Moyal brackets and the Weyl quantization

Caliceti, Emanuela; Graffi, Sandro

doi:10.1002/9781118855300.ch4

Cited by 5 publications

(5 citation statements)

References 38 publications

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“…Firstly, whenever the energies remain real even after the phase transition, a new Hamiltonian-dependent physical Hilbert space must be constructed (cf., e.g., Refs. [12] for some comments). Secondly, in the case of the loss of the reality of the energies, the picture of reality cannot be based on the safe and unitary PTQM theory.…”

Section: Discussionmentioning

confidence: 99%

“…All of these innovative, sophisticated treatments of quantum reality remained compatible with the quantum theory of textbooks. Still, the mere rearrangement of mathematical ingredients helped to resolve several old theoretical problems, say, in relativistic quantum mechanics [10] or in our understanding of the mechanisms of the quantum phase transitions [11,12].…”

Section: Introductionmentioning

confidence: 99%

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Admissible perturbations and false instabilities inPT-symmetric quantum systems

Znojil

2018

Phys. Rev. A

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One of the most characteristic mathematical features of the PT −symmetric quantum mechanics is the explicit Hamiltonian-dependence of its physical Hilbert space of states H = H(H). Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT −symmetric user-friendly family of J−parametric "discrete anharmonic" quantum Hamiltonians H = H( λ) is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of spectrum, λ ∈ D (physical) . In contradiction to conventional wisdom the systems are then shown stable with respect to admissible perturbations, inside the domain D (physical) , even in the immediate vicinity of the phase-transition boundaries ∂D (physical) .

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Admissible perturbations and false instabilities inPT-symmetric quantum systems

Znojil

2018

Phys. Rev. A

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“…The feasibility of such an alternative model-building strategy has been confirmed, e.g., by Caliceti et al [36] (cf. also a more recent review of the field in [41]). In essence, the latter authors revealed that in a number of specific toy models the conventional ansatz (45) may still serve as a productive constructive tool, yielding, at the small and real coupling constants, the real (i.e., energy-like) as well as complex (i.e., resonance-representing) lowlying spectra after standard resummation.…”

Section: A Detour To Meaningful Complex Spectramentioning

confidence: 98%

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

Znojil

2024

Symmetry

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Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.

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“…The resolution of the contradictions was provided by the quasi-Hermitian quantum theory [2,5,10,28]. In the framework of this theory it has been clarified that the energies may remain real even if the Hamiltonian itself appears non-selfadjoint.…”

Section: Matrix Models: Mathematical Appealmentioning

confidence: 99%

Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation

Znojil

Борисов

2018

Annals of Physics

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Three-parametric family of non-Hermitian but PT −symmetric six-by-six matrix Hamiltonians H (6) (x, y, z) is considered. The PT −symmetry remains spontaneously unbroken (i.e., the spectrum of the bound-state energies remains real so that the unitary-evolution stability of the quantum system in question is shown guaranteed) in a non-empty domain keywords quantum theory; non-Hermitian observables; discrete models of bound states; typology of instabilities; 2

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Moyal brackets and the Weyl quantization

Cited by 5 publications

References 38 publications

Admissible perturbations and false instabilities inPT-symmetric quantum systems

Admissible perturbations and false instabilities inPT-symmetric quantum systems

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

Two patterns of PT-symmetry breakdown in a non-numerical six-state simulation

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