Pith. sign in

REVIEW 3 major objections 3 minor 115 references

Classical theories of gravity produce entanglement

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A classical gravitational interaction can entangle two quantum masses by exchanging virtual matter particles, so at Planck-scale masses entanglement alone no longer proves gravity is quantum.

desk verdict A plausible but not airtight loophole in the BMV entanglement witness: the virtual-matter calculation is serious, but two unaddressed technical points keep the central claim conditional. read the letter →

arxiv 2510.19714 v3 pith:UTUM7ZDD submitted 2025-10-22 quant-ph

Joseph Aziz , Richard Howl This is my paper · ORCID
classification quant-ph
keywords classicalgravityquantumentanglementvirtualparticlescommunicationgravitationallyinducedsemiclassicalLOCCfieldtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the standard no-go result—classical gravity can only act through local operations and classical communication, hence cannot entangle—misses a process available when matter is treated as a quantum field. Even with a purely classical gravitational field that is identical in every branch of the superposition, the interaction Hamiltonian contains vertices that allow virtual matter particles to travel between the two masses. At fourth order in perturbation theory these virtual-matter exchanges yield a branch-dependent amplitude that creates spin entanglement. The effect scales differently from the quantum-gravity phase, so for low masses near-future experiments are unaffected, but for Planck-scale masses entanglement can no longer be treated as unambiguous evidence of quantum gravity.

What carries the argument

The central object is the fourth-order perturbative amplitude for virtual-matter exchange under a classical gravitational interaction, built from: (i) N-particle non-relativistic wavepacket states localized in spheres of radius R; (ii) a classical potential sourced by the expectation value of the matter energy-momentum tensor, hence identical in all superposition branches; and (iii) Feynman propagators of a massive complex scalar field connecting the two objects. The key identity is β^(4)_ij proportional to [∫ dx dy Φ(x)Φ(y) θ_{1i}(x)θ_{2j}(y)/|x−y|]^2, whose d_ij^{-2} branch dependence generates the entanglement. It does this by letting virtual matter particles traverse different distances

What would settle it

Recompute the final state without truncating to N-particle sectors, or measure spin negativity in a Planck-mass configuration where the predicted classical amplitude is about 0.1: if the number-transfer components destroy the spin witness, or no entanglement appears despite the predicted amplitude, the central claim is falsified.

Watch

Extended reading notes

Core claim

In a version of the two-superposed-masses experiment, with each mass described by an N-particle quantum-field-theory wavepacket and gravity by a classical gravitational potential that is the same in every branch, the leading entangling process is not the classical analogue of the graviton diagram but a fourth-order diagram in which the two objects exchange two virtual matter particles. The amplitude for branch (i,j) is β^(4)_ij ≈ (6/25 i G^2 m^2 M^3 R t / (ħ^3 d_ij))^2. Because the virtual matter propagator involves |x−y|, the distance between the spherical wavepackets depends on the branch, so β differs across branches even though the classical potential does not; this branch-dependence is

Load-bearing premise

The paper's spin-entanglement witness computes amplitudes from states that keep the particle number N fixed in each object; if the states |N±k⟩|N∓k⟩ that the same interaction creates are not truly negligible or are not traced out, they could carry which-branch information and spoil the predicted spin entanglement.

Editorial extensions

If this is right

  • For the near-term low-mass parameter regime (M≈10^-14 kg, t≈2 s), the classical-gravity entanglement amplitude is far below the quantum-gravity phase, so observing entanglement there retains its status as evidence for quantum effects in gravity.
  • At masses approaching the Planck mass and above, the classical amplitude reaches order 0.1 even at short times, so entanglement observation in that regime would not discriminate classical from quantum gravity.
  • The two amplitudes scale differently with mass, radius, and distance, so scanning these parameters can separate the classical-gravity contribution from the quantum-gravity contribution in a single experiment.
  • The LOCC-based theorems are not generic: classical gravity models with quantum-field-theoretic matter are not restricted to classical communication, because the matter sector itself provides a quantum channel.
  • In stochastic classical-gravity models with fundamental decoherence, the virtual-matter entanglement can still dominate when its generation rate exceeds the gravitational decoherence rate, so entanglement is not unique to semiclassical sourcing.

Reading between the lines

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

  • The same fourth-order virtual-matter mechanism should operate for any classical field coupled to quantum matter—for instance a classical electromagnetic potential—so the ability of a classical field to entangle is not peculiar to gravity; tabletop experiments may need to control classical electromagnetic backgrounds at the same perturbative order.
  • The material dependence of the classical amplitude (it scales with the constituent particle mass and radius) versus the quantum-gravity phase's independence of those parameters offers a direct experimental discriminator: changing the material at fixed total mass should change the classical contribution but not the quantum one.
  • The calculation truncates to fixed-N sectors; including the |N±k⟩|N∓k⟩ number-transfer channels could either introduce decoherence that suppresses the spin entanglement or open additional witnesses, so a full calculation is needed before Planck-scale experiments are designed.
  • If the effect is real, the experimental question shifts from 'does entanglement occur?' to 'does the entanglement amplitude scale as predicted?', making classical-sourcing models directly testable through the predicted dependence on mass, radius, time, and separation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper challenges the consensus that a classical gravitational interaction cannot entangle two quantum matter systems. Its central claim is that when matter is treated at QFT level, a classical (unquantized) gravitational interaction permits quantum communication through virtual matter, and this can generate entanglement in a Feynman-type experiment. The authors compute the fourth-order Dyson amplitude for two N00N-state mass distributions under semiclassical Einstein gravity and obtain β^(4)_RL ≈ [(6/25) G² m² M³ R t/(ℏ³ d_RL)]², concluding that for Planck-scale masses the observation of entanglement alone cannot be taken as evidence of quantum gravity. The paper includes extensive appendices on relativistic, momentum-space, stochastic, and boundary treatments.

Significance. If the technical concerns below are resolved, this is an important result. It provides a concrete, quantitative mechanism by which a local classical gravity model — semiclassical Einstein gravity — can produce gravitationally mediated entanglement, thereby sharpening the conditions under which entanglement-based quantum-gravity witnesses are valid. The strength of the paper is its explicit QFT calculation: the Dyson-series derivation is detailed and cross-checked in momentum space (Appendix B.3), a quantum-gravity counterpart is computed (Appendix B.6), and the final amplitude gives a falsifiable mass/time/distance scaling. The main weaknesses are the inconsistent sign convention in the central amplitude and an unquantified postselection assumption connecting the computed amplitude to the observable spin entanglement.

major comments (3)
  1. [Section 5.2, Eqs. (79)–(80), Fig. 6] There is an internal inconsistency in the definition of ϑ. Eq. (79) states β^(4)_ij = ( (6/25) i G² m² M³ R t / (ℏ³ d_ij) )². Since i² = −1, this is a negative real number, not a pure phase or positive parameter. Eq. (80) then sets β^(4)_RL =: ϑ and writes √ϑ = (6/25) G² m² M³ R t/(ℏ³ d_RL), treating ϑ as a positive real number. This is not a harmless notational slip: the sign and imaginary unit determine whether the branch amplitude α_RL is 1 + ϑ, 1 − ϑ, or 1 + i times a real quantity, which affects the entanglement witness and the ϑ/φ comparison in Fig. 6. The authors should either define ϑ := |β^(4)_RL| and adjust the text, or restore the i consistently and explain how a negative real branch amplitude enters the final state.
  2. [Footnote 39; Section 5.2, Eqs. (17)–(18)] The conversion of the computed amplitude β into an observable spin-entanglement witness assumes that the post-measurement state is the pure state (18) with support only on |N⟩_{1i}|N⟩_{2j}. However, the same interaction Hamiltonian (43) at fourth order also produces number-transfer sectors |N+k⟩_{1i}|N−k⟩_{2j} and |N−k⟩_{1i}|N+k⟩_{2j}. Footnote 39 dismisses these as 'not seen in the experiment' because of different Stern-Gerlach deflections, but no amplitude, overlap, or postselection error estimate is supplied. If these components are not cleanly separated from the |N,N⟩ port, or if they are traced out, the reduced spin state is not generically the pure entangled state used in the witness, and the claimed entanglement can disappear. This is load-bearing: it is the only step that turns the formal fourth-order amplitude into a concrete experimental prediction. Please quantify the discarde
  3. [Abstract and Section 6] The title and abstract state that 'classical theories of gravity' produce entanglement, but the detailed calculation in Section 5.2 is performed for one specific sourcing prescription: semiclassical Einstein gravity with the Newtonian potential given by the expectation value (63). Appendix B.10 discusses stochastic variants, and Appendix D reviews consistency issues, but the universal claim is broader than the demonstrated case. If the paper is intended as a counterexample to the claim that no local classical gravity can entangle, this should be stated explicitly; if it is intended as a general theorem, additional argument is needed to show that the virtual-matter mechanism survives across the full class of classical theories.
minor comments (3)
  1. [Fig. 6 caption] The caption reads 'The ratio ϑ/φ ... is identical to ϑ/φ.' The second occurrence is presumably meant to be something else (e.g., the ratio of quantum and classical effects).
  2. [Around Eq. (16)] The notation 'κj≠λj' for non-overlapping spheres is confusing; it should be 'κ≠λ' or 'i≠j' depending on which indices label the objects and branches.
  3. [Acknowledgements] The text says 'Note that this preprint has not undergone peer review. The Version of Record of this article is published in Nature'. These two statements are inconsistent in a submission context. If this is a reprint of a published article, that should be disclosed; if it is a fresh submission, the preprint boilerplate should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central amplitude is derived from the stated Hamiltonian via a standard Dyson-series calculation and is not fitted to the target conclusion.

full rationale

The paper's derivation chain is self-contained. The central object, β^(4)_RL, is obtained by expanding the interaction-picture unitary (9) in a Dyson series, taking the classical-gravity Hamiltonian (5)/(43), specifying the semiclassical sourcing Φ(x) from ⟨T00⟩ (63), and evaluating the fourth-order amplitude (53) with stated non-relativistic and long-time approximations to reach Eq. (79). No parameter is fitted to the claimed outcome, and the target statement (that classical gravity can generate entanglement) is not assumed among the inputs. The quantum-gravity comparison amplitude γ^(2)_ij is likewise re-derived from the graviton propagator, not imported as a conclusion. Self-citations appearing in the paper (e.g., refs. [19], [24], [33], [49]) provide experimental context or prior comparative results, but the load-bearing perturbative calculation does not reduce to those citations. The discussion of |N±k⟩|N∓k⟩ states in footnote 39 and Appendix B.5 is a possible operational limitation concerning postselection in the spin witness, not a circularity: it concerns whether the computed amplitude translates to an observable entangled spin state, not whether that amplitude was assumed. The scope gap between 'classical theories' in the title and the specific semiclassical model analyzed is a breadth/evidence limitation, not a circular dependence of the derivation on its conclusion.

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

The calculation has no fitted free parameters; all experimental parameters are inputs. It rests on the validity of QFT in curved spacetime and on a specific classical-gravity sourcing assumption (semiclassical Einstein gravity). The most fragile adopted axiom is the discarding of number-transfer final states in the entanglement witness.

assumptions (5)
  • domain assumption QFT in a fixed classical background metric (h_μν a c-number) is a valid description of a classical theory of gravity at low energies.
    Section 4, Eq. (5): the interaction Hamiltonian H_int^(CG) = -1/2 ∫ h_μν T̂^μν treats h as classical.
  • domain assumption The classical gravitational potential is sourced by the expectation value of the matter energy-momentum tensor and is identical in every superposition branch (semiclassical Einstein gravity).
    Section 5.2, Eqs. (62)-(64): Φ(x)=ΦC1+ΦC2 with each the average of L/R potentials; the same Φ in all branches is asserted as the meaning of classical gravity.
  • domain assumption Non-relativistic wavepacket approximations R ≫ ħ/(mc), t ≪ 2mR²/ħ, and ct ≫ d_ij hold and justify replacing ω_k by mc²/ħ and using delta-function time integrals.
    Section 5.1, around Eqs. (28) and (57)-(60).
  • ad hoc to paper Final states with |N±k⟩|N∓k⟩ matter-number transfers can be discarded when computing the spin entanglement witness.
    Eq. (19) projects onto |N⟩|N⟩; footnote [39] dismisses |N-k⟩|N+k⟩ states as not seen in the experiment.
  • standard math Standard QFT Wick contraction, Feynman propagators, and Dyson series are valid for the amplitudes.
    Used throughout Section 5 and Appendix B.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Classical theories of gravity produce entanglement." pith.science (2026). https://pith.science/paper/UTUM7ZDD

@misc{pith2026251019714,
  author       = {Pith},
  title        = {Pith review of: Classical theories of gravity produce entanglement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTUM7ZDD}},
  note         = {Machine review of arXiv:2510.19714}
}
read the original abstract

The unification of gravity and quantum mechanics remains one of the most profound open questions in science. With recent advances in quantum technology, an experimental idea first proposed by Richard Feynman is now regarded as a promising route to testing this unification for the first time. The experiment involves placing a massive object in a quantum superposition of two locations and letting it gravitationally interact with another mass. In modern versions of the experiment, if the two objects subsequently become entangled, this is considered unambiguous evidence that gravity obeys the laws of quantum mechanics. This conclusion derives from theorems that treat a classical gravitational interaction as a local interaction capable of only transmitting classical, not quantum, information. Here, we argue that the classical gravitational interaction can transmit quantum information, and thus generate entanglement through physically local processes. The effects are found to scale differently to the considered quantum gravity effect, providing information on the form of the experiment required to evidence the quantum nature of gravity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

115 extracted references · 37 canonical work pages

  1. [1]

    In: Rovelli, C

    Rovelli, C.: Notes for a brief history of quantum gravity. In: Rovelli, C. (ed.) Quantum Gravity. World Scientific, Singapore (2001). https://doi.org/10.1142/9789812777386 0059 . https://www.worldscientific.com/doi/10.1142/9789812777386 0059

    work page doi:10.1142/9789812777386 2001

  2. [2]

    Carlip, S.: Is quantum gravity necessary? Classical and Quantum Gravity25(15), 154010 (2008) https://doi.org/10.1088/0264-9381/25/15/154010

    work page doi:10.1088/0264-9381/25/15/154010 2008

  3. [3]

    In: DeWitt, C.M., Rickles, D

    Feynman, R.: The role of gravitation in physics. In: DeWitt, C.M., Rickles, D. (eds.) Chapel Hill Conference Proceedings, pp. 250–256. Edition Open Access, North Carolina (1957). https://doi. org/10.34663/9783945561294-00 .http://www.edition-open-sources.org/sources/5/index.html

    work page doi:10.34663/9783945561294-00 1957

  4. [4]

    Bose, S., Mazumdar, A., Morley, G.W., Ulbricht, H., Toroˇ s, M., Paternostro, M., Geraci, A.A., Barker, P.F., Kim, M.S., Milburn, G.: Spin Entanglement Witness for Quantum Gravity. Phys. Rev. Lett.119, 240401 (2017) https://doi.org/10.1103/PhysRevLett.119.240401

    doi:10.1103/physrevlett.119.240401 2017

  5. [5]

    Marletto, C., Vedral, V.: Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity. Phys. Rev. Lett.119, 240402 (2017) https://doi.org/10.1103/PhysRevLett.119.240402

    doi:10.1103/physrevlett.119.240402 2017

  6. [6]

    arXiv preprint arXiv:1311.4558 (2013) https://doi.org/https://arxiv.org/abs/1311.4558 [quant-ph]

    Kafri, D., Taylor, J.: A noise inequality for classical forces. arXiv preprint arXiv:1311.4558 (2013) https://doi.org/https://arxiv.org/abs/1311.4558 [quant-ph]

    arXiv 2013

  7. [7]

    Kafri, D., Taylor, J.M., Milburn, G.J.: A Classical Channel Model for Gravitational Decoherence. New J. Phys.16(6), 065020 (2014) https://doi.org/10/gknst9

    2014

  8. [8]

    Krisnanda, T., Zuppardo, M., Paternostro, M., Paterek, T.: Revealing Non-Classicality of Inac- cessible Objects. Phys. Rev. Lett.119(12), 120402 (2017) https://doi.org/10.1103/physrevlett. 119.120402 arXiv:1607.01140

    arXiv 2017

Show all 115 references
  1. [9]

    Marletto, C., Vedral, V.: Witnessing nonclassicality beyond quantum theory. Phys. Rev. D102, 086012 (2020) https://doi.org/10.1103/PhysRevD.102.086012

    2020 doi

  2. [10]

    Quantum6, 779 (2022) https://doi.org/10.22331/q-2022-08-17-779

    Galley, T.D., Giacomini, F., Selby, J.H.: A no-go theorem on the nature of the gravitational field beyond quantum theory. Quantum6, 779 (2022) https://doi.org/10.22331/q-2022-08-17-779

    2022 doi

  3. [11]

    arXiv preprint arXiv:2507.13201 (2025) https://doi.org/10.48550/ arXiv.2507.13201

    Ludescher, S.L., Loveridge, L.D., Galley, T.D., M¨ uller, M.P.: Gravity-mediated entanglement via infinite-dimensional systems. arXiv preprint arXiv:2507.13201 (2025) https://doi.org/10.48550/ arXiv.2507.13201

    2025 doi arXiv

  4. [12]

    Bose, S., Fuentes, I., Geraci, A.A., Khan, S.M., Qvarfort, S., Rademacher, M., Rashid, M., Toroˇ s, M., Ulbricht, H., Wanjura, C.C.: Massive quantum systems as interfaces of quantum mechan- ics and gravity. Rev. Mod. Phys.97, 015003 (2025) https://doi.org/10.1103/RevModPhys.97. 015003

    2025 doi

  5. [13]

    Cambridge University Press, Cambridge (2023)

    Huggett, N., Linnemann, N., Schneider, M.D.: Quantum Gravity in a Laboratory? Elements in the Foundations of Contemporary Physics. Cambridge University Press, Cambridge (2023). https://doi.org/10.1017/9781009327541

    2023 doi

  6. [14]

    Classical and Quantum Gravity36(3), 034001 (2019) https://doi.org/10.1088/ 1361-6382/aaf9ca 20

    Carney, D., Stamp, P.C.E., Taylor, J.M.: Tabletop experiments for quantum gravity: a user’s manual. Classical and Quantum Gravity36(3), 034001 (2019) https://doi.org/10.1088/ 1361-6382/aaf9ca 20

    2019

  7. [15]

    Science367(6480), 892–895 (2020) https://doi.org/10.1126/science.aba3993

    Deli´ c, U., Reisenbauer, M., Dare, K., Grass, D., Vuleti´ c, V., Kiesel, N., Aspelmeyer, M.: Cooling of a levitated nanoparticle to the motional quantum ground state. Science367(6480), 892–895 (2020) https://doi.org/10.1126/science.aba3993

    2020 doi

  8. [16]

    Nature591(7849), 225–228 (2021) https://doi.org/10.1038/ s41586-021-03250-7

    Westphal, T., Hepach, H., Pfaff, J., Aspelmeyer, M.: Measurement of gravitational coupling between millimetre-sized masses. Nature591(7849), 225–228 (2021) https://doi.org/10.1038/ s41586-021-03250-7

    2021

  9. [17]

    Nature631, 1–6 (2024) https://doi.org/10.1038/ s41586-024-07561-3

    Panda, C.D., Tao, M.J., Ceja, M., Khoury, J., Tino, G.M., M¨ uller, H.: Measuring gravitational attraction with a lattice atom interferometer. Nature631, 1–6 (2024) https://doi.org/10.1038/ s41586-024-07561-3

    2024

  10. [18]

    Science Advances7(22), 2879 (2021) https: //doi.org/10.1126/sciadv.abg2879

    Margalit, Y., Dobkowski, O., Zhou, Z., Amit, O., Japha, Y., Moukouri, S., Rohrlich, D., Mazumdar, A., Bose, S., Henkel, C., Folman, R.: Realization of a complete Stern-Gerlach interferometer: Toward a test of quantum gravity. Science Advances7(22), 2879 (2021) https: //doi.org...

    2021 doi

  11. [19]

    Classical and Quantum Gravity40(4), 047001 (2023) https://doi

    Christodoulou, M., Biagio, A.D., Howl, R., Rovelli, C.: Gravity entanglement, quantum reference systems, degrees of freedom. Classical and Quantum Gravity40(4), 047001 (2023) https://doi. org/10.1088/1361-6382/acb0aa

    2023 doi

  12. [20]

    arXiv:1907.08994 (2019) https://doi.org/10.48550/arXiv.1907.08994

    Marletto, C., Vedral, V.: Answers to a few questions regarding the BMV experiment. arXiv:1907.08994 (2019) https://doi.org/10.48550/arXiv.1907.08994

    1907 doi arXiv

  13. [21]

    Bose, S., Mazumdar, A., Schut, M., Toroˇ s, M.: Mechanism for the quantum natured gravitons to entangle masses. Phys. Rev. D105, 106028 (2022) https://doi.org/10.1103/PhysRevD.105. 106028

    2022 doi

  14. [22]

    (ed.): Conceptual Foundations of Quantum Field Theory

    Cao, T. (ed.): Conceptual Foundations of Quantum Field Theory. Cambridge University Press, Cambridge (1999). https://doi.org/10.1017/CBO9780511470813

    1999 doi

  15. [23]

    PRX Quantum2(1), 010325 (2021) https://doi.org/ 10/gkq6wg arXiv:2004.01189

    Howl, R., Vedral, V., Naik, D., Christodoulou, M., Rovelli, C., Iyer, A.: Non-Gaussianity as a signature of a quantum theory of gravity. PRX Quantum2(1), 010325 (2021) https://doi.org/ 10/gkq6wg arXiv:2004.01189

    2021 arXiv

  16. [24]

    Christodoulou, M., Di Biagio, A., Aspelmeyer, M., Brukner, ˇC., Rovelli, C., Howl, R.: Locally mediated entanglement in linearized quantum gravity. Phys. Rev. Lett.130, 100202 (2023) https://doi.org/10.1103/PhysRevLett.130.100202

    2023 doi

  17. [25]

    Studies in History and Philosophy of Science94, 31–46 (2022) https://doi.org/10.1016/j.shpsa.2022.04.003

    Wallace, D.: Quantum gravity at low energies. Studies in History and Philosophy of Science94, 31–46 (2022) https://doi.org/10.1016/j.shpsa.2022.04.003

    2022 doi

  18. [26]

    Scholarpedia12(4), 32997 (2017) https://doi.org/10.4249/scholarpedia.32997

    Donoghue, J.: Quantum gravity as a low energy effective field theory. Scholarpedia12(4), 32997 (2017) https://doi.org/10.4249/scholarpedia.32997

    2017 doi

  19. [27]

    Addison-Wesley, Reading, USA (1995)

    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Addison-Wesley, Reading, USA (1995). https://doi.org/10.1201/9780429503559

    1995 doi

  20. [28]

    arXiv preprint arXiv:1311.4558, (2023) https://doi.org/10.48550/arXiv.2305.05645

    Di Biagio, A., Howl, R., Brukner, ˇC., Rovelli, C., Christodoulou, M.: Circuit locality from relativistic locality in scalar field mediated entanglement. arXiv preprint arXiv:1311.4558, (2023) https://doi.org/10.48550/arXiv.2305.05645

    2023 doi arXiv

  21. [29]

    In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J

    Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies, pp. 125–189. Springer, Cham, Switzerland (2015). https...

    2015 doi

  22. [30]

    Cambridge University Press, Cambridge, UK (1995)

    Weinberg, S.: The Quantum Theory of Fields, Volume 1: Foundations, 1st edn. Cambridge University Press, Cambridge, UK (1995). https://doi.org/10.1017/CBO9781139644167

    1995 doi

  23. [31]

    Wiley, Hoboken, NJ (2010)

    Mandl, F., Shaw, G.: Quantum Field Theory, 2nd edn. Wiley, Hoboken, NJ (2010)

    2010

  24. [32]

    Journal of Mathematical Physics 55(7), 075201 (2014) https://doi.org/10.1063/1.4878920

    Derezi´ nski, J.: Quantum fields with classical perturbations. Journal of Mathematical Physics 55(7), 075201 (2014) https://doi.org/10.1063/1.4878920

    2014 doi

  25. [33]

    Howl, R., Penrose, R., Fuentes, I.: Exploring the unification of quantum theory and general relativity with a Bose–Einstein condensate. New J. Phys.21(4), 043047 (2019) https://doi.org/ 10.1088/1367-2630/ab104a

    2019 doi

  26. [34]

    (ed.) When Zeh meets Feynman: How to avoid the appearance of a classical World in gravity experiments, pp

    Aspelmeyer, M.: In: Kiefer, C. (ed.) When Zeh meets Feynman: How to avoid the appearance of a classical World in gravity experiments, pp. 85–95. Springer, Cham (2022). https://doi.org/ 10.1007/978-3-030-88781-0 5

    2022 doi

  27. [35]

    https://arxiv.org/abs/2504.03956

    Lapponi, A., Ferreri, A., Bruschi, D.E.: Gravitational redshift via quantized linear gravity (2025). https://arxiv.org/abs/2504.03956

    2025

  28. [36]

    Proceedings of the Physical Society

    Gupta, S.N.: Quantization of Einstein’s Gravitational Field: Linear Approximation. Proceedings of the Physical Society. Section A65(3), 161 (1952) https://doi.org/10.1088/0370-1298/65/3/ 301

    1952 doi

  29. [37]

    Gupta, S.N.: Supplementary conditions in the quantized gravitational theory. Phys. Rev.172, 1303–1307 (1968) https://doi.org/10.1103/PhysRev.172.1303

    1968 doi

  30. [38]

    Advances in Applied Clifford Algebras 28(5), 1–29 (2018) https://doi.org/10.1007/s00006-018-0904-5

    Pavˇ siˇ c, M.: Localized states in quantum field theory. Advances in Applied Clifford Algebras 28(5), 1–29 (2018) https://doi.org/10.1007/s00006-018-0904-5

    2018 doi

  31. [39]

    These states are considered not seen in the experiment, for example, due to their different interactions with a magnetic field of a Stern- Gerlach device compared to|N⟩ 1i|N⟩ 2j

    In the non-relativistic limit, ˆHI would in principle also allow states of the form|N−k⟩ 1i|N+k⟩ 2j and|N+k⟩ 1i|N−k⟩ 2j, wherekis some integer. These states are considered not seen in the experiment, for example, due to their different interactions with a magnetic field of a S...

  32. [40]

    Oxford University Press, Oxford (2008)

    Maggiore, M.: Gravitational Waves Volume 1: Theory and Experiments. Oxford University Press, Oxford (2008)

    2008

  33. [41]

    DeWitt, B.S.: Quantum Theory of Gravity. II. The Manifestly Covariant Theory. Phys. Rev. 162, 1195–1239 (1967) https://doi.org/10.1103/PhysRev.162.1195

    1967 doi

  34. [42]

    Alternatively, higher order terms can be considered

    Note that for the phase factor to be≈e −imcx0/ℏ we generally also requiret≲2mR 2/ℏunless further constraints are applied. Alternatively, higher order terms can be considered

  35. [43]

    The approximation is good forR≫ℏ/(mc) andt≲2mR 2/ℏas required also for (28) as detailed above

  36. [44]

    Annals of Physics35(2), 167–208 (1965) https://doi.org/10.1016/0003-4916(65)90077-1

    Ogievetsky, V.I., Polubarinov, I.V.: Interacting field of spin 2 and the einstein equations. Annals of Physics35(2), 167–208 (1965) https://doi.org/10.1016/0003-4916(65)90077-1

    1965 doi

  37. [45]

    Rham, C., Gabadadze, G.: Generalization of the fierz-pauli action. Phys. Rev. D82, 044020 (2010) https://doi.org/10.1103/PhysRevD.82.044020

    2010 doi

  38. [46]

    Such as gravitational self-interactions of the matter objects. 22

  39. [47]

    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A65, 032314 (2002) https://doi.org/10.1103/PhysRevA.65.032314

    2002 doi

  40. [48]

    npj Quantum Information6(1), 12 (2020) https://doi.org/10/ggz5q7 arXiv:1906.08808

    Krisnanda, T., Tham, G.Y., Paternostro, M., Paterek, T.: Observable Quantum Entangle- ment Due to Gravity. npj Quantum Information6(1), 12 (2020) https://doi.org/10/ggz5q7 arXiv:1906.08808

    2020 arXiv

  41. [49]

    arXiv:2304.00734 (2023) https://doi.org/10.48550/arXiv.2304.00734 [quant-ph]

    Howl, R., Cooper, N., Hackerm¨ uller, L.: Gravitationally-induced entanglement in cold atoms. arXiv:2304.00734 (2023) https://doi.org/10.48550/arXiv.2304.00734 [quant-ph]

    2023 doi arXiv

  42. [50]

    Physics Letters B792, 64–68 (2019) https://doi.org/10/gj6ssc

    Christodoulou, M., Rovelli, C.: On the Possibility of Laboratory Evidence for Quantum Superposition of Geometries. Physics Letters B792, 64–68 (2019) https://doi.org/10/gj6ssc

    2019

  43. [51]

    and Marshman, Ryan J

    van de Kamp, Thomas W. and Marshman, Ryan J. and Bose, Sougato and Mazumdar, Anupam: Quantum gravity witness via entanglement of masses: Casimir screening. Phys. Rev. A102, 062807 (2020) https://doi.org/10.1103/PhysRevA.102.062807

    2020 doi

  44. [52]

    Contemporary Physics64(2), 147–163 (2023) https://doi.org/ 10.1080/00107514.2023.2286074 https://doi.org/10.1080/00107514.2023.2286074

    Pedernales, J.S., Plenio, M.B.: On the origin of force sensitivity in tests of quantum gravity with delocalised mechanical systems. Contemporary Physics64(2), 147–163 (2023) https://doi.org/ 10.1080/00107514.2023.2286074 https://doi.org/10.1080/00107514.2023.2286074

    2023

  45. [53]

    Bengyat, O., Di Biagio, A., Aspelmeyer, M., Christodoulou, M.: Gravity-mediated entanglement between oscillators as quantum superposition of geometries. Phys. Rev. D110, 056046 (2024) https://doi.org/10.1103/PhysRevD.110.056046

    2024 doi

  46. [54]

    A Spin Entanglement Witness for Quantum Gravity

    Anastopoulos, C., Hu, B.-L.: Comment on “A Spin Entanglement Witness for Quantum Gravity” and on “Gravitationally Induced Entanglement Between Two Massive Particles Is Sufficient Evidence of Quantum Effects in Gravity”. https://doi.org/10.48550/arXiv.1804.11315

    doi arXiv

  47. [55]

    Classical and Quantum Gravity38(15), 155012 (2021) https: //doi.org/10.1088/1361-6382/ac0bf9

    Anastopoulos, C., Lagouvardos, M., Savvidou, K.: Gravitational effects in macroscopic quantum systems: a first-principles analysis. Classical and Quantum Gravity38(15), 155012 (2021) https: //doi.org/10.1088/1361-6382/ac0bf9

    2021 doi

  48. [56]

    A VS Quantum Science4(4), 045601 (2022) https://doi.org/10.1116/5.0101334

    Fragkos, V., Kopp, M., Pikovski, I.: On inference of quantization from gravitationally induced entanglement. A VS Quantum Science4(4), 045601 (2022) https://doi.org/10.1116/5.0101334

    2022 doi

  49. [57]

    ForF(k 0) to approximateF(γ) we also consider that it is approximately constant over changes of order 1/(ct)

    Here, we have assumed thatt≫ℏ/(mc 2) for the delta function approximation to be accurate, which is easily satisfied. ForF(k 0) to approximateF(γ) we also consider that it is approximately constant over changes of order 1/(ct). Given the spatial variation of Φ(x) it is possible...

  50. [58]

    See Appendix B.3 for a derivation of this result in momentum space

  51. [59]

    Nuclear Physics40, 353–356 (1963) https://doi.org/10

    Rosenfeld, L.: On quantization of fields. Nuclear Physics40, 353–356 (1963) https://doi.org/10. 1016/0029-5582(63)90279-7

    1963

  52. [60]

    Colloques Internationaux CNRS91(1), 15–29 (1962)

    Møller, C.: Les th´ eories relativistes de la gravitation. Colloques Internationaux CNRS91(1), 15–29 (1962)

    1962

  53. [61]

    Classical and Quantum Gravity 35(24), 245008 (2018) https://doi.org/10.1088/1361-6382/aaea20

    Kent, A.: Simple refutation of the Eppley–Hannah argument. Classical and Quantum Gravity 35(24), 245008 (2018) https://doi.org/10.1088/1361-6382/aaea20

    2018 doi

  54. [62]

    Physical Review A72(1), 012108 (2005) https: 23 //doi.org/10.1103/PhysRevA.72.012108

    Kent, A.: Nonlinearity without superluminality. Physical Review A72(1), 012108 (2005) https: 23 //doi.org/10.1103/PhysRevA.72.012108

    2005 doi

  55. [63]

    (eds.) Coupling Quantum Matter and Gravity, pp

    Giulini, D., Großardt, A., Schwartz, P.K.: In: Pfeifer, C., L¨ ammerzahl, C. (eds.) Coupling Quantum Matter and Gravity, pp. 491–550. Springer, Cham (2023). https://doi.org/10.1007/ 978-3-031-31520-6 16

    2023

  56. [64]

    Physics Letters A120(8), 377–381 (1987) https://doi.org/10.1016/0375-9601(87)90681-5

    Di´ osi, L.: A universal master equation for the gravitational violation of quantum mechanics. Physics Letters A120(8), 377–381 (1987) https://doi.org/10.1016/0375-9601(87)90681-5

    1987 doi

  57. [65]

    Tilloy, A., Di´ osi, L.: Sourcing semiclassical gravity from spontaneously localized quantum matter. Phys. Rev. D93, 024026 (2016) https://doi.org/10.1103/PhysRevD.93.024026

    2016 doi

  58. [67]

    Classical and Quantum Gravity 42(13), 135010 (2025) https://doi.org/10.1088/1361-6382/ade589

    Carney, D., Matsumura, A.: Classical-quantum scattering. Classical and Quantum Gravity 42(13), 135010 (2025) https://doi.org/10.1088/1361-6382/ade589

    2025 doi

  59. [68]

    Journal of Physics: Conference Series880(1), 012021 (2017) https://doi.org/10.1088/1742-6596/880/1/012021

    Helou, B., Chen, Y.: Extensions of Born’s rule to non-linear quantum mechanics, some of which do not imply superluminal communication. Journal of Physics: Conference Series880(1), 012021 (2017) https://doi.org/10.1088/1742-6596/880/1/012021

    2017 doi

  60. [69]

    Note that unlike with the classical gravity amplitudeβ (2) ij above, there are no other Feynman diagram amplitudes at fourth order or lower forβ (4) RL to combine with to make a separable state

  61. [70]

    Haine, S.A.: Searching for signatures of quantum gravity in quantum gases. New J. Phys.23(3), 033020 (2021) https://doi.org/10.1088/1367-2630/abd97d

    2021 doi

  62. [71]

    PRX Quantum2(3), 030330 (2021) https://doi.org/10/gmvfjc arXiv:2101.11629

    Carney, D., M¨ uller, H., Taylor, J.M.: Using an Atom Interferometer to Infer Gravitational Entanglement Generation. PRX Quantum2(3), 030330 (2021) https://doi.org/10/gmvfjc arXiv:2101.11629

    2021 arXiv

  63. [72]

    arXiv preprint arXiv:2302.07283 (2023) https://doi.org/10.48550/arXiv

    Oppenheim, J., Weller-Davies, Z.: Covariant path integrals for quantum fields back-reacting on classical space-time. arXiv preprint arXiv:2302.07283 (2023) https://doi.org/10.48550/arXiv. 2302.07283 [gr-qc]

    2023 doi arXiv

  64. [73]

    Academic press, San Diego (2007)

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic press, San Diego (2007)

    2007

  65. [74]

    With the last assumption dropped, we would ultimately obtain: β∝ M m2t ℏ3 Z d3xd 3yd 3z f(x)f ′∗(y)Φ(z)Φ(y) rzx rzy 1 (rzy −r zx +iϵ) 2 − 1 (rzy +r zx)2 , wherer zy :=|z−y|andr zx :=|z−x|. Assumimng long times and that Φ(k) does not support extremely high momentum modes, as we...

  66. [75]

    We are also assuming sufficiently long times that energy-momentum is respected at the vertices to a very good approximation

    Note that since Φ(x) is assumed constant with time, no energy flows into the vertices. We are also assuming sufficiently long times that energy-momentum is respected at the vertices to a very good approximation. 24

  67. [76]

    Note that here ˆϕ(x) need not be the effective quantum field of the particles of the matter objects

  68. [77]

    Page, D.N., Geilker, C.D.: Indirect Evidence for Quantum Gravity. Phys. Rev. Lett.47, 979–982 (1981) https://doi.org/10.1103/PhysRevLett.47.979

    1981 doi

  69. [78]

    Oppenheim, J.: A Postquantum Theory of Classical Gravity? Phys. Rev. X13, 041040 (2023) https://doi.org/10.1103/PhysRevX.13.041040

    2023 doi

  70. [79]

    Classical and Quantum Gravity 34(19), 193002 (2017) https://doi.org/10.1088/1361-6382/aa864f

    Bassi, A., Großardt, A., Ulbricht, H.: Gravitational decoherence. Classical and Quantum Gravity 34(19), 193002 (2017) https://doi.org/10.1088/1361-6382/aa864f

    2017 doi

  71. [80]

    Tilloy, A., Di´ osi, L.: Principle of least decoherence for Newtonian semiclassical gravity. Phys. Rev. D96, 104045 (2017) https://doi.org/10.1103/PhysRevD.96.104045

    2017 doi

  72. [81]

    Di´ osi, L.: Models for Universal Reduction of Macroscopic Quantum Fluctuations. Phys. Rev. A 40(3), 1165–1174 (1989) https://doi.org/10/fv2d9m

    1989

  73. [82]

    General Relativity and Gravitation 28, 581–600 (1996) https://doi.org/10/d52jm5

    Penrose, R.: On Gravity’s Role in Quantum State Reduction. General Relativity and Gravitation 28, 581–600 (1996) https://doi.org/10/d52jm5

    1996

  74. [83]

    Angeli, O., Carlesso, M.: Entanglement in markovian hybrid classical-quantum theories of gravity. Phys. Rev. D112, 024047 (2025) https://doi.org/10.1103/jzht-fbwt

    2025 doi

  75. [84]

    Trillo, D., Navascu´ es, M.: Di´ osi-Penrose model of classical gravity predicts gravitationally induced entanglement. Phys. Rev. D111, 121101 (2025) https://doi.org/10.1103/PhysRevD. 111.L121101

    2025 doi

  76. [85]

    Assuming that att= 0 we are able to start in a quantum superposition state

  77. [86]

    Journal of Physics: Confer- ence Series1275(1), 012038 (2019) https://doi.org/10.1088/1742-6596/1275/1/012038

    Andersen, T.C.: Quantum statistics in Bohmian trajectory gravity. Journal of Physics: Confer- ence Series1275(1), 012038 (2019) https://doi.org/10.1088/1742-6596/1275/1/012038

    2019 doi

  78. [87]

    D¨ oner, M.K., Großardt, A.: Is gravitational entanglement evidence for the quantization of space- time? Foundations of Physics52(5), 101 (2022) https://doi.org/10.1007/s10701-022-00619-0

    2022 doi

  79. [88]

    Mart ´ ın-Mart ´ ınez, E., Perche, T.R.: What gravity mediated entanglement can really tell us about quantum gravity. Phys. Rev. D108, 101702 (2023) https://doi.org/10.1103/PhysRevD.108. L101702

    2023 doi

  80. [89]

    Kent, A.: Should We Necessarily Treat Masses as Localized When Analysing Tests of Quantum Gravity? https://doi.org/10.48550/arXiv.2405.20514

    doi arXiv

  81. [90]

    Franzmann, G.: To Be or Not to Be, but Where? https://doi.org/10.48550/arXiv.2405.21031

    doi arXiv

  82. [91]

    Marchese, M.M., Pl´ avala, M., Kleinmann, M., Nimmrichter, S.: Newton’s laws of motion gen- erating gravity-mediated entanglement. Phys. Rev. A111, 042202 (2025) https://doi.org/10. 1103/PhysRevA.111.042202

    2025

  83. [92]

    Telali, E., Perche, T.R., Mart ´ ın-Mart ´ ınez, E.: Causality in relativistic quantum interactions without mediators. Phys. Rev. D111, 085005 (2025) https://doi.org/10.1103/PhysRevD.111. 085005

    2025 doi

  84. [93]

    Foundations of Physics7(1), 51–68 (1977) https://doi.org/10.1007/BF00715241

    Eppley, K., Hannah, E.: The necessity of quantizing the gravitational field. Foundations of Physics7(1), 51–68 (1977) https://doi.org/10.1007/BF00715241

    1977 doi

  85. [94]

    Physics Letters A143(1), 1–2 (1990) https://doi.org/10.1016/0375-9601(90)90786-N 25

    Gisin, N.: Weinberg’s non-linear quantum mechanics and supraluminal communications. Physics Letters A143(1), 1–2 (1990) https://doi.org/10.1016/0375-9601(90)90786-N 25

    1990 doi

  86. [95]

    Huggett, N., Callender, C.: Why quantize gravity (or any other field for that matter)? Philosophy of Science68(S3), 382–394 (2001) https://doi.org/10.1086/392923

    2001 doi

  87. [96]

    Physical Review D—Particles, Fields, Gravitation, and Cosmology73(6), 064025 (2006) https://doi.org/10.1103/ PhysRevD.73.064025

    Mattingly, J.: Why Eppley and Hannah’s thought experiment fails. Physical Review D—Particles, Fields, Gravitation, and Cosmology73(6), 064025 (2006) https://doi.org/10.1103/ PhysRevD.73.064025

    2006

  88. [97]

    Ballentine, L.E.: The Statistical Interpretation of Quantum Mechanics. Rev. Mod. Phys.42, 358–381 (1970) https://doi.org/10.1103/RevModPhys.42.358

    1970 doi

  89. [98]

    Rozema, L.A., Darabi, A., Mahler, D.H., Hayat, A., Soudagar, Y., Steinberg, A.M.: Violation of heisenberg’s measurement-disturbance relationship by weak measurements. Phys. Rev. Lett. 109, 100404 (2012) https://doi.org/10.1103/PhysRevLett.109.100404

    2012 doi

  90. [99]

    New Journal of Physics16(11), 115007 (2014) https://doi.org/10.1088/1367-2630/ 16/11/115007

    Bahrami, M., Großardt, A., Donadi, S., Bassi, A.: The Schr¨ odinger–Newton equation and its foundations. New Journal of Physics16(11), 115007 (2014) https://doi.org/10.1088/1367-2630/ 16/11/115007

    2014 doi

  91. [100]

    Physical Review D59(4), 043505 (1998) https: //doi.org/10.48550/arXiv.gr-qc/9703041

    Kent, A.: Causality in time-neutral cosmologies. Physical Review D59(4), 043505 (1998) https: //doi.org/10.48550/arXiv.gr-qc/9703041

    1998 doi arXiv

  92. [101]

    Weinberg’s Nonlinear Quantum Mechanics and Einstein-Podolsky- Rosen paradox

    Mielnik, B.: Comments on: “Weinberg’s Nonlinear Quantum Mechanics and Einstein-Podolsky- Rosen paradox”, by Joseph Polchinski. arXiv preprint quant-ph/0012041 (2000) https://doi. org/10.48550/arXiv.quant-ph/0012041

    2000 doi arXiv

  93. [102]

    Physical Review 135(3B), 849 (1964) https://doi.org/10.1103/PhysRev.135.B849

    Mead, C.A.: Possible connection between gravitation and fundamental length. Physical Review 135(3B), 849 (1964) https://doi.org/10.1103/PhysRev.135.B849

    1964 doi

  94. [103]

    Proceedings of the Royal Society A474(2220), 20180501 (2018) https://doi.org/10.1098/rspa.2018.0501

    Kent, A.: Testing causal quantum theory. Proceedings of the Royal Society A474(2220), 20180501 (2018) https://doi.org/10.1098/rspa.2018.0501

    2018

  95. [104]

    Nature Physics17(1), 74–78 (2021) https://doi

    Donadi, S., Piscicchia, K., Curceanu, C., Di´ osi, L., Laubenstein, M., Bassi, A.: Underground test of gravity-related wave function collapse. Nature Physics17(1), 74–78 (2021) https://doi. org/10.1038/s41567-020-1008-4

    2021 doi

  96. [105]

    Nature Communications14(1), 7910 (2023) https://doi.org/10.1038/s41467-023-43348-2

    Oppenheim, J., Sparaciari, C., ˇSoda, B., Weller-Davies, Z.: Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity. Nature Communications14(1), 7910 (2023) https://doi.org/10.1038/s41467-023-43348-2

    2023 doi

  97. [106]

    Physical Review D110(8), 084052 (2024)

    Di´ osi, L.: Classical-quantum hybrid canonical dynamics and its difficulties with special and general relativity. Physical Review D110(8), 084052 (2024)

    2024

  98. [107]

    Di´ osi, L.: Is there a relativistic Gorini-Kossakowski-Lindblad-Sudarshan master equation? Phys. Rev. D106, 051901 (2022) https://doi.org/10.1103/PhysRevD.106.L051901

    2022 doi

  99. [108]

    arXiv:2402.17844 (2024) https://doi.org/10.48550/arXiv.2402

    Grudka, A., Morris, T.R., Oppenheim, J., Russo, A., Sajjad, M.: Renormalisation of postquantum-classical gravity. arXiv:2402.17844 (2024) https://doi.org/10.48550/arXiv.2402. 17844

    2024 doi arXiv

  100. [109]

    Prince- ton University Press, Princeton, NJ, USA (2016)

    Penrose, R.: Fashion, Faith, and Fantasy in the New Physics of the Universe. Prince- ton University Press, Princeton, NJ, USA (2016). https://doi.org/10.1515/9781400880287 . http://www.jstor.org/stable/j.ctvc775bn

    2016 doi

  101. [110]

    Houghton Mifflin Harcourt, Boston, MA (2007)

    Smolin, L.: The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. Houghton Mifflin Harcourt, Boston, MA (2007)

    2007

  102. [111]

    Basic Books, New York, NY, USA (2006).https://www.basicbooks.com/titles/peter- woit/not-even-wrong/9780465092765/

    Woit, P.: Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical 26 Law. Basic Books, New York, NY, USA (2006).https://www.basicbooks.com/titles/peter- woit/not-even-wrong/9780465092765/

    2006

  103. [112]

    arXiv preprint hep-th/9711029 (1997) https://doi

    Schwarz, J.H.: The status of string theory. arXiv preprint hep-th/9711029 (1997) https://doi. org/10.48550/arXiv.hep-th/9711029

    1997 doi arXiv

  104. [113]

    Classical and Quantum Gravity22(19), 193 (2005) https://doi.org/10.1088/0264-9381/22/19/R01

    Nicolai, H., Peeters, K., Zamaklar, M.: Loop quantum gravity: an outside view. Classical and Quantum Gravity22(19), 193 (2005) https://doi.org/10.1088/0264-9381/22/19/R01

    2005 doi

  105. [114]

    Lami, L., Pedernales, J.S., Plenio, M.B.: Testing the quantumness of gravity without entangle- ment. Phys. Rev. X14, 021022 (2024) https://doi.org/10.1103/PhysRevX.14.021022 Appendix AFurther detail on entanglement from quantum gravity in QFT Here, we further discuss the first...

    2024 doi

  106. [115]

    FromZ(t), for example, the energy of the interacting vacuum can be extracted [27]

    ˆHI (x0 2)· · ·ˆHI (x0 n)|0⟩,(132) with ˆHI (t) the interaction picture version of ˆHint. FromZ(t), for example, the energy of the interacting vacuum can be extracted [27]. We consider the case that the finite walls are in a superposition of locations with the matter objects, ...

  107. [116]

    where gravitational waves are scattered off masses and then precisely analysed (indeed, as argued in [96], no physical experiment may be in this regime). 45 Without updating the states or measurement process [62, 68, 101], we can see how a superluminal signalling could, in the...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.