Signatures of noncommutativity in bar detectors of gravitational waves

Anirban Saha; Sukanta Bhattacharyya; Sunandan Gangopadhyay

REVIEW 3 major objections 5 minor 14 references

Signatures of noncommutativity in bar detectors of gravitational waves

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Keeping gravitational-wave terms to second order in the detector Hamiltonian reveals a new quantum transition that appears for linearly polarized waves and disappears for circularly polarized waves.

desk verdict New O(h^2) transition claim in an NC bar-detector model is undone by a dimensional error and a missing same-order two-step contribution; the polarization selection rule is amusing but unverified. read the letter →

arxiv 1908.06030 v1 pith:V23GZ3HU submitted 2019-08-15 gr-qc quant-ph

Sunandan Gangopadhyay , Sukanta Bhattacharyya , Anirban Saha This is my paper · ORCID

classification gr-qcquant-ph MSC 81Q0583C3581S10 PACS 04.30.-w04.80.Nn

keywords noncommutativespacetimegravitationalwavedetectorbartransitionprobabilityharmonicoscillatorsecond-orderperturbationpolarizationlengthscale

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 claims that a resonant bar detector, modeled as a two-dimensional harmonic oscillator in a noncommutative space, gains an additional gravitational-wave-induced transition when the interaction Hamiltonian is kept to second order in the wave amplitude h. The transition runs from the ground state to one of the perturbed second excited states and was absent in first-order calculations. It is present for linearly polarized gravitational waves and vanishes for circularly polarized waves. If correct, this transition offers a concrete signature of spatial noncommutativity and a way to distinguish wave polarizations.

What carries the argument

The machinery is the time-dependent interaction Hamiltonian V(t) built from the noncommutative harmonic oscillator, containing first-order terms in the wave amplitude and an explicit O($h^{2}$) term bilinear in the time derivatives of the metric perturbation. First-order time-dependent perturbation theory (Eq. 13) is applied to V(t), and matrix elements between the ground state and the three perturbed second excited states are computed. The noncommutative parameter enters through the dimensionless Lambda = m omega $\theta$ / (2 hbar), which shifts the resonance frequencies and weights the polarization-dependent amplitudes A, B, C, D.

What would settle it

Compute the second-order Dyson series for the O(h) part of the interaction and check whether iterated transitions (ground -> first excited -> $psi_2^{{(0)}}$) produce an O($h^{4}$) amplitude; if they do, the claimed transition probability is not uniquely due to the O($h^{2}$) term. Experimentally, a resonant bar detector seeing the omega-frequency resonance for a circularly polarized gravitational wave would contradict the paper's central polarization-dependent prediction.

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Extended reading notes

Core claim

The central claim is that the O($h^{2}$) terms in the interaction Hamiltonian of a noncommutative harmonic oscillator coupled to a gravitational wave produce a transition probability P_{0->2}^{(0)} from the ground state |0,0> to the perturbed second excited state $psi_2^{{(0)}}$, with a resonance at the oscillator frequency. The two other second-excited states receive no O($h^{2}$) contribution, so the first-order results are left unchanged. For linearly polarized periodic waves the new probability is nonzero and proportional to $h^{4}$, while for circularly polarized waves it is exactly zero. The paper thus identifies a purely gravity-induced effect that can simultaneously probe spatial noncommutativity and the polarization of the source.

Load-bearing premise

The calculation treats the full Hamiltonian, including its O($h^{2}$) part, as a single first-order perturbation in time-dependent perturbation theory, so any two-step processes built from two O(h) interactions are assumed to be negligible at the same order in $h^{4}$.

Editorial extensions

If this is right

Bar detectors could show a resonance at the oscillator frequency, in addition to the known resonances at 2 omega_+ and 2 omega_-, when irradiated by linearly polarized gravitational waves.
The absence of this new resonance for circularly polarized waves gives a direct polarization-discrimination test that does not rely on interferometric phase measurements.
The measured size of the new transition probability, if observed, would provide a quantitative handle on the noncommutative length scale sqrt(theta), estimated here to be of order 10^-20 m.
The O(h^2) transition survives for Gaussian burst waveforms and modulated Gaussian bursts, so the effect is not limited to monochromatic sources.
Because the new transition is purely gravity induced, it would distinguish genuine gravitational-wave excitation from local mechanical or thermal noise peaks at the same frequency.

Reading between the lines

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

A natural next check is whether iterated first-order interactions, where the oscillator passes through an intermediate first excited state, produce an O(h^4) amplitude into psi_2^{(0)}; if they do, the claimed isolation of the direct O(h^2) term would need revision.
The polarization dependence might be turned into a null test: a bar detector that sees the omega-resonance for circularly polarized waves would falsify the noncommutative prediction, whereas seeing it for linear waves would support it.
If the noncommutative scale is as large as the upper bound used here, resonant bar detectors operating at kHz frequencies may be competitive with tabletop noncommutativity searches, which typically probe much smaller length scales.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: the paper contains a real new observation, but the calculation supporting it is incomplete at the stated order, and the headline rate in Eq. (15) has a dimensional error on top. I would reject as written, though I would still send it to a referee rather than desk reject.

What is actually new: this extends the authors' earlier first-order calculation [42] to O(h^2) in the noncommutative harmonic-oscillator model of a bar detector. The claimed additional transition from the ground state to psi^(0)_2, and the difference between linear and circular polarization for that transition, are not in [42]. That selection rule is a clean algebraic fact and would be worth a footnote somewhere if it survives.

What the paper does well: the model setup is standard and follows their series; the perturbed states in Eq. (11) come from their previous work; and the authors are honest that this is an extension of their own program. The heavy self-citation is not by itself a problem for an incremental project.

Soft spots: (1) Eq. (15) does not match the Hamiltonian. The O(h^2) term in Eq. (14) is (hbar/4\omega) hdot^2 a^\dagger a^\dagger; the matrix element to the (unnormalized) psi^(0)_2 gives a prefactor like -i sqrt2/(4\omega), not -i\omega/(hbar sqrt2). The phase factor also has 2i\omega t/hbar where it should be 2i\omega t. So the advertised rate P0->2^(0) in Eq. (19) is not a trustworthy number.

(2) The bigger issue: they apply first-order time-dependent perturbation theory to the full V(t) and keep the O(h^2) piece of V. That misses the iterated O(h) path, ground -> psi1/psi2 -> psi0, which contributes to the same amplitude at O(h^2). Since the O(h) Hamiltonian couples the ground state to the other second excited states, and those states couple back to psi0 through the aa terms, there is no obvious symmetry killing that two-step term. The paper neither computes it nor argues it away, so the central claim is not established.

(3) The abstract's "strong possibility" of detecting noncommutativity is unsupported: even if the formula were right, P0->2^(0) scales as f0^4, i.e. h^4, and there is no SNR or rate estimate anywhere. The Lambda ~ 1.888 estimate is taken from their own earlier papers.

Useful for: people working in NC quantum mechanics and toy GW detector models, not for the general gr-qc reader. A competent referee could catch the missing Dyson term and the dimensional error, so the paper deserves a serious review rather than a desk rejection. My recommendation is reject, but the polarization observation, if repaired, could become a minor result in this literature.

Referee Report

3 major / 5 minor

Summary. The paper studies a two-dimensional harmonic oscillator as a model of a resonant bar gravitational-wave detector in the framework of noncommutative quantum mechanics. The authors extend their earlier first-order calculation in the gravitational-wave amplitude h by keeping terms up to O(h^2) in the interaction Hamiltonian. Their main new result is a transition amplitude C0->2(0) between the ground state and the perturbed second excited state psi_2^(0), induced by the O(h^2) terms (Eq. (15)). From this they compute transition probabilities for linearly polarized periodic gravitational waves (Eq. (19)), for circularly polarized waves (Eq. (24)), and for Gaussian burst waveforms (Eqs. (28)-(31)), and find that the new transition vanishes for circular polarization. They also estimate the noncommutative parameter Lambda ~ 1.9 for realistic bar parameters (Eq. (22)).

Significance. If the calculation were correct, the paper would provide a concrete, falsifiable prediction for how spatial noncommutativity affects bar-detector responses, including a polarization selection rule that could distinguish linear from circular gravitational-wave polarization. The numerical estimate that Lambda is of order unity for realistic bar detectors is striking and would motivate experimental searches. The paper also gives explicit formulas for several waveform templates. However, the central amplitude is dimensionally inconsistent, and the calculation omits second-order time-dependent perturbation theory contributions that are of the same order in h; therefore the quantitative predictions and the polarization selection rule are not currently established. These issues could in principle be addressed in a revision, but as it stands the paper's main claims are unsupported.

major comments (3)

[Section II, Eq. (15)] The amplitude C0->2(0) has the prefactor -i omega/(hbar sqrt(2)) multiplied by the integral of (h_dot_11^2 + h_dot_12^2). Since h_dot has dimension 1/T, the integral has dimension 1/T, while omega/hbar has dimension 1/(energy*time), so the product is not dimensionless. This dimensional inconsistency propagates into the transition probability P0->2(0) in Eq. (19) and the rates in Eq. (20), which also fail to be dimensionless. The formula must be rederived, and all derived probabilities and rates must be rechecked dimensionally before any quantitative conclusion can be drawn.
[Section II, Eqs. (13)-(14)] The transition amplitude C0->2(0) is computed by applying first-order time-dependent perturbation theory, Eq. (13), to the full interaction Hamiltonian V(t) = H1(t) that already contains O(h^2) terms. This gives only the direct matrix element of the O(h^2) term. It omits the second-order Dyson-series contribution from two O(h) interactions, which is also O(h^2) in amplitude (O(h^4) in probability). Because the O(h) part of H1 couples the ground state to psi_2^(1) and psi_2^(2), and those states couple back to psi_2^(0), there is no obvious symmetry that makes this two-step amplitude vanish. The paper neither computes this term nor argues for its absence, so the numerical value of P0->2(0) and the linear-versus-circular polarization distinction are not established.
[Section II, Eq. (15)] The phase factors in the transition amplitudes are written as e^{2i omega t / hbar} rather than e^{2i omega t}. With the exponent written with hbar in the denominator, the phase is dimensionful and inconsistent with Eq. (13), where the exponent is (i/hbar)(E_f - E_i)t = 2i omega t. This reinforces that the central formula has not been carefully checked.

minor comments (5)

[Section III, Eq. (20)] The first line of Eq. (20) is labeled as the rate for P0->2(1) but should be the rate for P0->2(0); these are distinct transitions.
[Section II, Eq. (11)] The perturbed states psi_2^(a) are written without explicit normalization factors; please specify the normalization.
[Section III, Eq. (22)] The numerical estimate Lambda = 1.888 treats omega as a frequency in kHz, but the text should clarify whether omega is the angular frequency or the cyclic frequency, since this changes the value.
[Section III, Eqs. (28) and (31)] The expressions for P0->2(0) contain factors such as omega^6 tau_g^4 that are not dimensionless; these are not probabilities as written, even after accounting for the delta-function regularization used elsewhere.
[Throughout] The notation epsilon+ and epsilon* in Eq. (3) is not consistently defined as functions of time; in Eq. (18) they are constants, while in Eq. (23) they become time-dependent. Please clarify the definitions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed second-order transition is obtained by direct matrix-element evaluation from the stated Hamiltonian, and the cited prior results are algebraic or numerical inputs rather than fitted to the prediction.

full rationale

The paper's new claim, the P0->2(0) transition induced by the O(h^2) terms in the interaction Hamiltonian, is obtained by direct substitution of the Hamiltonian from Eqs. (8) and (14) into the standard first-order time-dependent perturbation amplitude in Eq. (13). No parameter is fitted to the target probability, and no part of the target result is used to define the Hamiltonian. The perturbed states in Eq. (11), although cited to the authors' earlier paper [42], are fixed by the time-independent perturbation H2 in Eq. (10) and are parameter-free algebraic eigenstates; the citation is therefore not load-bearing in a circular sense. The numerical estimate Lambda = 1.888 in Eq. (22) is an arithmetic evaluation using an external bound on theta and reference bar-detector parameters, not a quantity fitted to the new transition probability; moreover, the new P0->2(0) itself does not depend on Lambda. The linear-versus-circular polarization distinction follows algebraically from the waveform: for circular polarization, hdot_11^2 + hdot_12^2 is constant, so its Fourier component at 2omega vanishes for omega > 0, forcing P0->2(0) = 0 in Eq. (24). The concern that iterated O(h) Dyson terms at the same order in h^4 are omitted is a correctness/completeness issue about the perturbation expansion, not a circularity, because the paper explicitly states that it uses the lowest-order amplitude in Eq. (13) and does not hide that approximation. Accordingly, no circular step can be exhibited under the required standard.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The computation imports the NCQM framework, the detector model, and the perturbed states from the authors' earlier papers, and the only numerical input is the NC parameter bound. The new transition is a direct matrix element of the O(h^2) term, but the chosen perturbative scheme leaves out iterated O(h) processes at the same order.

free parameters (1)

noncommutative parameter theta = |theta| approximately 4x10^-40 m^2 (upper bound from ref. 16)
Used in Eq. (22) to estimate Lambda approximately 1.888; the noncommutative signatures vanish if theta is much smaller.

assumptions (5)

domain assumption Noncommutative Heisenberg algebra [x_i,x_j]=i theta epsilon_ij and the Seiberg-Witten map (Eqs. 1, 6)
Defines the model of spatial noncommutativity and connects NC operators to commutative ones.
domain assumption TT-gauge geodesic deviation Hamiltonian for a 2D harmonic oscillator (Eqs. 2-5)
Models the bar detector phonon mode interacting with gravitational waves.
domain assumption Perturbed second excited states and energies quoted from ref. [42] (Eqs. 11-12)
The basis for the final states; no derivation is repeated in this paper.
ad hoc to paper First-order perturbation theory with V(t)=H1(t) captures the O(h^2)/O(h^4) transitions (Eqs. 13-15)
No justification is given for neglecting iterated O(h) terms that contribute at the same order in probability.
standard math delta(omega) = integral dt e^{i omega t} = T (Eq. 21)
Used to convert squared delta functions to transition rates; this is a finite-time regularization, not an exact identity.

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Cite this review

Pith. "Pith review of Signatures of noncommutativity in bar detectors of gravitational waves." pith.science (2026). https://pith.science/paper/V23GZ3HU

@misc{pith2026190806030,
  author       = {Pith},
  title        = {Pith review of: Signatures of noncommutativity in bar detectors of gravitational waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V23GZ3HU}},
  note         = {Machine review of arXiv:1908.06030}
}

abstract

The comparison between the noncommutative length scale $\sqrt{\theta}$ and the length variation $\delta L=h L$, detected in the GW detectors indicate that there is a strong possibility to detect the noncommutative structure of space in the GW detector set up. We therefore explore how the response of a bar detector gets affected due to the presence of noncommutative structure of space keeping terms upto second order in the gravitational wave perturbation ($h$) in the Hamiltonian. Interestingly, the second order term in $h$ shows a transition between the ground state and one of the perturbed second excited states that was absent when the calculation was restricted only to first order in $h$.

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Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

arXiv:1908.06030v1 [gr-qc] 15 Aug 2019 Signatures of noncommutativity in bar detectors of gravita tional waves Sunandan Gangopadhyay a, Sukanta Bhattacharyya b , Anirban Saha b a Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India b Department of Physics, West Bengal Stat...

work page Pith review arXiv 1908
[2]

(32) IV. CONCLUSION In this paper, we have extended our earlier calculations of the probabilities of transitions between the energy levels of a harmonic oscillator induced by gravitational waves in a spatial noncommutative framework including terms upto second order in h. We ﬁnd that apart from the usual transition 42, there is an additional transition be...

work page 1994
[28]

Quantum Mechanics: Funda- mentals

40 A. Saha, S. Gangopadhyay. Class. Quantum Grav. 33 (2016). 41 A. Saha, S. Gangopadhyay, S. Saha, Phys. Rev. D. 97 (2018) 044015. 42 S. Bhattacharyya, S. Gangopadhyay, A. Saha, Class. Quant. Grav. 36 (2019) 5, 055006. 43 K. Gottfried, T.M. Yan, “Quantum Mechanics: Funda- mentals”

work page 2016
[181]

Mocioiu, M

15 I. Mocioiu, M. Pospelov, and R. Roiban, Phys. Lett. B 489, 390 (2000). 16 S.M. Carroll, J.A. Harvey, V.A. Kosteleck´ y, C.D. Lane, T. Okamoto, Phys. Rev. Lett. 87 (2001). 141601. 17 O. Bertolami, J. G. Rosa, C. M. L. de Aragao, P. Casto- rina, D. Zappala, Phys. Rev. D 72 (2005) 025010. 18 A. Saha, Eur. Phys. J. C 51 199 (2007); ibid Phys. Rev. D 81 125...

work page 2000
[231]

Cerdonio et al , Class

30 M. Cerdonio et al , Class. Quant. Grav. 14 (1997)

work page 1997
[267]

Star Operation in Quantum Mechanics

6 L. Mezincescu, “Star operation in quantum mechanics”, [hep-th/0007046]. 7 B. Chakraborty, S. Gangopadhyay, A. Saha, Phys. Rev. D 70 (2004) 107707. 8 F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, A.G. Hazra, Phys. Rev. D 71 (2005) 085005. 9 F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, J. Gov- aerts, J. Phys. A 38 (2005)

work page Pith review arXiv 2004
[362]

Mauceli et al , Phys

27 E. Mauceli et al , Phys. Rev. D 54 (1996)

work page 1996
[977]

Seiberg, E

4 N. Seiberg, E. Witten, [JHEP 09 (1999) 032] 5 V.P. Nair, A.P. Polychronakos, Phys. Lett. B 505 (2001)

work page 1999

Show all 14 references

[1264]

28 D. G. Blair et al , Phys. Rev. Lett. 74 (1995)

1995
[1491]

Abrampvici et al , Science 256 (1992) 325; http://www.ligo.org

31 A. Abrampvici et al , Science 256 (1992) 325; http://www.ligo.org. 32 B. Caron et al , Class. Quant. Grav. 14 (1997) 1461; http://www.virgo.infn.it. 33 H. L¨ uck et al , Class. Quant. Grav. 14 (1997) 1471; http://www.geo600.uni-hannover.de. 34 M. Ando et al , Phys. Rev.Lett...

1992
[1908]

Astone et al , Astroparticle Physics 7 (1997)

29 P. Astone et al , Astroparticle Physics 7 (1997)

1997
[2008]

36 A. Saha, S. Gangopadhyay, Phys. Lett. B 681 , 96 (2009). 37 A. Saha, S. Gangopadhyay, S. Saha, Phys. Rev. D 83 (2011) 025004. 38 S. Gangopadhyay, A. Saha, S. Saha, Mod. Phys. Lett. A 28 (2013) 1350161. 39 S. Gangopadhyay, A. Saha, S. Saha, Gen.Rel.Grav. 47 (2015)

2009
[2010]

Astone et al , Phys

26 P. Astone et al , Phys. Rev. D 47 (2)(1993)

1993
[9849]

Gangopadhyay, F.G

10 S. Gangopadhyay, F.G. Scholtz, Phys. Rev. Lett. 102 (2009) 241602. 11 S. Bhattacharyya, S. Gangopadhyay, A. Saha, Euro. Phys. Lett. 120 (2017) no.3, 30005. 12 see for example R. J. Szabo, Class.Quant.Grav. 23 R199- R242, 2006 and the references therein. 13 P. Mukherjee, A. ...

2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

arXiv:1908.06030v1 [gr-qc] 15 Aug 2019 Signatures of noncommutativity in bar detectors of gravita tional waves Sunandan Gangopadhyay a, Sukanta Bhattacharyya b , Anirban Saha b a Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India b Department of Physics, West Bengal Stat...

work page Pith review arXiv 1908

[2] [2]

(32) IV. CONCLUSION In this paper, we have extended our earlier calculations of the probabilities of transitions between the energy levels of a harmonic oscillator induced by gravitational waves in a spatial noncommutative framework including terms upto second order in h. We ﬁnd that apart from the usual transition 42, there is an additional transition be...

work page 1994

[3] [28]

Quantum Mechanics: Funda- mentals

40 A. Saha, S. Gangopadhyay. Class. Quantum Grav. 33 (2016). 41 A. Saha, S. Gangopadhyay, S. Saha, Phys. Rev. D. 97 (2018) 044015. 42 S. Bhattacharyya, S. Gangopadhyay, A. Saha, Class. Quant. Grav. 36 (2019) 5, 055006. 43 K. Gottfried, T.M. Yan, “Quantum Mechanics: Funda- mentals”

work page 2016

[4] [181]

Mocioiu, M

15 I. Mocioiu, M. Pospelov, and R. Roiban, Phys. Lett. B 489, 390 (2000). 16 S.M. Carroll, J.A. Harvey, V.A. Kosteleck´ y, C.D. Lane, T. Okamoto, Phys. Rev. Lett. 87 (2001). 141601. 17 O. Bertolami, J. G. Rosa, C. M. L. de Aragao, P. Casto- rina, D. Zappala, Phys. Rev. D 72 (2005) 025010. 18 A. Saha, Eur. Phys. J. C 51 199 (2007); ibid Phys. Rev. D 81 125...

work page 2000

[5] [231]

Cerdonio et al , Class

30 M. Cerdonio et al , Class. Quant. Grav. 14 (1997)

work page 1997

[6] [267]

Star Operation in Quantum Mechanics

6 L. Mezincescu, “Star operation in quantum mechanics”, [hep-th/0007046]. 7 B. Chakraborty, S. Gangopadhyay, A. Saha, Phys. Rev. D 70 (2004) 107707. 8 F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, A.G. Hazra, Phys. Rev. D 71 (2005) 085005. 9 F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, J. Gov- aerts, J. Phys. A 38 (2005)

work page Pith review arXiv 2004

[7] [362]

Mauceli et al , Phys

27 E. Mauceli et al , Phys. Rev. D 54 (1996)

work page 1996

[8] [977]

Seiberg, E

4 N. Seiberg, E. Witten, [JHEP 09 (1999) 032] 5 V.P. Nair, A.P. Polychronakos, Phys. Lett. B 505 (2001)

work page 1999