Advances in High Energy Physics

Volume 2018, Article ID 7906536, 12 pages

https://doi.org/10.1155/2018/7906536

## Hermitian–Non-Hermitian Interfaces in Quantum Theory

Nuclear Physics Institute of the CAS, 250 68 Řež, Czech Republic

Correspondence should be addressed to Miloslav Znojil; zc.sac.fju@lijonz

Received 10 January 2018; Accepted 18 February 2018; Published 24 April 2018

Academic Editor: Saber Zarrinkamar

Copyright © 2018 Miloslav Znojil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlaps* alias* interfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

#### 1. Introduction

In virtually any representation of quantum theory, the states can be perceived as constructed in a suitable user-friendly Hilbert space . By a number of authors [1–4], it has been recommended to enhance the flexibility of the formalism by making use of an ad hoc, quantum-system-adapted physical inner product in , that is, by an introduction of a nontrivial, stationary metric operator . All of the other, relevant “physical” operators of the observables in (i.e., say, representing a coordinate or representing the energy, etc.) must be then chosen, in Diedonné’s terminology [5], quasi-Hermitian,These observables become Hermitian if and only if we reach the conventional textbook limit with . Otherwise, our candidates for the observables remain manifestly non-Hermitian in our friendly Hilbert space . The latter space (with artificial ) must be declared, therefore, auxiliary and unphysical, . Only the reincorporation of the amended metric will reinstall the space as physical, .

In certain very promising recent high-energy physics applications of the formalism, say, in neutrino physics [6, 7], people usually restrict attention to the special form of , where is parity while denotes charge. In such a setting the stationarity of the theory represents a serious obstacle for experimentalists, mainly because the adiabatic changes and tuning of the parameter-dependence of the observables may lead to multiple counterintuitive no-go theorems [3, 8, 9]. At the same time, the new degree of the kinematical freedom represented by the nontrivial metric may find its efficient use, say, in the manipulations leading to the experimental realizations of various quantum phase transitions in the theory [6, 10–16] as well as in the laboratory [17].

The consistent mathematical formulation of the theories with innovative and traditional proved to be truly challenging [18]. In practice, the main source of difficulties can be seen in the “smearing” feature of the use of generic [19]. Jones noticed that “we have to start with ” (i.e., with ) “because that is how the potential is defined” [20]. His analysis was then aimed at the search for natural interfaces (*alias* operational connections) between the hypothetical non-Hermitian dynamics (using ) and the available experimental setups (at ).

In a way based on a detailed study of certain overrestricted family of models (for purely technical reasons the interaction potentials were kept local), Jones arrived, not too surprisingly, at a heavily sceptical conclusion that the theory cannot be unitary. In his own words “the only satisfactory resolution of the dilemma is to treat the non-Hermitian potential as an effective one, and [] work in the standard framework of quantum mechanics, accepting that this effective potential may well involve the loss of unitarity” [21].

Jones’ conclusions were partially opposed and weakened in [19, 22, 23] where the assumption of “starting with ” (i.e., of our working with the potentials which are local in ) has been shown unfounded (because the value of the lowercase is not observable) and misleading (because one need not give up the unitarity in general). At the same time, the underlying, deep, and important conceptual problem of the possible existence of suitable Hermitian–non-Hermitian interfaces remained open.

An affirmative answer will be given in what follows. In order to formulate the problem more clearly we will have to recall, in the next section, a few well-known aspects of forming a nontrivial feasible contact and of a smooth transition between several versions of quantum dynamics. In the subsequent sections, we shall then point out that a formal key to the realization of the project of construction of the smooth interfaces lies in the properties of the inner-product metric operators which have to degenerate smoothly, in their turn, to the trivial limit . Furthermore, in Section 5 several technical aspects of such a general interface-construction recipe will be illustrated by an elementary toy model-Hamiltonian example admitting a nonnumerical and nonperturbative analytical treatment. Some of the possible impacts upon quantum phenomenology will finally be mentioned in Section 6.

#### 2. Quantum Dynamics in Schrödinger Picture

During the birth of quantum theory, its oldest (namely, the Heisenberg’s “matrix”) picture was quickly followed by Schrödinger’s “wave-function” formulation which proved less intuitive but more economical [24]. The conventional, “textbook”* alias* “Hermitian” Schrödinger picture (HSP, [25]) was later complemented by its “non-Hermitian” Schrödinger picture (NHSP) alternative (cf. the works by Dyson [1] or by nuclear physicists [2]). In this direction, the recent wave of new activities was inspired by Carl Bender with coauthors [4, 10–12]. The emerging, more or less equivalent innovated versions of the NHSP description of quantum dynamics were characterized as “quantum mechanics in pseudo-Hermitian representation” [3] or as “quantum mechanics in the Dyson’s three-Hilbert-space formulation” [1, 26], and so forth [18].

The availability of the two alternative representations of the laws of quantum evolution in Schrödinger picture inspired Jones to ask the above-cited questions about the existence of an “overlap of their applicability” in an “interface” [27]. His interest was predominantly paid to the scattering [20] and his answers were discouraging [21]. In papers [22, 23, 28] we opposed his scepticism. We argued that the difficulties with the HSP-NHSP interface may be attributed to the ultralocal, point-interaction toy model background of his methodical analysis. We introduced certain weakly nonlocal interactions and via their constructive description we reopened the possibility of practical realization of a smooth transition between the Hermitian and non-Hermitian theoretical treatment of scattering experiments.

Now we intend to return to the challenge of taking advantage of the specific merits of* both* of the respective HSP and NHSP representations inside their interface. We shall only pay attention to the technically less complicated quantum systems with bound states. Our old belief in the existence, phenomenological relevance, and, perhaps, even fundamental-theory usefulness of a domain of coexistence of alternative Schrödinger picture descriptions of quantum dynamics will be given an explicit formulation supported by constructive arguments and complemented by elementary, analytically solvable illustrative examples.

##### 2.1. The Concept of Hidden Hermiticity

An optimal formulation of quantum theory is, obviously, application-dependent [24]. Still, the so-called Schrödinger picture seems exceptional. Besides historical reasons this is mainly due to the broad applicability as well as maximal economy of the complete description of quantum evolution using the single Schrödinger equationWhenever the evolution is assumed unitary, the generator (called Hamiltonian) must be, due to Stone’s theorem [29], self-adjoint in ,Recently it has been emphasized that even in the unitary evolution scenario the latter Hamiltonian-Hermiticity constraint may be omitted or, better, circumvented. The idea, dating back to Dyson [1], relies upon a suitable preconditioning of wave functions. This induces the replacement of the “Hermitian,” lowercase Schrödinger equations (2) + (3) by their “non-Hermitian” uppercase alternativeThe preconditioning operator is assumed to be invertible but nonunitary, [2]. Thus, the standard textbook version of the Schrödinger picture splits into its separate Hermitian and non-Hermitian versions (cf. influential reviews [3, 4] and/or mathematical commentaries in [18]).

The slightly amended forms of Dyson’s version of the NHSP formalism proved successful in phenomenological applications, for example, in nuclear physics [2]. As we already indicated, the “non-Hermitian” philosophy of (4) was made widely popular by Bender [4]. Its appeal seems to result from the observation that the nonunitarity of makes the respective geometries in the two Hilbert spaces and mutually nonequivalent. As a consequence, the uppercase Hamiltonian acting in and entering the upgrade (4) of Schrödinger equation becomes manifestly non-self-adjoint* alias* non-Hermitian in ,Still, it is obvious that* both* the NHSP version (4) of Schrödinger equation* and* its HSP predecessor (2) represent* the same* quantum dynamics.

##### 2.2. The Choice between the HSP and NHSP Languages

Several reviews in monograph [18] may be recalled for an extensive account of multiple highly nontrivial mathematical details of the NHSP formalism. In applications, quantum physicists take it for granted, nevertheless, that we have a choice between the* two* alternative descriptions of the standard* unitary* evolution of wave functions. People are already persuaded that the basic mathematics of the HSP and NHSP constructions is correct and that the two respective Schrödinger equations are, for any practical purposes, equally reliable.

The accepted abstract HSP-NHSP equivalence still does not mean that the respective practical ranges of the two recipes are the same. The preferences really* depend* very strongly on the quantum system in question. Thus, the choice of the HSP language is made whenever the corresponding self-adjoint Hamiltonian possesses the most common form of superposition of a kinetic energy term with a suitable* local-interaction* potential,Similarly, the recent impressive success of the NHSP phenomenological models is almost exclusively related to the use of the non-Hermitian local-interaction Hamiltonianswhich are only required to possess the strictly real spectra of energies [3, 4].

##### 2.3. The Concept of the HSP-NHSP Interface

The two local-interaction operators (6) and (7) should be perceived as just the two illustrative elements of the two respective general families and of the eligible, that is, practically tractable and sufficiently user-friendly HSP and NHSP Hamiltonians. In a way influenced by this exemplification one has a natural tendency to assume that the latter two families are distinct and clearly separated, nonoverlapping [4],During the early stages of testing and weakening such as a priori assumption, Jones [27] introduced the concept of an interface as a potentially nonempty set of Hamiltonians,Basically, he had in mind a domain of a technically feasible and phenomenologically consistent interchangeability of the two pictures. He also outlined some of the basic features and possible realizations of such a Hermitian/non-Hermitian interface in [20]. Incidentally, the continued study of the problem made him more sceptical [21]. In a way based on a detailed analysis of a schematic though, presumably, generic toy model local-interaction Hamiltonianhe came to the conclusion that the merits of families and are really specific and that, in the case of scattering at least, their respective domains of applicability really lie far from each other, that is, . Even at the most favorable parameters and couplings, in his own words, “the physical picture [] changes drastically when going from one picture to the other” [21].

In our first paper [28] on the subject, we pointed out that Jones’ discouraging “no-interface” conclusions remain strongly model-dependent. For another, weakly nonlocal choice of , we encountered a much less drastic effect of the interchange of the mathematically equivalent Schrödinger equations (2) and (4) upon the predicted physical outcome of the scattering (see also the related footnote added in [21]). In our subsequent papers [22, 23] we further amended the model and demonstrated that in the context of scattering the overlaps may be nonempty. We showed that there may exist the sets of parameters for which the causality as well as the unitarity would be guaranteed for* both* of the Hamiltonians in (3) and (5). Thus, Jones’ ultimate recommendations of giving up the scattering models in and/or of “accepting …the loss of unitarity” while treating any “non-Hermitian scattering potential as an effective one” [21] may be requalified as oversceptical (cf. also [3]).

#### 3. Repulsion of Eigenvalues

The presentation of our results is to be preceded by a compact summary of some of the key specific features of spectra in the separate HSP and NHSP frameworks. This review may be found complemented, in Appendix A, by a brief explanation why the NHSP Hamiltonians which are* non-Hermitian* (though only in an auxiliary,* unphysical* Hilbert space) still do generate the* unitary* evolution (naturally, via wave functions in another, nonequivalent, physical Hilbert space).

Quantum dynamics of the one-dimensional motion described by an ordinary differential local-interaction Hamiltonian (6) is a frequent target of conceptual analyses. These models stay safely inside Hermitian class but still a brief summary of some of their properties and simplifications will facilitate a compact clarification of the purpose of our present study.

##### 3.1. Discrete Coordinates

The kinetic plus interaction structure of models (6) reflects their classical physics origin. It may also facilitate the study of bound states, say, by the perturbation-theory techniques [21] and/or by the analytic-construction methods [30]. Still, for our present purposes it is rather unfortunate that any transition to the hidden-Hermiticity language of the alternative model-building family would be counterproductive. One of the main obstacles of a hidden-Hermiticity reclassification of model (6) is technical because the associated Hamiltonians (5) are, in general, strongly non-local [31]. Another, subtler mathematical obstacle may be seen in the unbounded-operator nature of the kinetic energy (see [2] for a thorough though still legible explanation).

In [22, 23, 28], we proposed that one of the most efficient resolutions of at least some of the latter problems might be sought and found in the discretization of the coordinates. Thus, one replaces the real line of by a discrete lattice of grid points such that , with and with any suitable constant . This leads to the kinetic energy represented by the difference-operator LaplaceanIn parallel, one can argue that the sparse-matrix structure of this component of the Hamiltonian makes it very natural to replace also the strictly local (i.e., diagonal-matrix) interaction by its weakly nonlocal tridiagonal-matrix generalization [32].

##### 3.2. Elementary Example

Once we restrict our attention to the analysis of bound states, the above-mentioned doubly infinite tridiagonal matrices may be truncated yielding an by matrix Hamiltonian. Let us assume here that the latter matrix varies with a single real coupling strength and with a single real parameter modifying the interaction,This will enable us to assume that our parameters can vary, typically, with time (i.e., and/or ) and that, subsequently, also the energy levels of our quantum system form a set which can, slowly or quickly, vary. Thus, at a time of preparation of a Gedankenexperiment the energy of our system may be selected as equal to one of the real and time-dependent eigenvalues of our Hamiltonian . Naturally, the latter operator represents a quantum observable and must be self-adjoint in the underlying physical Hilbert space .

The first nontrivial tridiagonal matrix (12) with may represent, for example, a schematic quantum system with Hermitian-matrix interactionThe spectrum of energies may be then easily calculated and was sampled in Figure 1. The parameter-dependence of the energies seems to be such that they avoid “collisions.” As long as we choose , that is, Hamiltonianthe quadruplet of the energy eigenvalues becomes available also in the closed formThis formula explains not only the hyperbolic shapes of the curves in Figure 1 but also their closest-approach values and at .