Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'

As a part of issue 'Foundational Structures in Quantum Theory' the paper 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory' by Abhijeet Alase, Salini Karuvade and Carlo Maria Scandolo [1] fitted very well the scope of the volume. In a rigorous mathematical style it offered the readers an interesting material confirming the compatibility between the three recent conceptual innovations of quantum theory. Still, we believe that the authors' coverage of the subject deserves a few addenda, mainly because in loc. cit., the deeply satisfactory nature of the mathematical analysis seems to be accompanied by a perceivably less careful presentation of its implications in the context of the theoretical quantum physics.

Our first addendum is motivated by the last sentence of the abstract in [1]. It states that 'our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently'. Indeed, it is rather unfortunate that this statement diverts attention from the very interesting main mathematical message of the paper (viz. from the rigorous confirmation of compatibility between the three alternative versions of quantum theory) to its much less satisfactory physical contextualization. The impression is further strengthened by the last paragraph of the whole text where we read that 'in conclusion, neither PT-symmetry nor quasi-Hermiticity of observables leads to an extension of standard quantum mechanics.' Certainly, non-specialists could be mislead to interpret such a conclusion wrongly, as a disproof of usefulness of what is usually called PT-symmetric quantum theory (PTQT, an approach which is briefly reviewed in section 2.1 of loc. cit.) or of the so called quasi-Hermitian quantum theory (QHQT, cf its compact review in the subsequent section 2.2 of loc. cit.). The misunderstanding seems completed by the combination of the very first sentence of the abstract with the very last sentence of the text: at the beginning of the abstract we are told that 'PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory' (which is not too relevant at present), while the final question reads 'what possible constraints, if any, could lead to such a meaningful extension' [1].

The main weakness of such a 'theory-extension' motivation and of the 'physical' framing of paper [1] is that the original purpose of 'relaxing the Hermiticity constraint on Hamiltonians' (as proposed, by Bender with Boettcher, in their enormously influential letter [2]) was almost immediately shown overambitious and unfulfilled (see, e.g. the Mostafazadeh's 2010 very mathematical and detailed criticism and explanation 'that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory' [3]). Thus, the authors of [1] only come with their 'aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory' too late. For more than twelve years the answer is known to be negative [4].

Naturally, nobody claims that the PTQT itself is not useful. Nobody could also deny the relevance and the novelty of the mathematical results presented in paper [1]. It is only a pity that its authors did not better emphasize how well their analysis fits the subject of the special issue, especially due to their innovative turn of attention to the so called general probabilistic theories (GPT, cf their compact outline in section 2.3 of [1]).

Paradoxically, in the GPT context one immediately identifies the second weakness of the paper. It lies in a surprisingly short list of the GPT-approach-representing references. In the paper the list just incorporates the eight newer papers [5–12] (all of them published after the year 2000) plus a single older, Foulis-coauthored 1970 paper [13]. Not quite expectedly, the list of references does not contain any Gudder's results—after all, paper [1] is a part of the special issue which is explicitly declared to honor his contribution to the field. Thus, one would expect, for example, a reference to his later review papers [14, 15] where he formulated one of the key GPT-related mathematical theses that 'a physical system S under experimental investigation and governed by a scientific theory (which may be subject to modification in the light of new experimental evidence) is represented by a CB-effect algebra'. An equally unexpected gap in the references also concerns the absence of the Foulis' pioneering, effect-algebras introducing 1994 paper with Bennet [16], or his comparatively recent review [17]. Indeed, both of these papers sought and offered operational foundations and gained insight into the GPT-motivating relationship between quantum theory and classical probability theory (this was emphasized also in [5], etc).

What is an even worse omission is that the list of references does not contain any other subject-related studies like, e.g. paper [18] in which the predecessors of the present authors considered, explicitly, the PTQT-GPT relationship, having reconfirmed that 'from the standpoint of (generalized) effect algebra theory both representations of our quantum system coincide'. Similarly, the QHQT-GPT relationship may be found studied in paper [19] in which the mathematically fairly advanced analysis incorporated even the fairly realistic quantum models using unbounded operators. Indeed, the rate of the progress is striking, especially when one recalls just a few years younger report [14] in which the 'separable complex Hilbert space' is assumed to be just 'of dimension 1 or more'.

At present, it makes sense to accept the fact that in spite of the robust nature of the existing 'standard' formulations of quantum theory and, in particular, of the quantum mechanics of unitary systems, there still exist differences in the practical applicability of their various specific implementations. The motivation of the diversity is that 'no (particular) formulation produces a royal road to quantum mechanics' [20]. In some sense this implies that the concept of the 'extension' of the existing quantum theory is vague. The apparently minor technical differences between the current alternative formulations of quantum mechanics (as sampled, in [20], on elementary level) could happen to lead to 'decisive extensions' in the future.

A good illustrative example can be provided even within the current stationary forms of QHQT. Indeed, even in this framework the formalism can really be declared equivalent to its standard textbook form. Still, the equivalence can be confirmed only under certain fairly detailed and specific mathematical assumptions (see [21]). These assumptions are, even in the abstract context of functional analysis, far from trivial [22]. Paradoxically, even the popular physical quantum models of Bender and Boettcher [2] have been later found not  to belong to the 'admissible', QHQT-compatible class (see, e.g. [23, 24] for the corresponding subtle details). Thus, in spite of their manifest and unbroken PT-symmetry, even these originally proposed benchmark models still wait for a 'meaningful extension' of their fully consistent GPT interpretation.

In our third, last addendum we are now prepared to reopen the vague but important question of what the words of 'extension' of the 'standard' quantum theory could, or do, really mean. On one side, it is known and widely accepted that the various existing formulations of quantum theory 'differ dramatically in mathematical and conceptual overview, yet each one makes identical  predictions for all experimental results' [20]. On the other side, such a rigidity of the theory is far from satisfactory. For example, a suitable future amendment of quantum theory would be necessary for a still absent clarification of the concept of quantum gravity [25].

For the sake of brevity let us skip here the discussion of the parallel questions concerning the PT-symmetric quantum models. This being said we believe that even the QHQT formalism itself did not say its last word yet. Indeed, our optimism concerning its potential 'theory extension status' is based on the recent fundamental clarifications of its scope and structure. First of all, it became clear that in the QHQT descriptions of unitary systems it is sufficient to distinguish just between their representations in the 'generalized Schródinger picture' (GSP, stationary and best presented, by our opinion, in reviews [3, 21, 26]) and in its non-stationary 'non-Hermitian interaction picture' alternative (NIP, [27, 28]). Using this terminology one immediately reveals that the QHQT-related considerations of paper [1] just cover the GSP approach. In other words, the physical inner-product metric (denoted by symbol η) is perceived there as strictly time-independent. This means that in the GSP language one can easily identify the (stationary) generator G of the evolution of the wave functions with the ('observable-energy') Hamiltonian H (which has real spectrum and is, by assumption, $\eta-$quasi-Hermitian).

The situation becomes different after the extension of the QHQT approach to the non-stationary, NIP dynamical regime. In this case we will denote the inner-product metric by another dedicated symbol $\Theta = \Theta(t)$ as introduced in the first description of NIP in [29]. What is important is that the observable-energy operator $H = H(t)$ will get split in the sum of the two auxiliary operators G(t) and $\Sigma(t)$. As long as they are both neither observable nor Θ-quasi-Hermitian in general, we will exclusively assign the name of the Hamiltonian to the instantaneous energy operator H (with real spectrum), adding a word of warning that a different, less consequent terminology is often used by some other authors (see, e.g. [30, 31]). Even though neither the spectrum of G(t) nor the spectrum of $\Sigma(t)$ is real in general, the introduction of these operators endows the NIP formalism with an additional flexibility, capable, as we believe, of opening the new horizons in the contemporary quantum physics: in the context of relativistic quantum mechanics, for example, such a hypothetical 'theory-extension' possibility has been discussed, in detail, in [27]. For the purposes of a potentially new approach to the problem of the unitary-evolution models of quantum phase transitions in many-body context, the formalism has slightly been adapted in [32]. Last but not least, our very recent paper [28] has been devoted to the possible use of the NIP evolution equations in a Wheeler–DeWitt-equation-based schematic model of Big Bang in the context of quantum gravity and cosmology. In this spirit, therefore, certain sufficiently realistic NIP-based models could easily happen to acquire an 'extended quantum mechanics' status, perhaps, in the nearest future.

The key subject discussed in paper [1] was the question of the possible extension of the scope of quantum theory in general, and of the realization of such an ambitious project, in the respective PTQT and QHQT theoretical frameworks, in particular. In our present commentary we reminded the readers, marginally, of the existence of several older, comparably sceptical conclusions as available in the related literature (see section 2 for details). In section 3 we then added a few similar broader-context-emphasizing remarks on the mathematical, GPT-related aspects of the results of [1]. Still, the core of our present message (as presented in the longest section 4) concerned physics. We pointed out that at present, the question of the possible extension of the scope of the standard quantum theory should be considered open even in the narrower PTQT and QHQT frameworks.

In support of the latter statement we mentioned that

• even for the stationary and, apparently, most elementary PTQT potentials (sampled, say, by the most popular $V(x) = \textrm{i}x^3$), the widespread initial optimism and intuitive 'nothing new' understanding of their physical meaning and mathematical background have both recently been shattered by their more rigorous mathematical analysis;
• one can hardly say 'nothing new' even in a mathematically much better understood stationary QHQT alias  GSP framework where, typically, the use of certain stronger assumptions enables one to circumvent the obstacles revealed by rigorous mathematics. Indeed, even in the GSP framework one can search for an entirely new physics. Typically, a non-standard phenomenology becomes described by the QHQT models in an infinitesimally small vicinity of the so called exceptional points: paper [33] offers an illustrative sample of the quantum systems which cannot be described by the standard quantum theory;
• in fact, our return to optimism and expectation that the QHQT may be a 'fundamentally innovative' theory found its most explicit formulation in section 4. Briefly we exposed there an enormous growth of the flexibility of the QHQT approach after its ultimate non-stationary NIP generalization. In some sense, the emphasis put upon the deeply promising conceptual nature of such a flexibility can be read as the deepest core of our present comment and message.

No new data were created or analysed in this study.