In a way inspired by the brief 2002 note "The challenge of nonhermitian structures in physics" by Ramirez and Mielnik (with the text most easily available via arXiv: quant- ph/0211048 ) the situation in the theory is briefly summarized here as it looks twelve years later. Our text has three parts. In the first one we briefly mention the pre-history (dating back to the Freeman Dyson's proposal of the non-Hermitian-Hamiltonian method in 1956 and to its subsequent successful "interacting boson model" applications in nuclear physics) and, first of all, the amazing recent progress reached, in the stationary case, using, in essence, an inversion of the Dyson's approach. The impact on the latter idea upon abstract quantum physics is sampled, first of all, by the reference to papers by Bender et al. (who made the non-Hermitian model-building popular under the nickname of parity-times-time-reflection- symmetric alias PT-symmetric quantum mechanics) and by Mostafazadeh (who reinterpreted PT-symmetry as P-pseudo-Hermiticity). In the second part of our review the emphasis is shifted to the newest, non-stationary upgrade of the formalism which we proposed in the year 2009 and which is characterized by the simultaneous participation of a triplet of Hilbert spaces H in the representation of a single quantum system. In the third part of the review we finally emphasize that the majority of applications of our three-Hilbert-space (THS) recipe is still ahead of us because the enhancement of the flexibility is necessarily accompanied by an enhancement of the technical difficulties. An escape out of the technical trap is proposed to be sought in a restriction of attention to quantum models living in finite-dimensional Hilbert spaces H . As long as the use of such spaces is so typical for the quantum-control considerations, we conclude with conjecture that the THS formalism should start searching for implementations in the field of quantum control.
In a way inspired by the brief 2002 note "The challenge of nonhermitian structures in physics" by Ramirez and Mielnik (with the text most easily available via arXiv: quant- ph/0211048 ) the situation in the theory is briefly summarized here as it looks twelve years later. Our text has three parts. In the first one we briefly mention the pre-history (dating back to the Freeman Dyson's proposal of the non-Hermitian-Hamiltonian method in 1956 and to its subsequent successful "interacting boson model" applications in nuclear physics) and, first of all, the amazing recent progress reached, in the stationary case, using, in essence, an inversion of the Dyson's approach. The impact on the latter idea upon abstract quantum physics is sampled, first of all, by the reference to papers by Bender et al. (who made the non-Hermitian model-building popular under the nickname of parity-times-time-reflection- symmetric alias PT-symmetric quantum mechanics) and by Mostafazadeh (who reinterpreted PT-symmetry as P-pseudo-Hermiticity). In the second part of our review the emphasis is shifted to the newest, non-stationary upgrade of the formalism which we proposed in the year 2009 and which is characterized by the simultaneous participation of a triplet of Hilbert spaces H in the representation of a single quantum system. In the third part of the review we finally emphasize that the majority of applications of our three-Hilbert-space (THS) recipe is still ahead of us because the enhancement of the flexibility is necessarily accompanied by an enhancement of the technical difficulties. An escape out of the technical trap is proposed to be sought in a restriction of attention to quantum models living in finite-dimensional Hilbert spaces H . As long as the use of such spaces is so typical for the quantum-control considerations, we conclude with conjecture that the THS formalism should start searching for implementations in the field of quantum control.
In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian is PT -symmetric. Here P is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo-Hermiticity or, what is the same, self-adjoint operators in Kre˘ın spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019).
We review the methodology to theoretically treat parity-time- (PT-) symmetric, non-Hermitian quantum many-body systems. They are realized as open quantum systems withPTsymmetry and couplings to the environment which are compatible.PT-symmetric non-Hermitian quantum systems show a variety of fascinating properties which single them out among generic open systems. The study of the latter has a long history in quantum theory. These studies are based on the Hermiticity of the combined system-reservoir setup and were developed by the atomic, molecular, and optical physics as well as the condensed matter physics communities. The interest of the mathematical physics community inPT-symmetric, non-Hermitian systems led to a new perspective and the development of the elegant mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics, which do not make any reference to the environment. In the mathematical physics research, the focus is mainly on the remarkable spectral properties of the Hamiltonians and the characteristics of the corresponding single-particle eigenstates. Despite being non-Hermitian, the Hamiltonians can show parameter regimes, in which all eigenvalues are real. To investigate emergent quantum many-body phenomena in condensed matter physics and to make contact to experiments one, however, needs to study expectation values of observables and correlation functions. One furthermore, has to investigate statistical ensembles and not only eigenstates. The adoption of the concepts ofPT-symmetric and biorthogonal quantum mechanics by parts of the condensed matter community led to a controversial status of the methodology. There is no consensus on fundamental issues, such as, what a proper observable is, how expectation values are supposed to be computed, and what adequate equilibrium statistical ensembles and their corresponding density matrices are. With the technological progress in engineering and controlling open quantum many-body systems it is high time to reconcile the Hermitian with thePT-symmetric and biorthogonal perspectives. We comprehensively review the different approaches, including the overreaching idea of pseudo-Hermiticity. To motivate the Hermitian perspective, which we propagate here, we mainly focus on the ancilla approach. It allows to embed a non-Hermitian system into a larger, Hermitian one. In contrast to other techniques, e.g. master equations, it does not rely on any approximations. We discuss the peculiarities ofPT-symmetric and biorthogonal quantum mechanics. In these, what is considered to be an observable depends on the Hamiltonian or the selected (biorthonormal) basis. Crucially in addition, what is denoted as an 'expectation value' lacks a direct probabilistic interpretation, and what is viewed as the canonical density matrix is non-stationary and non-Hermitian. Furthermore, the non-unitarity of the time evolution is hidden within the formalism. We pick up several model Hamiltonians, which so far were either investigated from the Hermitian perspective or from thePT-symmetric and biorthogonal one, and study them within the respective alternative framework. This includes a simple two-level, single-particle problem but also a many-body lattice model showing quantum critical behavior. Comparing the outcome of the two types of computations shows that the Hermitian approach, which, admittedly, is in parts clumsy, always leads to results which are physically sensible. In the rare cases, in which a comparison to experimental data is possible, they furthermore agree to these. In contrast, the mathematically elegantPT-symmetric and biorthogonal approaches lead to results which, are partly difficult to interpret physically. We thus conclude that the Hermitian methodology should be employed. However, to fully appreciate the physics ofPT-symmetric, non-Hermitian quantum many-body systems, it is also important to be aware of the main concepts ofPT-symmetric and biorthogonal quantum mechanics. Our conclusion has far reaching consequences for the application of Green function methods, functional integrals, and generating functionals, which are at the heart of a large number of many-body methods. They cannot be transferred in their established forms to treatPT-symmetric, non-Hermitian quantum systems. It can be considered as an irony of fate that these methods are available only within the mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
Abstract I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.1
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.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.