Mutual Compatibility/Incompatibility of Quasi-Hermitian Quantum Observables

Abstract

In the framework of quasi-Hermitian quantum mechanics, the eligible operators of observables may be non-Hermitian, $A_j\ne A_j^{†}$, $j=1,2,\dots ,K$. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric $\mathsf{\Theta }\ne I$, guaranteeing the quasi-Hermiticity $A_j^{†}\mathsf{\Theta }=\mathsf{\Theta }{A}_j$. The task is easy at $K=1$ because there are many eligible metrics $\mathsf{\Theta }=\mathsf{\Theta }\left(A_1\right)$. In our paper, the next case with $K=2$ is analyzed. The criteria of the existence of a shared metric, $\mathsf{\Theta }=\mathsf{\Theta }(A_1,A_2)$, are presented and discussed.

1. Introduction

It used to be widely accepted that a consistent picture of a unitary quantum system can only be based on our knowledge of its observable quantities represented by a set of operators $A_1$, $A_2$, …that are all self-adjoint in a suitable Hilbert space (cf., e.g., [1]). More than thirty years ago, Scholtz, Geyer, and Hahne [2] pointed out that such a statement may lead to misunderstandings because, in some fairly realistic models, the operators of observables could be defined as acting, simultaneously, in two different Hilbert spaces, viz., in spaces ${\mathcal{H}}_{mathematical}$ and ${\mathcal{H}}_{physical}$.

By construction, the latter two spaces have to coincide as the sets of ket vectors $|\psi \rangle$. Still, they may be unitarily non-equivalent because they may differ by the use of the two alternative forms of the respective inner products. Thus, in order to avoid misunderstandings, one has to speak, first, about a maximally user-friendly inner product that specifies a manifestly unphysical Hilbert space ${\mathcal{H}}_{mathematical}$. At the same time, one also has to speak about the second correct physical inner product that must be used to specify the other presumably user-unfriendly Hilbert space of state ${\mathcal{H}}_{physical}$, which is still the only one that provides their correct probabilistic interpretation.

The technical details of the idea may be found in [2]. A decisive advantage of keeping the two spaces different has been found in the possibility of reducing the difference to the mere redefinition of the inner product,

$${\langle {\psi }_a|{\psi }_b\rangle }_{mathematical}\to {\left({\psi }_a\right|{\psi }_b)}_{physical}\equiv \langle {\psi }_a\left|\mathsf{\Theta }\right|{\psi }_b{\rangle }_{mathematical}.$$

(1)

The amended (often called physical) inner-product metric $\mathsf{\Theta }\ne I$ should be a positive-definite and bounded operator that is self-adjoint in ${\mathcal{H}}_{mathematical}$ and has a bounded inverse [3]. Under these assumptions, the “obligatory” requirement of Hermiticity of $A_j$s in ${\mathcal{H}}_{physical}$ can be re-expressed, in the preferred representation space ${\mathcal{H}}_{mathematical}$, as the quasi-Hermiticity [4] alias $\mathsf{\Theta }$—pseudo-Hermiticity [5],

$$A_j^{†}\mathsf{\Theta }=\mathsf{\Theta }{A}_j,\forall j.$$

(2)

Needless to add that the two Hilbert spaces would coincide in the limit of $\mathsf{\Theta }\to I$. Within the generic theory with $\mathsf{\Theta }\ne I$, nevertheless, the “hidden Hermiticity” constraint (2) will still imply the reality of the spectrum.

In most applications, the model-building process proceeds in the opposite direction. In the manner sampled in [2] or [6], one is just given a set $\left\{A_j\right\}$ of the candidates for observables. All of these operators have to be treated, in ${\mathcal{H}}_{mathematical}$, as non-Hermitian, $A_j\ne A_j^{†}$. For all of them, the reality of the spectrum is a merely necessary, but not sufficient, condition of their observability and mutual compatibility. One has to prove the mathematical consistency of the theory by showing that all of the preselected operators of observables share the quasi-Hermiticity property (2). A clarification of these items is also the main purpose of our present paper.

In our preceding study [7] of the problem, we showed that, even when one decides to work with the mere doublet of the “input” operators $A_1$ and $A_2$, the acceptable inner-product metric need not exist at all. The essence of such a no-go theorem was that the two “arbitrary” operators, $A_1$ and $A_2$, had to be quasi-Hermitian with respect to a single subscript-independent metric $\mathsf{\Theta }$. This simply imposes too many constraints upon $\mathsf{\Theta }$ in general.

In what follows, we will outline a continuation of the latter study. We will introduce and describe an efficient strategy for the construction of $\mathsf{\Theta }$ whenever it does exist.

2. Self-Adjoint Hamiltonians in Quasi-Hermitian Form

The innovative two-Hilbert-space reformulation of quantum theory as offered in [2] may be called quasi-Hermitian quantum mechanics (QHQM). In its original formulation, unfortunately, most of the mathematical as well as phenomenological roots of such a theory seemed to be rather specific, aimed just at nuclear physicists. Indeed, the authors themselves were nuclear physicists, so they decided to illustrate the merits of the formalism via a rather complicated many-particle example. In the broader physics community, therefore, the more abstract theoretical message as provided in [2] remained more or less unnoticed.

In the words of a recent review paper [5], “the main problem with this formalism is that it is generally very difficult to implement” (cf. p. 1217 in loc. cit.). A properly amended version of the idea of QHQM appeared, fortunately, just a few years later.

2.1. Quantum Models with Single Quasi-Hermitian Observable

An ultimate and successful version of the QHQM approach to quantum systems has been proposed by Bender with co-authors [8,9]. Their upgrade of the theory has been simpler, better motivated, better presented, and much more widely accepted. Almost immediately, the modified theory (currently known under a nickname of $\mathcal{PT}$-symmetric quantum mechanics [10]) became truly popular, partly also due to the methodologically very fortunate restriction of attention to specific quantum systems characterized by the use

(i)

of a single non-Hermitian operator $A_1$ serving phenomenological purposes and yielding, typically, bound-state energies as its eigenvalues;

(ii)

of the most common formulation, the theory called Schrödinger picture, in which the observable plays the role of quantum Hamiltonian, $A_1=H$ [5];

(iii)

of the mere ordinary-differential Hamiltonians $H=-d^2/d{x}^2+V\left(x\right)$ in which even the local interaction potentials themselves were kept maximally elementary;

(iv)

of an auxiliary antilinear parity times time-reversal symmetry ($\mathcal{PT}$-symmetry). This made the Hamiltonians tractable as self-adjoint in an ad hoc Krein space [11].

All of these well-invented simplifications contributed to the enormous growth in popularity of the non-Hermitian representations of quantum systems exhibiting $\mathcal{PT}$-symmetry. What followed was a highly productive and fruitful transfer of the related mathematics to several non-quantum areas of physics (cf., e.g., [12]). Nevertheless, we believe that it is now time to remove or weaken at least some of the above-listed simplifications while turning attention

(i)

to the models using at least two observables, i.e., not only $A_1$ (Hamiltonian) but also $A_2$ (in [7], the second operator was interpreted as representing the coordinate);

(ii)

to the possibilities of description of systems in the quasi-Hermitian Schrödinger picture of Ref. [13], in the quasi-Hermitian interaction picture of Ref. [14], or in the quasi-Hermitian Heisenberg picture of Ref. [15];

(iii)

to some alternative simplifications of dynamics using, e.g., the discretizations of coordinates allowing the systems to live in a finite-dimensional Hilbert space ${\mathcal{H}}^{\left(N\right)}$;

(iv)

to the systems with fewer symmetries (and, in particular, without $\mathcal{PT}$-symmetry) and to the constructions making use, at $N<\infty$, of the methods of linear algebra.

2.2. The Concept of Dyson Map

During the history of development of QHQM, as outlined, say, in [16], people proposed several simplifications treating the inner-product metric $\mathsf{\Theta }$ as a product of some other auxiliary operators. The first attempt can be traced back to Dyson’s paper [17] in which both of the above-mentioned Hilbert spaces, ${\mathcal{H}}_{mathematical}$ and ${\mathcal{H}}_{physical}$ (with their shared ket-vector elements denoted by the same symbol $|\psi \rangle$), were introduced as auxiliary, complementing just the usual third Hilbert space ${\mathcal{H}}_{standard}$ of traditional textbooks (here, we follow [18] and introduce, for the sake of clarity, the different “curved” ket symbol $|\psi \succ$ for its elements).

The essence of the problem was that the convergence of the routine variational calculations of the bound-state energy spectra of a certain multi-fermioninc quantum system appeared to be, in the latter representation space, too slow.

The reason was the existence of fermion–fermion correlations. With this in mind, Dyson managed to accelerate the convergence via a map between states $|\psi \succ \in {\mathcal{H}}_{standard}$ and $|\psi \rangle \in {\mathcal{H}}_{mathematical}$,

$$|\psi \succ =\mathsf{\Omega }|\psi \rangle .$$

(3)

Although the mapping was designed to mimic the correlations, its conventional unitary form did not work. Success has only been achieved when the operators $\mathsf{\Omega }$ were chosen non-unitarily, i.e., such that

$${\mathsf{\Omega }}^{†}\mathsf{\Omega }=\mathsf{\Theta }\ne I.$$

(4)

This was precisely the reason why Dyson had to replace the space ${\mathcal{H}}_{mathematical}$ by its amendment ${\mathcal{H}}_{physical}$. Indeed, only the latter Hilbert space could have been declared physically and unitarily equivalent to its standard textbook alternative. Indeed, the emergent equivalence appeared to be due to the coincidence of all of the inner products of states before and after the mapping,

$$\prec {\psi }_a|{\psi }_b\succ =\langle {\psi }_a\left|\mathsf{\Theta }\right|{\psi }_b\rangle .$$

(5)

In nuclear physics, Dyson’s map (3) found a number of applications [19], mainly because the complicated fermionic statistics (reflecting the Pauli principle in ${\mathcal{H}}_{standard}$) were, by construction, simplified and replaced by the bosonic-representation alternative in both ${\mathcal{H}}_{mathematical}$ and ${\mathcal{H}}_{physical}$. This was another benefit that simplified the calculations performed in the most friendly Hilbert space ${\mathcal{H}}_{mathematical}$. Simultaneously, with the knowledge of a (preselected) factor-operator $\mathsf{\Omega }$, it was entirely straightforward to interpret the manifestly non-Hermitian Hamiltonian $A_1\equiv H\ne H^{†}$ as isospectral with its transform

$$\mathsf{\Omega }H{\mathsf{\Omega }}^{-1}=\mathfrak{h}.$$

(6)

Due to the validity of the quasi-Hermiticity (2) of $H=A_1$, the latter operator $\mathfrak{h}$ is self-adjoint in ${\mathcal{H}}_{standard}$ and isospectral with H.

3. The Problem of Compatibility

In 1956, Dyson introduced the mapping (3) + (4), interpreted as an auxiliary, non-unitary transformation of correlated fermions into interacting bosons. He managed to represent fermionic (so-called Fock) space ${\mathcal{H}}_{standard}$ in unphysical bosonic ${\mathcal{H}}_{mathematical}$ as well as in the correct bosonic ${\mathcal{H}}_{physical}$. All this was just the technically motivated change in representation, with the difference between the latter two Hilbert spaces only involving the fermion-correlation-reflecting change in the inner product in the two interacting-boson representations of the original fermionic system.

In 1992, the authors of [2] extended these considerations and concluded that, under certain conditions, the role of the observables of a conventional quantum system could be played by a set $A_j$ of manifestly non-Hermitian but bounded operators possessing real and discrete spectra. For illustration, they chose a system characterized by triplet, $A_1$, $A_2$, and $A_3$, certain linearly independent preselected non-Hermitian candidates for the observables (cf. section Nr. 3 in loc. cit.). Such a choice was truly impressive because it admitted even the emergence of quantum phase transitions. Later, as we already mentioned, such a direction of research was found to be difficult and, more or less completely, abandoned. The subsequent developments in the field led to the preference of studies of less-realistic quantum models, with their dynamics controlled by the mere single non-Hermitian observable.

3.1. Quantum Models with Two Quasi-Hermitian Observables

Until recently, most of the applications of the QHQM approach remained restricted to the systems characterized by the observability of the mere single operator $A_1=H$ representing the Hamiltonian. In our present paper, we are going to return to the origins. We will assume more realistic dynamical input knowledge of more than one non-Hermitian observable, keeping in mind the well-known fact that such knowledge is essential and needed, after all, even for an unambiguous reconstruction of the correct physical metric $\mathsf{\Theta }$.

In 2017, we already initiated such a return to the origins in [7]. We managed to show there that, in the general multi-observable setting, even the first additional linearly independent operator candidate $A_2$ for a non-Hamiltonian observable cannot be arbitrary. In a way inspired by a few specific realistic examples, we demonstrated that an unrestricted choice in the two linearly independent operators $A_1$ and $A_2$ need not be acceptable as representing a pair of observables of a single quantum system in general.

For the purposes of the proof of the latter statement, it has been sufficient to use perturbation techniques. We showed that, even in the specific “almost Hermitian” models with two preselected “almost Hermitian” candidates $A_1$ and $A_2$ for observables and under the related small-non-Hermiticity-reflecting ansatz

$$\mathsf{\Theta }=I+ϵF+\mathcal{O}\left({ϵ}^2\right)$$

(7)

one arrives at the constraints (in fact, as a consequence of Equation (2)) for which a nontrivial first-order-metric solution F need not exist at all.

In our present continuation of the latter study, we will drop the “almost Hermiticity” assumptions as over-restrictive and redundant. We will analyze the “mutual compatibility/incompatibility” problem in its full generality. For methodological reasons as well as for the sake of brevity of our text, we will only assume that the dimension N of the relevant Hilbert spaces is finite.

3.2. A New Trick: Two Auxiliary Standard Spaces

In Section 2, we worked, in effect, with a triplet of Hilbert spaces, viz., with ${\mathcal{H}}_{standard}$ (known from the conventional textbooks and, in the above-mentioned Dyson’s example, “fermionic”) and with ${\mathcal{H}}_{mathematical}$ and ${\mathcal{H}}_{physical}$ (in the Dyson’s example, both of them were “bosonic”). The former one was related to the other two by the Dyson map $\mathsf{\Omega }$ (cf. Equation (3)), while the “physicality” of ${\mathcal{H}}_{physical}$ was due to relation (5).

There existed just two versions of the observable Hamiltonian (cf. Equation (6)). More precisely, the representation $\mathfrak{h}$ of the observable Hamiltonian in ${\mathcal{H}}_{standard}$ was, necessarily, diagonalizable. In some cases, one could even work, directly, with its most elementary diagonal-matrix form (cf., e.g., Eq. Nr. 5 in [20]).

With the turn in attention to the quantum models characterized by the pairs of observables $A_1$ and $A_2$, the overall theoretical scenario becomes different. First of all, once we fix a Dyson map, we may still consider the pair of analogues of Equation (6), viz., mappings $A_1\to {\mathfrak{a}}_1$ and $A_2\to {\mathfrak{a}}_2$. Now, the point is that we can only diagonalize one of the resulting operators ${\mathfrak{a}}_j$. For this reason, we will rather insist on their diagonality and achieve this goal by the use of two different Dyson maps,

$${\mathsf{\Omega }}_1{A}_1{\mathsf{\Omega }}_1^{-1}={\mathfrak{a}}_1,{\mathsf{\Omega }}_2{A}_2{\mathsf{\Omega }}_2^{-1}={\mathfrak{a}}_2$$

(8)

This leads to the two different images of states in ${\mathcal{H}}_{standard}$ (cf. Equation (3)) and, therefore, to the two different pictures of physics or, after a re-wording, to the two different versions of the physical Hilbert space of textbooks.

In the light of the definition of metric (4) and the inner-product equivalence (5), we would also have the two versions of the correct Hilbert space ${\mathcal{H}}_{physical}$, which would be, naturally, mutually incompatible. There is only one way of avoiding the difficulty: given the two diagonalization requirements (8), we must make use of the “hidden” ambiguity of the respective inner-product metrics, as explained in our older paper [18].

4. Problem of Compatibility/Incompatibility

The assumption of diagonality of the two above-mentioned real matrices ${\mathfrak{a}}_1$ (with non-vanishing elements ${\left({\mathfrak{a}}_1\right)}_{nn}=a_n^{\left[1\right]}$) and ${\mathfrak{a}}_2$ (with ${\left({\mathfrak{a}}_2\right)}_{nn}=a_n^{\left[2\right]}$) enables us to replace Equation (8) by its equivalent Hermitian-conjugate matrix-equation alternative

$$A_1^{†}{\mathsf{\Omega }}_1^{†}={\mathsf{\Omega }}_1^{†}{\mathfrak{a}}_1,A_2^{†}{\mathsf{\Omega }}_2^{†}={\mathsf{\Omega }}_2^{†}{\mathfrak{a}}_2.$$

(9)

Moreover, we may treat both of the matrices ${\mathsf{\Omega }}_j^{†}$ with $j=1$ or $j=2$ as concatenations of eigenvectors of the respective matrices $A_j^{†}$. Hence, once we accept the notation convention of [18] and once we mark these eigenvectors by a doubled ket symbol, i.e., once we replace 〉 by $\rangle \rangle$, we will be able to re-read every column of Equation (9) as a relation that has the very standard form of a Schrödinger-equation-like eigenvalue problem,

$$A_1^{†}|{\psi }_m^{\left[1\right]}\rangle \rangle =|{\psi }_m^{\left[1\right]}\rangle \rangle a_m^{\left[1\right]},A_2^{†}|{\psi }_m^{\left[2\right]}\rangle \rangle =|{\psi }_m^{\left[2\right]}\rangle \rangle a_m^{\left[2\right]},m=1,2,\dots ,N.$$

(10)

As a consequence, both of the concatenations of the column vectors will have the obvious form composed of the eigenvectors obtained by the solution of Equation (11),

$${\mathsf{\Omega }}_j^{†}=\left\{\right|{\psi }_1^{\left[j\right]}\rangle \rangle ,|{\psi }_2^{\left[j\right]}\rangle \rangle ,\dots ,|{\psi }_N^{\left[j\right]}\rangle \rangle \},j=1,2.$$

Once we decided to deal, in the present paper, just with the pairs of observables $A_1$ and $A_2$, the latter “ketket” eigenvectors had to form just the two different families of eigenvectors of $A_j^{†}$ with $j=1$ or $j=2$ in general. What is essential is that they may be calculated by standard methods, and that they may then be used to define the metric via Equation (4).

4.1. Residual Freedom

Once we are given the two dynamics-representing operators $A_1$ and $A_2$, which are defined as non-Hermitian in the finite-dimensional Hilbert space ${\mathcal{H}}_{mathematical}^{\left(N\right)}$, relation (8) may be re-written, equivalently, as the two entirely conventional matrix eigenvalue problems,

$$A_1^{†}{\mathsf{\Omega }}_1^{†}={\mathsf{\Omega }}_1^{†}{\mathfrak{a}}_1,A_2^{†}{\mathsf{\Omega }}_2^{†}={\mathsf{\Omega }}_2^{†}{\mathfrak{a}}_2.$$

(11)

The present eigenvectors form the columns of matrices ${\mathsf{\Omega }}_j^{†}$. Standard methods may be used to reconstruct both the latter two matrices and the corresponding diagonal matrices ${\mathfrak{a}}_j$ of eigenvalues that are all real and, say, non-degenerate.

Now, our most important observation is that every (i.e., the $n$-th) column of the benchmark solutions ${\mathsf{\Omega }}_1^{†}$ and/or ${\mathsf{\Omega }}_2^{†}$ of the two respective eigenvalue problems (11) is only defined up to an arbitrary (i.e., real or complex) non-vanishing multiplication factor ${\kappa }_1^{\left[n\right]}\ne 0$ or ${\kappa }_2^{\left[n\right]}\ne 0$, respectively. Thus, once we keep these factors just real and positive, and once we let these factors form the two respective diagonal matrices ${\mathfrak{c}}_1$ and ${\mathfrak{c}}_2$, we reveal that every initial arbitrarily normalized solution matrix ${\mathsf{\Omega }}_j^{†}$ is not unique and can be modified as follows,

$${\mathsf{\Omega }}_j^{†}\to {\mathsf{\Omega }}_j^{†}{\mathfrak{c}}_j,j=1,2.$$

(12)

Using such a transformation, therefore, one obtains all of the acceptable solutions that are parametrized by the respective real, positive, and diagonal matrices ${\mathfrak{c}}_j$. The $n$-th column of our arbitrarily normalized and fixed initial solution ${\mathsf{\Omega }}_j^{†}$ of the respective Equation (11) is being multiplied by an entirely arbitrary subscript-dependent constant ${\kappa }_j^{\left[n\right]}={\left({\mathfrak{c}}_j\right)}_{n,n}\ne 0$. These values may even be complex because, in a way pointed out in [18], there is no constraint imposed upon the variability in the diagonal elements of ${\mathfrak{c}}_j$ with $j=1,2$. Here, without any loss of generality, we will use just the real and positive values of these parameters.

Similarly, the eligible metrics defined by Formula (4) also become, for the same reason, ambiguous,

$${\mathsf{\Theta }}_j={\mathsf{\Theta }}_j\left(I\right)={\mathsf{\Omega }}_j^{†}{\mathsf{\Omega }}_j\to {\mathsf{\Theta }}_j={\mathsf{\Theta }}_j\left({\mathfrak{c}}_j\right)={\mathsf{\Omega }}_j^{†}{\mathfrak{c}}_j^2{\mathsf{\Omega }}_j,j=1,2.$$

(13)

At the same time, as long as such a characterization of the ambiguity is exhaustive, we may now formulate our present main conclusion.

Lemma 1.

The two N by N matrix candidates $A_1$ and $A_2$ for observables can be declared compatible if and only if there exist positive diagonal matrices ${\mathfrak{c}}_1$ and ${\mathfrak{c}}_2$ such that

$${\mathsf{\Theta }}_1\left({\mathfrak{c}}_1\right)={\mathsf{\Theta }}_2\left({\mathfrak{c}}_2\right).$$

(14)

4.2. Criteria

In a way not noticed in [7], the test and verification of coincidence (14) are facilitated by the solution of Equation (11). This enables us to rewrite Equation (14) as relation

$${\mathsf{\Omega }}_1^{†}{\mathfrak{c}}_1^2{\mathsf{\Omega }}_1={\mathsf{\Omega }}_2^{†}{\mathfrak{c}}_2^2{\mathsf{\Omega }}_2$$

(15)

in which one can only vary the $2N$ diagonal matrix elements of ${\mathfrak{c}}_1^2$ and ${\mathfrak{c}}_2^2$.

One of the consequences of the set of constraints (15) is that we can immediately eliminate half of the variable parameters since we can write, say,

$${\mathfrak{c}}_2^2=M^{†}{\mathfrak{c}}_1^{2}M,M={\mathsf{\Omega }}_1{\mathsf{\Omega }}_2^{-1}.$$

(16)

Both sides of this relation are Hermitian matrices that have two parts. The diagonal parts are elementary and provide simply an explicit definition of the diagonal matrix ${\mathfrak{c}}_2^2$ in terms of the known matrix M and the variable diagonal matrix ${\mathfrak{c}}_1^2$.

The $N-$plet of the real and positive variables in ${\mathfrak{c}}_1^2$ is then still constrained by the rest of relation (16), which can be read, at the larger N, as an over-determined set of $N(N-1)/2$ constraints. They may be written, say, as a multiplet of linear equations

$${\left(M^{†}{\mathfrak{c}}_1^{2}M\right)}_{m,n}=0,\forall m<n.$$

(17)

In practice, the implementation of such a criterion would be only straightforward in the two-dimensional case since at $N=2$ we would just have a single constraint (17) imposed upon two real parameters. In general, in contrast, the two arbitrarily chosen independent matrices $A_1$ and $A_2$ will not be mutually compatible at $N\ge 4$ (and more so when $N=\infty$) since $N(N-1)/2>N$. In these cases, one will have to rely on the purely numerical techniques.

For some purposes, a less pragmatic alternative approach to the problem could be based on our following final result.

Lemma 2.

The two N by N matrix candidates $A_1$ and $A_2$ for observables can be declared compatible (i.e., simultaneously quasi-Hermitian) if and only if the $2N-$parametric matrix

$$\mathcal{U}({\mathfrak{c}}_1,{\mathfrak{c}}_2)={\mathfrak{c}}_{1}M{\mathfrak{c}}_2^{-1}$$

(18)

is unitary.

Proof. 

The unitarity is equivalent to requirement (16). □

One of the consequences of the latter observation would be constructive. At a given Hilbert-space dimension N, indeed, one could choose an arbitrary unitary matrix $\mathcal{U}$ and, having recalled Equation (18), one would be able to define the matrix M. After its pre-multiplication by some $2N$ arbitrary constants, a more or less arbitrary factorization would yield, in the light of Equation (16), the doublet of matrices ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$. In terms of any one of them, one could define the “shared” metric $\mathsf{\Theta }$. An arbitrary choice in the two diagonal matrices, ${\mathfrak{a}}_1$ and ${\mathfrak{a}}_2$, would then finally lead to the construction of the compatible pair of operators, $A_1$ and $A_2$, via the respective relations (8).

In the opposite direction of the constructive analysis, the process of implementation of the unitarity criterion (18) could be based on the most general unitary-matrix ansatz $\mathcal{U}$ for the left-hand side of Equation (18). Thus, in the simplest example with $N=2$, such an ansatz would be four-parametric,

$$\mathcal{U}=exp\left[\begin{array}{cc}i\alpha & i\gamma +\delta \\ i\gamma -\delta & i\beta \end{array}\right].$$

(19)

This means that, at any given two-by-two dynamical-input-information matrix $M^{\left(2\right)}$, one could search for the four unknown parameters ${\kappa }_j^{\left[n\right]}$ with $j=1,2$ and $n=1,2$ via the solution of the set of constraints (18), forming a quadruplet of coupled nonlinear algebraic equations.

5. Example

One of the expected outcomes of our present methodological study can be seen in its possible relevance in the theory of quantum gravity. In this context, indeed, many conventional quantization methods cease to be applicable because they rely upon the availability of a classical “background” represented, typically, by space or space-time. In the case of a gravitational field, in contrast, one also has to quantize the space-time itself, in principle at least [21]. As a consequence, a consistent quantum theory of gravity must be, in Ashtekar’s words, “background-independent” [22]. In this sense, a minimal number of the candidates for observables $A_j$ is two, and the demonstration of their mathematical compatibility is of deep and fundamental relevance indeed.

In a narrower setting of quantum cosmology, in addition, one has to quantize classical scenarios in which one often encounters several different forms of singularities. Thus, typically, the latter candidates $A_j$ have to admit the existence of the singularities represented, in the Hilbert-space operator context, by so-called Kato’s [23] exceptional points (EPs). In a realistic setting, these operators even have to be non-stationary and non-Hermitian, and, in parallel, the physical Hilbert-space inner-product metric itself also has to be non-stationary and non-Hermitian.

In spite of all of these technical challenges, the practical applicability of the general theory has still been demonstrated and illustrated, in the existing literature, by a number of simple physical examples. A few particularly suitable examples of such a type were described in Ref. [24].

From the purely phenomenological point of view, one of the important merits of the latter classes of models is that all of them admit the non-Hermitian exceptional-point degeneracy [23]. This means that all of them could be understood as mimicking the Big-Bang-like singularity in quantum cosmology. The additional methodological merit is that all of these families of solvable models are represented by matrices $A_j$ of arbitrary dimensions N.

As we already mentioned above, nevertheless, we do not need the large-matrix models (because the conditions of their mutual compatibility become over-restrictive at $N\ge 4$) or the minimal models with $N=2$ (for them, the parametric freedom survives). Thus, we decided to pick up, for our present illustration purposes, just the one-parametric sets of the $N=3$ matrices such that

$${A_1}^{†}=\left[\begin{array}{ccc}-2& \sqrt{2-2s^2}& 0\\ -\sqrt{2-2s^2}& 0& \sqrt{2-2s^2}\\ 0& -\sqrt{2-2s^2}& 2\end{array}\right]$$

(20)

and

$${A_2}^{†}=\left[\begin{array}{ccc}-2ia& \sqrt{2}& 0\\ \sqrt{2}& 0& \sqrt{2}\\ 0& \sqrt{2}& 2ia\end{array}\right].$$

(21)

They seem totally different (notice that the former one is asymmetric but real, while the latter one is complex-symmetric). Moreover, as long as we can quickly determine their diagonal partners of Equation (8), say,

$${\mathbf{a}}_{\mathbf{1}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& -2s& 0\\ 0& 0& 2s\end{array}\right]$$

(22)

and

$${\mathbf{a}}_{\mathbf{2}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& -2\sqrt{-a^2+1}& 0\\ 0& 0& 2\sqrt{-a^2+1}\end{array}\right]$$

(23)

we may also find closed one-parametric forms of the arbitrarily normalized related matrices

$${\mathsf{\Omega }}_1^{†}=\left[\begin{array}{ccc}-2+2s^2& 2s+1+s^2& -2s+1+s^2\\ -2\sqrt{2-2s^2}& \sqrt{2-2s^2}\left(s+1\right)& -\sqrt{2-2s^2}\left(-1+s\right)\\ -2+2s^2& 1-s^2& 1-s^2\end{array}\right]$$

(24)

plus, similarly, matrix ${\mathsf{\Omega }}_2^{†}$ (which appeared, incidentally, just a bit too large for its display in print).

In the next step of illustration, we will fix parameters $s=1/2$ and $a=1/2$, yielding a simpler formula for

$${\mathsf{\Omega }}_1^{†}=\left[\begin{array}{ccc}-3/2& 9/4& 1/4\\ -\sqrt{6}& 3/4\sqrt{6}& 1/4\sqrt{6}\\ -3/2& 3/4& 3/4\end{array}\right]$$

(25)

as well as a more comfortably printable formula for

$${{\mathsf{\Omega }}_2}^{†}=\left[\begin{array}{ccc}3/2& 3/8+3/8i\sqrt{3}& 3/8-3/8i\sqrt{3}\\ 3/4i\sqrt{2}& -3/8\sqrt{2}\sqrt{3}-3/8i\sqrt{2}& 3/8\sqrt{2}\sqrt{3}-3/8i\sqrt{2}\\ -3/2& 3/4& 3/4\end{array}\right].$$

(26)

What remains to be completed is just the reconstruction of the two entirely general alternative three-parametric metrics of Equation (14) or (15). With abbreviations ${\left({\mathfrak{c}}_1\right)}_{nn}^2=x_n$ and ${\left({\mathfrak{c}}_2\right)}_{nn}^2=y_n$, these N by N identities are tractable as linear equations that have, ultimately, the following explicit solution defined in terms of a single variable $\varrho$, viz.,

$${\mathit{x}}_{\mathit{1}}=-1/12\varrho +1/12i\sqrt{3}\varrho ,$$

$${\mathit{x}}_{\mathit{2}}=1/9\varrho ,$$

$${\mathit{x}}_{\mathit{3}}=\varrho ,$$

$${\mathit{y}}_{\mathit{1}}={\displaystyle \frac{15}{16}}\varrho -3/16i\sqrt{3}\varrho ,$$

$${\mathit{y}}_{\mathit{2}}=-3/16\varrho -{\displaystyle \frac{9}{16}}i\sqrt{3}\varrho ,$$

$${\mathit{y}}_{\mathit{3}}=3/8\varrho +3i\sqrt{3}\varrho .$$

Obviously, this solution exists, but it is not real and positive. We may conclude that there exists no Hilbert-space metric that would make both models (20) and (21) (with real spectra) mutually compatible, i.e., simultaneously quasi-Hermitian.

In conclusion, let us add that, conceptually, it would be entirely straightforward to also test the compatibility of the triplets or multiplets of certain preselected operators $A_j$ with real spectra and with $j=1,2,\dots ,K$. In such cases, we could merely replace the two-term requirement (15) with its $K-$term generalization. The number of constraints will then grow quickly with K since the $(K-1)-$plet of Equation (17) (i.e., as many as $N(N-1)(K-1)/2$ formally independent conditions) will have to be imposed upon the mere $N-$plet of variable parameters.

6. Summary

In our present paper, we emphasized the importance of a possible guarantee of mutual compatibility between any two different candidates, $A_i$ and $A_2$, for the observables, and we explained how such a guarantee could be provided in practice.

Before the formulation of our results, we emphasized that, first of all, the non-Hermiticity of $A_1\ne A_1^{†}$ and $A_2\ne A_2^{†}$ is just an artifact of our decision to perform the mathematical operations (e.g., localizations for spectra, etc.) in a preselected and, by assumption, manifestly unphysical but by far the user-friendliest Hilbert space ${\mathcal{H}}_{mathematical}$. Secondly, we pointed out that there is a large difference between the quantum systems and states living in the finite- or infinite-dimensional Hilbert spaces, and, for the sake of brevity, we restricted our attention just to the former family.

In the first step of our constructive resolution of the problem of compatibility of $A_1$ with $A_2$, we decided to treat the inner-product metric $\mathsf{\Theta }$ as an operator product of Dyson map $\mathsf{\Omega }$ with its conjugate (cf. Equation (4) above).

In the second step, we noticed and made use of the close connection between the conjugate operator ${\mathsf{\Omega }}^{†}$ and an un-normalized (or, more precisely, “zeroth”, arbitrarily normalized) set of vectors defined as eigenvectors of the conjugate of any one of the observables in question. This led quickly to our ultimate main result (cf. Lemma 2). Due to the generality of the latter result, we only have to remind the readers that, at present, the QHQM-inspired use of the Hermitian operators is widely accepted [5,10,25] and employed in methodological studies [16] as well as in fairly diverse phenomenological considerations [6,12,26,27].

In the narrower framework of the hiddenly unitary quantum theory, a key to the mathematical consistency of the corresponding description of physical reality has been found in the well-known fact that a typical operator $A_j$ that is non-Hermitian in a preselected (and, presumably, sufficiently “user-friendly”) Hilbert space ${\mathcal{H}}_{mathematical}$ can still be reinterpreted as self-adjoint in some other “amended” Hilbert space ${\mathcal{H}}_{physical}$. A decisive reason for the feasibility of such a model-building strategy lies in the possibility of the amendment, ${\mathcal{H}}_{mathematical}\to {\mathcal{H}}_{physical}$, of the space, which can be mediated by the mere upgrade (1) of the inner product in ${\mathcal{H}}_{mathematical}$ (to be compared with Eq. Nr. (2.2) of review [2], where the authors used symbol T in place of our present $\mathsf{\Theta }$).

The latter form of the representation of a user-unfriendly Hilbert space ${\mathcal{H}}_{physical}$ in its user-friendly alternative ${\mathcal{H}}_{mathematical}$ has been found to be a decisive technical merit of the innovation. The formalism could be called quantum mechanics in a quasi-Hermitian representation because all of the candidates $A_j$ for the observables had to satisfy the same quasi-Hermiticity constraint (2) in ${\mathcal{H}}_{mathematical}$ (cf. also Eq. Nr. (2.1e) and the related comments in [2]).

Without any danger of confusion, one may narrow the scope of our present considerations by dealing with the first nontrivial scenario in which one is given a quasi-Hermitian Hamiltonian $H(=A_1)$ and another observable $Q(=A_2)$, treated very formally (and in a way inspired by our preceding rather realistic QHQM study [7]) as a “spatial coordinate”. A generalization to the models with multiplets of observables is obvious.

Independently of the physical motivation used in [7], the necessity of considering at least two operators of observables emerged, most urgently, in the potential applications of QHQM in quantum cosmology [28]. In such a context, indeed, a mathematically consistent treatment of quantum gravity necessarily requires a nontrivial dynamical nature of the space. In the related literature, the authors usually speak about the “background-independent quantization” of gravity; see, e.g., the truly comprehensive monograph [29].

The complex nature of the problems of quantum gravity lies already far beyond the scope of our present paper, of course. For our present methodological purposes, it was sufficient to work just with the Hilbert spaces of a finite dimension $N<\infty$, ${\mathcal{H}}^{\left(N\right)}$. Naturally, a transition to the models with $N=\infty$ might still be highly nontrivial: as a word of warning, let us recall, pars pro toto, a sample of the emergent subtleties in [3] or [21].