsf{\Omega }}_1$ of Ref. [10] to any integer N was achieved there in the form which is recalled here in

Table 2. Eligible versions of the hybrid NSP Hamiltonians. At any N and j, the related hybrid inner-product metrics are defined in terms of products ${\mathsf{\Omega }}_{\left[j\right]}={\mathsf{\Omega }}_N{\mathsf{\Omega }}_{N-1}\dots {\mathsf{\Omega }}_{j+1}$ (cf. [11]).

j	Hamiltonian	Physical Metric	Quasi-Hermiticity
0	$H=H_0$ (non-hybrid case)	$\mathsf{\Theta }={\mathsf{\Theta }}^{\left[0\right]}={\mathsf{\Omega }}_{\left[0\right]}^{†}{\mathsf{\Omega }}_{\left[0\right]}$	$H^{†}\mathsf{\Theta }=\mathsf{\Theta }H$
1	$H_1={\mathsf{\Omega }}_1{H}_0{\mathsf{\Omega }}_1^{-1}$	${\mathsf{\Theta }}^{\left[1\right]}={\mathsf{\Omega }}_{\left[1\right]}^{†}{\mathsf{\Omega }}_{\left[1\right]}$	$H_1^{†}{\mathsf{\Theta }}^{\left[1\right]}={\mathsf{\Theta }}^{\left[1\right]}{H}_1$
2	$H_2={\mathsf{\Omega }}_2{H}_1{\mathsf{\Omega }}_2^{-1}$	${\mathsf{\Theta }}^{\left[2\right]}={\mathsf{\Omega }}_{\left[2\right]}^{†}{\mathsf{\Omega }}_{\left[2\right]}$	$H_2^{†}{\mathsf{\Theta }}^{\left[2\right]}={\mathsf{\Theta }}^{\left[2\right]}{H}_2$
...	...	...	...
$N$ − $2$	$H_{N-2}={\mathsf{\Omega }}_{N-2}{H}_{N-3}{\mathsf{\Omega }}_{N-2}^{-1}$	${\mathsf{\Theta }}^{[N-2]}={\mathsf{\Omega }}_{[N-2]}^{†}{\mathsf{\Omega }}_{[N-2]}$	$H_{N-2}^{†}{\mathsf{\Theta }}^{[N-2]}={\mathsf{\Theta }}^{[N-2]}{H}_{N-2}$
$N$ − $1$	$H_{N-1}={\mathsf{\Omega }}_{N-1}{H}_{N-2}{\mathsf{\Omega }}_{N-1}^{-1}$	${\mathsf{\Theta }}^{[N-1]}={\mathsf{\Omega }}_N^{†}{\mathsf{\Omega }}_N$	$H_{N-1}^{†}{\mathsf{\Theta }}^{[N-1]}={\mathsf{\Theta }}^{[N-1]}{H}_{N-1}$
N	$\mathfrak{h}={\mathsf{\Omega }}_N{H}_{N-1}{\mathsf{\Omega }}_N^{-1}\equiv H_N$	I (Hermitian theory)	n.a., $\mathfrak{h}={\mathfrak{h}}^{†}$

. The parameter $j=0,1,\dots ,N$ numbers there, are the alternative formulations of the unitary quantum theory in its mathematical (i.e., quasi-Hermitian) representation in $\mathcal{F}$.

In Table 2, the basic features of the resulting general hybridized QHQM theory are given a recurrent form which complements the recipe of Ref. [11]. In the language of mathematics, just a more systematic partial transfer of the information about dynamics from the Hamiltonian to the metric is performed. This means that every choice of j specifies a different physics-representing Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$.

Such double-indexed symbols denoting the individual Hilbert spaces of interest were initially introduced, with a more abstract motivation in [11]. Our present $(N+1)-$plet of the physical Hilbert spaces can be found there to be just a subset of a larger multiplet arranged in a triangular-shaped lattice (cf. the diagram Nr. 32 in [11]). Incidentally, all of the other Hilbert-space elements ${\mathcal{R}}_k^{\left[j\right]}$ of the lattice with $k>0$ appeared manifestly unphysical.

For the state-vector elements of our present physical $k=0$ Hilbert spaces, it appeared necessary to use a specific, amended Dirac’s bra-ket notation convention. Its basic features are summarized in

Table 3. Kets and bras of the physical Hilbert spaces ${\mathcal{R}}_0^{\left[j\right]}$ as represented in $\mathcal{F}$.

j	ket	bra	Hamiltonian
0	${|}_{\left[0\right]}\psi \rangle \equiv |\psi \rangle$	${\langle \langle }_{\left[0\right]}{\psi |=\langle }_{\left[0\right]}\psi |{\mathsf{\Theta }}^{\left[0\right]}\equiv \langle \langle \psi |$	$H_0={\sum }_{\cdot }{|}_{\left[0\right]}\cdot \rangle E_{\cdot }{\langle \langle }_{\left[0\right]}\cdot |\equiv H$
1	${|}_{\left[1\right]}\psi \rangle ={\mathsf{\Omega }}_1{|}_{\left[0\right]}\psi \rangle$	${\langle \langle }_{\left[1\right]}{\psi |=\langle }_{\left[1\right]}\psi |{\mathsf{\Theta }}^{\left[1\right]}$	$H_1={\sum }_{\cdot }{|}_{\left[1\right]}\cdot \rangle E_{\cdot }{\langle \langle }_{\left[1\right]}\cdot |$
2	${|}_{\left[2\right]}\psi \rangle ={\mathsf{\Omega }}_2{|}_{\left[1\right]}\psi \rangle$	${\langle \langle }_{\left[2\right]}{\psi |=\langle }_{\left[2\right]}\psi |{\mathsf{\Theta }}^{\left[2\right]}$	$H_2={\sum }_{\cdot }{|}_{\left[2\right]}\cdot \rangle E_{\cdot }{\langle \langle }_{\left[2\right]}\cdot |$
...	...	...	...
$N-1$	${|}_{[N-1]}\psi \rangle ={\mathsf{\Omega }}_{N-1}{|}_{[N-2]}\psi \rangle$	${\langle \langle }_{[N-1]}{\psi |=\langle }_{[N-1]}\psi |{\mathsf{\Theta }}^{[N-1]}$	$H_{N-1}={\sum }_{\cdot }{|}_{[N-1]}\cdot \rangle E_{\cdot }{\langle \langle }_{[N-1]}\cdot |$
N	${|}_{\left[N\right]}\psi \rangle ={\mathsf{\Omega }}_N{|}_{[N-1]}\psi \rangle \equiv |\psi \succ$	${\langle \langle }_{\left[N\right]}{\psi |=\langle }_{\left[N\right]}\psi |\equiv \prec \psi |$	$H_N={\sum }_{\cdot }{|}_{\left[N\right]}\cdot \rangle E_{\cdot }{\langle \langle }_{\left[N\right]}\cdot |\equiv \mathfrak{h}$

. In particular, using an additional biorthonormality assumption, a formal spectral representation of the Hamiltonians also becomes available.

In spite of the full phenomenological equivalence of the formulations of the theory at the different choices of j (i.e., in spite of the $j-$independence of the predictions of the theory) the feasibility of the practical mathematical search for the solutions of Schrödinger equations living in the respective $j-$superscripted Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ may vary quite strongly with j. A persuasive illustration of the phenomenon was provided, via several toy-model examples, in [10,11,12]. In most cases, unfortunately, an optimal representation index j may only be specified a posteriori.

In our present methodical considerations it makes sense to consider, therefore, both N and j as quantities, an optimal choice of which crucially depends on a preselected Hamiltonian. In the stationary NSP interpretation of all of the three QHQM Table 1, Table 2 and Table 3 the individual operators of the Hilbert-space metric were also assumed time-independent. Nevertheless, the same Hilbert-space-classification pattern can be equally well used in the more general, non-stationary quantum theory of our present interest.

4. Non-Stationary Systems and Composite Coriolis Forces

In the non-stationary theory the first feature to notice is the emergence of the quantum Coriolis forces. As a consequence, in a way paralleling the derivation of Equation (7), once we admit the manifest time-dependence of the separate factor operators ${\mathsf{\Omega }}_n={\mathsf{\Omega }}_n\left(t\right)$ and of their products ${\mathsf{\Theta }}^{\left[j\right]}={\mathsf{\Theta }}^{\left[j\right]}\left(t\right)$, the related Hamiltonians $H_j\left(t\right)$ themselves lose their role of the generators of the evolution of the states in ${\mathcal{R}}_0^{\left[j\right]}$ at $j=0,1,\dots ,N$.

4.1. Schrödinger Equations for Vectors ${|}_{\left[j\right]}\psi \left(t\right)\rangle$

The basic details of the necessary upgrade of the theory may be explained even without reference to the factorization ansatz (2). We may set $N=1$ and re-derive immediately the time-dependent Schrödinger Equation (7) containing not only the general NIP Hamiltonian $H\left(t\right)$ but also the general NIP Coriolis force of Equation (8). Naturally. such a version of the non-stationary NIP formalism may already be found described in the literature (cf., e.g., [19]). Now, our attention has to be redirected to the non-stationary factorization-related theories which are characterized by the choice of a suitable larger $N\ge 2$. Due to such a decision, the two existing and sufficiently well known QHQM model-building strategies (corresponding to the choices of $j=0$ and $j=N$ in Table 2) have to be complemented by all of the remaining eligible options with $0<j<N$.

After a non-stationary reinterpretation of symbols in Table 2 and Table 3 let us now fix the value of N and describe the consequences. In a preparatory step we only have to choose $j=N$ and rewrite the conventional Schrödinger Equation (3) using an innovated, more explicit notation,

$$\mathrm{i}\frac{d}{dt}{|}_{\left[N\right]}\psi \left(t\right)\rangle =H_N{\left(t\right)|}_{\left[N\right]}{\psi \left(t\right)\rangle ,|}_{\left[N\right]}\psi \left(t\right)\rangle \in {\mathcal{R}}_0^{\left[N\right]}.$$

(10)

The operator $H_j\left(t\right)$ with maximal $j=N$ is exceptional by being self-adjoint not only in the $N-$th physical Hilbert space ${\mathcal{R}}_0^{\left[N\right]}$ but also in our “mathematical”, auxiliary representation space $\mathcal{F}$. The “trivial” conventional quantum mechanics of textbooks is obtained. The knowledge of the Hamiltonian and of the initial value of ${|}_{\left[N\right]}\psi \left(t\right)\rangle$ at $t=0$ enables us to reconstruct the state vector ${|}_{\left[N\right]}\psi \left(t\right)\rangle$ at any subsequent time $t>0$.

Once we turn attention to the first nontrivial reformulation of the NIP theory in the next physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ with $j=N-1$, we may recall relation

$$H_N^{†}\left(t\right)=H_N\left(t\right)={\mathsf{\Omega }}_N\left(t\right)H_{N-1}\left(t\right){\mathsf{\Omega }}_N^{-1}\left(t\right).$$

(11)

Having, in parallel, formula ${|}_{\left[N\right]}\psi \left(t\right)\rangle ={\mathsf{\Omega }}_N\left(t\right){|}_{[N-1]}\psi \left(t\right)\rangle$ and introducing the following family of the auxiliary tilded operators

$${\tilde{\mathsf{\Sigma }}}_j\left(t\right)=\frac{\mathrm{i}}{{\mathsf{\Omega }}_j\left(t\right)}{\dot{\mathsf{\Omega }}}_j\left(t\right),{\dot{\mathsf{\Omega }}}_j\left(t\right)=\frac{d}{dt}{\mathsf{\Omega }}_j\left(t\right),j=N,N-1,\dots ,2,1,$$

(12)

we may define the initial Coriolis force ${\mathsf{\Sigma }}_N\left(t\right)={\tilde{\mathsf{\Sigma }}}_N\left(t\right)$ and rewrite the conventional Hermitian Schrödinger Equation (10) of textbooks as another equation living in the neighboring, modified Hilbert space ${\mathcal{R}}_0^{[N-1]}$ with ket-vector elements ${|}_{[N-1]}\psi \left(t\right)\rangle$,

$$\mathrm{i}\frac{d}{dt}{|}_{[N-1]}\psi \left(t\right)\rangle =G_{N-1}\left(t\right){|}_{[N-1]}\psi \left(t\right)\rangle ,G_{N-1}\left(t\right)=H_{N-1}\left(t\right)-{\mathsf{\Sigma }}_N\left(t\right).$$

(13)

In the special case of $N=1$ the new generator $G_{N-1}\left(t\right)$ of the evolution of the kets in ${\mathcal{R}}_0^{[N-1]}$ coincides with operator $G\left(t\right)$ of Equation (7) of course.

Whenever $N>1$, the $j=N-1$ version (13) of the NIP Schrödinger equation is new. It differs not only from its conventional $j=N$ predecessor (10) but also from its ultimate QHQM $j=0$ alternative (7). In a way paralleling the stationary case, the corresponding version of quantum theory may be assigned the acronym “hybrid NIP” alias “HIP”, therefore.

It is worth adding that in a way illustrated by an elementary toy model in [12], the spectrum of the HIP generators $G_j\left(t\right)$ with $j<N$ may be complex irrespectively of the possible unitarity of the evolution of the physical quantum system in question. In other words, even in the unitary dynamical regime the “false non-stationary Hamiltonian” operators $G_j\left(t\right)$ with $j<N$ cannot be interpreted as quasi-Hermitian.

Our comment was formulated in the $j=N-1$ context but it applies to the HIP theory at any smaller j. In particular, in order to move from $j=N-1$ to $j=N-2$ at $N>1$ we may set ${|}_{[N-1]}\psi \left(t\right)\rangle ={\mathsf{\Omega }}_{N-1}\left(t\right){|}_{[N-2]}\psi \left(t\right)\rangle$. This enables us to convert quasi-Hermitian Equation (13) into a new, modified form of Schrödinger equation living in the next physical Hilbert space,

$$\mathrm{i}\frac{d}{dt}{|}_{[N-2]}\psi \left(t\right)\rangle =G_{N-2}\left(t\right){|}_{[N-2]}\psi \left(t\right)\rangle ,G_{N-2}\left(t\right)=H_{N-2}\left(t\right)-{\mathsf{\Sigma }}_{N-1}\left(t\right)$$

(14)

where ${|}_{[N-2]}\psi \left(t\right)\rangle \in {\mathcal{R}}_0^{[N-2]}$ and where

$${\mathsf{\Sigma }}_{N-1}\left(t\right)={\tilde{\mathsf{\Sigma }}}_{N-1}\left(t\right)+{\mathsf{\Omega }}_{N-1}^{-1}\left(t\right){\mathsf{\Sigma }}_N\left(t\right){\mathsf{\Omega }}_{N-1}\left(t\right).$$

(15)

This result indicates that and how the construction can be continued iteratively.

Theorem 1.

In the $j-$th physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ with $j=N-k$, the Schrödinger equation controlling the time-evolution of the state-vector kets ${|}_{[N-k]}\psi \left(t\right)\rangle \in {\mathcal{R}}_0^{[N-k]}$ has the form

$$\mathrm{i}\frac{d}{dt}{|}_{[N-k]}\psi \left(t\right)\rangle =G_{N-k}\left(t\right){|}_{[N-k]}\psi \left(t\right)\rangle ,G_{N-k}\left(t\right)=H_{N-k}\left(t\right)-{\mathsf{\Sigma }}_{N-k+1}\left(t\right)$$

(16)

where formula

$${\mathsf{\Sigma }}_{N-k+1}\left(t\right)={\tilde{\mathsf{\Sigma }}}_{N-k+1}\left(t\right)+{\mathsf{\Omega }}_{N-k+1}^{-1}\left(t\right){\mathsf{\Sigma }}_{N-k+2}\left(t\right){\mathsf{\Omega }}_{N-k+1}\left(t\right)$$

(17)

defines the necessary composite Coriolis force recursively.

Proof. 

The proof is elementary, provided by the text preceding the theorem. □

4.2. Schrödinger Equations for Vectors ${|}_{\left[j\right]}\psi \left(t\right)\rangle \rangle$

Let us now turn consider the formulae listed in the bra-vector column of Table 3. For the sake of brevity, we will not add and write the time-dependence arguments $\left(t\right)$ explicitly, bearing still in mind the generic non-stationarity of the underlying NIP and HIP formulations of the theory.

As long as all of our mathematical manipulations are assumed represented in the unique and time-independent auxiliary (i.e., unphysical but user-friendly) Hilbert space $\mathcal{F}$, we may amply use the fact that in such a space the inner-product metric is trivial. This means that the Hermitian conjugation converts every state-vector bra ${\langle \langle }_{\left[j\right]}\psi \left(t\right)|$ into its dual, twice-marked ket-vector representation ${|}_{\left[j\right]}\psi \left(t\right)\rangle \rangle$, slightly easier to deal with. Thus, at $j=N$, $N-1$, …we have

$${|}_{\left[N\right]}{\psi \rangle \rangle =|}_{\left[N\right]}{\psi \rangle =|\psi \succ ,|}_{[N-1]}\psi \rangle \rangle ={\mathsf{\Omega }}_N^{†}{|}_{\left[N\right]}{\psi \rangle \rangle ,|}_{[N-2]}\psi \rangle \rangle ={\mathsf{\Omega }}_{N-1}^{†}{|}_{[N-1]}\psi \rangle \rangle$$

(18)

etc. The elementary form of these recurrences enables us to prove not only that Equation (10) may be assigned its trivial conjugate partner

$$\mathrm{i}\frac{d}{dt}{|}_{\left[N\right]}\psi \left(t\right)\rangle \rangle =H_N^{†}{\left(t\right)|}_{\left[N\right]}{\psi \left(t\right)\rangle \rangle ,|}_{\left[N\right]}\psi \left(t\right)\rangle \rangle \in {\mathcal{R}}_0^{\left[N\right]}$$

(19)

but also that such a partnership extends, in recurrent manner, to all of the other relevant physical Hilbert spaces.

Theorem 2.

In the $j-$th physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ with $j=N-k$, the time-evolution of the twice-marked state-vectors ${|}_{[N-k]}\psi \left(t\right)\rangle \rangle \in {\mathcal{R}}_0^{[N-k]}$ is controlled by Schrödinger equation

$$\mathrm{i}\frac{d}{dt}{|}_{[N-k]}\psi \left(t\right)\rangle \rangle =G_{N-k}^{†}{\left(t\right)|}_{[N-k]}\psi \left(t\right)\rangle \rangle ,k=0,1,\dots ,N.$$

(20)

Proof. 

One of the easiest ways for the construction of the set of Equation (20), may start from the relation

$${|}_{[N-k]}\psi \left(t\right)\rangle \rangle ={\mathsf{\Omega }}_{N-k+1}^{†}{\left(t\right)|}_{[N-k+1]}\psi \left(t\right)\rangle \rangle$$

(21)

which follows immediately from the definitions listed in the third column of Table 3. □

5. Operator Evolution Equations

In the preceding section, we explained that the stationary NSP formulae of Table 2 may be assigned their very natural non-stationary NIP analogues. Now, we have to add that at every index j of the physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}\left(t\right)$, the assumption of the time-dependence of the related physical inner-product metric ${\mathsf{\Theta }}^{\left[j\right]}\left(t\right)$ is connected with the generic time-dependence of the energy-observable $H_j\left(t\right)$ (i.e., in our present terminology, of the Hamiltonian). Simultaneously, it is necessary to imagine that the time-dependence of the same metric must be also compatible with the possible time-variability of any other admissible observable, say, $A_j\left(t\right)$.

5.1. Heisenberg Equations

Necessarily [2,5], the latter operator must have a real spectrum and it must be quasi-Hermitian,

$$A_j^{†}\left(t\right){\mathsf{\Theta }}^{\left[j\right]}\left(t\right)={\mathsf{\Theta }}^{\left[j\right]}\left(t\right)A_j\left(t\right).$$

(22)

By construction, the standard probabilistic physical interpretation of the theory is in fact in a one-to-one correspondence with the validity of the latter postulate. Naturally, the non-stationarity of the metric leads to certain conceptual complications (cf. [19] or Theorem Nr. 2 in [5]). One of the most serious ones is that the mathematics of the evolution becomes practically prohibitively complicated whenever the operator $A_j\left(t\right)$ of interest (which is, by assumption, non-Hermitian in our manipulation space $\mathcal{F}$) remains time-dependent even after a return to its conventional self-adjoint representation in Schrödinger picture with trivial metric at $j=N$.

For this reason, we will postulate in what follows, that at $j=N$ (i.e., in effect, in Schrödinger picture) the observable of interest becomes time-independent, $A_N\left(t\right)=A_N\left(0\right)=\mathfrak{a}$. Under this assumption we may recollect that

$${\mathsf{\Omega }}_N\left(t\right)A_{N-1}\left(t\right)=\mathfrak{a}{\mathsf{\Omega }}_N\left(t\right)$$

(23)

(cf. the last line of Table 2). Thus, in an almost complete parallel with our preceding recursive considerations we may differentiate relation (23). The result

$$\mathrm{i}\frac{d}{dt}{A}_{N-1}\left(t\right)=A_{N-1}\left(t\right){\mathsf{\Sigma }}_N\left(t\right)-{\mathsf{\Sigma }}_N\left(t\right)A_{N-1}\left(t\right).$$

(24)

can be perceived as the desired $j=N-1$ operator-evolution equation in which the role of the generator (i.e., of a “false Hamiltonian” [12]) is played by precisely the same Coriolis force as defined by Equation (12) above.

In the same spirit we may differentiate the whole set of the identities listed in Table 2,

$${\mathsf{\Omega }}_{N-k}\left(t\right)A_{N-k-1}\left(t\right)=A_{N-k}\left(t\right){\mathsf{\Omega }}_{N-k}\left(t\right),k=1,2,\dots ,N-1.$$

(25)

The description of the HIP dynamics becomes completed.

Theorem 3.

In the $j-$th physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ with $j=N-k$, the time-evolution of any operator $A_j\left(t\right)$ representing an observable is controlled by Heisenberg equation

$$\mathrm{i}\frac{d}{dt}{A}_{N-k}\left(t\right)=A_{N-k}\left(t\right){\mathsf{\Sigma }}_{N-k+1}\left(t\right)-{\mathsf{\Sigma }}_{N-k_1}\left(t\right)A_{N-k}\left(t\right).$$

(26)

The role of the generator of evolution is played by the composite Coriolis force of Equation (17).

5.2. Evolution Equations for Density Matrices

In the NIP review [19] we considered, in our present terminology, only the specific and non-hybrid $j=N$ theory. We pointed out that besides the knowledge of the Hamiltonian and metric, the description of dynamics also requires the factorization of the metric $\mathsf{\Theta }\left(t\right)={\mathsf{\Omega }}^{†}\left(t\right)\mathsf{\Omega }\left(t\right)$ yielding the definition (8) of the Coriolis force $\mathsf{\Sigma }\left(t\right)$.

From a purely technical point of view, it is necessary to first distinguish between the stationary NSP theory (in which $\mathsf{\Theta }\left(t\right)=\mathsf{\Theta }\left(0\right)$ is time-independent) and the non-stationary NIP theory (in which the Coriolis forces emerge). In both of these cases, a key technical role is played in our manifestly unphysical representation space $\mathcal{F}$, by the respective two formulae

$$|{\psi }^{\left(NSP\right)}\left(t\right)\rangle \rangle ={\mathsf{\Theta }}^{\left(NSP\right)}\left(0\right)|\psi \left(t\right)\rangle \mathrm{and}|{\psi }^{\left(NIP\right)}\left(t\right)\rangle \rangle ={\mathsf{\Theta }}^{\left(NIP\right)}\left(t\right)|\psi \left(t\right)\rangle$$

(27)

(cf. the first line of Table 3). Clearly, in both of these formulae it would make sense to replace the difficult task of the construction of the operator (i.e., in finite dimensions, of an N by N array of matrix elements of the metric) by the construction of the mere vector (i.e., in finite dimensions, of the N components of $\left|\psi \right(t)\rangle \rangle$).

The latter observation sounds paradoxical; however, its clarification is easy. This is because at any j, the physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}\left(t\right)$ is always represented in the same auxiliary mathematical space $\mathcal{F}$. In other words, it is sufficient to recall Table 3 and to work simply with the sets of the $j-$dependent biorthonormalized bases in $\mathcal{F}$.

In such a biorthonormal basis setting, it is only necessary to remind the users that even the pure state of the quantum system in question is represented by the pair of the ket vectors or, more formally, by the elementary projector

$${\pi }_{\psi }\left(t\right)=|\psi \left(t\right)\rangle \frac{1}{\langle \langle \psi \left(t\right)\left|\psi \right(t)\rangle }\langle \langle \psi \left(t\right)|$$

(28)

(cf. a more extensive commentary is also available in [23]). Incidentally, such an observation simplifies transition to a statistical version of QHQM where the states become represented by the non-Hermitian and time-dependent version of the density matrix (cf., e.g., [24]),

$$\varrho \left(t\right)=\sum _k|{\psi }^{\left(k\right)}\left(t\right)\rangle \frac{p_k}{\langle \langle {\psi }^{\left(k\right)}\left(t\right)|{\psi }^{\left(k\right)}\left(t\right)\rangle }\langle \langle {\psi }^{\left(k\right)}\left(t\right)|.$$

(29)

Here, the “weights” or “probabilities” of the $k-$superscripted individual pure-state components have to be normalized to one, ${\sum }_k{p}_k=1$.

Let us now turn our attention to the consequences. First of all, let us once more choose the non-hybrid NIP option with $j=N$, i.e., let us assume that $\mathsf{\Theta }=\mathsf{\Theta }\left(t\right)$ (needed, first of all, in the definition (27)) can vary with time. In such a case, the Coriolis force $\mathsf{\Sigma }\left(t\right)$ of Equation (8) starts playing, suddenly, a decisive dynamical role. By the differentiation of the operator of metric $\mathsf{\Theta }\left(t\right)={\mathsf{\Omega }}^{†}\left(t\right)\mathsf{\Omega }\left(t\right)$, one obtains, first of all, the law

$$\mathrm{i}\frac{d}{dt}\mathsf{\Theta }\left(t\right)=\mathsf{\Theta }\left(t\right)\mathsf{\Sigma }\left(t\right)-{\mathsf{\Sigma }}^{†}\left(t\right)\mathsf{\Theta }\left(t\right)$$

(30)

controlling its evolution in time. In light of our previous comment, nevertheless, it makes sense to circumvent the solution of such an equation for operators in favor of the incomparably more economical solution of Schrödinger Equation (20), i.e.,

$$\mathrm{i}\frac{d}{dt}|\psi \left(t\right)\rangle \rangle =G^{†}\left(t\right)|\psi \left(t\right)\rangle \rangle ,G^{†}\left(t\right)=H^{†}\left(t\right)-{\mathsf{\Sigma }}^{†}\left(t\right).$$

(31)

The latter equation has the form of the adjoint partner of Equation (7) in $\mathcal{F}$. It is important to notice that, as a consequence, the overlaps $\langle \langle \psi \left(t\right)\left|\psi \right(t)\rangle$ remain constant in time.

Once we recall the definitions of states (28) or (29) (with constant weights $p_k\ne p_k\left(t\right)$), the explicit construction of the metric (i.e., of the operator which is, due to relation (27), implicitly present in $\left|\psi \right(t)\rangle \rangle$) may be declared redundant [19]. As the most important byproduct of such a simplification, we finally deduced in [19], the equation

$$\mathrm{i}{\partial }_t\varrho \left(t\right)=G\left(t\right)\varrho \left(t\right)-\varrho \left(t\right)G\left(t\right)$$

(32)

which formally controls the evolution of the non-hybrid density matrices in Liouvillean picture.

On this background, the transition to the hybrid formalism is immediate.

Theorem 4.

In the $j-$th physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}$ with $0\le j\le N$ the evolution of the mixed state represented by the density matrix

$${\varrho }_j\left(t\right)=\sum _k{|}_{\left[j\right]}{\psi }^{\left(k\right)}\left(t\right)\rangle \frac{p_k^{\left[j\right]}}{\langle {\langle }_{\left[j\right]}{\psi }^{\left(k\right)}\left(t\right){|}_{\left[j\right]}{\psi }^{\left(k\right)}\left(t\right)\rangle }{\langle \langle }_{\left[j\right]}{\psi }^{\left(k\right)}\left(t\right)|$$

(33)

is prescribed by the evolution equation

$$\mathrm{i}{\partial }_t{\varrho }_j\left(t\right)=G_j\left(t\right){\varrho }_j\left(t\right)-{\varrho }_j\left(t\right)G_j\left(t\right).$$

(34)

Proof. 

It is sufficient to recall the Schrödinger equations of the Theorems 1 and 2 controlling the time-evolution of the respective state-vector kets ${|}_{\left[j\right]}\psi \left(t\right)\rangle$ and ${|}_{\left[j\right]}\psi \left(t\right)\rangle \rangle$. Subsequently, one only has to recall the triviality of the conventional Hermitian conjugation in the unphysical representation space $\mathcal{F}$. □

6. An Exactly Solvable Illustrative Example of Applicability of the FNIP Formulation of Quantum Mechanics

One of the best non-stationary illustrative examples supporting and motivating the applicability of the present factorization-based NIP technique may be found in Ref. [13]. In the methodical setting, the latter model parallels its stationary predecessor of Ref. [25]. Hence, its main merit lies in the fact that in the factorization ansatz (2) the number N of the factors (which partially depends on how one counts them)is, in any case, nontrivial, $N>2$.

From a mathematical perspective, the latter model is in fact exceptional because both the initial, dynamical input information about the system and the ultimate, Hamiltonian-operator-based information about the probabilistic interpretation of the unitary quantum system in question are carried by the local potentials entering the respective linear, ordinary differential operators.

In the literature, the task is most often found feasible just in the stationary version of the formalism (for details see, say, the recent review paper [5]). In the context we recommended, in [11], just an innovative approach to the Hermitizations in which one has a choice between N alternative stationary representations of the system.

In the more general non-stationary case (see its introduction in [8]), the emerging larger set of the necessary evolution equations often becomes almost prohibitively complicated (see also the discussion of this point in Section 4 above). For this reason, any simplification of the task (including the one based on the non-stationary extension of the alternative-representation idea of Ref. [11]) may be expected to play a key role in applications.

In the paper by [13], the search for an exactly solvable illustrative example of the general non-stationary NIP Schrödinger Equation (i.e., in our present notation, of the ordinary differential equation of the form (7)) was based on a suitable ansatz (2) where the Dyson map has been factorized using $N=4$. Its separate factors were chosen in the elementary exponential-operator forms

$${\mathsf{\Omega }}_4\left(t\right)=e^{\alpha \left(t\right)x},{\mathsf{\Omega }}_3\left(t\right)=e^{\beta \left(t\right)p^3},{\mathsf{\Omega }}_2\left(t\right)=e^{i\gamma \left(t\right)p^2},{\mathsf{\Omega }}_1\left(t\right)=e^{i\delta \left(t\right)p}$$

(35)

where p denoted the momentum and where the optional coefficient functions $\alpha \left(t\right)$, $\beta \left(t\right)$, $\gamma \left(t\right)$ and $\delta \left(t\right)$ of time t were merely assumed sufficiently smooth and real.

In terms of these functions the Coriolis force of Equation (7) was then easily evaluated to acquire closed form

$$\mathsf{\Sigma }\left(t\right)=ix\dot{\alpha }+i\dot{\beta }{p}^3-(3\dot{\alpha }\beta +\dot{\gamma })p^2-(2i\gamma \dot{\alpha }+\dot{\delta })p-i\delta \dot{\alpha }.$$

This being specified the authors of [13] decided to analyze the family of the unitary quantum systems defined by a specific generator possessing the following “wrong-sign” anharmonic-oscillator form,

$$G\left(t\right)=G(z,t)=p^2+\frac{m\left(t\right)}{4}{z}^2-\frac{g\left(t\right)}{16}{z}^4.$$

Both the mass term $m\left(t\right)$ and the coupling $g\left(t\right)$ were assumed real and smooth while the choice of a suitable complex contour of z was inspired by Jones and Mateo [26], i.e., specified by formula $z=-2i\sqrt{1+ix}$ where $x\in \mathbb{R}$. This made the Schrödinger equation defined on a real line of the “coordinate” x (entering also ansatz (35)) so that the straightforward reference to the definition of the observable Hamiltonian $H\left(t\right)=G\left(t\right)+\mathsf{\Sigma }\left(t\right)$ (denoted, in [13], by a tilded symbol $\tilde{H}\left(t\right)$) led to an immediate evaluation of the isospectral partner Hamiltonian $\mathfrak{h}\left(t\right)$ which has to be, by definition, self-adjoint in $\mathcal{L}$.

The latter condition (i.e., constraint (4)) appeared to specify all of the functions $\alpha \left(t\right)$, $\beta \left(t\right)$, $\gamma \left(t\right)$ and $\delta \left(t\right)$ (which defined the preselected and factorized Dyson map) in terms of a real integration constant $c_1$ and of the two functions $m\left(t\right)$ and $g\left(t\right)$ which characterized the preselected generator $G\left(t\right)$ (all of these relations were made explicit by Eq. Nr. (2.11) in [13]).

The latter result became feasible due to a certain auxiliary functional constraint as given by relation Nr. (2.12) in [13]. As a consequence, a remarkable reparametrization of $G\left(t\right)$ has been obtained in which $m\left(t\right)$ and $g\left(t\right)$ appeared reparametrized in terms of another integration constant $c_2$ and of an “arbitrarily reparametrized time” $\sigma \left(t\right)$ and its derivatives,

$$g\left(t\right)=\frac{1}{4{\sigma }^3\left(t\right)},m\left(t\right)=\frac{4c_2+{\dot{\sigma }}^2\left(t\right)-2\sigma \left(t\right)\ddot{\sigma }\left(t\right)}{4{\sigma }^2\left(t\right)}.$$

(36)

In terms of the latter “new time” function $\sigma \left(t\right)$ is was finally possible to write down an explicit (albeit discouragingly lengthy) formula for the “conventional textbook” Hamiltonian $\mathfrak{h}\left(t\right)$ in $\mathcal{L}$ (cf. formula Nr. (2.14) or, in the special case with $c_1=c_2=0$, formula Nr. (2.15) in [13]).

In this manner the Hermitization $H\left(t\right)\to \mathfrak{h}\left(t\right)$ has been completed. Nevertheless, the authors of paper [13] also developed an alternative version of the Hermitization in which one needs to use another ansatz (2) with a larger $N>4$ and in which the result of the isospectral mapping $H\left(t\right)\to \widehat{\mathfrak{h}}\left(t\right)$ leads to a less exotic and fully conventional form of the self-adjoint Hamiltonian $\widehat{\mathfrak{h}}\left(t\right)=p_y^2+\mathfrak{v}\left(y\right)$. The kinetic energy term $p^2$ is well separated here from the local and confining double well potential given, in loc. cit., by formula Nr. (2.19).

7. Discussion

The strength of our present FNIP approach to the quantum mechanics of closed and unitary systems lies in its potential phenomenological applicability as illustrated by the natural emergence of the factorization of the Dyson map in the analytically solvable model of the preceding section. From an alternative, purely pragmatic, and numerically oriented point of view, another persuasive motivation and support for the use of the flexible FNIP menu has been provided via a schematic two-state model in Ref. [12], where strictly one of the representations of the system in question appeared most user-friendly and, hence, optimal and, from the perspective of maximal economy, unique. Naturally, the formal and mathematical preferences of such a type will always be strongly physics-dependent.

7.1. The Occurrence of Non-Stationary Non-Hermitian Observables in Applications

In the context of physics, there exist many quantum systems of interest in which the non-stationary and non-Hermitian versions of observables play a key role. In addition to the highly abstract cosmological problems and besides the methodical challenges related to the quantization of gravity (in which a key role is played by the non-stationary and non-Hermitian but Hermitizable Wheeler–DeWitt Equation [27]), the need for the use of the same, very general class of non-stationary operators also occurs in the gauge field models [28], or in the atomic and molecular systems [29,30,31], or in the field-theory-mimicking spin systems exposed to a time-dependent external field [32], etc. One of the oldest successful application of a realistic non-Hermitian Hamiltonian of a closed, unitary quantum system may be found in the Dyson’s phenomenologically motivated study [7] devoted to the phenomenon of ferromagnetism. The main idea lies within the decision of treatment of a conventional bound-state wave function $\left|\psi \right(t)\succ$ living in a conventional physical Hilbert space $\mathcal{L}$ as a result of action (5) of a suitable operator $\mathsf{\Omega }\left(t\right)$ upon a (presumably, simpler) element $\left|\psi \right(t)\rangle$ of a (presumably, user-friendlier) auxiliary Hilbert space $\mathcal{F}$. Naturally, the use of ansatz (5) can be perceived as a more or less routine separation and sharing of the degrees of freedom hidden in $\left|\psi \right(t)\succ$ between a “simplified state vector” $\left|\psi \right(t)\rangle$ and an educated guess of some of the most relevant “correlations” $\mathsf{\Omega }\left(t\right)$.

In the latter sense, the Dyson’s study may be perceived as paralleled by several analogous constructive techniques involving, e.g., the so-called interacting boson model philosophy used in nuclear physics [33,34,35], or the flexible coupled cluster method of atomic and molecular physics (CCM, cf. Refs. [36,37]), etc.

Even in the Dyson’s times, the potential efficiency of ansatz (5) was not unknown. Typically, it was used in the traditional Hartree–Fock approach to many-body systems where $\left|\psi \right(t)\rangle$ is usually chosen equal to an elementary determinant [1]). Nevertheless, the essence of the Dyson-inspired innovation manifested in the non-unitarity of the mapping (cf. (6)). Naturally, even with the stationary version of such an ansatz using constant operators $\mathsf{\Omega }\left(t\right)=\mathsf{\Omega }\left(0\right)=\mathsf{\Omega }$ and $\mathsf{\Theta }\left(t\right)=\mathsf{\Theta }\left(0\right)=\mathsf{\Theta }$ the work with ansatz (5) appeared to lead to the emergence of multiple serious technical challenges and obstacles. Their concise list may be found, e.g., on p. 1216 of review [5]. This was likely the main reason why even the applications of the simpler, stationary NSP version of the formalism long remained restricted just to some specific physical systems, say, in nuclear physics [38]. The situation only changed after Bender along with Boettcher [39] turned the attention of the community to the subset of quasi-Hermitian models exhibiting the parity-time symmetry ($\mathcal{PT}$-symmetry, cf. also Appendix A for more comments).

7.2. A Competition between Alternative Formulations of Quantum Mechanics

In the conventional studies of unitary quantum systems, one mostly has a choice between the theory in Schrödinger picture (SP) and in Heisenberg picture (HP). Occasionally, one encounters also the third methodical option called intermediate picture (IP, cf., e.g., p. 321 in [1]). The criteria of the choice are usually pragmatic and purpose-dependent. Thus, the SP approach appears mathematically most economical, especially when all of the observables of interest remain stationary. In contrast, the HP “representation” is attractive due to its closest formal analogy with classical mechanics. In comparison, the most general IP formulation of quantum mechanics has traditionally been perceived as a mere technical complement and interpolation between the two fundamental extremes.

In the methodically innovated and generalized QHQM framework of Scholtz et al. [2], all of the three (viz, SP, HP and IP) approaches appeared to play equally fundamental roles (cf. [3,5,6]). In the present paper, we paid attention to the specific non-Hermitian IP (NIP) philosophy emphasizing that the main source of its relevance lies in its flexibility. Indeed, the NIP formalism assumes the knowledge of certain non-Hermitian operators of observables with real spectra (a.k.a. quasi-Hermitian operators [40]) which are, in principle at least, Hermitizable (cf. [4,8,19,41,42]). This means that the NIP formalism is fully compatible with the standard textbook postulates of quantum theory.

Such a Hermitization “involves the construction of a metric” [2]. In the older, NSP stationary approach the physical Hilbert-space metric $\mathsf{\Theta }$ must necessarily (or at least de facto, up to an overall multiplication factor) remain stationary (cf., e.g., Theorem 2 in [5]). Currently, it is widely accepted that the same metric-stationarity requirement may be also encountered beyond the non-Hermitian SP setting, viz., in the non-Hermitian HP approach [43].

The methodical relevance and practical importance of the recent mathematical developments is currently encouraging an increased popularity of non-Hermitian formalism beyond the strictly specified domain of physics of the closed and hiddenly Hermitian (i.e., NSP or NIP) quantum systems. Pars pro toto, let us mention the enormous productivity of the transfer of the related innovative mathematics, say, to the quantum theory of open systems [44] or to the even less standard nonlinear effective theories and methods [45,46].

7.3. Three Most Important Hilbert Spaces of the Theory

More than half a century ago, during their numerical study of a certain realistic many-body quantum system, Freeman Dyson [7] revealed that an optimalization of his computations can be achieved via a suitable modification of the standard diagonalization techniques of textbooks. The efficiency of some of his calculations appeared enhanced after they replaced the conventional (and time-independent) self-adjoint Hamiltonian $\mathfrak{h}$ by its isospectral avatar H of Equation (7) where the operator $\mathsf{\Omega }$ was not required unitary. Later on, the judicious choice of $\mathsf{\Omega }$ also succeeded in a decisive simplification of some other realistic calculations [38].

From a purely methodical perspective, it was not too difficult to keep in mind that the self-adjointness of $\mathfrak{h}$ in $\mathcal{L}$ was equivalent to the $\mathsf{\Theta }-$quasi-Hermiticity of H in $\mathcal{F}$. Such an observation implies that one might have a choice between the direct diagonalization of $\mathfrak{h}$ and the preconditioning $\mathfrak{h}\to H$. At the same time, such a freedom of decision only emerges when the manifestly self-adjoint Hamiltonian $\mathfrak{h}$ appears much more complicated than H. One of the reasons is that in contrast to the entirely elementary Hermitian-conjugation rule in $\mathcal{L}$, the use of its metric-dependent NSP alternative requires certain additional and, for some specific models of quantum dynamics, truly nontrivial mathematical analysis [47,48,49,50].

The introduction of a non-trivial QHQM metric $\mathsf{\Theta }\ne I$ had to mediate, first of all, an (indirect) return of the Hamiltonian H to the desirable, unitarity-compatible ultimate mathematical status of a “hidden” Hermiticity. Thus, one could treat its quasi-Hermiticity as a constraint in which H plays the role of an input information about dynamics (unconstrained by the over-restrictive Hermiticity requirement [39]) while the metric is interpreted as one of the available Hermitization tools [51].

The motivation of the split and the main reason for the introduction of another space has been found to lie in the related possibility of a sufficient (i.e., in practice, drastic) simplification of the inner product. Thus, in $\mathcal{F}\equiv {\mathcal{H}}_{unphysical}$ one can use the Dirac’s bra-ket convention and write ${\langle {\psi }_1|{\psi }_2\rangle }_{\left(unphysical\right)}=\langle {\psi }_1|{\psi }_2\rangle$. In parallel, the “correct physical” inner product in ${\mathcal{H}}_{physical}$ can still be represented, in the preferable and decisively user-friendlier auxiliary “mathematical” Hilbert space ${\mathcal{H}}_{unphysical}$, by a closed formula ${\langle {\psi }_1|{\psi }_2\rangle }_{\left(physical\right)}=\langle {\psi }_1\left|\mathsf{\Theta }\right|{\psi }_2\rangle$.

The costs of the simplification of the Hilbert space ${\mathcal{H}}_{physical}\to {\mathcal{H}}_{unphysical}$ must be acceptable. Among them, the best visible item to check is that the (obligatory) self-adjointness of the observables in ${\mathcal{H}}_{physical}$ has to be replaced by the respective quasi-Hermiticity constraints [40]. In this manner, the QHQM formalism can be briefly characterized by the triplet of the Hilbert spaces according to the following diagram,

(37)

One can summarize that any ‘dynamical input” knowledge of a suitable set of operators characterizing the observable aspects of a quantum system in ${\mathcal{H}}_{unphysical}$ can lead to a consistent theoretical Hermitization interpretation in two ways, viz, via a return to ${\mathcal{H}}_{textbook}$, or via a transition to ${\mathcal{H}}_{physical}$.

Perhaps, the former reconstruction recipe has seldom been found feasible. Still, its rather exceptional exemplification was offered by Buslaev and Grecchi [25] (see also Appendix A below). They were able to reconstruct a self-adjoint textbook isospectral partner $\mathfrak{h}$ (acting in ${\mathcal{H}}_{textbook}=L^2(\mathbb{R}$) of a preselected non-Hermitian operator H (introduced and defined as acting in ${\mathcal{H}}_{unphysical}$) using strictly analytic means.

7.4. Hybrid Versions of the Formalism

The currently increasing popularity of the name “non-Hermitian” is partially undeserved because its mathematically more correct version would be “quasi-Hermitian” (cf. [40]). The reason is that one can still easily restore the probabilistic physical-state meaning of the kets $\left|\psi \right(t)\rangle$ by the mere ad hoc change of the “false but standard” inner product $\langle {\psi }_a|{\psi }_b\rangle$ in $\mathcal{F}$. The point is that the restoration (with its discovery dating back to review [2]) is straightforward because it is entirely sufficient to introduce another, third Hilbert space (say, ${\mathcal{H}}_{physical}$) in which the ket-vector elements remain the same (i.e., we can equally well write $|\psi \rangle \in {\mathcal{H}}_{physical}$) and we only have to amend the inner product properly.

In review [2], we read that in the quasi-Hermitian NSP framework “the normal quantum-mechanical interpretation …involves the construction of a metric (if it exists)”. In other words, for a return to ${\mathcal{H}}_{physical}$ it is sufficient to keep working in $\mathcal{F}$ (with its conventional Dirac’s bra-ket notation and the current, manifestly unphysical inner product $\langle {\psi }_a|{\psi }_b\rangle ={\langle {\psi }_a|{\psi }_b\rangle }_{\mathcal{F}}$) and just introduce its correct physical upgrade in ${\mathcal{H}}_{physical}$.

A new and promising compromise between the use of the trivial and nontrivial physical Hilbert-space metric has recently been found in a new, “hybrid” reformulation of the formalism of quantum mechanics in [10]. In it one combines a partial, intermediate Hermiticity-violating preconditioning of the Hamiltonian $\mathfrak{h}\to H_1$ (cf. equation Nr. 13 in [10]) with a partial simplification $\mathsf{\Theta }\to {\mathsf{\Theta }}_2$ of the Hermitian-conjugation-mediating operator alias Hilbert-space inner-product metric (cf. equation Nr. 14 in [10]). An explicit, fairly persuasive demonstration of the (possible) optimality of the compromise has been also provided via an illustrative two-state quantum model (cf. equations Nr. 24 and 25 in [10]).

In a more mathematical spirit, let us add that in our preparatory studies [11,52] a fairly complicated notation was needed to explain the idea of an iterative Hermitization at $N>0$. We had to introduce sophisticated superscripts marking the individual non-equivalent Hermitian conjugations, e.g., in the auxiliary sequence of the Hilbert spaces ${\mathcal{R}}_k^{\left[0\right]}$ numbered by the subscript. In the present paper we managed to simplify the discussion by assuming that all of the relevant mathematical entities (i.e., the vectors and the operators) are defined as represented in the single, manifestly unphysical but still mathematically optimal working Hilbert space ${\mathcal{R}}_N^{\left[0\right]}$ alias $\mathcal{F}$ with the conventional trivial inner product.

This change of notation allowed us to summarize the essence of the hybrid NSP formalism in Table 3. After our transition to the non-stationary NIP framework it was then sufficient to add there the indication of the time-dependence of the maps ${\mathsf{\Omega }}_j={\mathsf{\Omega }}_j\left(t\right)$ and of the metrics ${\mathsf{\Theta }}^{\left[j\right]}={\mathsf{\Theta }}^{\left[j\right]}\left(t\right)$. In comparison with the above-mentioned older studies we also postulated there that at every j the kets and bras form a bicomplete and biorthonormal basis in $\mathcal{F}$. Thus, even the more mathematically rigorous readers may still accept Table 3 as corresponding to the quantum models for which the physical Hilbert space in question is finite-dimensional. Whenever one needs to work with $dim{\mathcal{R}}_0^{\left[j\right]}=\infty$, a complementary reading may be sought, say, in the dedicated reviews in [6].

In all three Table 1, Table 2 and Table 3, one could replace the word “Hamiltonian” with the name of any other observable. Nevertheless, our knowledge of the Hamiltonian $H\left(t\right)$ plays a key role in the theory because it enables us to recall the Stone theorem [16] and to relate the Hermiticity of the Hamiltonian $\mathfrak{h}\left(t\right)$ guaranteeing the unitarity of the evolution to the mathematically equivalent condition of the quasi-Hermiticity of the Hamiltonian $H\left(t\right)$ in $\mathcal{F}$.

8. Summary

Dyson’s long-ago formulated idea [7] of the description of a unitary quantum system using a manifestly non-Hermitian version of its Hamiltonian $H=H^{†}$ was treated, for a long time, as a special mathematical trick rather than a deeper innovation of the theory. The change only came when some of the basic conceptual problems were shown tractable using several additional mathematical assumptions: Among them, the most important ones were the restrictions of the class of the admissible observables to the operators which have to be, e.g., bounded (cf. Scholtz et al. [2]) or pseudo-Hermitian alias $\mathcal{PT}-$symmetric (see [39] and reviews [3,5]).

After these developments, it became widely accepted that the resulting QHQM theory offers a fully consistent theoretical framework and picture of the evolution of unitary quantum systems. In practice, the broadening of the scope of the theory was accompanied by a steady development of its technical aspects which made the formalism applicable not only to the models using the stationary inner products (in the NSP framework) but also in the non-stationary NIP scenario.

In our present paper, we paid attention to the latter methodical context and we proposed its efficient simplification based on the general $N-$term factorization of the underlying Dyson transformation operators $\mathsf{\Omega }\left(t\right)$. We may summarize that our present main result may be seen in the explicit description of the consequences of the latter factorization ansatz. We revealed that in contrast to the initial scepticism as expressed in our preparatory study [12], the resulting new FNIP formulation of quantum mechanics still remains formally transparent and user-friendly.

The core reason behind the survival of the simplicity of our formalism may be seen in the discovery of the comparatively elementary formulae which define the operators (which we called quantum Coriolis forces) in a recurrent way. This enabled us to precisely show that the complexity of the set of the underlying evolution equations (in which the Coriolis force forms the essential ingredient) remains acceptable and that it basically does not grow with N. Moreover, from the theoretical perspective we may say that the main new feature of the formalism is that it enables its user to make a choice of the correct physical Hilbert space ${\mathcal{R}}_0^{\left[j\right]}\left(t\right)$ out of the menu of $N+1$ eligible candidates such that $0\le j\le N$ (here we included also $j=0$ which would only mark the conventional Hermitian theory).

Our new formalism incorporates several existing versions of the non-stationary non-Hermitian quantum mechanics. Besides the trivial Hermitian theory with $j=0$, we have to mention the NIP theory where there is no hybrid option because $N=1$. Similarly, only a single hybrid option was found tractable in [12] so that no variable index j was needed either. In this sense, the variability of $j-$options in our present quantum theory of bound states can be interpreted as a new degree of flexibility, i.e., a new method of searching for an optimality of the practical quantum model-building.