Master equation for systems interacting with linearized gravity

Angelo Bassi; Anirudh Gundhi; Oliviero Angeli

Master equation for systems interacting with linearized gravity

T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two masses in a thermal graviton bath obey a master equation whose noise decoheres proper-distance superpositions and whose dissipation reproduces classical gravitational-wave emission.

desk verdict The paper is solid; the supposedly hand-picked unitary transformation is effectively unique at O(G), and the classical bremsstrahlung match is a genuine external check. read the letter →

arxiv 2608.06121 v1 pith:LD2U7QOG submitted 2026-08-06 quant-ph gr-qc

Oliviero Angeli , Anirudh Gundhi , Angelo Bassi This is my paper · ORCID

classification quant-phgr-qc PACS 03.65.Yz04.30.-w

keywords gravitationaldecoherencelinearizedgravityopenquantumsystemsmasterequationFerminormalcoordinatesproperdistancewavesBrownianmotion

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 derives the open quantum master equation for two masses interacting with a thermal bath of linearized gravitational waves, working in variables that refer directly to the measurable proper distance between the masses. It shows that the canonical momentum produced by the textbook Lagrangian in Fermi normal coordinates secretly depends on the gravitational field that is later traced away, so the reduced dynamics expressed in those variables would not be physically meaningful. The fix is a unitary transformation that moves the environment-dependent part of the system momentum into the gravitational field, yielding a clean system–environment split. To leading order in Newton's constant, the resulting master equation has a dissipative sector that reproduces the classical power loss from gravitational bremsstrahlung and a noise sector that destroys coherence between superpositions of different proper distances at a rate set by the potential-energy difference relative to the Planck energy. When the two masses sit at a fixed baseline $l_0$ with small quantum fluctuations, the equation reduces to a quantum-Brownian-motion-type master equation whose decoherence rate grows with the baseline length.

What carries the argument

The load-bearing object is the unitary transformation $\hat{T}=e^{i\hat{F}/\hbar}$ with $\hat{F}=m\sqrt{\pi G/c^2}\,\hat{\xi}'^2\hat{p}'_h(0,t)$. It acts as a phase-space rotation that absorbs the environment-dependent piece of the system momentum into the gravitational field, so the transformed momentum $\hat{p}_\xi$ depends only on the mass positions and velocities and commutes with the gravitational variables; this is what makes the partial trace physically meaningful. The Hamiltonian produced by this transformation couples the mass quadrupole $\hat{I}=m\hat{\xi}^2$ to the gravitational-wave amplitude through $\nabla^2\hat{h}$ at the origin, and the environment has a supra-ohmic $\omega^5$ spectrum. The master equation (40) is obtained by integrating out the thermal graviton bath, with the noise and dissipation kernels derived from the two-point function of the gravitational field; in the small-fluctuation limit $\hat{\xi}\to l_0 + \delta\hat{\xi}$ the same procedure gives the linear master equation (50). All observable predictions follow from these equations.

What would settle it

Compute the reduced master equation, in the position basis, starting from a different unitary transformation that still makes the system momentum independent of the environment (for example, with the generator $\hat{F}$ replaced by another function of $\hat{\xi}'$ and $\hat{p}'_h$ that satisfies the same physical requirements); if the predicted decoherence rate for a superposition of two proper distances changes, the master equation is not unique.

Watch

Extended reading notes

Core claim

The central claim is that the correct reduced dynamics of two masses in a linearized-gravity environment is governed by the Hamiltonian (32) and the Markovian master equation (40), derived to first order in $G$ from the influence-functional formalism after applying the unitary transformation (23). In this representation the system momentum $\hat{p}_\xi$ is a genuine system observable that commutes with the gravitational variables, so the reduced density matrix yields predictions for position, velocity, and energy. The dissipative part of the master equation matches the classical gravitational-bremsstrahlung law with time-averaged power loss $-(16/15)\, G I^2 \omega^6/c^5$ for $I = m l_0^2$, and the noise part suppresses coherences between states of different mass quadrupole $m\xi^2$, i.e., different proper distances, at the rate $\Gamma = 2\omega^4 \Delta I^2 k_BT/(E_P^2 \hbar)$, equivalently $\Gamma = 8(\Delta V/E_P)^2 k_BT/\hbar$.

Load-bearing premise

The entire derivation rests on the hand-picked unitary transformation in Eq. (23) that removes the gravitational field from the system momentum; the paper does not prove this choice is the only physically correct system–environment split, so if a different valid split yields different reduced dynamics, the master equation would not be a unique prediction.

Editorial extensions

If this is right

The dissipative sector reproduces the classical gravitational-bremsstrahlung energy loss of an oscillating two-mass system, so the master equation passes a classical consistency check.
Superpositions of two masses with different proper separations lose coherence exponentially, at a rate set by the difference in their gravitational potential energy relative to the Planck energy.
For small fluctuations around a baseline $l_0$, the decoherence rate scales as $l_0^2$, so the farther apart the masses, the faster their spatial superposition decoheres under graviton noise.
In the limit $\omega\to 0$ (no harmonic confinement) the leading-order decoherence rate vanishes; graviton-scattering decoherence would appear only at order $G^2$, beyond the perturbative range of this master equation.
The nonlinear master equation drives the system toward a stationary thermal state with energy $k_BT$, with a relaxation rate $\gamma_R = 16(\hbar\omega/E_P)^2 k_BT/\hbar$.

Reading between the lines

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

Beyond the paper: the $l_0^2$-dependent decoherence rate offers a concrete experimental signature distinguishing graviton-induced decoherence from ordinary environmental noise, since blackbody radiation and gas collisions do not scale with baseline in the same tidal way.
Beyond the paper: the same unitary-transformation strategy could be applied to other derivative-coupled open quantum systems (for instance, atoms coupled to the electromagnetic field in gauges where the conjugate momentum is field-dependent), and the resulting master equations may differ from the naive ones.
Beyond the paper: because the master equation (40) is non-Gaussian and its moment hierarchy does not close, numerical simulation of the full equation for an interferometric setup would test how well the semiclassical truncation used for the energy-loss and thermalization estimates holds.
Beyond the paper: the derivation assumes a thermal graviton state; replacing it with the squeezed vacuum expected from inflationary cosmology would modify the noise kernel, and the decoherence rate could be enhanced, which is an immediate extension of the presented calculation.

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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

Read the paper and the reader's take. I think the reader's weakest-assumption is the one thing that does not bite. The unitary transformation in Eq. (23) looks hand-picked, but it is fixed to O(G) by the demand that the velocity operator be expressible in system observables alone. A general generator F = m sqrt(pi G/c^2) f(xi') p'_h(0) works only if f'(xi)=2xi; the residual constant shifts the field by a c-number and drops out of the reduced dynamics. So the central system-environment split is not a free choice.

What is genuinely new: the identification of the problem with standard canonical quantization for gravitational open systems (the system momentum inherits the traced-out field), and the unitary transformation that fixes it. The resulting master equation (40) is the first to my knowledge written in terms of the proper-distance variables with a clean S-E split. The dissipative sector matching classical gravitational bremsstrahlung after the full polarization treatment (Appendix H) is a strong external check. Eq. (57), the linearized Caldeira-Leggett-type equation with decoherence rate proportional to the baseline length l0^2, is a real new result and physically sensible (tidal effect of GWs).

The derivation itself is careful. Appendices B-E are worked out in detail; the Omega_max^5 divergence cancels between the Hamiltonian and the dissipation kernel; the dependence on the cutoff is reduced to terms that are argued—reasonably—to be subdominant under the stated constraints. The paper also correctly notes that S-matrix scattering decoherence is O(G^2) and outside the perturbative order. Citing the authors' own earlier work for the electrodynamic analogue is fine; the needed pieces are reproduced here.

Soft spots: the thermal graviton state is a benchmark, not the physical environment. The real graviton field is closer to vacuum or squeezed; the paper says this, but it means the concrete rates (54)-(57) should not be read as quantitative predictions without knowing T. Also, the full master equation contains many cutoff-dependent terms and the argument they are safe relies on Planck-scale reasoning plus xi << lambda; that is plausible but not fully airtight. The decoherence analysis isolates a few dominant terms from a non-Gaussian master equation, which is honest but leaves the full dynamics unsolved.

Bottom line: this is a serious, technically sound paper for the gravitational decoherence community. It deserves a serious referee and, assuming the numerics in the appendices check out, publication. I'd bring it to reading group and would cite the linearized result.

Referee Report

0 major / 4 minor

Summary. The paper develops an open-quantum-system description of two masses coupled to a thermal bath of linearized gravitational waves. Working in Fermi normal coordinates so that the system variable is the proper distance, it shows that the canonical momenta obtained from the standard Lagrangians are unsuitable for a reduced description, and introduces a unitary transformation that removes the environment dependence from the system velocity. The resulting O(G) Hamiltonian (32) is then used to derive a non-Markovian and, in the long-time/high-temperature limit, a Markovian master equation (40). The dissipative sector is shown to reproduce classical gravitational bremsstrahlung after the full polarization treatment, and the noise sector gives decoherence rates for superpositions of different proper distances (55), and length-dependent rates in the linearized fluctuation limit (57). The calculations are supported by detailed appendices covering regularization, the Hamiltonian transformation, Dirac quantization, the noise and dissipation kernels, the Markovian limit, semiclassical dynamics, moment equations, and the full polarization analysis.

Significance. If correct, the paper resolves a genuine ambiguity in gravitational decoherence analyses: the choice of system-environment split changes the reduced dynamics, and the authors provide a physically motivated criterion, namely that the system velocity should be expressible in terms of system observables alone. The master equation is corroborated by three internal checks: exact cancellation of the Omega_max^5 divergence, reproduction of the classical quadrupole bremsstrahlung formula after the 8/15 polarization factor, and reduction to a Caldeira-Leggett-type equation with a baseline-length-dependent decoherence rate. The explicit regularization and the detailed appendices make the derivation reproducible. The stated limitations, a thermal graviton bath, a UV cutoff, a leading-order-in-G treatment, and a simplified polarization treatment in the main text, are appropriate for the claimed result.

minor comments (4)

[Sec. VIII, Eqs. (55) and (57)] These rates are presented as the paper's decoherence predictions, but they are computed in the simplified single-polarization setup. Appendix H shows that the full polarization treatment multiplies the gravitational kernels by 8/15, so the quoted rates should either include this factor or be explicitly labeled as order-one estimates, since otherwise Eqs. (55) and (57) read as the final full-polarization rates.
[Sec. IV, Eq. (23)] The choice of the unitary generator F is motivated by the QED analogy but not proved unique. A short argument that the condition f'(xi)=2xi fixes the generator up to a c-number shift of the field would remove the appearance of a hand-picked transformation and strengthen the central system-environment decomposition.
[After Eq. (41)] The discussion of cutoff-dependent contributions addresses the unitary sector, but the last line of Eq. (40) contains f(Omega_max,omega,T), which also has Omega_max^4 and Omega_max^3 pieces. A sentence explaining why this term does not dominate the decoherence estimates, or at least quantifying its expected size relative to Eq. (54), would be helpful for readers concerned about the physical role of the cutoff.
[Appendix F and Sec. VII] There are a few typographical errors: 'brehmsstralung' in Appendix F should be 'bremsstrahlung', and 'envirnmental' in Sec. VII should be 'environmental'. In addition, Eq. (55) states the position-basis solution rho_t(xi,xi') = e^{-Gamma t} rho_0(xi,xi') twice in consecutive lines.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the unitary split is fixed by the velocity-observability requirement, and the master equation is checked against independent classical and prior results.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. The unitary transformation in Eq. (23) is not imported as a forced result: the paper verifies explicitly that the transformed momentum gives an environment-independent velocity, Eqs. (25)-(28), and the paper does not claim uniqueness of the split, explicitly stating that no specific physical meaning is attached to it. The self-citations to Refs. [19] and [35] are motivational and technical templates, not load-bearing: Ref. [19] motivates the functional form of the generator, but the paper independently checks the relevant commutation and velocity relations, while Ref. [35] supplies the Feynman-Vernon structure, with the gravitational master equation stated explicitly in Eq. (35) and the kernels computed from the gravitational two-point function in Appendix D and Eqs. (36)-(37). The dissipative benchmark is not fitted: the prefactor 8/15 in the full polarization treatment of Appendix H is obtained by explicit integration over polarization tensors, and the resulting energy loss Eq. (43) is matched to the classical formula rather than imposed. The decoherence rates in Eqs. (55) and (57) are computed from the thermal noise kernel, not obtained by adjusting parameters to prior estimates. The manuscript states its own limitations, such as the thermal-state benchmark in Sec. VI, the UV cutoff and Fermi-normal-coordinate truncation in Eq. (6), and the simplified single-polarization treatment in the main text with the full treatment in Appendix H; these restrict the regime of validity but do not constitute circular reasoning.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard quantum mechanics, the dipole-truncated Fermi coordinate metric, the TT gauge, a factorized thermal initial state, a long-time Markovian limit, and a hand-chosen unitary transformation. No new particles or forces are introduced; the only free parameter is the UV cutoff Omega_max, which is physically constrained.

free parameters (1)

Omega_max (UV cutoff frequency)
Introduced to regularize delta functions (Appendix A); constrained by the dipole approximation Omega_max << 2 pi c / xi and quantum gravity lambda_min >> l_P; not fitted to data.

assumptions (6)

standard math Standard canonical quantization with Weyl operator ordering
Used to promote the classical Hamiltonian (30) to the operator Hamiltonian (31).
domain assumption Fermi normal coordinate metric truncated at second order in the geodesic deviation
The interaction Hamiltonian (14) and the UV cutoff (6) follow from Eq. (4), valid when all graviton wavelengths satisfy xi << lambda.
domain assumption Transverse-traceless gauge for the metric perturbation
Simplifies the Riemann tensor expression (9) and the free graviton action (11).
domain assumption Initially factorized system-environment state, environment Gaussian with zero mean
Standard open-quantum-system assumption stated in Sec. VI; enables the Feynman-Vernon influence functional treatment.
domain assumption Long-time Markovian limit t >> 1/Omega_max and t >> hbar/(k_B T)
Needed to obtain the simple kernels (38) and the Markovian master equation (40).
domain assumption The chosen unitary transformation (Eq. 23) yields the physically meaningful system-environment decomposition
Motivated by the QED analysis in Ref. [19]; without it the velocity operator would depend on environmental degrees of freedom.

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Cite this review

Pith. "Pith review of Master equation for systems interacting with linearized gravity." pith.science (2026). https://pith.science/paper/LD2U7QOG

@misc{pith2026260806121,
  author       = {Pith},
  title        = {Pith review of: Master equation for systems interacting with linearized gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LD2U7QOG}},
  note         = {Machine review of arXiv:2608.06121}
}

abstract

We investigate the open quantum dynamics of a system of two masses interacting with an environment of linearized gravitational waves. We formulate the analysis in terms of the observable proper distance between the two masses, and show that the canonical variables obtained from the standard Lagrangian, expressed in terms of the Fermi normal coordinates, are not suitable for an effective description of the system. We resolve this issue through a unitary transformation that provides a physically meaningful system--environment decomposition and derive the master equation to leading order in $G$. Its dissipative sector reproduces the classical energy loss due to gravitational-wave emission, while the noisy contributions suppress coherences between states with different mass quadrupole, or effectively, different proper separations. In the regime where the proper distance can be described by considering small quantum fluctuations around an average distance $l_0$, the dynamics reduces to a Caldeira--Leggett-type equation, with a decoherence rate dependent on the baseline length $l_0$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 48 canonical work pages

[1]

ˆF , " ˆF , ˆp2 ξ 2m ## =− i 2ℏ πmG c2 ˆp2 h(0, t) h ˆξ2, n ˆξ,ˆpξ oi = +2π mG c2 ˆp2 h(0, t)ˆξ2 .(B5) The interaction Hamiltonian, instead, will transform as i ℏ

The master equation is well–defined mathematically, but it not useful from the physical point of view, being a function of ˆp′ 1, which cannot be experimentally measured in the open quantum system scenario Eq. (21) represents. In summary, the challenge that one faces in describ- ing open system dynamics starting from the classical ac- tion (14), is that u...

work page 2011
[2]

(C10) and (C11) give the following Poisson brackets {ϕ1(x), ϕ2(y)}=δ 3(x−y)− 4mπG c2 ξ2δ3(x)δ3(y)̸= 0, (C12) which is inconsistent withϕ 1(x) =ϕ 2(x) = 0

However, Eqs. (C10) and (C11) give the following Poisson brackets {ϕ1(x), ϕ2(y)}=δ 3(x−y)− 4mπG c2 ξ2δ3(x)δ3(y)̸= 0, (C12) which is inconsistent withϕ 1(x) =ϕ 2(x) = 0. This motivates the introduction of the so-called Dirac brackets {·,·} D which are constructed so that{ϕ 1, ϕ2}D = 0, and is thus consistent with the constraintsϕ 1 =ϕ 2 = 0. In terms of th...

work page
[3]

Noise kernel The real part of the correlation function is precisely the noise kernel Re[α(τ)] =N(τ) introduced in the main text and it has contributions coming from both thermal 14 and vacuum fluctuations. The former are finite and can be expressed as NT (τ) = 1 2π m2G c5 Z ∞ 0 dωω 5nT (ω) cosωτ = 1 2π m2G c5 Z ∞ 0 dω ω5 cosωτ eℏω/kbT −1 = 1 2 m2G c5ℏ kbT...

work page
[4]

Just as the vac- uum noise kernelN 0 it requires regularization

Dissipation kernel The dissipation kernelDis instead related to the imag- inary component ofα(τ) by D(τ) =−2θ(τ) Imα(τ) = θ(τ) 2π m2G c5 Z ∞ 0 dωω 5 sinωτ(D9) whereθis Heaviside theta function. Just as the vac- uum noise kernelN 0 it requires regularization. If the timescales of interest are large with respect to the in- verse of the cutoffϵ, we can use t...

work page
[5]

Dissipation kernel From Eq. (35) of the main text we have that the con- tributions due to the dissipation kernel are of the form ∂t ˆρ= i 2ℏ Z t 0 dτD(τ) h ˆξ2, n ˆξ2(−τ),ˆρ oi (E1) Therefore one needs to evaluate i 2ℏ Z dτD(τ)ξ 2(−τ) =− i 4ℏ m2G c5 Z t 0 dτ d5 dτ 5 δ(τ) ˆξ2(−τ) (E2) The integral can be evaluated by repeated integration by parts. The boun...

work page
[6]

Thermal noise kernel The contributions from the noise kernel (D6), given the similar distributional form to the dissipation ker- 15 nel (D10), can be calculated along the same lines: Z dτN T (τ) ˆξ2(−τ) = m2GkbT 2c5ℏ Z dτ d4 dτ 4 δ(τ) ˆξ2(−τ) = m2GkbT 2c5ℏ × h −δ (2)(τ) d dτ ˆξ2(τ)−δ(τ) d3 dτ 3 ˆξ2(τ) + 1 2 d4 dτ 4 ˆξ2(τ) i τ=0 = m2GkbT 2c5ℏ × h 4Ωmaxω2 π...

work page
[7]

ˆξ2, " ˆξ2 + ˆp2 ξ m2ω2 ,ˆρt ## − f2(t) 2ℏ

V acuum fluctuations kernel The contributions coming from the vacuum fluctua- tions are more involved to compute given the complicated time dependence Eq. (D8). However, rewriting the free solution ˆξ2(−τ) as ˆξ2(−τ) = 1 2 ˆξ2 + ˆp2 ξ m2ω2 − 1 2mω n ˆξ,ˆpξ o sin(2ωτ) + 1 2 ˆξ2 − ˆp2 ξ m2ω2 cos(2ωτ),(E7) allows us to clearly identify three distinct contrib...

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[8]

Appendix F: Semiclassical dynamics An exact treatment of the dynamics (40) is technically challenging

F ull master equation Putting together all the contributions just calculated we obtain the full Markovian master equation ∂t ˆρ(t) =−i ℏ " ˆp2 ξ 2m + 1 2 mω2 ˆξ2 − m2GΩmaxω2 πc5 (Ω2 max + 2ω2) ˆξ4 + GΩ3 max πc5 n ˆξ2, p2 ξ o ,ˆρ(t) # − i ℏ GΩmax πc5 Ω2 max + 2ω2 h ˆξ2, ˆp2 ξ,ˆρ i − i ℏ 2mGω4 c5 h ˆξ2, nn ˆξ,ˆpξ o ,ˆρ(t) oi − 1 ℏ mG πc5 kbT ℏ Ω3 max + 2kbT...

work page

Show all 57 references

[9]

(F3) reproduces the standard classical result [48], apart from the numerical pre-factor

We see that Eq. (F3) reproduces the standard classical result [48], apart from the numerical pre-factor. The discrepancy is entirely accounted for by the sim- plified treatment of the gravitational wave polarizations adopted in the main text. When the polarization modes are pr...
[10]

Anastopoulos, Quantum theory of nonrelativistic par- ticles interacting with gravity, Physical Review D54, 1600 (1996)

C. Anastopoulos, Quantum theory of nonrelativistic par- ticles interacting with gravity, Physical Review D54, 1600 (1996)

1996
[11]

Breuer, E

H.-P. Breuer, E. G¨ okl¨ u, and C. L¨ ammerzahl, Metric fluc- tuations and decoherence, Classical and Quantum Grav- ity26, 105012 (2009)

2009
[12]

Anastopoulos and B

C. Anastopoulos and B. Hu, A master equation for grav- itational decoherence: probing the textures of spacetime, Classical and Quantum Gravity30, 165007 (2013)

2013
[13]

Blencowe, Effective field theory approach to gravi- tationally induced decoherence, Physical review letters 111, 021302 (2013)

M. Blencowe, Effective field theory approach to gravi- tationally induced decoherence, Physical review letters 111, 021302 (2013)

2013
[14]

Oniga and C

T. Oniga and C. H.-T. Wang, Quantum gravitational de- coherence of light and matter, Phys. Rev. D93, 044027 (2016)

2016
[15]

Oniga and C

T. Oniga and C. H.-T. Wang, Quantum coherence, ra- diance, and resistance of gravitational systems, Physical Review D96, 084014 (2017)

2017
[16]

Asprea, G

L. Asprea, G. Gasbarri, and A. Bassi, Gravitational de- coherence: A general nonrelativistic model, Physical Re- view D103, 104041 (2021)

2021
[17]

Asprea, A

L. Asprea, A. Bassi, H. Ulbricht, and G. Gasbarri, Grav- itational decoherence and the possibility of its interfer- ometric detection, Physical Review Letters126, 200403 (2021)

2021
[18]

A. R. Malcolm, Z.-W. Wang, B. Sharmila, and A. Datta, Signatures of a gravitational quantum vacuum on the dynamics of massive particles, Physical Review D111, 106022 (2025)

2025
[19]

Parikh, F

M. Parikh, F. Wilczek, and G. Zahariade, Signatures of the quantization of gravity at gravitational wave detec- tors, Physical Review D104, 046021 (2021)

2021
[20]

Parikh, F

M. Parikh, F. Wilczek, and G. Zahariade, The noise of gravitons, International Journal of Modern Physics D29, 2042001 (2020)

2020
[21]

Cho and B.-L

H.-T. Cho and B.-L. Hu, Non-markovian quantum mas- ter and fokker-planck equation for gravitational systems and gravitational decoherence, Physical Review D112, 046021 (2025)

2025
[22]

Toroˇ s, A

M. Toroˇ s, A. Mazumdar, and S. Bose, Loss of coherence and coherence protection from a graviton bath, Physical Review D109, 084050 (2024)

2024
[23]

Sharifian, M

M. Sharifian, M. Zarei, M. Abdi, N. Bartolo, and S. Matarrese, Open quantum system approach to the gravitational decoherence of spin-1/2 particles, Physical Review D109, 043510 (2024)

2024
[24]

Anastopoulos and B.-L

C. Anastopoulos and B.-L. Hu, Gravitational decoher- ence: A thematic overview, A VS Quantum Science4, 10.1116/5.0077536 (2022)

2022 doi
[25]

Bassi, A

A. Bassi, A. Großardt, and H. Ulbricht, Gravitational decoherence, Classical and Quantum Gravity34, 193002 (2017)

2017
[26]

J. P. Paz and W. H. Zurek, Environment-induced deco- herence, classicality, and consistency of quantum histo- ries, Physical Review D48, 2728 (1993)

1993
[27]

Lagrangian and hamiltonian approach to electrodynam- ics, the standard lagrangian and the coulomb gauge, in Photons and Atoms(John Wiley & Sons, Ltd, 1997) Chap. 2, pp. 79–168

1997
[28]

Gundhi, O

A. Gundhi, O. Angeli, and A. Bassi, From equivalent la- grangians to inequivalent open quantum system dynam- ics, Phys. Rev. Res.8, 013133 (2026)

2026
[29]

R. P. Feynman and F. L. Vernon Jr, The theory of a gen- eral quantum system interacting with a linear dissipative system, Annals of physics281, 547 (2000)

2000
[30]

E. A. Calzetta and B.-L. B. Hu,Nonequilibrium quantum field theory(Cambridge University Press, 2009)

2009
[31]

A. O. Caldeira and A. J. Leggett, Path integral approach to quantum brownian motion, Physica A: Statistical me- chanics and its Applications121, 587 (1983)

1983
[32]

Parikh and F

M. Parikh and F. Setti, Quantum-gravitational noise cor- relation in nearby detectors, Physical Review D111, 046004 (2025)

2025
[33]

Kanno, J

S. Kanno, J. Soda, and J. Tokuda, Noise and decoherence induced by gravitons, Physical Review D103, 044017 (2021)

2021
[34]

Cho and B.-L

H.-T. Cho and B.-L. Hu, Quantum noise of gravitons and stochastic force on geodesic separation, Physical Review D105, 086004 (2022)

2022
[35]

Moreira and L

T. Moreira and L. C` eleri, Graviton induced decoherence of a composite particle, arXiv preprint arXiv:2504.09578 10.48550/arXiv.2504.09578 (2025)

2025 doi arXiv
[36]

Cho and B.-L

H.-T. Cho and B.-L. Hu, Graviton noise on tidal forces and geodesic congruences, Physical Review D107, 084005 (2023)

2023
[37]

F. K. Manasse and C. W. Misner, Fermi normal coordi- nates and some basic concepts in differential geometry, Journal of mathematical physics4, 735 (1963)

1963
[38]

Marzlin, Fermi coordinates for weak gravitational fields, Physical Review D50, 888 (1994)

K.-P. Marzlin, Fermi coordinates for weak gravitational fields, Physical Review D50, 888 (1994)

1994
[39]

Rakhmanov, Fermi-normal, optical, and wave- synchronous coordinates for spacetime with a plane gravitational wave, Classical and Quantum Gravity31, 085006 (2014)

M. Rakhmanov, Fermi-normal, optical, and wave- synchronous coordinates for spacetime with a plane gravitational wave, Classical and Quantum Gravity31, 085006 (2014)

2014
[40]

Maggiore,Gravitational waves: Volume 1: Theory and experiments, Vol

M. Maggiore,Gravitational waves: Volume 1: Theory and experiments, Vol. 1 (Oxford university press, 2008)

2008
[41]

S. M. Carroll,Spacetime and geometry(Cambridge Uni- versity Press, 2019)

2019
[42]

P. A. Dirac,Lectures on quantum mechanics(Courier Corporation, 2013)

2013
[43]

E. T. Whittaker,A treatise on the analytical dynamics of particles and rigid bodies(CUP Archive, 1964) p. 266

1964
[44]

Gundhi and A

A. Gundhi and A. Bassi, Motion of an electron through vacuum fluctuations, Physical Review A107, 062801 (2023)

2023
[45]

Breuer and F

H.-P. Breuer and F. Petruccione,The theory of open quantum systems(OUP Oxford, 2002)

2002
[46]

Schlosshauer,Decoherence and the quantum-to- classical transition(Springer, 2007)

M. Schlosshauer,Decoherence and the quantum-to- classical transition(Springer, 2007)

2007
[47]

L. P. Grishchuk and Y. V. Sidorov, On the quantum state of relic gravitons, Classical and Quantum Gravity6, L161 (1989)

1989
[48]

Martin, Inflationary perturbations: The cosmologi- cal schwinger effect, inInflationary Cosmology, Lecture Notes in Physics, Vol

J. Martin, Inflationary perturbations: The cosmologi- cal schwinger effect, inInflationary Cosmology, Lecture Notes in Physics, Vol. 738 (Springer, 2008) pp. 193–241

2008
[49]

B. L. Hu, J. P. Paz, and Y. Zhang, Quantum brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise, Physical Re- view D45, 2843 (1992)

1992
[50]

B. Hu, J. P. Paz, and Y. Zhang, Quantum brownian motion in a general environment. ii. nonlinear coupling and perturbative approach, Physical Review D47, 1576 21 (1993)

1993
[51]

Ford and R

G. Ford and R. O’connell, Exact solution of the hu-paz- zhang master equation, Physical Review D64, 105020 (2001)

2001
[52]

M. R. Gallis and G. N. Fleming, Environmental and spontaneous localization, Physical Review A42, 38 (1990)

1990
[53]

Joos and H

E. Joos and H. D. Zeh, The emergence of classical proper- ties through interaction with the environment, Zeitschrift f¨ ur Physik B Condensed Matter59, 223 (1985)

1985
[54]

J. F. Donoghue, M. M. Ivanov, and A. Shk- erin, Epfl lectures on general relativity as a quan- tum field theory, arXiv preprint arXiv:1702.00319 10.48550/arXiv.1702.00319 (2017)

2017 doi arXiv
[55]

Pitkin, S

M. Pitkin, S. Reid, S. Rowan, and J. Hough, Gravi- tational wave detection by interferometry (ground and space), Living reviews in relativity14, 5 (2011)

2011
[56]

C. A. Galv˜ ao and N. A. Lemos, On the quantization of constrained generalized dynamics, Journal of mathemat- ical physics29, 1588 (1988)

1988
[57]

Eq.(3.319) [31]

See e.g. Eq.(3.319) [31]

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

ˆF , " ˆF , ˆp2 ξ 2m ## =− i 2ℏ πmG c2 ˆp2 h(0, t) h ˆξ2, n ˆξ,ˆpξ oi = +2π mG c2 ˆp2 h(0, t)ˆξ2 .(B5) The interaction Hamiltonian, instead, will transform as i ℏ

The master equation is well–defined mathematically, but it not useful from the physical point of view, being a function of ˆp′ 1, which cannot be experimentally measured in the open quantum system scenario Eq. (21) represents. In summary, the challenge that one faces in describ- ing open system dynamics starting from the classical ac- tion (14), is that u...

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[2] [2]

(C10) and (C11) give the following Poisson brackets {ϕ1(x), ϕ2(y)}=δ 3(x−y)− 4mπG c2 ξ2δ3(x)δ3(y)̸= 0, (C12) which is inconsistent withϕ 1(x) =ϕ 2(x) = 0

However, Eqs. (C10) and (C11) give the following Poisson brackets {ϕ1(x), ϕ2(y)}=δ 3(x−y)− 4mπG c2 ξ2δ3(x)δ3(y)̸= 0, (C12) which is inconsistent withϕ 1(x) =ϕ 2(x) = 0. This motivates the introduction of the so-called Dirac brackets {·,·} D which are constructed so that{ϕ 1, ϕ2}D = 0, and is thus consistent with the constraintsϕ 1 =ϕ 2 = 0. In terms of th...

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[3] [3]

Noise kernel The real part of the correlation function is precisely the noise kernel Re[α(τ)] =N(τ) introduced in the main text and it has contributions coming from both thermal 14 and vacuum fluctuations. The former are finite and can be expressed as NT (τ) = 1 2π m2G c5 Z ∞ 0 dωω 5nT (ω) cosωτ = 1 2π m2G c5 Z ∞ 0 dω ω5 cosωτ eℏω/kbT −1 = 1 2 m2G c5ℏ kbT...

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[4] [4]

Just as the vac- uum noise kernelN 0 it requires regularization

Dissipation kernel The dissipation kernelDis instead related to the imag- inary component ofα(τ) by D(τ) =−2θ(τ) Imα(τ) = θ(τ) 2π m2G c5 Z ∞ 0 dωω 5 sinωτ(D9) whereθis Heaviside theta function. Just as the vac- uum noise kernelN 0 it requires regularization. If the timescales of interest are large with respect to the in- verse of the cutoffϵ, we can use t...

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[5] [5]

Dissipation kernel From Eq. (35) of the main text we have that the con- tributions due to the dissipation kernel are of the form ∂t ˆρ= i 2ℏ Z t 0 dτD(τ) h ˆξ2, n ˆξ2(−τ),ˆρ oi (E1) Therefore one needs to evaluate i 2ℏ Z dτD(τ)ξ 2(−τ) =− i 4ℏ m2G c5 Z t 0 dτ d5 dτ 5 δ(τ) ˆξ2(−τ) (E2) The integral can be evaluated by repeated integration by parts. The boun...

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[6] [6]

Thermal noise kernel The contributions from the noise kernel (D6), given the similar distributional form to the dissipation ker- 15 nel (D10), can be calculated along the same lines: Z dτN T (τ) ˆξ2(−τ) = m2GkbT 2c5ℏ Z dτ d4 dτ 4 δ(τ) ˆξ2(−τ) = m2GkbT 2c5ℏ × h −δ (2)(τ) d dτ ˆξ2(τ)−δ(τ) d3 dτ 3 ˆξ2(τ) + 1 2 d4 dτ 4 ˆξ2(τ) i τ=0 = m2GkbT 2c5ℏ × h 4Ωmaxω2 π...

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[7] [7]

ˆξ2, " ˆξ2 + ˆp2 ξ m2ω2 ,ˆρt ## − f2(t) 2ℏ

V acuum fluctuations kernel The contributions coming from the vacuum fluctua- tions are more involved to compute given the complicated time dependence Eq. (D8). However, rewriting the free solution ˆξ2(−τ) as ˆξ2(−τ) = 1 2 ˆξ2 + ˆp2 ξ m2ω2 − 1 2mω n ˆξ,ˆpξ o sin(2ωτ) + 1 2 ˆξ2 − ˆp2 ξ m2ω2 cos(2ωτ),(E7) allows us to clearly identify three distinct contrib...

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[8] [8]

Appendix F: Semiclassical dynamics An exact treatment of the dynamics (40) is technically challenging

F ull master equation Putting together all the contributions just calculated we obtain the full Markovian master equation ∂t ˆρ(t) =−i ℏ " ˆp2 ξ 2m + 1 2 mω2 ˆξ2 − m2GΩmaxω2 πc5 (Ω2 max + 2ω2) ˆξ4 + GΩ3 max πc5 n ˆξ2, p2 ξ o ,ˆρ(t) # − i ℏ GΩmax πc5 Ω2 max + 2ω2 h ˆξ2, ˆp2 ξ,ˆρ i − i ℏ 2mGω4 c5 h ˆξ2, nn ˆξ,ˆpξ o ,ˆρ(t) oi − 1 ℏ mG πc5 kbT ℏ Ω3 max + 2kbT...

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