Classical Gravity Coupled to Quantum Matter

Updated 11 November 2025

Classical gravity coupled to quantum matter is a framework where a smooth, deterministic gravitational field interacts with fully quantum matter, leading to unique conceptual challenges.
The models use hybrid master equations with irreversible dynamics to capture decoherence and diffusion trade-offs, ensuring complete positivity in the classical-quantum system.
Experimental predictions focus on measurable decoherence rates and classical noise thresholds that critically differentiate hybrid models from semiclassical and fully quantum gravitational theories.

Classical gravity coupled to quantum matter refers to any framework in which the gravitational field is fundamentally classical—typically described by smooth metric fields and deterministic phase-space variables—while the matter sector is taken as fully quantum, i.e., governed by Hilbert-space or field-theoretic quantum dynamics. This scenario sits at the intersection of general relativity and quantum theory and confronts the conceptual and mathematical challenges arising from reconciling a non-classical source for the very structure of spacetime while retaining a classical notion of causality and geometry.

1. Theoretical Frameworks and the GPT No-Go Theorem

The most theory-independent treatment employs the language of Generalized Probabilistic Theories (GPTs), which generalizes both classical and quantum theories into a convex-operational setting. In this approach, each physical system is characterized by a real vector space of states, a convex state space, a set of effects (linear functionals), and a semigroup of allowed transformations. Systems are composed via tensor products, with classical systems identified as n-simplexes and quantum systems as irreducible, nonclassical state spaces.

The central result is a no-go theorem: no reversible, information-preserving coupling can exist between a fully classical gravitational field and a genuinely quantum matter system that allows for nontrivial back-reaction (Galley et al., 2023). Formally, if $G$ is a classical system and $M$ a fully non-classical GPT system (irreducible, no non-disturbing measurements), and if they interact via a reversible transformation $T$ with nonzero information flow from $M$ to $G$ , contradiction follows. One of the following must fail:

(i) The matter is not fully quantum (i.e., is reducible/partly classical).
(ii) The interaction is not reversible (e.g., is fundamentally stochastic or dissipative).
(iii) The matter does not back-react on the field (i.e., no information about $M$ ever flows to $G$ ).

This generalizes and underpins multiple prior analyses demonstrating the mathematical impossibility of constructing a deterministic, quantum theory respecting all standard axioms for a classical gravity/quantum matter system.

2. Concrete Realizations: Irreversible Hybrid Dynamics

A wide range of recent research implements viable classical–quantum couplings within irreversible open-system frameworks. These generically take the form of stochastic hybrid master equations or diffusive Langevin-type equations on phase space and quantum states: - Hybrid canonical formalism: The quantum-classical hybrid density is evolved via a dynamical equation combining Dirac (commutator) and Poisson brackets, with additional decoherence and diffusion terms mandated by complete positivity. The total generator is non-reversible due to these terms, which reflect the essential unavoidability of noise and irreversibility in any classical–quantum back-reaction (Diósi, 2011, Diósi, 11 Apr 2024).

The explicit quantum master equation for the reduced density reads, for Newtonian gravity,

$\frac{d}{dt}\hat{\rho}_Q = -\frac{i}{\hbar}[\hat{H}_Q + \hat{H}_G, \hat{\rho}_Q] - \frac{G}{4\hbar} \int d^3r d^3s \frac{1}{|r-s|} [\hat{f}(r), [\hat{f}(s), \hat{\rho}_Q]],$

where $\hat{f}(r)$ is the mass-density operator. The decoherence term irreversibly suppresses quantum superpositions between states with macroscopically different mass distributions.

Classical stochastic field/Lindblad dynamics: In the Oppenheim et al. framework, the state is an operator-valued distribution $\rho_{CQ}(z)$ on the gravitational phase space, evolving under a linear, completely positive (CP) and trace preserving (TP) master equation (Oppenheim, 2018). The general form includes both unitary quantum evolution and a generalized classical Markov process on $z$ as well as decoherence terms arising from the stochastic back-action of quantum matter on the field.
Continuous measurement-plus-feedback models (e.g. Kafri–Taylor–Milburn, Tilloy–Diósi): These operationally simulate gravity as a classical LOCC (local operations, classical communication) channel; the effect is a minimal, inescapable momentum/position diffusion, with a well-defined lower bound on the noise strength to prevent the creation of entanglement or violation of the no-signalling principle (Angeli et al., 22 Jan 2025, Donadi et al., 2022).

3. Structure of Hybrid Master Equations and Irreversibility

All consistent classical–quantum gravity models derive a hybrid master equation for the joint state of matter and classical field:

The classical degrees of freedom evolve stochastically (diffusion) in response to quantum measurement back-action.
The quantum degrees of freedom suffer irreversible decoherence, typically in the mass-density basis.
Complete positivity imposes a trade-off: suppression of quantum coherence (decoherence rate $D_0$ ) necessitates a minimum amount of classical field diffusion (diffusion coefficient $D_2$ ), saturating inequalities of the form $4 D_2 \geq D_0^{-1}$ (Oppenheim et al., 2022, Layton et al., 2023).

In practical terms, any experimental attempt to prolong quantum coherence in such a hybrid system mandates an increase in the classical gravitational field's diffusive noise, which can be independently tested in sensitive accelerometer or force-noise experiments (Angeli et al., 22 Jan 2025).

Key mathematical ingredients:

Double-commutator decoherence superoperators (e.g. $[\hat{f}(r), [\hat{f}(s), \hat{\rho}_Q]]$ ), typically with kernels $\sim G / |r-s|$ (Diósi–Penrose structure).
Fokker-Planck terms for the field diffusion.
Consistency conditions on kernels and coefficients imposed by CP-TP (Lindblad) requirements.

4. Experimental Consequences and Observational Signatures

Irreversible hybrid models make distinct, falsifiable predictions:

Decoherence of quantum matter: For instance, mass-superpositions are predicted to decay at rates set by the gravitational self-energy of the superposed configurations (Diósi–Penrose rate) (Donadi et al., 2022, Layton et al., 2023).
Fundamental classical diffusion noise: Any classical-only gravitational interaction must induce momentum or displacement diffusion in mesoscopic oscillators at rates $\gamma \ge Gm^2/(\hbar d^3)$ (for two masses $m$ separated by $d$ ) (Angeli et al., 22 Jan 2025, Kryhin et al., 2023).
Decoherence–diffusion trade-off: Prolonged matter coherence or the non-observation of excess noise in force measurements places stringent limits on hybrid gravity models. Recent bounds from molecular interferometry and ultra-sensitive Cavendish experiments already exclude significant classes of continuous-noise irreversibly coupled models, with future interferometric and cryogenic experiments poised to close the remaining parameter space (Layton et al., 2023, Oppenheim et al., 2022).
Absence of gravitationally-induced entanglement via classical gravity: All classical hybrid models preclude the mediation of entanglement—thus, confirmed observation of gravitationally-induced entanglement (via the so-called BMV witness protocols) would refute them.

5. Comparison to Semiclassical and Quantum Approaches

Semiclassical gravity (Schrödinger–Newton, $G_{\mu\nu} = 8\pi G \langle \hat{T}_{\mu\nu} \rangle$ ) is deterministic and nonlinear but fails to yield a CP-TP evolution for the quantum state and is nonviable in the presence of entanglement or for mixed states. It also suffers from conceptual inconsistencies (superluminal signalling, average-field pathologies) (Giulini et al., 2022, Großardt, 2017).
Irreversible hybrid models avoid these pathologies by enforcing linearity on the full classical–quantum state and accepting irreversibility and noise as essential structural features for consistency.
Fully quantum gravity circumvents the no-go and irreversibility constraints: if one insists on a reversible (unitary, information-preserving) interaction with back-reaction, the gravitational field is forced to possess quantum (superposed) degrees of freedom—thereby recovering standard quantum field theory in curved spacetime in the appropriate limit (Galley et al., 2023).

6. Mathematical and Physical Limitations

All fundamentally irreversible classical gravity–quantum matter couplings are currently rigorously formulated only in the Newtonian/weak-field approximation. Markovian extensions to special or general relativity suffer from incurable UV divergences, as locality and covariance force delta-correlated noise kernels which produce infinite heating and energy production rates (divergences $\delta(0)$ ). Attempts with smeared kernels break general covariance (Diósi, 11 Apr 2024).
Covariant path-integral approaches have been employed for scalar (Nordström) and quadratic gravity, leading to renormalizable, CP, and diffeomorphism-invariant formulations, but full spin-2 GR generalization remains an open question (Grudka et al., 27 Feb 2024, Oppenheim et al., 10 Jan 2024).
Markovianity and a time-local master equation structure appear fundamentally incompatible with full Lorentz invariance. Non-Markovian or fundamentally different stochastic process structures may be necessary for a relativistically robust hybrid theory.

7. Outlook and Conceptual Implications

The irreversibility of any theory coupling classical gravity to quantum matter is now established on operational and mathematical grounds. Accepting this implies:

If gravity is classical: One must embrace stochastic, open-system dynamics at the fundamental level for the combined matter-field system. The resulting theory is experimentally testable, with unique predictions for decoherence and field-diffusion rates. Large regions of parameter space for such theories are already under empirical pressure.
If reversibility is fundamental: The gravitational field must possess non-classical (quantum) degrees of freedom; attempts to mediate reversible quantum back-action without quantizing gravity are mathematically inconsistent.
Experimental discrimination: Ultra-sensitive measurements of macroscopic coherence, force-noise, and cross-correlation spectra between masses provide sharp tools for distinguishing between fundamentally classical and quantum models of gravity.

This synthesis provides a rigorous, comprehensive foundation for understanding the limitations and possibilities of hybrid classical gravity–quantum matter theories, motivating continued theoretical refinement and focused precision experiments (Galley et al., 2023, Oppenheim, 2018, Oppenheim et al., 2022, Diósi, 11 Apr 2024, Angeli et al., 22 Jan 2025).