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REVIEW 4 major objections 3 minor 50 references

Does Cosmology require Hermiticity in Quantum Mechanics?

T0 review · 4 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Cosmology squeezes non-Hermitian quantum effects to near zero

desk verdict A cleanly written but under-derived template: the bounds constrain invented parameters rather than Hermiticity, and the inflationary mode equation has a sign error. read the letter →

arxiv 2602.05973 v1 pith:SAMF5U3V submitted 2026-02-05 astro-ph.CO gr-qchep-phhep-thquant-ph

Oem Trivedi , Alfredo Gurrola This is my paper · ORCID
classification astro-ph.COgr-qchep-phhep-thquant-ph
keywords non-HermitianquantummechanicscosmologyWheeler-DeWittequationcosmologicalconstraintsstructuregrowthinflationaryperturbationsspatialflatnessmodifiedgravity
topics Dark Energy
open problems Dark Energy
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

This paper asks whether cosmology can test a basic axiom of quantum mechanics: that the generator of time evolution is Hermitian, keeping probabilities conserved. By extending the Wheeler–DeWitt equation with an anti-Hermitian term, the authors show that any such term would act as a source or sink for primordial fluctuations and for the growth of cosmic structure. Comparing the resulting predictions with inflation, distance, and growth data, they derive stringent bounds that suppress non-Hermiticity across cosmic history — the effective friction and gain rates must be far below the Hubble scale. The conclusion is that the semiclassical branch describing our universe behaves as if Hermiticity holds, and that Hermiticity may be an emergent property rather than a fundamental axiom. The same data allow more room for non-Hermitian effects if gravity is modified beyond general relativity.

What carries the argument

The central object is the non-Hermitian Wheeler–DeWitt operator Ĥ_NH = Ĥ_H + iΓ̂_H, where Γ̂_H is an anti-Hermitian functional on superspace; the semiclassical Born–Oppenheimer expansion converts Γ̂_H into effective non-unitary rates for perturbations, parametrized as friction γ(t), background source ξ, curvature bias ξ_K, and inflationary damping α(N). These phenomenological couplings carry the argument from the fundamental postulate to observable power spectra and growth factors.

What would settle it

A future dataset requiring a constant growth-sector friction with |γ₀|/H₀ of order 0.1 — e.g., a persistent σ8 tension that cannot be removed by modified gravity — or a measured running of the spectral index dα/dN of order unity would contradict the derived bounds and falsify the central claim.

Watch

Extended reading notes

Core claim

Starting from a non-Hermitian Wheeler–DeWitt constraint, Ĥ_NH = Ĥ_H + iΓ̂_H acting on the wave function of the universe, the authors carry out a semiclassical Born–Oppenheimer reduction in which the anti-Hermitian functional Γ̂_H leaks into the perturbation sector as an effective gain/loss generator. In the late universe this appears as a friction term γ(t) in the matter growth equation, a source Q(t) in the dark-energy continuity equation, and a curvature bias ξ_K in the evolution of Ω_K; during inflation it appears as a damping rate α(N) for Mukhanov–Sasaki modes. Each of these leaves a characteristic imprint — an exponential envelope on the power spectrum, a shift in σ8, a distortion of H

Load-bearing premise

The bounds are only about Hermiticity if the invented couplings γ(t), ξ, and α(N) genuinely encode the underlying anti-Hermitian functional Γ̂_H — a mapping the paper itself says is model dependent, not derived.

Editorial extensions

If this is right

  • If the central claim is right, any fundamental non-Hermitian component of quantum mechanics must be confined to the deep ultraviolet or to regimes where spacetime is not semiclassical; it cannot persist along the branch we inhabit.
  • Cosmological observations become a new, infrared test of quantum foundations: even tiny non-unitary rates, integrated over gigayears, would destroy the observed consistency between geometric and growth probes.
  • Spatial flatness joins inflation and structure formation as a Hermiticity witness — an unbiased weighting of different curvature sectors is an observational requirement, not just a theoretical preference.
  • In modified gravity theories with higher-curvature terms, cosmological constraints on non-Hermiticity weaken: the data bounds combinations of γ, H(z), and G_eff rather than Hermiticity alone, so conclusions about Hermiticity depend on the assumed gravitational EFT.
  • The bounds suggest that Hermiticity or effective unitarity is an emergent property selected by the semiclassical consistency of our universe, not an axiom that must be imposed at the fundamental level.

Reading between the lines

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

  • If the same non-Hermitian functional also biases other measure-zero sectors of superspace, the flatness argument generalizes: any anti-Hermitian source that distinguishes between fine-tuned alternatives (e.g., homogeneity, isotropy) should be similarly suppressed — a testable prediction for quantum-cosmology models.
  • The proposed growth-sector bound can be sharpened with future surveys by measuring the growth index directly; a detection of a redshift-dependent growth index inconsistent with GR would force a re-evaluation of the mapping between Γ̂_H and γ(t).
  • One could reinterpret the anti-Hermitian term as a coarse-grained environmental influence rather than a fundamental departure; under that reading the bounds constrain possible non-unitary decoherence rates in quantum gravity, connecting to open-quantum-system descriptions of cosmology.
  • The modified-gravity relaxation hints at a degeneracy: a model with a non-Hermitian sector and a different G_eff(z) can mimic a pure GR universe; disentangling them requires joint fits of growth and expansion that the paper does not perform.
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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 / 3 minor

Summary. The paper proposes a non-Hermitian extension of the Wheeler–DeWitt framework by replacing the Hermitian constraint with \(\hat{H}_{\rm NH}=\hat{H}_{\rm H}+i\hat{\Gamma}_{\rm H}\) (Eqs. (10)–(11)). It claims that a semiclassical Born–Oppenheimer reduction turns the anti-Hermitian component into observable effects: a friction term \(\gamma(t)\) in the growth equation (Eq. (16)), a source term \(\xi\) in the dark-energy continuity equation (Eq. (30)), a curvature bias \(\xi_K\) in \(\Omega_K\) evolution (Eq. (35)), and an inflationary damping rate \(\alpha(N)\) (Eq. (45)). It then asserts constraints \(|\gamma_0|/H_0\lesssim 2\varepsilon\), \(|\xi|\ll1\), \(|\xi_K|\ll1\), and \(\alpha\ll1\), concluding that cosmology strongly suppresses non-Hermiticity. The central claim is not established: the link from \(\hat{\Gamma}_{\rm H}\) to the phenomenological parameters is asserted rather than derived, and the constraints are largely restatements of assumed observational tolerances.

Significance. If a rigorous semiclassical reduction from a concrete \(\hat{\Gamma}_{\rm H}\) to the proposed proxies existed, this would be an interesting and creative new arena for testing non-Hermitian quantum mechanics with cosmological data. The paper is clearly written and transparent about some of its limitations, acknowledging that the mapping is model-dependent and that the parametrizations are 'minimal' or 'conservative.' However, the current manuscript does not deliver such a reduction; it provides a dictionary of possible phenomenological couplings without showing that they follow from the postulated non-Hermitian Wheeler–DeWitt constraint. Because the central claim is unsupported, the significance is presently programmatic rather than established.

major comments (4)
  1. [§3 (Eqs. (10)–(16))] The bridge from the non-Hermitian Wheeler–DeWitt equation (Eq. (11)) to the growth-sector friction \(\gamma(t)\) is asserted, not derived. The text states that Eq. (16) is 'the most conservative late-time parametrization' and that 'the mapping between \(\gamma(t)\) and \(\hat{K}(t)\) is model dependent.' No concrete \(\hat{\Gamma}_{\rm H}\) is specified, and the Born–Oppenheimer reduction that would produce \(\hat{K}(t)\) and then the growth equation is only sketched. The same applies to \(\xi\) in Eq. (30), \(\xi_K\) in Eq. (35), and \(\alpha(N)\) in Eq. (45). Without a definite mapping, the derived bounds constrain these invented parameters, not Hermiticity itself. This is a load-bearing gap: the abstract's claim that the paper derives constraints on non-Hermiticity is not supported.
  2. [§3–§5 (Eqs. (27)–(29), (30), (35)–(37), (45)–(49))] The claimed 'derived' bounds are circular. In Eq. (27) the paper assumes \(|\Delta\sigma_8/\sigma_8|\lesssim\varepsilon\), where \(\varepsilon\) is an input tolerance, and Eq. (28) then yields \(|\int\gamma dt|\lesssim2\varepsilon\), leading to Eq. (29) \(|\gamma_0|/H_0\lesssim2\varepsilon\). This is a restatement of the assumption. Similarly, Eq. (30) defines a source \(Q=\xi H\rho_{\rm DE}\) and then immediately says \(|\xi|\ll1\) is required by 'the empirical success of precision distance measures'; Eq. (37) imposes \(|\xi_K|\ll1\) directly from \(|\Omega_K|\ll1\); and Eqs. (47)–(48) impose \(\alpha\ll1\), \(d\alpha/dN\ll1\) because large distortions are 'observationally excluded.' No dataset or quantitative \(\varepsilon\) is ever used. The abstract's phrase 'strong infrared constraints' is thus an overstatement: the constraints are self-consistency conditions on the chosen proxies,
  3. [§5 (Eqs. (39)–(42))] The inflationary derivation contains an algebraic inconsistency. From the Hamiltonian in Eq. (39), the Heisenberg equations in Eqs. (40)–(41) give \(v'=\pi+\Gamma v\) and \(\pi'=-\omega^2 v+\Gamma\pi\). Eliminating \(\pi\) yields \(v''-2\Gamma v'+(\omega^2-\Gamma'+\Gamma^2)v=0\), not Eq. (42), which has \(-\Gamma^2\) and also has a sign convention that does not follow from these equations. If the explicit \(i\) in Eq. (39) is retained, the resulting equation is complex and not Eq. (42). Consequently the damping envelope in Eq. (44) and the bounds on \(\alpha\) in Eqs. (46)–(49) are not reliably derived from the stated Hamiltonian.
  4. [§2 (Eq. (13))] The interpretation of the anti-Hermitian term as damping or gain rests on the standard inner product in Eq. (13). The introduction notes that a positive metric operator \(\eta\) can restore unitary, norm-preserving evolution even for non-Hermitian Hamiltonians. The paper never specifies which inner product is used along the semiclassical branch or why the standard one is the physically relevant one. If the \(\eta\)-metric is the correct description, the same formal non-Hermiticity could be unobservable, and all constraints derived here would lose their physical meaning. This is a load-bearing convention that is left unflagged.
minor comments (3)
  1. [§3 (background equations)] The text repeatedly writes 'FLR W' instead of 'FLRW' (e.g., 'the background geometry is well described by an FLR W spacetime'). Please correct.
  2. [§5 (Eqs. (45) and (50))] The notation \(\alpha\) is used for the dimensionless non-Hermitian inflationary rate (Eq. (45)) and later for the \(R^2\) coupling in the modified-gravity action (Eq. (50)). This clash is confusing; please use a different symbol for one of them.
  3. [General] The paper says it 'confronts' the framework with observational data, but no actual data, likelihood, or posterior is used. If a revision is pursued, a concrete dataset with a specified \(\varepsilon\) would be needed to make the constraints quantitative.

Circularity Check

4 steps flagged · score 7.0 of 10

Late-time and inflationary 'bounds' on non-Hermiticity are restatements of the assumed agreement with the Hermitian baseline; the bridge from Γ̂H to the bounded proxies is admitted to be model-dependent.

  1. self definitional [Eqs. (15)-(16) and (27)-(29)]
    "In the presence of an anti-Hermitian contribution in the emergent dynamics, the most conservative late-time parametrization ... is to add a real friction term γ(t) ˙δm ... The mapping between γ(t) and K̂(t) is model dependent ... Requiring that the non-unitary damping or gain does not spoil the observed near-consistency between growth observables and geometric probes motivates the bound |Δσ8/σ8|≲ε ... If γ is approximately constant over a characteristic duration Δt of order the Hubble time, Δt ∼ H₀⁻¹ then this becomes |γ₀|/H₀ ≲ 2ε."

    The output bound is the input tolerance rewritten. Eq. (27) gives Δσ8/σ8 = −½∫γ dt; imposing |Δσ8/σ8| ≲ ε immediately yields |∫γ| ≲ 2ε and, for constant γ, |γ0|/H0 ≲ 2ε. Moreover, γ is introduced as a phenomenological friction whose relation to the original anti-Hermitian generator K̂(t) is explicitly admitted to be model-dependent, so the step does not actually constrain non-Hermiticity in Eq. (11).

  2. self definitional [After Eq. (30)]
    "Taking the minimal scaling Q = ξHρ_DE implies ρ_DE ∝ a^{−3(1+w)+ξ} and thus an order ξ distortion of H(z) relative to the Hermitian baseline, so the empirical success of precision distance measures requires |ξ| ≪ 1 over the probed redshift range."

    ξ is defined as the fractional distortion of the dark-energy density/expansion history relative to the Hermitian baseline. Saying that precision distance measurements require |ξ| ≪ 1 is exactly the statement that the Hermitian baseline fits the data. No independent relation to Γ̂H or K̂(t) is supplied, so the 'constraint' is the assumed concordance renamed.

2 more flagged steps
  1. self definitional [Eqs. (35)-(37)]
    "Projecting this effect into the semiclassical late-time dynamics, a curvature dependent non-Hermitian contribution may be parametrized phenomenologically as an effective evolution equation for Ω_K, dΩ_K/dlna = −2Ω_K + ξ_K ... even a small constant ξ_K drives Ω_K toward an asymptotic value of order ξ_K and requiring |Ω_K| ≪ 1 today implies |ξ_K| ≪ 1."

    The evolution equation is posited with an arbitrary source ξ_K, and the solution asymptotes to Ω_K ≈ ξ_K/2. The derived inequality |ξ_K| ≪ 1 is therefore just the assumed input |Ω_K| ≪ 1 expressed through the new parameter. ξ_K is not derived from Eq. (11); it is a label for whatever drives Ω_K, so observed flatness is re-branded as a non-Hermitian bound.

  2. renaming known result [Eqs. (39)-(49)]
    "In the non-Hermitian extension, the reduced quadratic Hamiltonian for each Fourier mode may be written as H_k = 1/2(π_k² + ω_k² v_k²) + i Γ_k(τ) 1/2(v_kπ_k + π_kv_k) ... Defining the dimensionless non-Hermitian rate α(N) ≡ Γ(N)/H(N) ... Consistency with Eqs. (47), (48) and (49) therefore requires α ≪ 1 and dα/dN ≪ 1 during horizon exit."

    α is defined through Γ/H and Eq. (47) shows a shift in n_s−1 of −2α. The 'bound' α ≪ 1 is exactly the assumption that the observed spectrum is close to the Hermitian slow-roll result; without specifying (n_s−1)⁰ or using actual data, this is the near-Hermiticity premise restated. Additionally, with [v,π]=i the stated Heisenberg equations (40)-(42) do not follow from Eq. (39), so the formal reduction is asserted rather than derived.

full rationale

The paper's central claim is that cosmology derives strong infrared constraints on non-Hermiticity, but the actual derived inequalities are constraints on invented phenomenological coefficients γ, ξ, ξ_K and α, not on the original operator Γ̂H or K̂(t). The bridge is explicitly conceded: 'The mapping between γ(t) and K̂(t) is model dependent.' In each sector the 'prediction' is a restatement of the input: |Δσ8/σ8| ≲ ε gives |γ0|/H0 ≲ 2ε; Q=ξHρ_DE gives |ξ|≪1 from the Hermitian fit; dΩ_K/dlna = −2Ω_K+ξ_K gives |ξ_K|≪1 from observed flatness; and α≪1 follows only by assuming the Hermitian spectral index is close to the observed one. No actual dataset or numerical tolerance ε is used, so the bounds are self-consistency conditions on the chosen parametrizations. The paper also flags the 'most conservative' and 'minimal scaling' choices, and the η-metric unitary restoration mentioned in the introduction is never applied, making the damping/gain interpretation convention-dependent. The inflationary derivation is additionally internally inconsistent: the claimed Heisenberg equations do not follow from the stated Hamiltonian with standard canonical commutation relations. There is no self-citation problem here; the circularity is internal to the derivation chain. Taken together, the central 'strong constraints' reduce to the assumed agreement with Hermitian cosmology, so the score is 7.

Assumptions & free parameters 6 free parameters · 8 assumptions · 3 invented entities

The central contribution is carried almost entirely by paper-introduced assumptions: the non-Hermitian WdW postulate (Eq. 11), the model-dependent semiclassical mappings to γ, ξ_K, α (Eqs. 16, 35, 39, 45), and the implicit standard inner product (Eq. 13). The observable 'constraints' then restate those choices as smallness conditions. No external benchmark, dataset, or machine-checked step appears.

free parameters (6)
  • γ(t) — growth-sector friction/gain rate = constrained |∫γ dt| ≲ 2ε; never derived from Γ̂_H
    Introduced in Eq. (16) as 'the most conservative late-time parametrization' of the anti-Hermitian generator; the paper concedes the mapping is model dependent. All late-time quantitative content flows through this proxy.
  • ε — observational tolerance on Δσ₈/σ₈ = never set (symbolic)
    Defined after Eq. (27) as the allowed shift |Δσ₈/σ₈| ≲ ε. The headline bound |γ₀|/H₀ ≲ 2ε (Eq. 29) is exactly 2× this assumed input; no survey value is quoted.
  • ξ — source scaling in the dark-energy continuity equation = constrained |ξ| ≪ 1
    From the ansatz Q = ξHρ_DE (Eq. 30) chosen as 'minimal scaling'; the constraint 'precision distance measures require |ξ| ≪ 1' restates the observational premise.
  • ξ_K — curvature bias in Ω_K evolution = constrained |ξ_K| ≪ 1
    Phenomenological proxy in Eq. (35) for a curvature-dependent U_I; Eq. (37) says Ω_K small today implies ξ_K small — the premise restated as a conclusion.
  • α(N) = Γ(N)/H(N) — dimensionless inflationary non-Hermitian rate = constrained α ≪ 1, dα/dN ≪ 1
    Defined in Eq. (45); shifts n_s by -2α (Eq. 47) and r exponentially (Eq. 49), so the bound is set by the observed spectral indices being near scale-invariant.
  • Γ_k(τ) — mode-dependent non-Hermitian coupling in the quadratic action = unconstrained; sign flipped to −Γ for damping
    The iΓ_k anticommutator in Eq. (39) is a chosen operator ordering; Eqs. (40)-(42) with real damping follow only with a real (non-i) coupling under standard Heisenberg evolution — an apparent sign inconsistency.
assumptions (8)
  • ad hoc to paper Non-Hermitian Wheeler-DeWitt equation Ĥ_NH Ψ = 0 with Ĥ_NH = Ĥ_H + iΓ̂_H (Eqs. 10-11) is the quantum implementation of the classical constraint.
    Postulated by analogy with Eq. (2); no derivation from any underlying theory and no quantization scheme fixing the form of Γ̂_H.
  • domain assumption WKB/Born-Oppenheimer semiclassical reduction Ψ ≃ e^{iS₀/ℏ}ψ with an emergent time variable (Eq. 13).
    Standard quantum-cosmology technique; the paper relies on it to convert the constraint equation into an effective Schrödinger equation.
  • ad hoc to paper The anti-Hermitian component induces the specific observable proxies: γ(t) friction (Eq. 16), ξ_K bias (Eq. 35), α(N) damping (Eq. 43).
    Called 'the most conservative parametrization'; the paper states the mapping between γ(t) and K̂(t) is model dependent, so the proxies are assumed rather than derived.
  • domain assumption Norm evolution uses the naive inner product: d⟨ψ|ψ⟩/dt = -(2/ℏ)⟨ψ|K̂|ψ⟩ (Eq. 13).
    The η-metric restoration of unitarity discussed in the introduction is never applied; the damping/gain interpretation presumes this unflagged convention.
  • domain assumption Standard subhorizon growth equation (Eq. 14) with scale-independent factorization δ_m = D(t)δ_m(t_i) and P(k,t) = D²(t)P(k,t_i).
    Baseline for the growth sector; no scale-dependent growth or modified clustering is considered beyond the single friction proxy.
  • ad hoc to paper WdW current non-conservation dJ_a/da = (2/ℏ)U_I|Ψ|² (Eq. 34) biases probability flow toward k ≠ 0 branches.
    Interpretive step: differential amplification of curvature sectors is asserted to translate into late-time Ω_K bias through the ansatz of Eq. (35).
  • standard math Perturbation-theory approximations: linearization in ε (Eq. 20), quasi-static u̇ ≈ 0 (Eqs. 23-24), slowly-varying γ and α, |Γ_k| ≪ ω_k.
    Standard expansion assumptions, but their validity over gigayear integrations is never checked quantitatively.
  • domain assumption Higher-curvature action αR² (Eq. 50) with its scalar-tensor representation can absorb non-Hermitian effects through G_eff (Eq. 54).
    Illustrative beyond-GR discussion using standard f(R) equivalence; the claimed relaxation of bounds is qualitative, not derived.
invented entities (3)
  • Γ̂_H — anti-Hermitian constraint operator on superspace
    purpose: Parametrize departures from Hermiticity in the Wheeler-DeWitt constraint (Eq. 10).
    No falsifiable handle: the paper never relates Γ̂_H directly to any observable. The constrained proxies (γ, α, ξ_K) connect to it only through asserted, model-dependent mappings; since Γ̂_H can be tuned arbitrarily small, the 'constraint' is vacuous.
  • K̂(t) — emergent anti-Hermitian generator in the light sector
    purpose: Source/sink of norm in the semiclassical Schrödinger picture (Eq. 13); the origin of γ(t) and α(N).
    Its existence and form are inferred, not constructed, from Γ̂_H; the paper states the mapping is model dependent.
  • Γ_k anticommutator coupling in the inflationary quadratic Hamiltonian (Eq. 39)
    purpose: Produce damping of primordial fluctuations during horizon exit.
    Chosen operator ordering with an apparent i-sign mismatch against Heisenberg equations (40)-(41) under the standard rule; no independent constraint.

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Cite this review

Pith. "Pith review of Does Cosmology require Hermiticity in Quantum Mechanics?." pith.science (2026). https://pith.science/paper/SAMF5U3V

@misc{pith2026260205973,
  author       = {Pith},
  title        = {Pith review of: Does Cosmology require Hermiticity in Quantum Mechanics?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAMF5U3V}},
  note         = {Machine review of arXiv:2602.05973}
}
read the original abstract

We explore the consequences of allowing non-Hermitian structures in quantum cosmology by extending the Wheeler DeWitt framework beyond strictly Hermitian dynamics. Using a controlled semiclassical reduction, we show how anti Hermitian contributions propagate into both early universe primordial fluctuations and late-time structure growth as effective damping or gain terms. Confronting this framework with inflationary observables, growth of structure and the observed near flatness of the universe, we derive strong infrared constraints that suppress non Hermiticity across cosmic history. We demonstrate that these bounds are mutually consistent between early and late-time probes and can be partially relaxed in theories beyond General Relativity. Our results establish cosmology as a novel arena for testing foundational aspects of quantum mechanics and suggest that Hermiticity may emerge dynamically along the semiclassical branch describing our universe.

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Works this paper leans on

50 extracted references · 1 linked inside Pith

  1. [1]

    B¨ ohm,Quantum mechanics: foundations and applica- tions(Springer Science & Business Media, 2013)

    A. B¨ ohm,Quantum mechanics: foundations and applica- tions(Springer Science & Business Media, 2013)

    2013

  2. [2]

    Zettili, Quantum mechanics: concepts and applica- tions, (2009)

    N. Zettili, Quantum mechanics: concepts and applica- tions, (2009)

    2009

  3. [3]

    J. J. Sakurai and J. Napolitano,Modern quantum me- chanics(Cambridge University Press, 2020)

    2020

  4. [4]

    D. J. Griffiths and D. F. Schroeter,Introduction to quan- tum mechanics(Cambridge university press, 2018). 8

    2018

  5. [5]

    Shankar,Principles of quantum mechanics(Springer Science & Business Media, 2012)

    R. Shankar,Principles of quantum mechanics(Springer Science & Business Media, 2012)

    2012

  6. [6]

    R. J. Scherrer,Quantum mechanics: an accessible intro- duction(World Scientific, 2024)

    2024

  7. [7]

    Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

    N. Moiseyev,Non-Hermitian quantum mechanics(Cam- bridge University Press, 2011)

    2011

  8. [8]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics69, 249 (2020)

    2020

Show all 50 references
  1. [9]

    Hatano and D

    N. Hatano and D. R. Nelson, Localization transitions in non-hermitian quantum mechanics, Physical review let- ters77, 570 (1996)

    1996

  2. [10]

    Jones-Smith and H

    K. Jones-Smith and H. Mathur, Relativistic non- hermitian quantum mechanics, Physical Review D89, 125014 (2014)

    2014

  3. [11]

    Gopalakrishnan and M

    S. Gopalakrishnan and M. J. Gullans, Entanglement and purification transitions in non-hermitian quantum me- chanics, Physical review letters126, 170503 (2021)

    2021

  4. [12]

    Hatano and D

    N. Hatano and D. R. Nelson, Vortex pinning and non- hermitian quantum mechanics, Physical Review B56, 8651 (1997)

    1997

  5. [13]

    C. M. Bender, Making sense of non-hermitian hamiltoni- ans, Reports on Progress in Physics70, 947 (2007)

    2007

  6. [14]

    Longhi, Optical realization of relativistic non- hermitian quantum mechanics, Physical review letters 105, 013903 (2010)

    S. Longhi, Optical realization of relativistic non- hermitian quantum mechanics, Physical review letters 105, 013903 (2010)

    2010

  7. [15]

    K. A. Jones-Smith,Non-Hermitian quantum mechanics, Ph.D. thesis, Case Western Reserve University (2010)

    2010

  8. [16]

    Krejˇ ciˇ r ´ ık, P

    D. Krejˇ ciˇ r ´ ık, P. Siegl, M. Tater, and J. Viola, Pseu- dospectra in non-hermitian quantum mechanics, Journal of mathematical physics56(2015)

    2015

  9. [17]

    Cui and Y

    X.-D. Cui and Y. Zheng, Geometric phases in non-hermitian quantum mechanics, Physical Review A—Atomic, Molecular, and Optical Physics86, 064104 (2012)

    2012

  10. [18]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Reviews of Modern Physics93, 015005 (2021)

    2021

  11. [19]

    Gardas, S

    B. Gardas, S. Deffner, and A. Saxena, Non-hermitian quantum thermodynamics, Scientific reports6, 23408 (2016)

    2016

  12. [20]

    A. S. Matsoukas-Roubeas, F. Roccati, J. Cornelius, Z. Xu, A. Chenu, and A. del Campo, Non-hermitian hamiltonian deformations in quantum mechanics, Jour- nal of High Energy Physics2023, 1 (2023)

    2023

  13. [21]

    C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meis- ter, Faster than hermitian quantum mechanics, Physical Review Letters98, 040403 (2007)

    2007

  14. [22]

    Cao and S.-P

    K. Cao and S.-P. Kou, Statistical mechanics for non- hermitian quantum systems, Physical Review Research 5, 033196 (2023)

    2023

  15. [23]

    P. R. Giri and P. Roy, Non-hermitian quantum mechan- ics in non-commutative space, The European Physical Journal C60, 157 (2009)

    2009

  16. [24]

    C.-Y. Ju, A. Miranowicz, G.-Y. Chen, and F. Nori, Non- hermitian hamiltonians and no-go theorems in quantum information, Physical Review A100, 062118 (2019)

    2019

  17. [25]

    C.-Y. Ju, A. Miranowicz, Y.-N. Chen, G.-Y. Chen, and F. Nori, Emergent parallel transport and curvature in hermitian and non-hermitian quantum mechanics, Quan- tum8, 1277 (2024)

    2024

  18. [26]

    Bojowald, Quantum cosmology: a review, Reports on Progress in Physics78, 023901 (2015)

    M. Bojowald, Quantum cosmology: a review, Reports on Progress in Physics78, 023901 (2015)

    2015

  19. [27]

    Bojowald, Loop quantum cosmology, Living Reviews in Relativity11, 4 (2008)

    M. Bojowald, Loop quantum cosmology, Living Reviews in Relativity11, 4 (2008)

    2008

  20. [28]

    D. L. Wiltshireet al., An introduction to quantum cos- mology, Cosmology: the Physics of the Universe , 473 (1996)

    1996

  21. [29]

    Ashtekar and P

    A. Ashtekar and P. Singh, Loop quantum cosmology: a status report, Classical and Quantum Gravity28, 213001 (2011)

    2011

  22. [30]

    S. W. Hawking, Quantum cosmology., Three Hundred Years of Gravitation , 631 (1987)

    1987

  23. [31]

    Gell-Mann and J

    M. Gell-Mann and J. B. Hartle, Quantum mechanics in the light of quantum cosmology, inFoundations of Quan- tum Mechanics in the Light of New Technology: Selected Papers from the Proceedings of the First through Fourth International Symposia on Foundations of Quantum Me- chanic...

    1996

  24. [32]

    Bojowald,Quantum cosmology(Springer, 2011)

    M. Bojowald,Quantum cosmology(Springer, 2011)

    2011

  25. [33]

    Vilenkin, Predictions from quantum cosmology, Phys- ical Review Letters74, 846 (1995)

    A. Vilenkin, Predictions from quantum cosmology, Phys- ical Review Letters74, 846 (1995)

    1995

  26. [34]

    Calcagni,Classical and quantum cosmology(Springer, 2017)

    G. Calcagni,Classical and quantum cosmology(Springer, 2017)

    2017

  27. [35]

    R. L. Arnowitt, S. Deser, and C. W. Misner, Dynamical Structure and Definition of Energy in General Relativity, Phys. Rev.116, 1322 (1959)

    1959

  28. [36]

    B. S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev.160, 1113 (1967)

    1967

  29. [37]

    R. L. Arnowitt, S. Deser, and C. W. Misner, The Dynam- ics of general relativity, Gen. Rel. Grav.40, 1997 (2008), arXiv:gr-qc/0405109

    1997 arXiv

  30. [38]

    Ashtekar, Loop quantum cosmology: an overview, General Relativity and Gravitation41, 707 (2009)

    A. Ashtekar, Loop quantum cosmology: an overview, General Relativity and Gravitation41, 707 (2009)

    2009

  31. [39]

    Vilenkin, Quantum cosmology and the initial state of the universe, Physical Review D37, 888 (1988)

    A. Vilenkin, Quantum cosmology and the initial state of the universe, Physical Review D37, 888 (1988)

    1988

  32. [40]

    Vilenkin, Approaches to quantum cosmology, Physical Review D50, 2581 (1994)

    A. Vilenkin, Approaches to quantum cosmology, Physical Review D50, 2581 (1994)

    1994

  33. [41]

    Banerjee, G

    K. Banerjee, G. Calcagni, M. Martin-Benito,et al., Intro- duction to loop quantum cosmology, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 8, 016 (2012)

    2012

  34. [42]

    J. J. Halliwell, Decoherence in quantum cosmology, Phys- ical Review D39, 2912 (1989)

    1989

  35. [43]

    J. J. Halliwell, Introductory lectures on quantum cosmol- ogy, inQuantum cosmology and baby universes(World Scientific, 1991) pp. 159–243

    1991

  36. [44]

    Chataignier, C

    L. Chataignier, C. Kiefer, and P. Moniz, Observations in quantum cosmology, Classical and Quantum Gravity40, 223001 (2023)

    2023

  37. [45]

    Vilenkin, Classical and quantum cosmology of the starobinsky inflationary model, Physical Review D32, 2511 (1985)

    A. Vilenkin, Classical and quantum cosmology of the starobinsky inflationary model, Physical Review D32, 2511 (1985)

    1985

  38. [46]

    P. V. Moniz,Quantum Cosmology-The Supersymmetric Perspective-Vol. 1: Fundamentals, Vol. 803 (Springer, 2010)

    2010

  39. [47]

    Paw lowski and A

    T. Paw lowski and A. Ashtekar, Positive cosmological constant in loop quantum cosmology, Physical Review D—Particles, Fields, Gravitation, and Cosmology85, 064001 (2012)

    2012

  40. [48]

    Linde, Inflation and quantum cosmology, Physica Scripta1991, 30 (1991)

    A. Linde, Inflation and quantum cosmology, Physica Scripta1991, 30 (1991)

    1991

  41. [49]

    Anninos, C

    D. Anninos, C. Baracco, and B. M¨ uhlmann, Remarks on 2d quantum cosmology, Journal of Cosmology and Astroparticle Physics2024(10), 031

  42. [50]

    Pinto-Neto and J

    N. Pinto-Neto and J. Fabris, Quantum cosmology from the de broglie–bohm perspective, Classical and Quantum Gravity30, 143001 (2013)

    2013

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