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Diffeomorphism invariant classical-quantum path integrals for Nordström gravity
Open accessAbstract
When classical degrees of freedom and quantum degrees of freedom are consistently coupled, the former diffuse, while the latter undergo decoherence. Here, we construct a theory of quantum matter fields and Nordström gravity in which the space-time metric is treated classically. The dynamics is constructed via the classical-quantum path integral and is completely positive, trace preserving (CPTP) and respects the classical-quantum split. The weak-field limit of the model matches the Newtonian limit of the full covariant path integral, but it is easier to show that the theory is both diffeomorphism invariant and CPTP and has the appropriate classical limit. Published by the American Physical Society 2024
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TL;DR
The paper constructs a diffeomorphism-invariant classical-quantum path integral for Nordström gravity, with a classical conformal metric coupled to quantum matter. It shows the theory is completely positive, trace-preserving, preserves the classical-quantum split, and has a Newtonian limit matching earlier weak-field CQ gravity results.
Key findings
- A diffeomorphism-invariant CQ path integral for Nordström gravity is constructed by treating the conformal factor as the stochastic gravitational potential and imposing conformal flatness via a delta functional/Lagrange multiplier.
- The dynamics are completely positive and trace-preserving, and with the saturated decoherence-diffusion trade-off \(4D_0 = D^{-1}_2\), the path integral preserves purity of the quantum state conditioned on the classical degrees of freedom.
- In the Newtonian \(c \to \infty\) limit, the theory gives the Newtonian interaction on average and matches the Newtonian limit of [33], while necessarily including diffusion away from the Newtonian solution lower-bounded by the decoherence rate into mass eigenstates.
- The construction demonstrates no tension between diffeomorphism invariance, stochastic CQ dynamics, and the classical-quantum split, though Nordström gravity remains a toy model that does not describe all gravitational phenomena.
Methods
- The authors use the classical-quantum path integral formalism, writing a CQ action of the form that guarantees complete positivity, with classical and quantum branches and a Feynman-Vernon-like decoherence term.
- They specialize to Nordström gravity, where the classical degree of freedom is the conformal factor \(\Phi\) in \(g_{\mu\nu}=e^{2\Phi/c^2}\eta_{\mu\nu}\), and impose conformal flatness \(C^\mu{}_{\nu\rho\sigma}=0\) through a Lagrange multiplier.
- They take the Newtonian \(c\to\infty\) limit with pressureless dust matter, recovering a randomly sourced Poisson equation and the same decoherence-diffusion relation as in prior Newtonian CQ gravity work.
Limitations
- The paper states that Nordström gravity is a toy model: it lacks the ability to describe some gravitational phenomena, does not predict light bending, fails to predict the correct time-delay factor, and misses subleading corrections to static test-particle acceleration.
- The theory is diffeomorphism invariant but not background independent; it has a preferred frame given by the requirement that the metric be conformally flat.
- The paper notes that in the \(c\to\infty\) limit the path integral becomes discontinuous, a remnant of the approximation, and that the action is suggestive of an effective theory requiring higher-derivative kinetic terms for normalization.
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0 Valid · 4 Not valid · 51 Not checked · 74 Own results and interpretations
In the paper
“Nevertheless, we recover the same Newtonian limit behavior as in [33], suggesting that the resolution of the constraints and the resulting dynamics was correct.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > I. INTRODUCTION
What the sources say
- Not matchedcites [33]
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In the paper
“The dynamics is constructed via the classical-quantum path integral and is completely positive, trace preserving (CPTP), and respects the classical-quantum split.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > Abstract
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“In the Newtonian c → ∞ limit we find that the theory gives rise to the Newtonian interaction on average.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > I. INTRODUCTION
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“The weak field limit of the model matches the Newtonian limit of the full covariant path integral but it is easier to show that the theory is both diffeomorphism invariant, CPTP, and has the appropriate classical limit.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > Abstract
plain- Asserted: passes — The authors put it forward as true.
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A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Any path integral with an action of the form in Equation ( 3) is completely positive [31] and the scalar gravity CQ theory we consider in Section IV is of this type.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > II. PATH INTEGRAL FOR CQ DYNAMICS
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Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > II. PATH INTEGRAL FOR CQ DYNAMICS
“the ϕ -field is the ket-field, while the ϕ + field is the bra-field.”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > II. PATH INTEGRAL FOR CQ DYNAMICS
“ϱ(z, ϕ + , ϕ -, t) |ϕ + ⟩ ⟨ϕ -|”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > CQ Nordstrom
“- √ -gD 0 (x) 8 T [ϕ + m ] -T [ϕ - m ] 2”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > CQ Nordstrom
“- cD 0 (x)e 6Φ c 2 8 m + (x) -m -(x) 2”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > CQ Nordstrom
“- √ -gc 8 D 0 (x) 1152 π 2 G 2 N R - 24πG N c 4 T [ϕ + m , ϕ - m ] 2”
§ Diffeomorphism invariant classical-quantum path integrals for Nordstrom gravity > CQ Nordstrom
“- c 7 D 0 (x) 192 π 2 G 2 N -e Φ c 2 □e Φ c 2 + 4πG N e 6Φ c 2 c 2 m(x) 2”
Citations
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- 10.1103/physrevlett.119.240402 · DOI
- 10.1103/physrevd.34.470 · DOI · Paper page
Cited by
Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity
Oppenheim, J., Sparaciari, C., Šoda, B., & Weller-Davies, Z. (2023) · Nature Communications · doi:10.1038/s41467-023-43348-2 · Cited by 98
Self-citationThe open effective field theory of inflation
Salcedo, S. A., Colas, T., & Pajer, E. (2024) · Journal of High Energy Physics · doi:10.1007/jhep10(2024)248 · Cited by 80
Covariant Path Integrals for Quantum Fields Backreacting on Classical Space-Time
Oppenheim, J., & Weller-Davies, Z. (2026) · Physical Review X · doi:10.1103/2rcd-dzcf · Cited by 34
Self-citationLayton, I., & Oppenheim, J. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.020331 · Cited by 30
Self-citationAnomalous contribution to galactic rotation curves due to stochastic spacetimePreprint
Oppenheim, J., & Russo, A. (2024) · arXiv (Cornell University) · doi:10.48550/arxiv.2402.19459 · Cited by 16
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Grudka, A., Morris, T. R., Oppenheim, J., Russo, A., & Sajjad, M. (2024) · ePrints Soton (University of Southampton) · doi:10.48550/arxiv.2402.17844 · Cited by 15
Self-citationEmergence of phantom cold dark matter from spacetime diffusion
Oppenheim, J., Panella, E., & Pontzen, A. (2026) · Physical review. D/Physical review. D. · doi:10.1103/7whh-9j22 · Cited by 7
Self-citationRestoring the second law to classical-quantum dynamics
Layton, I., & Miller, H. J. D. (2026) · Physical Review Research · doi:10.1103/kqy2-1vmx · Cited by 6
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Weller-Davies, Z. (2024) · arXiv (Cornell University) · doi:10.48550/arxiv.2402.17024 · Cited by 3
Self-citationA quantum oscillator interacting with a classical oscillatorPreprint
Sajjad, M., Russo, A., Arcos, M., Grudka, A., & Oppenheim, J. (2024) · arXiv (Cornell University) · doi:10.48550/arxiv.2403.07479 · Cited by 3
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Oppenheim, J., & Sajjad, M. (2026) · arXiv (Cornell University) · doi:10.48550/arxiv.2605.05375 · Cited by 3
Self-citationMcGaugh, S. (2024) · doi:10.59350/6hx0s-bd890
Metadata
- Authors
- Oppenheim, J., Russo, A., & Weller-Davies, Z.
- Affiliations
- University College London (GB)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2024-07-03
- Volume
- 110
- Issue
- 2
- ISSN
- 2470-0010 (ISSN-L)
- 2470-0029
- OpenAlex
- W4400283784
- Semantic Scholar
- 2b131611deb73f7635d23b5d2beec4d23bf082bb
- Cited by
- 12
- References
- 77
- Citations by year
- 2026: 4 · 2024: 7
- FWCI
- 0.68
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Physical review. D/Physical review. D. (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Engineering and Physical Sciences Research Council
- National Science Foundation (PHY11-25915)
- Simons Foundation
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- diffeomorphism invariance, decoherence, weak-field limit, Newtonian limit, classical limit, quantum gravity
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