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arxiv: 2605.05375 · v1 · submitted 2026-05-06 · ✦ hep-th · gr-qc

Recognition: unknown

Stochastic modes in postquantum classical gravity

Jonathan Oppenheim , Muhammad Sajjad
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:10 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords stochastic gravityclassical gravityquantum matter couplingmetric fluctuationspositive semi-definite actionLISA Pathfinder noisedecoherence boundslinearized gravity
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0 comments X

The pith

Consistency in coupling classical spacetime to quantum matter requires spacetime to evolve stochastically.

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

The paper establishes that a covariant theory treating gravity classically while coupling it to quantum fields cannot remain deterministic. Linearizing the metric fluctuations around flat space and decomposing them into scalar, vector, and tensor parts identifies dynamical stochastic modes: a spin-2 field and a spin-0 scalar that diffuse according to their wave equations. The effective action on these modes is shown to be positive semi-definite, satisfying a necessary condition for classical consistency, and explicit two-point functions are computed for the Newtonian potential. These predictions are compared directly to excess noise in LISA Pathfinder data and to decoherence rates, yielding bounds on the theory's two dimensionless coupling constants.

Core claim

Starting from the classical-quantum path integral, linearization around Minkowski space followed by scalar-vector-tensor decomposition yields stochastic modes consisting of a classical spin-2 field and a spin-0 scalar, both diffusing around their respective wave equations, together with non-dynamical vector and scalar fields; the action is positive semi-definite on the dynamical modes, the two-point function of Newtonian potential fluctuations matches forms testable against LISA Pathfinder noise, and consistency is verified across Onsager-Machlup, Martin-Siggia-Rose, and stochastic differential equation formulations.

What carries the argument

The stochastic modes identified through scalar-vector-tensor decomposition of linearized metric fluctuations, which enforce the randomness required by mathematical consistency.

If this is right

  • The power spectral density of Newtonian potential fluctuations directly constrains one combination of the two dimensionless couplings via LISA Pathfinder excess noise.
  • Bounds on stochastic gravitational wave energy density in an FLRW background constrain the second combination of couplings.
  • The effective action for matter distributions shows that decoherence experiments are limited by fluctuations in both the Newtonian potential and the curvature perturbation.
  • The pure gravity sector is consistent across the Onsager-Machlup action, the Martin-Siggia-Rose form, and the stochastic differential equation formulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If these stochastic modes are present, precision gravitational sensors could search for a specific noise spectrum distinct from both quantum gravity and classical noise sources.
  • The positivity requirement on the action could restrict the allowed range of couplings in cosmological models containing quantum matter.
  • Extending the same decomposition to backgrounds with curvature might reveal how the stochasticity scales near black holes or during inflation.
  • The non-dynamical vector and scalar fields could still contribute to effective matter interactions at short distances even if they carry no propagating degrees of freedom.

Load-bearing premise

Linearizing the metric around Minkowski space and applying the scalar-vector-tensor decomposition fully captures the stochastic dynamics without missing nonlinear or background-dependent effects that could change mode identification or positivity.

What would settle it

An experimental measurement of deterministic evolution in the Newtonian potential or gravitational wave modes at the amplitude predicted by the two-point function, or a direct detection of negative values in the power spectral density of metric fluctuations.

read the original abstract

We study fluctuations of the metric in the postquantum theory of classical gravity, a covariant theory which couples a classical spacetime with quantum matter fields. Mathematical consistency requires spacetime to evolve stochastically. Starting from the classical-quantum path integral, we linearize around Minkowski space and perform a scalar-vector-tensor decomposition, identifying the stochastic modes: a classical spin-2 field and spin-0 scalar, both diffusing around their respective wave equations. There is also a non-dynamical vector and scalar field. These are related to the degrees of freedom found in quadratic gravity, but here interpreted as stochastic contributions to spacetime. We show that the action is positive semi-definite (PSD) on all dynamical modes, which is a necessary condition for the theory to consistently treat spacetime classically. We compute the two-point function and power spectral density corresponding to fluctuations of the Newtonian potential, and compare it to the excess noise found in LISA Pathfinder. This sets a bound on one combination of the two dimensionless coupling constants of the theory, while bounds on the stochastic gravitational wave energy density in a FLRW background constrain another combination. We derive the effective action for matter distributions, and find that bounds from decoherence experiments are constrained by fluctuations in the Newtonian potential $\Phi$ and the curvature perturbation $\psi$. Finally, we show consistency between different formulations of the pure gravity theory, the Onsager-Machlup form of the action, the Martin-Siggia-Rose form, and that given by stochastic differential equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that mathematical consistency of the classical-quantum path integral in postquantum classical gravity requires spacetime to evolve stochastically. Starting from the path integral, the authors linearize the metric fluctuations around Minkowski space, perform a scalar-vector-tensor decomposition, and identify dynamical stochastic modes: a classical spin-2 graviton and a spin-0 scalar, both satisfying diffusive wave equations, along with non-dynamical vector and scalar fields. They demonstrate that the quadratic action is positive semi-definite on the dynamical modes, compute the two-point function and power spectral density of Newtonian potential fluctuations, and use LISA Pathfinder noise data plus decoherence bounds to constrain the two dimensionless coupling constants. Consistency is shown between the Onsager-Machlup, Martin-Siggia-Rose, and stochastic differential equation formulations of the pure gravity sector.

Significance. If the central results hold, the work supplies a concrete mechanism by which consistency between classical gravity and quantum matter forces stochastic metric fluctuations, with the PSD property providing a necessary condition for a consistent classical treatment. Explicit two-point functions and direct comparison to LISA Pathfinder excess noise yield falsifiable bounds on the couplings, while the effective action for matter distributions links to decoherence phenomenology. The explicit verification of equivalence among three stochastic formulations is a technical strength that enhances internal coherence.

major comments (2)
  1. [§3] §3 (linearization and SVT decomposition): The necessity of stochastic evolution is asserted for the full theory, yet the PSD property and mode identification are derived exclusively for the quadratic action after linearization around Minkowski. No explicit check is performed on cubic or higher-order terms in the metric fluctuations that could generate sign-indefinite contributions on the spin-2 or spin-0 sectors, which would undermine the claim that consistency requires stochasticity beyond the linearized regime.
  2. [§4] §4 (PSD proof): The demonstration that the action is positive semi-definite is restricted to the dynamical modes after SVT decomposition; it is not shown whether the non-dynamical vector and scalar fields remain non-propagating or could mix with dynamical modes at higher order, potentially affecting the overall consistency argument for classical spacetime.
minor comments (2)
  1. [§2] The two dimensionless coupling constants are introduced in the abstract and used for bounds in §5, but their explicit definitions and normalization conventions should be stated in the introduction or §2 to aid readability.
  2. [§5] The comparison of the Newtonian potential two-point function to LISA Pathfinder data in §5 would benefit from a brief statement of the precise frequency window and noise model assumptions employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and have revised the paper accordingly to clarify the scope of our results.

read point-by-point responses

  1. Referee: [§3] §3 (linearization and SVT decomposition): The necessity of stochastic evolution is asserted for the full theory, yet the PSD property and mode identification are derived exclusively for the quadratic action after linearization around Minkowski. No explicit check is performed on cubic or higher-order terms in the metric fluctuations that could generate sign-indefinite contributions on the spin-2 or spin-0 sectors, which would undermine the claim that consistency requires stochasticity beyond the linearized regime.

    Authors: We agree that the explicit demonstration of the positive semi-definiteness and the identification of the stochastic modes are performed at the linearized level around Minkowski space. The assertion that mathematical consistency requires stochastic evolution originates from the structure of the classical-quantum path integral in the full theory, but the detailed analysis of the action's properties is carried out after linearization. Higher-order terms in the metric fluctuations are not examined in this work, as our focus is on the leading-order stochastic fluctuations and their observational implications. We have revised the manuscript in the introduction and section 3 to more precisely state that our results on the necessity of stochasticity and the PSD property hold within the linearized approximation, and that extending this to the full nonlinear theory is left for future investigation. revision: yes

  2. Referee: [§4] §4 (PSD proof): The demonstration that the action is positive semi-definite is restricted to the dynamical modes after SVT decomposition; it is not shown whether the non-dynamical vector and scalar fields remain non-propagating or could mix with dynamical modes at higher order, potentially affecting the overall consistency argument for classical spacetime.

    Authors: In the linearized theory, the vector and scalar fields are indeed non-dynamical following the SVT decomposition, and the PSD property is verified specifically for the dynamical spin-2 and spin-0 modes. We do not claim that these non-dynamical fields remain non-propagating at all orders; our analysis is confined to the quadratic action. We have updated section 4 to include a statement clarifying that the non-dynamical nature and the PSD proof apply to the linear regime, and that potential mixing at higher orders would require a separate nonlinear analysis, which is outside the present scope. This clarification ensures the consistency argument is appropriately bounded to the linearized fluctuations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via path integral and external constraints

full rationale

The paper starts from the classical-quantum path integral, linearizes around Minkowski, performs SVT decomposition to identify dynamical modes (spin-2 and spin-0), and demonstrates that the quadratic action is positive semi-definite on those modes as a consistency requirement for classical spacetime. Coupling bounds are extracted from independent external data (LISA Pathfinder excess noise and decoherence experiments), not from internal fits. No quoted step reduces a claimed prediction or necessity result to a definition, a fitted parameter, or a self-citation chain; the central consistency argument retains independent content from the explicit linear analysis and external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the prior postquantum classical gravity framework (assumed consistent), the validity of linearization, and the interpretation of quadratic-gravity-like degrees of freedom as stochastic. Two dimensionless coupling constants are introduced and bounded rather than derived.

free parameters (1)
  • two dimensionless coupling constants
    The abstract states that LISA Pathfinder and FLRW bounds constrain combinations of these two constants; they are not derived from first principles but fitted or bounded by data.
axioms (2)
  • domain assumption The postquantum classical gravity theory is mathematically consistent only when spacetime evolves stochastically.
    Invoked in the opening sentence as a requirement for consistency of the classical-quantum coupling.
  • domain assumption Linearization around Minkowski space plus SVT decomposition fully captures the stochastic dynamics.
    Used to identify the modes; no justification given in abstract for neglecting nonlinear terms.

pith-pipeline@v0.9.0 · 5562 in / 1507 out tokens · 26680 ms · 2026-05-08T16:10:26.876933+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 30 canonical work pages · 2 internal anchors

  1. [1]

    Stochastic quantum dynamics and relativity,

    N. Gisin, “Stochastic quantum dynamics and relativity,” Helv. Phys. Acta62, 363–371 (1989)

    1989

  2. [2]

    Indirect evidence for quantum gravity,

    Don N Page and CD Geilker, “Indirect evidence for quantum gravity,” Physical Review Letters47, 979 (1981)

    1981

  3. [3]

    On the interaction between classical and quantum systems,

    Philippe Blanchard and Arkadiusz Jadczyk, “On the interaction between classical and quantum systems,” Physics Letters A175, 157–164 (1993)

    1993

  4. [4]

    Diosi, Quantum dynamics with two planck constants and the semiclassical limit, arXiv preprint quant-ph/9503023 (1995)

    Lajos Diosi, “Quantum dynamics with two Planck constants and the semiclassical limit,” (1995), arXiv:quant-ph/9503023

    work page arXiv 1995

  5. [5]

    Coupling classical and quantum variables using continuous quantum measurement theory,

    Lajos Di´ osi and Jonathan J Halliwell, “Coupling classical and quantum variables using continuous quantum measurement theory,” Physical review letters81, 2846 (1998)

    1998

  6. [6]

    A classical channel model for gravitational decoherence

    D Kafri, J M Taylor, and G J Milburn, “A classical channel model for gravitational decoherence,” New Journal of Physics 16, 065020 (2014), arXiv:1401.0946 [quant-ph]

    work page Pith review arXiv 2014

  7. [7]

    Sourcing semiclassical gravity from spontaneously localized quantum matter.Phys

    Antoine Tilloy and Lajos Di´ osi, “Sourcing semiclassical gravity from spontaneously localized quantum matter,” Physical Review D93(2016), 10.1103/physrevd.93.024026, arXiv:1509.08705 [quant-ph]

    work page doi:10.1103/physrevd.93.024026 2016

  8. [8]

    A Postquantum Theory of Classical Gravity?,

    Jonathan Oppenheim, “A postquantum theory of classical gravity?” Physical Review X13, 041040 (2023), arXiv:1811.03116, arXiv:1811.03116

    work page arXiv 2023

  9. [9]

    Oppenheim, C

    Jonathan Oppenheim, Carlo Sparaciari, Barbara ˇSoda, and Zachary Weller-Davies, “The two classes of hybrid classical- quantum dynamics,” (2022), arXiv:2203.01332 [quant-ph]

    work page arXiv 2022

  10. [10]

    Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity,

    Jonathan Oppenheim, Carlo Sparaciari, Barbara ˇSoda, and Zachary Weller-Davies, “Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity,” Nature Communications14, 7910 (2023), arXiv:2203.01982, arXiv:2203.01982

    work page arXiv 2023

  11. [11]

    Any consistent coupling between classical gravity and quantum matter is fundamentally irreversible,

    Thomas D. Galley, Flaminia Giacomini, and John H. Selby, “Any consistent coupling between classical gravity and quantum matter is fundamentally irreversible,” Quantum7, 1142 (2023), arXiv:2301.10261 [quant-ph]

    work page arXiv 2023

  12. [12]

    Principle of least decoherence for newtonian semiclassical gravity,

    Antoine Tilloy and Lajos Di´ osi, “Principle of least decoherence for newtonian semiclassical gravity,” Physical Review D 96, 104045 (2017)

    2017

  13. [13]

    A healthier semi-classical dynamics,

    Isaac Layton, Jonathan Oppenheim, and Zachary Weller-Davies, “A healthier semi-classical dynamics,” Quantum8, 1565 (2024), arXiv:2208.11722 [quant-ph]

    work page arXiv 2024

  14. [14]

    Covariant path integrals for quantum fields back-reacting on classical space-time,

    Jonathan Oppenheim and Zachary Weller-Davies, “Covariant path integrals for quantum fields back-reacting on classical space-time,” arXiv:2302.07283 (2023), arXiv:2302.07283

    work page arXiv 2023

  15. [15]

    Renormalisation of postquantum-classical gravity,

    Andrzej Grudka, Jonathan Oppenheim, Andrea Russo, and Muhammad Sajjad, “Renormalisation of postquantum-classical gravity,” (2024), arXiv:2402.17844 [hep-th]

    work page arXiv 2024

  16. [16]

    Decoherence of quantum fields induced by classical gravity,

    J. Oppenheim, C. Sparaciari, ˇSoda, and Z. Weller-Davies, “Decoherence of quantum fields induced by classical gravity,” (2022), manuscript in preparation

    2022

  17. [17]

    Current experimental upper bounds on spacetime diffusion,

    Martijn Janse, Dennis G Uitenbroek, Loek van Everdingen, Jaimy Plugge, Bas Hensen, and Tjerk H Oosterkamp, “Current experimental upper bounds on spacetime diffusion,” arXiv preprint arXiv:2403.08912 (2024), arXiv:2403.08912

    work page arXiv 2024

  18. [18]

    Spin entanglement witness for quantum gravity,

    Sougato Bose, Anupam Mazumdar, Gavin W Morley, Hendrik Ulbricht, Marko Toroˇ s, Mauro Paternostro, Andrew A Geraci, Peter F Barker, MS Kim, and Gerard Milburn, “Spin entanglement witness for quantum gravity,” Physical review letters119, 240401 (2017)

    2017

  19. [19]

    Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity

    C. Marletto and V. Vedral, “Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity,” Phys. Rev. Lett.119, 240402 (2017), arXiv:1707.06036 [quant-ph]

    work page Pith review arXiv 2017

  20. [20]

    Tabletop experiments for quantum gravity: a user’s manual,

    Daniel Carney, Philip CE Stamp, and Jacob M Taylor, “Tabletop experiments for quantum gravity: a user’s manual,” Classical and Quantum Gravity36, 034001 (2019)

    2019

  21. [21]

    Testing the quantumness of gravity without entanglement,

    Ludovico Lami, Julen S. Pedernales, and Martin B. Plenio, “Testing the quantumness of gravity without entanglement,” Physical Review X14(2024), 10.1103/physrevx.14.021022, arXiv:2302.03075 [quant-ph]

    work page doi:10.1103/physrevx.14.021022 2024

  22. [22]

    Distinguishable Consequence of Classical Gravity on Quantum Matter,

    Serhii Kryhin and Vivishek Sudhir, “Distinguishable consequence of classical gravity on quantum matter,” Phys. Rev. Lett.134, 061501 (2025), arXiv:2309.09105 [gr-qc]

    work page arXiv 2025

  23. [23]

    Quantum interference of large organic molecules,

    Stefan Gerlich, Sandra Eibenberger, Mathias Tomandl, Stefan Nimmrichter, Klaus Hornberger, Paul Fagan, Jens T¨ uxen, 21 Marcel Mayor, and Markus Arndt, “Quantum interference of large organic molecules,” Nature communications2, 263 (2011)

    2011

  24. [24]

    In-depth analysis of lisa pathfinder performance results: Time evolution, noise projection, physical models, and implications for lisa,

    M Armano, H Audley, J Baird, P Binetruy, M Born, D Bortoluzzi, E Castelli, A Cavalleri, A Cesarini, V Chiavegato, et al., “In-depth analysis of lisa pathfinder performance results: Time evolution, noise projection, physical models, and implications for lisa,” Physical Review D110, 042004 (2024)

    2024

  25. [25]

    The weak field limit of quantum matter back-reacting on classical spacetime,

    Isaac Layton, Jonathan Oppenheim, Andrea Russo, and Zachary Weller-Davies, “The weak field limit of quantum matter back-reacting on classical spacetime,” (2023)

    2023

  26. [26]

    Geoff Penington, public communicationhttps://twitter.com/quantum_geoff/status/1765886036956299595(2024)

    work page arXiv 2024

  27. [27]

    Diffusion in the stochastic Klein-Gordon equation,

    Jonathan Oppenheim and Emanuele Panella, “Diffusion in the stochastic Klein-Gordon equation,” (2025), arXiv:2511.10738 [gr-qc]

    work page arXiv 2025

  28. [28]

    Emergence of phantom cold dark matter from spacetime diffusion,

    Jonathan Oppenheim, Emanuele Panella, and Andrew Pontzen, “Emergence of phantom cold dark matter from spacetime diffusion,” Physical Review D (2024), to appear, arXiv:2407.13820 [gr-qc]

    work page arXiv 2024

  29. [29]

    Testing post-quantum gravity with gravitational waves,

    Yuki Hirotani and Akira Matsumura, “Testing post-quantum gravity with gravitational waves,” (2026), arXiv:2603.29230 [gr-qc]

    work page arXiv 2026

  30. [30]

    Minimal noise in non-quantized gravity,

    Giuseppe Fabiano, Tomohiro Fujita, Akira Matsumura, and Daniel Carney, “Minimal noise in non-quantized gravity,” (2026), arXiv:2603.26075 [quant-ph]

    work page internal anchor Pith review arXiv 2026

  31. [31]

    Diffeomorphism invariant classical-quantum path integrals for nordstrom gravity,

    Jonathan Oppenheim, Andrea Russo, and Zachary Weller-Davies, “Diffeomorphism invariant classical-quantum path integrals for nordstrom gravity,” (2024), arXiv:2401.05514 [gr-qc]

    work page arXiv 2024

  32. [32]

    Is there a relativistic gorini-kossakowski-lindblad-sudarshan master equation?

    Lajos Di´ osi, “Is there a relativistic gorini-kossakowski-lindblad-sudarshan master equation?” Physical Review D106(2022), 10.1103/physrevd.106.l051901, arXiv:2206.05701 [quant-ph]

    work page doi:10.1103/physrevd.106.l051901 2022

  33. [33]

    Classical-quantum scattering,

    Daniel Carney and Akira Matsumura, “Classical-quantum scattering,” (2024), arXiv:2412.04839 [hep-th]

    work page arXiv 2024

  34. [34]

    Path integrals and the indefiniteness of the gravitational action,

    G. W. Gibbons, S. W. Hawking, and M. J. Perry, “Path integrals and the indefiniteness of the gravitational action,” Nuclear Physics B138, 141–150 (1978)

    1978

  35. [35]

    The path integral measure, conformal factor problem and stability of the ground state of quantum gravity,

    Pawel O. Mazur and Emil Mottola, “The path integral measure, conformal factor problem and stability of the ground state of quantum gravity,” Nuclear Physics B341, 187–212 (1990)

    1990

  36. [36]

    Marolf,Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals,JHEP07(2022) 108 [2203.07421]

    Donald Marolf, “Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean saddles from Lorentzian path integrals,” Journal of High Energy Physics2022, 108 (2022), arXiv:2203.07421

    work page arXiv 2022

  37. [37]

    In-depth analysis of lisa pathfinder performance results: Time evolution, noise projection, physical models, and implications for lisa,

    M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, V. Chiavegato, A.M. Cruise, D. Dal Bosco, K. Danzmann, M. De Deus Silva, I. Diepholz, G. Dixon, R. Dolesi, L. Ferraioli, V. Ferroni, E.D. Fitzsimons, M. Freschi, L. Gesa, D. Giardini, F. Gibert, R. Giusteri, C. Grimani, J. Grzymisch, I. Harrison, M...

    work page doi:10.1103/physrevd.110.042004 2024

  38. [38]

    Renormalization of higher-derivative quantum gravity,

    Kellogg S Stelle, “Renormalization of higher-derivative quantum gravity,” Physical Review D16, 953 (1977)

    1977

  39. [39]

    Quadratic gravity,

    Alberto Salvio, “Quadratic gravity,” Frontiers in Physics (2018)

    2018

  40. [40]

    Aspects of quadratic gravity,

    Luis Alvarez-Gaume, Alex Kehagias, Costas Kounnas, Dieter L¨ ust, and Antonio Riotto, “Aspects of quadratic gravity,” Fortschritte der Physik64, 176–189 (2016)

    2016

  41. [41]

    On the degrees of freedom ofr 2 gravity in flat spacetime,

    Anamaria Hell, Dieter Lust, and George Zoupanos, “On the degrees of freedom ofr 2 gravity in flat spacetime,” (2023), arXiv:2311.08216 [hep-th]

    work page arXiv 2023

  42. [42]

    Stochastic Gravity: Theory and Applications,

    B. L. Hu and E. Verdaguer, “Stochastic Gravity: Theory and Applications,” Living Rev. Rel.11, 3 (2008), arXiv:0802.0658 [gr-qc]

    work page arXiv 2008

  43. [43]

    Stochastic gravity,

    J. W. Moffat, “Stochastic gravity,” Physical Review D56, 6264–6277 (1997), arXiv:gr-qc/9610067 [gr-qc]

    work page arXiv 1997

  44. [44]

    A path integral for completely positive classical-quantum dynamics,

    Jonathan Oppenheim and Zachary Weller-Davies, “A path integral for completely positive classical-quantum dynamics,” (2022), manuscript in preparation

    2022

  45. [45]

    The theory of a general quantum system interacting with a linear dissipative system,

    R.P Feynman and F.L Vernon, “The theory of a general quantum system interacting with a linear dissipative system,” Annals of Physics24, 118–173 (1963)

    1963

  46. [46]

    Fluctuations and irreversible processes,

    L. Onsager and S. Machlup, “Fluctuations and irreversible processes,” Phys. Rev.91, 1505–1512 (1953)

    1953

  47. [47]

    Masud Chaichian and Andrey Demichev,Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics, Series in Mathematical and Computational Physics (CRC Press, 2018)

    2018

  48. [48]

    Quantum theory of gravity. ii. the manifestly covariant theory,

    Bryce S. DeWitt, “Quantum theory of gravity. ii. the manifestly covariant theory,” Phys. Rev.162, 1195–1239 (1967)

    1967

  49. [49]

    Classical gravity with higher derivatives,

    KS Stelle, “Classical gravity with higher derivatives,” General Relativity and Gravitation9, 353–371 (1978)

    1978

  50. [50]

    Angular momentum content of a virtual graviton,

    Duane A. Dicus and Scott Willenbrock, “Angular momentum content of a virtual graviton,” Physics Letters B609, 372–376 (2005). [51]α≥0 throughout

    2005

  51. [51]

    Dynamical structure and definition of energy in general relativity,

    R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical structure and definition of energy in general relativity,” Phys. Rev. 116, 1322–1330 (1959)

    1959

  52. [52]

    The stochastic wave equation with fractional noise: A random field approach,

    Raluca M. Balan and Ciprian A. Tudor, “The stochastic wave equation with fractional noise: A random field approach,” Stochastic Processes and their Applications120, 2468–2494 (2010)

    2010

  53. [53]

    Republication of: The dynamics of general relativity,

    Richard Arnowitt, Stanley Deser, and Charles W Misner, “Republication of: The dynamics of general relativity,” General 22 Relativity and Gravitation40, 1997–2027 (2008)

    1997

  54. [54]

    Anomalous contribution to galactic rotation curves due to stochastic spacetime,

    Jonathan Oppenheim and Andrea Russo, “Anomalous contribution to galactic rotation curves due to stochastic spacetime,” arXiv preprint (2024), arXiv:2402.19459 [gr-qc]

    work page arXiv 2024

  55. [55]

    arXiv e-prints , keywords =

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), “Upper Limits on the Isotropic Gravitational-Wave Background from the first part of LIGO, Virgo, and KAGRA’s fourth Observing Run,” (2025), arXiv:2508.20721 [gr-qc]

    work page arXiv 2025

  56. [56]

    Muhammad Sajjad Jonathan Oppenheim, Manuscript in preparation

  57. [57]

    Canonical quantum gravity and the problem of time,

    Chris J. Isham, “Canonical quantum gravity and the problem of time,” (1992), arXiv:gr-qc/9210011 [gr-qc]

    work page internal anchor Pith review arXiv 1992

  58. [58]

    Time and interpretation of quantum gravity proc. 4th canadian conf. on general relativity and relativistic astrophyscis,

    K Kuchar, “Time and interpretation of quantum gravity proc. 4th canadian conf. on general relativity and relativistic astrophyscis,” (1992)

    1992

  59. [59]

    On a lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,

    Hans-Karl Janssen, “On a lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties,” Z Physik B23, 377–380 (1976)

    1976

  60. [60]

    TECHNIQUES DE RENORMALISATION DE LA TH ´EORIE DES CHAMPS ET DYNAMIQUE DES PH´ENOM`ENES CRITIQUES,

    C de Dominicis, “TECHNIQUES DE RENORMALISATION DE LA TH ´EORIE DES CHAMPS ET DYNAMIQUE DES PH´ENOM`ENES CRITIQUES,” J. Phys., Colloq.37, C1–247–C1–253 (1976)

    1976

  61. [61]

    Statistical dynamics of classical systems,

    Paul Cecil Martin, ED Siggia, and HA Rose, “Statistical dynamics of classical systems,” Physical Review A8, 423 (1973)

    1973

  62. [62]

    T¨ auber,Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, 2014)

    Uwe C. T¨ auber,Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, 2014)

    2014

  63. [63]

    [65]G ρσ˜hρσ = 4(˜Φ△ψ+ ˜ψ((△ −3∂ 2 0)ψ− △Φ)

    This is equivalent to adding the term−βg µν to the deWitt metric and setting all scalar potentials apart fromψ, ϕto zero. [65]G ρσ˜hρσ = 4(˜Φ△ψ+ ˜ψ((△ −3∂ 2 0)ψ− △Φ)

  64. [64]

    The equations of motion also clearly differ from the Einstein equations for the scalar sector

  65. [65]

    Without introducing additional fields as in the Stueckelberg trick

  66. [66]

    K. J. Barnes, Ph.D. thesis (1963), unpublished

    1963

  67. [67]

    Lagrangian theory for neutral massive spin-2 fields,

    RJ Rivers, “Lagrangian theory for neutral massive spin-2 fields,” Il Nuovo Cimento (1955-1965)34, 386–403 (1964)

    1955

  68. [68]

    On ghost-free tensor lagrangians and linearized gravitation,

    Peter Van Nieuwenhuizen, “On ghost-free tensor lagrangians and linearized gravitation,” Nuclear Physics B60, 478–492 (1973)

    1973

  69. [69]

    Quantum Gravity with Higher Derivative Terms,

    J. Julve and M. Tonin, “Quantum Gravity with Higher Derivative Terms,” Nuovo Cim. B46, 137–152 (1978). 23 Appendix A: The Onsager Machlup and the Janssen–DeDominicis path integrals In this section we motivate the stochastic gravitational path integral, and introduce an alternate formalism, the Janssen-de-Dominicis (JD) [60],[61] or Martin-Siggia-Rose (MSR...

    1978

  70. [70]

    The metric,gmay be considered as a point in the Wheeler-DeWitt superspace, and the measure constructed on the space of the deformations at that given point

    The Gravitational Path Integral Measure In the section, we review the path integral measure for the gravitational action, largely following the presentation in [35]. The metric,gmay be considered as a point in the Wheeler-DeWitt superspace, and the measure constructed on the space of the deformations at that given point. The space of metrics at a point (b...

  71. [71]

    Choice of discretization Here we show that the choice of discretization does not matter, mainly based off the presentation in [63]. Consider the following Langevin equation ϕi+1 −ϕ i ϵ =αF i+1 + (1−α)F i +ξ i (A41) where 0≤α≤1 andF i is the force att i, and note that the choice of discretization is equivalent to setting the value of the Heaviside function...

  72. [72]

    It is convenient to do so, as working with second order SDEs requires one to compute a Jacobian that is non-lower triangular

    Second Order SDEs and Heat Baths Since the stochastic path integral in either of its iterations (OM or JD) is formulated using Stochastic differential equations that are first order in time, let us explicitly write down the second order equations as first order ones. It is convenient to do so, as working with second order SDEs requires one to compute a Ja...

  73. [73]

    Pole Prescription Here we discuss the pole-prescriptions for the two-point function, as justified in [27]. The pole-prescription for the infinite time two-point function is given by 1 p4 = 1 (k2 −(ω−iϵ) 2)(k2 −(ω+iϵ) 2) (A51) We can obtain this directly from the action (A13), where the limits of integration are from±∞, and we can work in momentum space: ˆ...

  74. [74]

    Calculational Details for the Spectral Acceleration For the two point function, we assume that it is the convolution of two retarded Green’s functions: ⟨ΦΦ⟩= Z ∞ −∞ d3z Z tf 0 dz0Gret(x−z)G ret(y−z) = Z ∞ −∞ d3z Z tf 0 dz0 Z d4p (2π)4 Z d4p′ (2π)4 G(p)G(p′) (A57) where G(p)G(p′) = 1 8α(α−3β) (8(α−3β) k2k′2 + 4ω2ω′2(α−3β) k2k′2p2p′2 + (−2α+ 8β) p2p′2 ) (A5...

  75. [75]

    Spin Projectors in Linearized Gravity The Barnes–Rivers spin projection operators provide a standard decomposition of symmetric rank-2 tensor fields in momentum space [68, 69], and have been widely used in higher-derivative gravity theories [70, 71]

  76. [76]

    Transverse and Longitudinal Projectors Given a momentumk µ withk 2 =k µkµ, define the transverse and longitudinal rank-2 projectors: θµν =η µν − kµkν k2 ,(A80) ωµν = kµkν k2 ,(A81) satisfyingθ µν +ω µν =η µν,θ µαθρν =θ µν,ω µαωρν =ω µν,θ µαωρν = 0. 34

  77. [77]

    Space-time Representation We may write the following in space-time as θµν =η µν − ∂µ∂ν 22 ,(A82) ωµν = ∂µ∂ν 22 ,(A83) where the expressions above are to be understood more properly as the Fourier transform of the momentum-space projectors

  78. [78]

    Barnes–Rivers Spin Projection Operators The identity operator on the space of symmetric rank-2 tensors is Iµν,ρσ = 1 2 ηµρηνσ +η µσηνρ .(A84) It decomposes into four orthogonal spin projectors: Spin-2 (transverse traceless): P (2) µν,ρσ = 1 2 θµαθνβ +θ µβθνα − 1 3 θµν θρσ.(A85) Spin-1 (transverse vector): P (1) µν,ρσ = 1 2 θµαωνβ +θ µβωνα +θ ναωµβ +θ νβ ω...