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REVIEW 4 major objections 5 minor 45 references

Response of a uniformly accelerated Unruh-DeWitt detector in polymer quantization

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A uniformly accelerated Unruh-DeWitt detector in polymer quantization exhibits a non-thermal response governed by an intrinsic regulator ε≈2.16, in place of the infinitesimal Fock regulator.

desk verdict Direct numerical computation of polymer Unruh-DeWitt response is new and plausible, but ε≈2.16 is a post-hoc fit, not a derived regulator—needs a convergence study and a tempered claim. read the letter →

arxiv 1909.01027 v1 pith:WNVFHGUS submitted 2019-09-03 hep-th quant-ph

Gopal Sardar , Subhashish Banerjee This is my paper · ORCID
classification hep-thquant-ph
keywords polymerquantizationUnruh-DeWittdetectorUnruheffecttwo-pointfunctioniepsilonregulatorthermalspectrumloopquantumgravityMathieufunctions
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

An Unruh-DeWitt detector is a two-level quantum system that serves as a thermometer for quantum fields: when accelerated uniformly through a vacuum, its transition rate reveals whether the vacuum looks thermal. This paper studies that rate when the scalar field is quantized with polymer quantization, the method used in loop quantum gravity, instead of the usual Fock quantization. By numerically computing the field's two-point function along the detector's Rindler trajectory, the authors find that the standard "$i\epsilon$" regulator of Fock-space calculations appears automatically in polymer quantization, with a finite dimensionless value $\epsilon\approx 2.16$ rather than an infinitesimal one. Because this regulator is large, the detector's transition rate no longer follows the Planck thermal spectrum at high acceleration, so the paper concludes that polymer structure imprints a non-thermal signature on the Unruh effect. The authors also show that a spatially smeared detector behaves similarly, and that the extracted $\epsilon$ is independent of the polymer length scale.

What carries the argument

The argument runs through the dimensionless two-point function, $\tilde{G} = 4\pi^2 l_\star^2 G$, built from the mode matrix element $D_k(\Delta t) = \sum_n |b_{4n+3}|^2 e^{-i \Delta E_{4n+3}\Delta t}$. In polymer quantization the energy gaps $\Delta E_{4n+3}/|k|$ and the coefficients $b_{4n+3}$ come from the Mathieu-characteristic spectrum of the polymer harmonic oscillator, and numerically $|b_7|^2/|b_3|^2$ stays below about $0.006$ over the whole mode range, so the paper keeps only the dominant $b_3$ term and evaluates the integral with the extra regulator set to zero. Comparing this numerical $\tilde{G}$ with the Fock closed form $1/[-(\Delta t - i\epsilon)^2 + |\Delta x|^2]$ yields the fitted value $\epsilon\approx 2.16$; the fitted ansatz then supplies the transition rate that deviates from Planck. Thus the central object is the two-point function comparison itself, not a direct amplitude calculation.

What would settle it

The decisive check is to compute the polymer two-point function $\tilde{G}$ at high resolution over a wide range of $\Delta t$ with $\Delta x=0$ and over a wide range of $\Delta x$ with $\Delta t=0$, then test whether $\tilde{G}\,[-(\Delta t - i\,2.16)^2 + |\Delta x|^2]$ equals 1 within numerical error at every point; any systematic deviation falsifies the single-regulator ansatz and the resulting non-thermal spectrum.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that in polymer quantization the two-point function of a massless scalar field along a uniformly accelerated trajectory is, to numerical accuracy, the Fock-space two-point function with the standard regulator replaced by a single finite imaginary time shift: $\tilde{G} = 1/[-(\Delta t - i\,2.16)^2 + |\Delta x|^2]$, with intervals in units of the polymer scale $l_\star$. Feeding this ansatz into the detector response gives a transition rate that matches the numerically computed polymer rate and departs from the Planck distribution $R_\omega = \omega/(e^{2\pi\omega/a}-1)$ at low $\omega/a$, i.e., high acceleration. The paper takes this as evidence that the regulator is not a bookkeeping device but a generic, physical cutoff emerging from polymer quantization, with a value that does not depend on the polymer length scale. For a spatially smeared detector, the transition rate is found to be more sensitive to the detector-size parameter $\delta$ than to the regulator $\epsilon$, and for small $\delta$ and $\epsilon$ the Fock and polymer results agree.

Load-bearing premise

The result stands on the numerical fit that the polymer two-point function equals the Fock two-point function with a single imaginary time shift of 2.16; if the true polymer correlation function has any additional structure beyond that one shift, the claimed generic regulator and the non-thermal transition rate built on it would not follow.

Editorial extensions

If this is right

  • If $\epsilon\approx 2.16$ is generic, a uniformly accelerated detector in polymer vacuum will show a transition rate that deviates from the Planck spectrum at high acceleration ($a l_\star \sim 1$), with the deviation growing as acceleration increases.
  • The Unruh effect becomes, in principle, a probe of Planck-scale structure: a measured non-thermal departure at high acceleration would be a signature of polymer-type quantization, and the extracted $\epsilon$ would be a universal number independent of the polymer scale.
  • Because the regulator value does not depend on the polymer length scale, the same dimensionless $\epsilon$ should appear for any polymer-quantized massless scalar field, making the prediction robust across choices of the polymer scale.
  • For spatially smeared detectors, small detector size $\delta$ and small regulator $\epsilon$ affect the transition rate in similar ways, so experiments seeking the Unruh effect must separate finite-detector-size artifacts from genuine Planck-scale signatures.
  • Because the deviation from thermality grows as acceleration grows, testing the prediction would require accelerations comparable to the inverse polymer length scale, the Planck-scale probing window the paper identifies.

Reading between the lines

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

  • An untested consequence of the single-regulator fit is that the polymer vacuum along a Rindler trajectory is indistinguishable from a Fock vacuum with imaginary time shifted by $2.16\,l_\star$; if that is exact, no single Unruh temperature can be assigned, because the absorption-to-emission detailed-balance ratio would not be $e^{-\omega/T}$ at any fixed $T$.
  • The paper does not compute how higher $b_{4n+3}$ terms shift the fitted $\epsilon$; its own Fig. 2 shows $|b_7|^2/|b_3|^2\lesssim 0.006$ over the mode range, so a calculation including that term would test robustness with modest numerical effort.
  • If the claim is generic across field content, the same numerical comparison could be repeated for a vector or spinor field, or in lower spacetime dimensions; a change in the fitted $\epsilon$ would mean the 'generic' regulator is actually species- or dimension-dependent.
  • The paper leaves open whether $\epsilon\approx 2.16$ is tied to the point $g\approx 0.26$ where the polymer energy-gap ratio $\Delta E/|k|$ is minimal; testing whether the regulator tracks that minimum under different polymer scales would connect the detector response to the underlying oscillator spectrum.
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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 studies the response of an Unruh-DeWitt detector uniformly accelerated through the vacuum of a massless scalar field quantized with polymer (loop) quantization. For Fock quantization the authors reproduce the standard thermal Unruh response using the i-epsilon regularization. For polymer quantization they use the polymer harmonic-oscillator spectrum from Mathieu functions, truncate the mode matrix element D_k to its dominant b3 term (Eq. 49), and compute the two-point function and detector transition rate numerically. The central claim is that the polymer two-point function behaves like the Fock two-point function with a single imaginary regulator of finite value epsilon approximately 2.16, and that this finite regulator leads to a non-thermal transition rate at high accelerations. The paper also compares point-like and spatially smeared detectors.

Significance. If the central claim were established, the result would be significant: it would show that polymer quantization, a canonical quantization method related to loop quantum gravity, introduces a generic finite regulator of order the polymer scale into the Unruh effect, producing a non-thermal spectrum for Planck-scale accelerations. The paper has real strengths: it constructs the polymer two-point function numerically, checks the dominance of the b3 coefficient, compares against Fock results, and includes a smeared-detector analysis. The direct numerical transition rate in Fig. 6 is an independent computation and appears non-thermal. However, the headline claim that epsilon approximately 2.16 is a generic regulator is currently supported only by an unvalidated ansatz fitted to the same numerical data whose rate is then reproduced; this is a load-bearing gap that requires substantial additional support.

major comments (4)
  1. [Section VI A 3 (ansatz for the polymer two-point function)] The value epsilon approximately 2.16 is introduced by an ansatz, G~ = 1/[-(Delta t - i 2.16)^2 + |Delta x|^2], after inspecting the numerically computed polymer two-point function along the slices Delta t = 0 and Delta x approximately 0. No fitting procedure, goodness-of-fit, residuals, or error bars are reported. This is load-bearing because the abstract and Section VIII state that the regulator epsilon is generic in polymer quantization with a finite value approximately 2.16. The agreement between the direct polymer transition rate and the rate computed from the ansatz in Fig. 6 is expected if the ansatz is fitted to the two-point function, since the rate is a weighted integral of that same function; the agreement is therefore partly by construction. To support the claim, the authors should report quantitative residuals over the full (Delta t, |Delta x|) plane, and in particular along the Rindler trajectory where Delta t and |Delta x| vary jointly, and should test whether the residual structure is within numerical error.
  2. [Section VI A 1, Eq. (49), and Fig. 2] The truncation D_k approximately |b3|^2 exp(-i Delta E3 Delta t) is justified by the plot of |b7|^2/|b3|^2, but the amplitude ratio alone does not bound the integrated contribution of the omitted terms. The omitted terms carry phase factors exp[-i(Delta E_{4n+3}-Delta E_3) Delta t], whose arguments grow with Delta t; for Delta t = 75 and moderate g this phase difference is not negligible. A convergence test that includes the b7 term, or an explicit bound on the resulting error in G~ and in the transition rate, is needed before the numerical two-point function can be considered accurate. In addition, the numerical integrations in Eq. (52) use gmin = 10^-3 and gmax = 10^3 with no sensitivity study; the claimed accuracy of Fig. 6 depends on these choices.
  3. [Sections VI A 4 and VIII (length-scale independence)] The statement that the value epsilon approximately 2.16 does not depend on the polymer length scale l* is asserted without a scan over l* or an analytic argument. Because the computation is formulated entirely in units of l* (Eq. 51), the dimensionless number 2.16 is automatically invariant under rescaling in the presented plots. If the claim of genericity is intended physically, the authors need to show results for different polymer scales or provide a derivation showing that the physical regulator is exactly 2.16 l*.
  4. [Section VI A 4 and Eq. (22)] The transition rate is computed at finite observation time a tau = 15, but the proof of exponential decay of transients, Eq. (41), is derived only in Fock quantization. In polymer quantization the transient behavior has not been analyzed, so the non-thermal distortion visible at low omega~ in Fig. 6 could in principle receive a finite-time contribution. The authors should verify that the plotted rate is stable when a tau is increased, or otherwise quantify the transient error for the polymer computation.
minor comments (5)
  1. [Abstract and Section VI A 3] The units of epsilon approximately 2.16 should be stated explicitly; from the scaling (Eq. 51) the physical regulator is 2.16 l*, and this is not clear from the abstract.
  2. [Section VIII] There is a typographical repetition in 'it is observed that the the regulator epsilon'; also, 'the the' appears in the summary sentence of Section VIII.
  3. [Figures 4 and 5] The figures showing the two-point function have no numerical error bars or tolerance information; a short statement about the estimated numerical accuracy would help, especially because the claim epsilon approximately 2.16 is based on matching these curves.
  4. [Section VI A 3] The phrase 'for all possible spacetime intervals' overstates the numerical evidence, which covers the finite range |Delta t|, |Delta x| <= 75 in units of l*.
  5. [Section VII] The comparison between the regulator epsilon and the smearing scale delta in Figs. 7 and 8 uses only a few values; a brief explanation of why these values are representative, rather than a systematic scan, would improve the presentation.

Circularity Check

2 steps flagged · score 6.0 of 10

Partial circularity: ε≈2.16 is fitted to the polymer two-point function and then used as the explanatory regulator; the ansatz transition-rate match is by construction, though the direct polymer rate independently shows non-thermality.

  1. fitted input called prediction [Section VI A 3, 'Two-point function' (after Eq. (52), Figs. 4-5 discussion)]
    "Analyzing these properties of ˜G and comparing with the standard form obtained from Fock quantization, we may conclude that in polymer quantization, there is an imaginary constant factor associated with ∆t which comes due to the standard “iϵ” regularization in Fock quantization. In polymer quantization this ϵ≈ 2.16 whereas in Fock quantization, the limit ϵ→ 0 is taken at the end of the computation. Therefore, in analogy with the Fock space two-point function, we can make an ansatz of the polymer two-point function as ˜G = 1/[−(∆t−i 2.16)2 +|∆x|2]."

    The finite value 2.16 is not derived from the polymer Hamiltonian; it is inferred by comparing the numerically computed polymer two-point function with the Fock form, including the coincidence limit G~(0,0)≈0.21, which fixes 1/ε²≈0.21. The paper itself calls the resulting expression an 'ansatz'. The abstract then elevates this fitted constant to a claimed result: 'Numerically, we show here that the regulator ε is generic in polymer quantization ... with a finite value ε≈2.16.' This is a fitted input presented as a predicted/generic regulator, with no derivation, error bar, or goodness-of-fit. The nonlinear polymer energy gap ΔE3(g) (Eqs. 45/47) makes the assumed Lorentz-invariant single-pole form an extra assumption, not a consequence of the polymer two-point function.

  2. fitted input called prediction [Section VI A 4, 'Unruh effect' (Fig. 6 discussion)]
    "We can see that in polymer quantization there is a non-thermal transition rate (dot dashed green line) which closely matches with the transition rate of the detector using the ansatz of the polymer-two-point function (red line). Therefore, we may conclude that the large value of the regulator ϵ≈ 2.16 plays a crucial role for the non-thermal transition rate."

    The red-line rate is computed from the ansatz whose parameter ε=2.16 was fitted to reproduce the polymer two-point function used for the green-line rate. The transition rate is a functional of the two-point function (Eq. 22), so agreement between the two rates is by construction rather than independent confirmation. The causal claim that 'the large value of the regulator ε≈2.16' is responsible for non-thermality reduces to the fitted parameter; only the directly computed polymer rate (green line) is independent evidence of non-thermality.

full rationale

Most of the derivation is self-contained: the polymer two-point function is computed numerically from the polymer spectrum and coefficients (Eqs. 42-53), and the Fock thermal result is derived by contour integration (Eqs. 36-40). The direct polymer transition rate in Fig. 6 is therefore an independent numerical result and is non-thermal, so the paper's core observation is not globally circular. However, the abstract's headline claim—that polymer quantization yields a generic finite regulator ε≈2.16—is not derived: Section VI A 3 selects ε≈2.16 by matching the numerical two-point function (including G~(0,0)≈0.21) to the Fock form and explicitly introduces the resulting expression as an 'ansatz'. No derivation from the polymer Hamiltonian, no error estimate, and no out-of-sample test is given. The subsequent agreement between the ansatz-based transition rate and the direct polymer rate (Fig. 6) is automatic because the ansatz was fitted to the correlator that defines that rate. Thus the explanatory claim (ε≈2.16 causes non-thermal spectrum) is partially circular/fitted-input-as-prediction, while the raw non-thermal spectrum retains independent content. No load-bearing self-citation chain was found. Score 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim is built on the standard Unruh-DeWitt formalism and the polymer scalar-field spectrum from the literature. The paper adds two nontrivial assumptions of its own: keeping only the b3 term in the field-mode sum, and the Fock-form ansatz with a fitted finite regulator. The numerical integration cutoffs and the choice of comparison regulators are also hand-chosen.

free parameters (3)
  • epsilon_fit (polymer regulator) = 2.16 (dimensionless, in units of polymer length)
    Obtained by comparing numerically computed polymer and Fock two-point functions; presented as the main result. No error bar, no derivation from the polymer Hamiltonian. Used in the ansatz and in the transition-rate comparison. Section VI A 3.
  • Fock comparison epsilon = 0.01
    Hand-chosen small regulator used for the Fock transition rate in Figures 6-8; the paper states the physical result is the ε→0 limit, but no convergence check with this value is reported. Section VI A 4.
  • g integration limits = g_min = 10^-3, g_max = 10^3
    Numerical cutoffs for the mode integral replacing 0 and infinity in Eq. (52); no convergence study is reported. Section VI A 3.
assumptions (4)
  • domain assumption The detector response rate is determined by the Fourier transform of the Wightman function along the trajectory (Eqs. 20-24).
    Standard Unruh-DeWitt detector model in QFT; adopted without proof.
  • domain assumption Polymer quantization of the scalar field is represented by the Mathieu-function energy spectrum and the superselected states of Refs. [35] and [37] (Eqs. 42-48).
    Borrowed from prior literature; the paper does not re-derive the polymer spectrum.
  • ad hoc to paper D_k is approximated by the single term |b3|^2 exp(-i ΔE3 Δt), neglecting all higher coefficients (Eq. 49).
    Justified only by a numerical plot of |b7/b3|^2; no full sum is evaluated and no sensitivity test is shown for the integrated two-point function.
  • ad hoc to paper The polymer two-point function has exactly the Fock form with a single imaginary regulator ε≈2.16 (ansatz in Section VI A 3).
    Introduced after inspecting the numerical data; not derived, and the fitted value has no uncertainty.

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Cite this review

Pith. "Pith review of Response of a uniformly accelerated Unruh-DeWitt detector in polymer quantization." pith.science (2026). https://pith.science/paper/WNVFHGUS

@misc{pith2026190901027,
  author       = {Pith},
  title        = {Pith review of: Response of a uniformly accelerated Unruh-DeWitt detector in polymer quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNVFHGUS}},
  note         = {Machine review of arXiv:1909.01027}
}
abstract

If an Unruh-DeWitt detector moves with a uniform acceleration in Fock-space vacuum, then the transition rate of the detector is proportional to the thermal spectrum. It is well known that the transition rate of the detector crucially depends on the two-point function along the detectors trajectory and in order to compute it the standard "$i \epsilon$" regularization is used for Fock space. Numerically, we show here that the regulator $\epsilon$ is generic in polymer quantization, the quantization method used in \emph{loop quantum gravity} with a finite value $\epsilon \approx 2.16$, which leads to non-thermal spectrum for the uniformly accelerated detector. We also discuss the response of a spatially smeared detector.

Figures

Figures reproduced from arXiv: 1909.01027 by the authors.

Figure 1
Figure 1. FIG. 1: Contour used to evaluate [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The solid red line represents numerically evaluated [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: In Fock quantization, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Figure depicts the scaled two-point function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The sub-fig (a) represents the real-part and (b) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The transition rate of the Unruh-DeWitt detector [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Transition rate of the Unruh-DeWitt detector along [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Transition Rate of the Unruh-DeWitt detector along [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reference graph

Works this paper leans on

45 extracted references · 28 canonical work pages

  1. [1]

    equation (31)

    Matrix element Dk(∆t) The two-point function is completely understood from the matrix element Dk(∆t), cf. equation (31). In poly- mer quantization, the matrix element (29) can be ex- pressed as Dpoly k (∆t) = ∑ n |b4n+3|2e−i∆E4n+3∆t, (50) = |b3|2e−i∆E3∆t [ 1 +|b7|2 |b3|2e−i(∆E7−∆E3)∆t +... ] . In Fock quantization, only the first term is non-vanishing. How...

    work page

  2. [2]

    For the purpose of comparison, we de- note the coefficientb3 asbk and the corresponding energy gap ∆E3 as ∆Ek also for polymer quantization

    Coefficient bk and energy gap ∆Ek As discussed earlier, there is only one non-vanishing coefficient b1 in Fock quantization which can be ex- pressed in terms of dimensionless parameter g as|bk|2≡ |b1|2 = 1 2(l⋆/g). For the purpose of comparison, we de- note the coefficientb3 asbk and the corresponding energy gap ∆E3 as ∆Ek also for polymer quantization. Fig- ur...

    work page

  3. [3]

    Two-point function In order to facilitate the numerical computation, we scale the two-point function as G(∆t, ∆x) = 1 4π2l2⋆ ˜G , (51) where ˜G is dimensionless. Taking into account the stan- dard regulator ϵ, and with the help of equations (30) and (31), the dimensionless two-point function can be expressed as ˜Gϵ = ∫ gmax gmin dgT (g,|∆x|) e−ip(g)−ϵg, (...

    work page

  4. [4]

    iϵ” REGULARIZATION Up to this point our study of the Unruh effect involves a point-like detector. In order to regularize the two-point function, the standard “ iϵ

    Unruh effect In order to numerically compute the transition rate of the Unruh-DeWitt detector along the Rindler trajectory Rω(τ, 0) (Eq. 22), we have taken aτ = 15 and a = 1 in the units of l−1 ⋆ and the regulator is ϵ = 0 for poly- mer and ϵ = 0.01 for Fock quantization. The Fig. 6 exhibits the transition rate of the detector with a scaling ˜R with respec...

    work page

  5. [5]

    Takagi, Progress of Theoretical Physics Supplement 88, 1 (1986)

    S. Takagi, Progress of Theoretical Physics Supplement 88, 1 (1986)

    1986

  6. [6]

    S. A. Fulling, Phys.Rev. D7, 2850 (1973)

    work page 1973

  7. [7]

    Unruh, Phys.Rev

    W. Unruh, Phys.Rev. D14, 870 (1976)

    work page 1976

  8. [8]

    L. C. Crispino, A. Higuchi, and G. E. Matsas, Rev.Mod.Phys. 80, 787 (2008), arXiv:0710.5373

    arXiv 2008

Show all 45 references
  1. [9]

    De Bi` evre and M

    S. De Bi` evre and M. Merkli, Classical and Quantum Gravity 23, 6525 (2006)

    2006

  2. [10]

    Omkar, S

    S. Omkar, S. Banerjee, R. Srikanth, and A. K. Alok, 9 0 0.12 0.24 0.36 1.5 3 4.5 Transition rate R~ ω~ δ = 0.01 ε = 0.01 without regulator FIG. 8: Transition Rate of the Unruh-DeWitt detector along the Rindler trajectory in polymer quantization, where al⋆ =

  3. [11]

    Longhi and R

    P. Longhi and R. Soldati, Phys.Rev. D83, 107701 (2011), arXiv:1101.5976

    2011 arXiv

  4. [12]

    P. C. W. Davies, J. Phys. A8, 609 (1975)

    1975

  5. [13]

    N. D. Birrell and P. C. W. Davies, Quantum fields in curved space, 7 (Cambridge university press, 1984)

    1984

  6. [14]

    Banerjee, A

    S. Banerjee, A. K. Alok, and S. Omkar, Eur. Phys. J. C76, 437 (2016), arXiv:1511.03029

    2016 arXiv

  7. [15]

    Chiou, Phys

    D.-W. Chiou, Phys. Rev. D97, 124028 (2018), arXiv:1605.06656

    2018 arXiv

  8. [16]

    The solid red, dashed blue and dot dashed green lines represent the results obtained from polymer quantization with the detector size and ϵ regulators δ = 0.01, ϵ = 0; δ = 0, ϵ = 0.01 and δ =ϵ = 0, respectively. Quant. Inf. Comput. 16, 0757 (2016), arXiv:1408.1477

    2016 arXiv

  9. [17]

    Banerjee, A

    S. Banerjee, A. K. Alok, S. Omkar, and R. Srikanth, JHEP 02, 082 (2017), arXiv:1603.05450

    2017 arXiv

  10. [18]

    Nicolini and M

    P. Nicolini and M. Rinaldi, Phys.Lett. B695, 303 (2011), arXiv:0910.2860

    2011 arXiv

  11. [19]

    Padmanabhan, Rept.Prog.Phys

    T. Padmanabhan, Rept.Prog.Phys. 73, 046901 (2010), arXiv:0911.5004

    2010 arXiv

  12. [20]

    Agullo, J

    I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Phys.Rev. D77, 124032 (2008), arXiv:0804.0513

    2008 arXiv

  13. [21]

    W. G. Unruh and R. M. Wald, Phys. Rev. D29, 1047 (1984)

    1984

  14. [22]

    Alkofer, G

    N. Alkofer, G. D’Odorico, F. Saueressig, and F. Verstee- gen, Phys. Rev. D94, 104055 (2016), arXiv:1605.08015

    2016 arXiv

  15. [23]

    B. S. DeWitt, in General Relativity: An Einstein Cente- nary Survey (1980), pp. 680–745

    1980

  16. [24]

    K. J. Hinton, Journal of Physics A: Mathematical and General 16, 1937 (1983)

    1983

  17. [25]

    Schlicht, Class

    S. Schlicht, Class. Quant. Grav. 21, 4647 (2004), gr- qc/0306022

    2004

  18. [26]

    Louko and A

    J. Louko and A. Satz, Class. Quant. Grav. 23, 6321 (2006), gr-qc/0606067

    2006 arXiv

  19. [27]

    Langlois, Annals Phys

    P. Langlois, Annals Phys. 321, 2027 (2006), gr- qc/0510049

    2006

  20. [28]

    Louko, JHEP 09, 142 (2014), arXiv:1407.6299

    J. Louko, JHEP 09, 142 (2014), arXiv:1407.6299

    2014 arXiv

  21. [29]

    Sriramkumar and T

    L. Sriramkumar and T. Padmanabhan, Class. Quant. Grav. 13, 2061 (1996), gr-qc/9408037

    1996 arXiv

  22. [30]

    Agullo, J

    I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, New J. Phys. 12, 095017 (2010), arXiv:1010.4004

    2010 arXiv

  23. [31]

    C. J. Fewster, B. A. Jurez-Aubry, and J. Louko, in 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astro- physics, and Relativistic Field Theories (MG14) Rome, Italy, July 12-18, 2015 (2015), arXiv:1511.00701

    2015 arXiv

  24. [32]

    Satz, Class

    A. Satz, Class. Quant. Grav. 24, 1719 (2007), gr- qc/0611067

    2007

  25. [33]

    Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007), ISBN 9781139467599

    T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007), ISBN 9781139467599

    2007

  26. [34]

    Hmmer, E

    D. Hmmer, E. Martin-Martinez, and A. Kempf, Phys. Rev. D93, 024019 (2016), arXiv:1506.02046

    2016 arXiv

  27. [35]

    Ashtekar, S

    A. Ashtekar, S. Fairhurst, and J. L. Willis, Class.Quant.Grav. 20, 1031 (2003), gr-qc/0207106

    2003 arXiv

  28. [36]

    Superselection rules are invoked to arrive at these π-periodic and π-antiperiodic states in v [37]

    cen and sen, respectively, the energy eigenstates can be expressed as ψ2n(v) = ce n(1/4g2,v )/√π, (43) ψ2n+1(v) = se n+1(1/4g2,v )/√π, wherev =πk √l⋆ +π/2. Superselection rules are invoked to arrive at these π-periodic and π-antiperiodic states in v [37]. 5 For low-energy mode...

  29. [37]

    Halvorson, Studies in history and philosophy of mod- ern physics 35, 45 (2004)

    H. Halvorson, Studies in history and philosophy of mod- ern physics 35, 45 (2004)

    2004

  30. [38]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski, Class.Quant.Grav. 21, R53 (2004), gr-qc/0404018

    2004 arXiv

  31. [39]

    Rovelli, Quantum Gravity , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004), ISBN 9780521837330

    C. Rovelli, Quantum Gravity , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004), ISBN 9780521837330

    2004

  32. [40]

    Rindler, Am.J.Phys

    W. Rindler, Am.J.Phys. 34, 1174 (1966)

    1966

  33. [41]

    G. M. Hossain, V. Husain, and S. S. Seahra, Phys.Rev. D82, 124032 (2010), arXiv:1007.5500

    2010 arXiv

  34. [42]

    Abramowitz and I

    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Applied mathematics series (Dover Publications, 1964), ISBN 9780486612720

    1964

  35. [43]

    J. F. Barbero G., J. Prieto, and E. J. Villaseor, Class.Quant.Grav. 30, 165011 (2013), arXiv:1305.5406

    2013 arXiv

  36. [44]

    P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Rev. Mod. Phys. 84, 1 (2012), arXiv:1103.0835

    2012 arXiv

  37. [45]

    Aspachs, G

    M. Aspachs, G. Adesso, and I. Fuentes, Phys. Rev. Lett. 105, 151301 (2010), arXiv:1007.0389

    2010 arXiv

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