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REVIEW 4 major objections 5 minor 37 references

Ghost-Free Quantisation of Higher Time-Derivative Theories via Non-Unitary Similarity Transformations

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A ghostly sector with bounded spectrum and non-normalisable states becomes physical under a non-unitary similarity transformation and a redefined inner product.

desk verdict A promising but formally unfinished proof-of-concept: the ghost-to-physical map works at the level of formal eigenfunctions, but the operator-domain and metric-positivity step needs real work before the claim is solid. read the letter →

arxiv 2506.21400 v1 pith:AME7FPIO submitted 2025-06-26 quant-ph hep-thmath-phmath.MP

Andreas Fring , Takano Taira , Bethan Turner This is my paper · ORCID
classification quant-phhep-thmath-phmath.MP
keywords highertime-derivativetheoriesghostproblemnon-unitarysimilaritytransformationspseudo-HermitianquantummechanicsPTsymmetryPais-Uhlenbeckoscillatornormalisableeigenstatesboundedspectrum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tackles the long-standing ghost problem in higher time-derivative theories, where quantisation either gives unbounded spectra or non-normalisable eigenstates. It proposes a new method: map the Hermitian 'ghostly' Hamiltonian to another Hermitian Hamiltonian via a non-unitary similarity transformation, then redefine the inner product with the metric $\rho = \eta^\dagger \eta$ so that the previously non-normalisable states become normalisable while the spectrum stays bounded below. The authors demonstrate the method on a concrete model related to the Pais-Uhlenbeck oscillator, showing that in a parameter region the $(1,1)$-sector, traditionally discarded, becomes a well-defined quantum system. If correct, the construction offers a consistent re-interpretation of HTDTs and a new way to build ghost-free quantum models.

What carries the argument

The central object is the non-unitary similarity transformation $\eta = \eta_2 \eta_1 \eta_0$ built from three explicitly defined operators: $\eta_0 = \exp(-\delta x^2/2 - \lambda y^2/2)$, $\eta_1 = \exp(\kappa p_x^2/2 + \xi p_y^2/2)$, and $\eta_2 = \exp(\mu p_x p_y + \tau x y)$. These are exponentials of quadratic generators; their adjoint actions are computed via the Baker-Campbell-Hausdorff formula, and $\eta_2$ is factorised into $SU(2)$ group elements through a Gauss decomposition. The transformation maps the Hermitian ghostly Hamiltonian $h_0$ to a Hermitian partner $h_3$, preserving the spectrum, while the new metric $\rho = \eta^\dagger \eta$ defines the inner product in which the formerly non-normalisable eigenstates become normalisable. The parameter choices in $\eta_1$ and $\eta_2$ are tuned to eliminate non-Hermitian terms and to render the transformed ground state normalisable.

What would settle it

Evaluate the new inner product $\langle \phi_0 | \rho | \phi_0 \rangle$ for the parameter set in figure 2 ($\nu=4$, $\Omega=-2$, $g=3$, $\lambda$ in the region $|\lambda|>2.13$). If it diverges or is not strictly positive, or if the metric $\rho = \eta_0^\dagger \eta_1^\dagger \eta_2^\dagger \eta_2 \eta_1 \eta_0$ has a negative eigenvalue, the claimed normalisability fails. Alternatively, check directly whether $e^{\xi p_y^2/2}$ maps $L^2(\mathbb{R}^2)$ into $L^2(\mathbb{R}^2)$ for $\xi>0$ in that region.

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Extended reading notes

Core claim

The central claim is that the sector of the ghostly Hamiltonian $h_0 = p_x^2 - p_y^2 + \nu^2 x^2 + \Omega y^2 + gxy$ with $\epsilon = \eta = 1$, which has a spectrum bounded from below but non-normalisable eigenstates under the standard $L^2$ inner product, becomes fully physical after the sequence of non-unitary transformations $\eta_0 \eta_1 \eta_2$. Specifically, the transformed Hamiltonian $h_3 = \eta_2 \eta_1 \eta_0 \, h_0 \, \eta_0^{-1} \eta_1^{-1} \eta_2^{-1}$ is Hermitian and isospectral to $h_0$, and with the metric $\rho = \eta_0^\dagger \eta_1^\dagger \eta_2^\dagger \eta_2 \eta_1 \eta_0$ the eigenstates, in particular the Gaussian ground state, are normalisable. The paper exhibits explicit parameter choices (e.g., $\nu=4$, $\Omega=-2$, $g=3$, $|\lambda|>2.13322$) where the normalisability condition $\alpha>0$, $\beta>0$, $alpha\beta-\gamma^2>0$ holds for the $(1,1)$-sector, while the spectrum remains bounded from below. The authors present this as a proof of concept that the ghost problem in HTDTs can be circumvented by re-interpreting the Hilbert-space structure, rather than discarding the offending sector.

Load-bearing premise

The argument assumes the non-unitary operators $\eta_1$ and $\eta_2$ are well-defined invertible maps on the quantum state space for the parameter ranges used, so that the similarity transformation truly preserves the spectrum and the new metric $\rho = \eta^\dagger \eta$ is positive definite; the paper does not prove these domain and positivity properties.

Editorial extensions

If this is right

  • The $(1,1)$-sector of the ghostly model, previously deemed unphysical, admits a consistent quantum description with a positive-definite inner product and a spectrum bounded from below.
  • The construction yields an isospectral Hermitian partner for a Pais-Uhlenbeck-related model, so the energy spectrum of the physical theory is unchanged while the states become normalisable.
  • The method extends pseudo/quasi-Hermitian quantum mechanics by mapping between two Hermitian Hamiltonians, not from a non-Hermitian one, broadening the class of tractable ghost problems.
  • For vanishing coupling $g=0$, the transformed Hamiltonian becomes a sum of two harmonic oscillators in the relevant parameter region, making the physical interpretation explicit.
  • The framework suggests a route to ghost-free quantisation of more general higher time-derivative theories, including field-theoretic extensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the metric $\rho$ is indeed positive, the construction can be read as a choice of a new physical Hilbert space for the ghostly theory; the non-uniqueness of $\eta$ might then be fixed by requiring a second observable to be Hermitian, a point the authors flag but do not resolve.
  • The Gaussian ansatz suggests the method could extend to HTDTs whose ground states are Gaussian, but the proof of concept does not establish how the transformation generalises to interacting field theories.
  • A sharper test of the method would be to examine the domain issues of the unbounded operators $\eta_1$ and $\eta_2$; the paper does not prove they are well-defined on $L^2(\mathbb{R}^2)$, so the preservation of the spectrum is an assumption that could fail.
  • Numerically scanning the full parameter space could determine whether the normalisable region $R_1$ is as large as suggested and whether similar regions exist for other branch choices.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a method to render physical the 'ghost sector' of a higher time-derivative theory, using a two-dimensional oscillator with an indefinite kinetic term as a concrete model. Starting from a Hermitian Hamiltonian h0 whose eigensystem is taken from earlier work, the authors construct a sequence of non-unitary similarity transformations η = η2 η1 η0 designed to map h0 to a new Hermitian Hamiltonian h3. They define a new inner product metric ρ = η†η and claim that in a specific parameter region the formerly non-normalisable eigenstates of the (1,1)-sector become normalisable, while the spectrum remains bounded from below. The paper presents explicit formulas for the transformed Hamiltonian and for the ground-state wavefunction, and shows numerically that the normalisability conditions can be satisfied in a proof-of-concept regime.

Significance. If the construction can be made rigorous, the paper would provide a concrete example of how pseudo-Hermitian techniques can be used to reinterpret ghostly sectors of higher time-derivative theories as physical, potentially extending the toolbox for ghost-free quantum models. The explicit formulas and parameter scans are useful and the connection to the Pais-Uhlenbeck oscillator is relevant. However, the central claim rests on operator-theoretic steps that are only handled formally, so the significance is contingent on closing those gaps.

major comments (4)
  1. [§2.3–2.4, Eqs. (2.7), (2.31)] The similarity operators η1 and η2 are unbounded, yet the paper never specifies their domains or proves that they define invertible maps on L²(R²) in the parameter regions used. For example, with δ=0, Ω=-2, and |λ|>√2, ξ = λ/(λ²+Ω) is positive, so η1 in momentum space is multiplication by exp(ξ p_y²/2), an unbounded operator with unbounded inverse. Likewise η2 = exp(μ p_x p_y + τ x y) is only formally defined. For unbounded operators, formal conjugation does not preserve spectra without domain conditions. The statement that h0 and h3 are isospectral and that ρ = η†η is a positive metric is therefore not established at the operator level. This is load-bearing for the paper's central claim that the ghost sector becomes a genuine Hilbert-space sector. Please provide a careful domain analysis, prove positivity of ρ on a dense domain containing the eigenstates, and establish (essential) self-adjointness of h3, or justify the formal manipulations by an approximation argument.
  2. [§2.4, Eqs. (2.29)–(2.30) and p.2, p.7] The normalisability check is performed only for the ground state ϕ3. The paper asserts that ground-state normalisability is sufficient because excited states inherit the Gaussian factor, but the excited states of the transformed Hamiltonian are never constructed. It is not shown that all excited states are square-integrable with respect to the new metric, nor that the metric is positive on the full sector. Since the physical interpretation requires normalisable eigenstates for the entire spectrum, this gap must be closed, for instance by writing the transformed excited states explicitly or by proving a general inheritance argument for the polynomial prefactors.
  3. [§2.4, Eqs. (2.21)–(2.22)] The claim that h3 'has lost its ghostly nature and possesses regions in parameter space where it is positive definite' is made without proof. Positive definiteness or at least boundedness from below of h3 is essential to the claim that the spectrum is bounded from below after the transformation. Please provide the missing derivation, or state precisely how this follows from isospectrality to h0 combined with the sector classification of [11].
  4. [§2.1, Eqs. (2.2)–(2.5)] The full eigensystem and the sector classification (ϵ, η) are imported from the authors' earlier work [11] without derivation or verification. The construction depends critically on the existence of a sector with bounded spectrum and non-normalisable states, and on the assertion that ground-state normalisability controls all excited states. If those results are not independently verified, the present proof of concept inherits any errors. Please either include a concise derivation of the needed parts of the eigensystem or state explicitly which results are assumed and why they are reliable.
minor comments (5)
  1. [Eq. (2.8)] Typo: the adjoint action of η0 on p_y should read η0 p_y η0^{-1} = p_y - iλ y, not p_x - iλ y.
  2. [Eqs. (2.2)–(2.3)] The notation for the spectrum is unclear: define the ranges of N and n precisely, and clarify whether the floor function is intended in the upper limit of n.
  3. [Fig. 2 and surrounding text] The definitions of the δ+ and δ− branches and the reason why only the δ+ branch yields a normalisable solution should be stated more explicitly.
  4. [§2.3, after Eq. (2.14)] The statement that for g→0 the Hamiltonian becomes a sum of two harmonic oscillators in a certain parameter regime should be quantified; for example, with δ=0 and Ω=-2 the regime is |λ|>√2.
  5. [References] Reference [11] is cited in final form without an arXiv identifier; providing the arXiv number would aid verification.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is an explicit non-unitary similarity transformation with direct normalisability checks; prior self-citations are inputs, not the derived conclusion.

full rationale

The paper's derivation chain is self-contained after importing the h0 eigensolution: the sector classification, ground state (2.5), and spectrum (2.2)-(2.3) are taken from the authors' prior paper [11], but that is an input, not the conclusion being derived. The new claim is that eta2 eta1 eta0 h0 eta0^{-1} eta1^{-1} eta2^{-1} = h3 is Hermitian and that the transformed ground state (2.29) satisfies the standard normalisability condition (2.6) in an explicit parameter region (figures 1-2). These checks are direct algebraic/numerical evaluations of formulas (2.17)-(2.19) and (2.30) with constraints (2.23); no quantity is fitted to the target conclusion, and no prediction is made from external data. The transformation parameters are tuned by hand, but tuning parameters to find a non-empty existence region in a proof of concept is not circularity. The self-citations [11] and [30] provide the starting model and its Pais-Uhlenbeck connection; they are not used as a substitute for the Hermiticity or normalisability checks. Limitation passages in sections 2.4 and 3 (explicit proof-of-concept status, non-uniqueness of eta, and open questions) are caveats, not admissions of circularity. The unboundedness of eta1 and eta2 and the unproven positivity of rho are mathematical gaps that would affect correctness, but they do not make the argument circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The construction rests on the prior eigensolution of h0, a small number of hand-tuned transformation parameters, and an unexamined domain assumption for unbounded similarity maps. No external data or observable predictions are involved.

free parameters (4)
  • delta = delta = 0 in Figure 1; delta = delta_+ = [sqrt(4 lambda^2 nu^2 + (lambda^2 + Omega)^2) - lambda^2 - Omega] / (2…
    Introduced in eta0 = exp(-delta x^2/2). Hand-chosen; the paper says it 'will merely distort the parameter regime without altering the qualitative behaviour'.
  • lambda = |lambda| > sqrt(2) for the g = 0 case; |lambda| > 2.13322 (region R1) for the g = 3 example
    Introduced in eta0 = exp(-lambda y^2/2). Central tuning parameter: normalisability of the formerly non-normalisable sector is achieved only for these large-|lambda| regions.
  • kappa and xi = kappa = delta / (delta^2 - nu^2), xi = lambda / (lambda^2 + Omega)
    Not free: fixed by the requirement that the non-Hermitian terms proportional to i p_x x and i p_y y in H2 vanish.
  • mu and tau = tau = (delta^2 - nu^2) arctanh(Theta) / (2 delta), mu = delta / [2 (delta^2 - nu^2)] arctanh(Theta), with |Theta| < 1
    Determined by delta, lambda and model parameters via the Hermiticity constraints for h3; not free once delta and lambda are chosen.
assumptions (6)
  • domain assumption The complete eigensystem of h0, including spectrum formulas (2.2)-(2.3) and ground state (2.5), is as stated in Ref. [11].
    The paper imports the solution from the authors' own prior work and does not re-derive it; the sector labels (epsilon, eta) and normalisability properties are load-bearing.
  • domain assumption Normalisability of all excited states is inherited from the ground state, as stated in Ref. [11].
    Used to extend the ground-state normalisability check to the whole spectrum; taken from the authors' previous paper.
  • domain assumption eta0, eta1 and eta2 are invertible operators on L2(R2) and the metric rho = eta^dagger eta is positive, despite eta1 and eta2 being unbounded for the parameter regions used.
    Not discussed in the paper; required for the similarity transformation to preserve the spectrum and for rho to define a valid inner product.
  • standard math The Gaussian normalisability conditions alpha > 0, beta > 0, alpha beta - gamma^2 > 0 applied to psi2 and phi3 are sufficient for physical normalisability.
    Standard Gaussian integral criterion; the paper uses it to read off normalisability from the wavefunction coefficients.
  • domain assumption The model h0 is related to the Pais-Uhlenbeck oscillator in the relevant parameter regime, per Ref. [30].
    Bridges the toy model to the named HTDT prototype; imported from a self-cited companion paper.
  • standard math The Baker-Campbell-Hausdorff formula and the su(2) Gauss decomposition of exp(mu S_- + tau S_+) are valid for the unbounded generators used.
    Used to compute adjoint actions (2.8)-(2.11) and to factor eta2 in (2.24).
invented entities (1)
  • Non-unitary similarity map eta = eta2 eta1 eta0 and associated metric rho = eta^dagger eta
    purpose: To redefine the inner product so that the formerly non-normalisable (1,1)-sector eigenstates become normalisable while preserving the bounded-below spectrum.
    The map and metric are constructed to achieve normalisability; no independent observable or external prediction is provided.

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Pith. "Pith review of Ghost-Free Quantisation of Higher Time-Derivative Theories via Non-Unitary Similarity Transformations." pith.science (2026). https://pith.science/paper/AME7FPIO

@misc{pith2026250621400,
  author       = {Pith},
  title        = {Pith review of: Ghost-Free Quantisation of Higher Time-Derivative Theories via Non-Unitary Similarity Transformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AME7FPIO}},
  note         = {Machine review of arXiv:2506.21400}
}
read the original abstract

We address the long-standing ``ghost problem" in higher time-derivative theories (HTDTs), where quantisation typically yields sectors with either unbounded spectra or non-normalisable eigenstates; both rendering the theory unphysical. We propose a novel method that preserves the bounded nature of the spectrum in one particular sector while restoring normalisability by employing a non-unitary similarity transformation. Inspired by techniques from pseudo/quasi-Hermitian PT-symmetric quantum mechanics, we construct a non-unitary map between two Hermitian Hamiltonians, converting ghostly sectors into physically viable ones. We demonstrate the feasibility of this approach using a concrete HTDT model, related to the Pais-Uhlenbeck oscillator, and show that the transformed system admits normalisable eigenstates and a spectrum bounded from below. This framework offers a consistent re-interpretation of HTDTs and extends the toolbox for constructing ghost-free quantum models.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

37 extracted references · 35 canonical work pages

  1. [11]

    Fring, T

    A. Fring, T. Taira, and B. Turner, Quantisations of exactly solvable ghostly models, J. Phys. A: Math. Theor. 58(23), 235301 (2025)

    work page 2025

  2. [1]

    Pais and G

    A. Pais and G. E. Uhlenbeck, On field theories with non-localized action, Phys. Rev. D 79(1), 145 (1950)

    work page 1950

  3. [2]

    K. S. Stelle, Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16(4), 953 (1977)

    work page 1977

  4. [3]

    A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91(1), 99–102 (1980)

    work page 1980

  5. [4]

    S. L. Adler, Einstein gravity as a symmetry-breaking effect in quantum field theory, Rev. Mod. Phys. 54(3), 729 (1982)

    work page 1982

  6. [5]

    A. V. Smilga, Spontaneous generation of the Newton constant in the renormalizable gravity theory, ITEP preprint 63 (1982) 8 pp, arXiv:1406.5613

    arXiv 1982

  7. [6]

    Modesto and I

    L. Modesto and I. L. Shapiro, Superrenormalizable quantum gravity with complex ghosts, Phys. Lett. B 755, 279–284 (2016)

    work page 2016

  8. [7]

    S. W. Hawking and T. Hertog, Living with ghosts, Phys. Rev. D 65(10), 103515 (2002)

    work page 2002

Show all 37 references
  1. [8]

    H. A. Weldon, Quantization of higher-derivative field theories, Ann. Phys. 305(2), 137–150 (2003)

    2003

  2. [9]

    R. P. Woodard, Ostrogradsky’s theorem on Hamiltonian instability, Scholarpedia 10(8), 32243 (2015)

    2015

  3. [10]

    Fring, T

    A. Fring, T. Taira, and B. Turner, Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories, Universe 10(5), 198 (2024)

    2024

  4. [12]

    T.-J. Chen, M. Fasiello, E. A. Lim, and A. J. Tolley, Higher derivative theories with constraints: Exorcising Ostrogradski’s Ghost, JCAP 2013(02), 042 (2013). 8 Ghost-Free Quantisation of HTDT via Non-Unitary Similarity Transformations

    2013

  5. [13]

    Salvio and A

    A. Salvio and A. Strumia, Quantum mechanics of 4-derivative theories, The EPJ C 76, 1–15 (2016)

    2016

  6. [14]

    Anselmi, Fakeons and Lee-Wick models, JHEP 2018(2) (2018)

    D. Anselmi, Fakeons and Lee-Wick models, JHEP 2018(2) (2018)

    2018

  7. [15]

    C. M. Bender and P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100(11), 110402 (2008)

    2008

  8. [16]

    Raidal and H

    M. Raidal and H. Veerm¨ ae, On the quantisation of complex higher derivative theories and avoiding the Ostrogradsky ghost, Nucl. Phys. B 916, 607–626 (2017)

    2017

  9. [17]

    A. V. Smilga, Ghost-free higher-derivative theory, Phys. Lett. B 632(2-3), 433–438 (2006)

    2006

  10. [18]

    Smilga, Benign ghosts in higher-derivative systems, in J

    A. Smilga, Benign ghosts in higher-derivative systems, in J. Phys.: Conf. Series, volume 2038, page 012023, IOP Publishing, 2021

    2021

  11. [19]

    Deffayet, S

    C. Deffayet, S. Mukohyama, and A. Vikman, Ghosts without runaway instabilities, Phys. Rev. Lett. 128(4), 041301 (2022)

    2022

  12. [20]

    Deffayet, A

    C. Deffayet, A. Held, S. Mukohyama, and A. Vikman, Global and local stability for ghosts coupled to positive energy degrees of freedom, JCAP 2023(11), 031 (2023)

    2023

  13. [21]

    V. E. D ´ ıez, J. G. Gaset Rif` a, and G. Staudt, Foundations of Ghost Stability, Fortschritte der Physik , 2400268 (2024)

    2024

  14. [22]

    Bolonek and P

    K. Bolonek and P. Kosinski, Hamiltonian Structures for Pais–Uhlenbeck Oscillator, Acta Physica Polonica B 36(6), 2115 (2005)

    2005

  15. [23]

    Damaskinsky and M

    E. Damaskinsky and M. Sokolov, Remarks on quantization of Pais–Uhlenbeck oscillators, J. Phys. A: Math. Gen. 39(33), 10499 (2006)

    2006

  16. [24]

    N. G. Stephen, On the Ostrogradski instability for higher-order derivative theories and a pseudo-mechanical energy, J. Sound Vibration 310(3), 729–739 (2008)

    2008

  17. [25]

    Mostafazadeh, A Hamiltonian formulation of the Pais–Uhlenbeck oscillator that yields a stable and unitary quantum system, Phys

    A. Mostafazadeh, A Hamiltonian formulation of the Pais–Uhlenbeck oscillator that yields a stable and unitary quantum system, Phys. Lett. A 375(2), 93–98 (2010)

    2010

  18. [26]

    Andrzejewski, Hamiltonian formalisms and symmetries of the Pais–Uhlenbeck oscillator, Nucl

    K. Andrzejewski, Hamiltonian formalisms and symmetries of the Pais–Uhlenbeck oscillator, Nucl. Phys. B 889, 333–350 (2014)

    2014

  19. [27]

    Andrzejewski, Conformal Newton–Hooke algebras, Niederer’s transformation and Pais–Uhlenbeck oscillator, Phys

    K. Andrzejewski, Conformal Newton–Hooke algebras, Niederer’s transformation and Pais–Uhlenbeck oscillator, Phys. Lett. B 738, 405–411 (2014)

    2014

  20. [28]

    Andrzejewski, A

    K. Andrzejewski, A. Galajinsky, J. Gonera, and I. Masterov, Conformal Newton–Hooke symmetry of Pais–Uhlenbeck oscillator, Nucl. Phys. B 885, 150–162 (2014)

    2014

  21. [29]

    Elbistan and K

    M. Elbistan and K. Andrzejewski, Various disguises of the Pais-Uhlenbeck oscillator, Nucl. Phys. B 994, 116327 (2023)

    2023

  22. [30]

    Felski, A

    A. Felski, A. Fring, and B. Turner, Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model, arXiv:2505.07869 (2025)

    2025 arXiv

  23. [31]

    C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243–5246 (1998). 9 Ghost-Free Quantisation of HTDT via Non-Unitary Similarity Transformations

    1998

  24. [32]

    Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int

    A. Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int. J. Geom. Meth. Mod. Phys. 7, 1191–1306 (2010)

    2010

  25. [33]

    C. M. Bender, P. E. Dorey, C. Dunning, A. Fring, D. W. Hook, H. F. Jones, S. Kuzhel, G. Levai, and R. Tateo, PT Symmetry: In Quantum and Classical Physics, World Scientific, Singapore (2019)

    2019

  26. [34]

    Fring, An introduction to PT-symmetric quantum mechanics-time-dependent systems, in J

    A. Fring, An introduction to PT-symmetric quantum mechanics-time-dependent systems, in J. Phys.: Conf. Series, v2448, p012002, IOP Publishing (2023)

    2023

  27. [35]

    F. G. Scholtz, H. B. Geyer, and F. Hahne, Quasi-Hermitian Operators in Quantum Mechanics and the Variational Principle, Ann. Phys. 213, 74–101 (1992)

    1992

  28. [36]

    B. C. Hall, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics (Springer, New York, 2015)

    2015

  29. [37]

    A. B. Klimov and S. M. Chumakov, A group-theoretical approach to quantum optics: models of atom-field interactions, (John Wiley & Sons, Weinheim, 2009). 10

    2009

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