Abstract This document is our reply to the Comment (Miloslav Znojil 2023 J. Phys. A: Math. Theor. 56 038001) on our recent work titled ‘The operational foundations of PT-symmetric and quasi-Hermitian quantum theory’. The original Comment consists of three addenda to our work. The first addendum claims that our work is ill-motivated as the motivating question, namely whether PT-symmetric quantum theory extends the standard quantum theory, was already answered in the literature. The second addendum points to some missing references in our work, and the third addendum suggests what constraints could lead to an extension of standard quantum theory. In our reply, we explain that the claim in the first addendum is a result of a misinterpretation of our motivating question. When interpreted correctly, the third addendum in the Comment in itself elaborates on why our motivating question is interesting and relevant. We also briefly comment on the prospects of an extension of standard quantum theory along the lines suggested in the third addendum. As our response to the second addendum, we explain our rationale behind citing certain references while leaving out others.
Abstract This document is our reply to the Comment (Miloslav Znojil 2023 J. Phys. A: Math. Theor. 56 038001) on our recent work titled ‘The operational foundations of PT-symmetric and quasi-Hermitian quantum theory’. The original Comment consists of three addenda to our work. The first addendum claims that our work is ill-motivated as the motivating question, namely whether PT-symmetric quantum theory extends the standard quantum theory, was already answered in the literature. The second addendum points to some missing references in our work, and the third addendum suggests what constraints could lead to an extension of standard quantum theory. In our reply, we explain that the claim in the first addendum is a result of a misinterpretation of our motivating question. When interpreted correctly, the third addendum in the Comment in itself elaborates on why our motivating question is interesting and relevant. We also briefly comment on the prospects of an extension of standard quantum theory along the lines suggested in the third addendum. As our response to the second addendum, we explain our rationale behind citing certain references while leaving out others.
Abstract In Alase et al (2022 J. Phys. A: Math. Theor. 55 244003), Alase et al wrote that ‘the constraint of quasi-Hermiticity on observables’ is not ‘sufficient to extend the standard quantum theory’ because ‘such a system is equivalent to a standard quantum system.’ Three addenda elucidating the current state of the art are found necessary. The first one concerns the project: in the related literature the original ‘aim of extending standard quantum theory’ has already been abandoned shortly after its formulation. The second comment concerns the method, viz., the study in ‘the framework of general probabilistic theories’ (GPT). It is noticed that a few other, mathematically consistent GPT-like theories are available. The authors do not mention, in particular, the progress achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We add that this approach already found its advanced realistic applications in the quasi-Hermitian models using the unbounded operators of observables acting in the infinite-dimensional Hilbert spaces. Thirdly, the ‘intriguing open question’ about ‘what possible constraints, if any, could lead to such a meaningful extension’ (in the future) is given an immediate tentative answer: the possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of the observables, provided only that one has in mind its recently developed non-stationary version.
Abstract PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
Yi-Chan Lee et al. claim (cf. Phys. Rev. Lett. 112, 130404 (2014)) that the “recent extension of quantum theory to non-Hermitian Hamiltonians” (which is widely known under the nickname of “PT-symmetric quantum theory”) is “likely false as a fundamental theory”. By their opinion their results “essentially kill any hope of PT-symmetric quantum theory as a fundamental theory of nature”. In our present text we explain that their toy-model-based considerations are misleading and that they do not imply any similar conclusions.
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
I point out that if one defines the operator UR(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator H but an operator H′ which differs from H if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963