Diósi–Penrose Model: Gravity-Driven Collapse

• The Diósi–Penrose model is a gravity-related collapse proposal where quantum superpositions of mass distributions are unstable due to a characteristic gravitational self-energy.
• Analytical formulations span Penrose’s spacetime-instability argument, Diósi’s stochastic dynamics, and semiclassical reinterpretations with detailed mathematical structure.
• Key challenges include resolving cutoff regularization, achieving relativistic completion, and ensuring thermodynamic consistency to validate the collapse mechanism.

Searching arXiv for papers on the Diósi–Penrose model and closely related work. The Diósi–Penrose model is a class of gravity-related objective-collapse proposals in which quantum superpositions of different mass distributions are unstable, with a characteristic lifetime set by a gravitational self-energy associated with the difference between the branches. In its standard heuristic form, the collapse time is estimated as $T \approx \hbar/\Delta E_G$, where $\Delta E_G$ measures the Newtonian self-energy of the mismatch between the two mass configurations (Gao, 2010). Across the literature, this basic idea appears in several distinct formulations: Penrose’s spacetime-instability argument, Diósi’s stochastic mass-density dynamics, semiclassical-gravity reinterpretations, relativistic generalizations, dissipative extensions, and recent open-system or filtering reformulations (Oosterkamp et al., 2013, Quandt-Wiese, 2017, Kassandrov et al., 2020, Gough et al., 24 Jan 2026). The model is technically influential because it offers a concrete gravity-motivated alternative to standard unitary evolution, but its status remains unsettled because of unresolved issues concerning regularization, relativistic completion, thermodynamic consistency, and empirical viability (Bahrami et al., 2014, Donadi et al., 2021, Figurato et al., 2024).

1. Historical and conceptual formulation

The basic Diósi–Penrose criterion concerns a superposition of two distinct mass distributions, or equivalently two distinct Newtonian gravitational fields. In Penrose’s formulation, the central estimate is

$T \approx \frac{\hbar}{\Delta E_G},$

with $\Delta E_G$ the gravitational self-energy of the difference between the two branches (Gao, 2010). One representation used in the literature is

$$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$

where $\Phi_1,\Phi_2$ are the Newtonian potentials of the two alternatives (Gao, 2010). Another standard expression writes the same scale as the self-energy of the density difference, for example

$$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$

which is also equivalent to a field-difference form involving $|\mathbf g_1-\mathbf g_2|^2$ (Quandt-Wiese, 2017, Quandt-Wiese, 2017).

Penrose’s motivation is that different mass distributions correspond semiclassically to different spacetime geometries, and therefore to an ambiguity in defining a single time-translation structure for the superposition (Gao, 2010). A complementary reconstruction argues that what “makes Penrosian wavefunction collapse tick” is the ambiguity of branch-dependent gravitational time dilation, which enters quantum phase evolution once a common time parameter becomes ill-defined (Oosterkamp et al., 2013). In that interpretation, the collapse timescale remains of Penrose type, but the physical emphasis shifts from an abstract self-energy to time-dilation-induced phase ambiguity (Oosterkamp et al., 2013).

A distinct line of development reinterprets the same energy scale in semiclassical gravity. For a two-branch superposition

$|\psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle,$

with branch mass densities $\rho_i(\mathbf x)$, the total energy in the shared classical potential acquires an additional term proportional to $\Delta E_G$0, where $\Delta E_G$1 is precisely the Diósi–Penrose energy (Quandt-Wiese, 2017). In this reading, the DP energy is the energetic cost of forcing branches with different preferred geometries to share one classical spacetime (Quandt-Wiese, 2017). This framework was further extended into a “temporally expanding spacetime” model, which preserves the same DP lifetime scale while embedding collapse into a moving-future-boundary ontology (Quandt-Wiese, 2017).

2. Dynamical realizations and mathematical structure

The most direct dynamical realization is Diósi’s stochastic mass-density model. In the regularized form used in recent analyses, the density operator obeys

$\Delta E_G$2

with dissipator

$\Delta E_G$3

(Figurato et al., 2024). The collapse-driving operator is the smeared mass density $\Delta E_G$4, typically regularized as

$\Delta E_G$5

(Figurato et al., 2024). The cutoff $\Delta E_G$6 is necessary because the unsmeared Newton kernel is divergent for pointlike constituents (Bahrami et al., 2014, Figurato et al., 2024).

For rigid bodies, the center-of-mass dynamics becomes translation covariant,

$\Delta E_G$7

with a form factor exhibiting the amplification mechanism through a double sum over constituents (Figurato et al., 2024). When free motion can be neglected, spatial coherences decay approximately as

$\Delta E_G$8

where $\Delta E_G$9 is the DP gravitational mismatch energy for branch separation $T \approx \frac{\hbar}{\Delta E_G},$0 (Figurato et al., 2024).

Another important mathematical lineage is the Schrödinger–Newton self-gravity model,

$T \approx \frac{\hbar}{\Delta E_G},$1

which is often taken as a stationary self-localization model associated with the Diósi–Penrose idea (Kassandrov et al., 2020). Although Schrödinger–Newton is not a stochastic collapse law, it has been interpreted as providing the candidate self-gravitating states toward which localization might occur (Kassandrov et al., 2020). A relativistic embedding was constructed through coupled Dirac and Maxwell-like equations,

$T \approx \frac{\hbar}{\Delta E_G},$2

with the nonrelativistic limit reducing exactly to the spherical Schrödinger–Newton system (Kassandrov et al., 2020). That construction incorporates spin and clarifies the relation between active gravitational and inertial mass, while remaining a vector-gravity analogue rather than a full Einstein theory (Kassandrov et al., 2020).

A more recent reformulation recasts the DP stochastic equation as a quantum filter. In that approach, the Newton kernel

$T \approx \frac{\hbar}{\Delta E_G},$3

is diagonalized in Fourier modes, and the DP Lindbladian is realized as the unconditional dynamics of an open quantum system coupled to a bosonic field (Gough et al., 24 Jan 2026). Continuous measurement of an output quadrature yields a quantum Kushner–Stratonovich equation whose pure-state unraveling reproduces the DP stochastic law (Gough et al., 24 Jan 2026). This retains the Newtonian kernel, the mass-density coupling, and the same spatial noise covariance, but reinterprets the noise as an innovations process rather than a fundamental classical gravitational fluctuation (Gough et al., 24 Jan 2026).

3. Regularization, localization scale, and mass dependence

The ultraviolet problem is central to the Diósi model. Without smearing, the total transition rate diverges linearly in momentum cutoff because the kernel behaves as $T \approx \frac{\hbar}{\Delta E_G},$4 (Bahrami et al., 2014). This is why a coarse-graining or cutoff parameter $T \approx \frac{\hbar}{\Delta E_G},$5 must be introduced (Bahrami et al., 2014, Figurato et al., 2024). Historically, two scales recur in the literature. One is a nucleon Compton-wavelength-scale cutoff, regarded as “natural” in a nonrelativistic treatment; the other is a much larger phenomenological cutoff of order $T \approx \frac{\hbar}{\Delta E_G},$6, introduced to avoid severe heating (Bahrami et al., 2014). Experimental constraints now impose a lower bound $T \approx \frac{\hbar}{\Delta E_G},$7, excluding the natural parameter-free version tied to nuclear-wavefunction scales in matter (Donadi et al., 2021, Figurato et al., 2024).

The localization scale of self-gravitating stationary states is likewise a persistent issue. In the nonrelativistic Schrödinger–Newton picture, the characteristic width is

$T \approx \frac{\hbar}{\Delta E_G},$8

which becomes

$T \approx \frac{\hbar}{\Delta E_G},$9

when written in Planck units (Poveda et al., 2021). For ordinary particles this is enormous compared with microscopic localization scales. In the relativistic Dirac–graviMaxwell embedding, the characteristic ground-state width is

$\Delta E_G$0

with $\Delta E_G$1 and $\Delta E_G$2, so for a nucleon the width is macroscopically huge on particle scales (Kassandrov et al., 2020). This leads to the conclusion that a nucleon cannot be localized sufficiently by self-gravity alone to model realistic wavefunction reduction (Kassandrov et al., 2020).

A separate quasi-relativistic correction based on the Grave de Peralta parametrization modifies the standard width to

$\Delta E_G$3

showing that the self-localization width collapses to zero as $\Delta E_G$4 and becomes undefined for larger masses (Poveda et al., 2021). That result is interpreted not as literal zero-size collapse, but as a breakdown of the nonrelativistic self-gravity picture near the Planck scale (Poveda et al., 2021).

At the mesoscopic and macroscopic level, geometry and dimensionality matter strongly. In the center-of-mass DP energy for composite objects,

$\Delta E_G$5

the double sum over constituent pairs makes collapse highly sensitive to packing, dimensionality, and branch separation (Figurato et al., 2024). For a $\Delta E_G$6-dimensional crystal in the asymptotic regime analyzed there,

$\Delta E_G$7

so three-dimensional bodies collapse much more effectively than thin two-dimensional plates (Figurato et al., 2024).

4. Microscopic versus coarse-grained mass density

An important refinement is the distinction between a smooth averaged mass density and the actual microscopic distribution concentrated in nuclei. For solids in quantum superpositions with very small displacements, taking nuclear granularity into account can increase the Diósi–Penrose energy substantially relative to a smeared-mass treatment (Quandt-Wiese, 2017). In that analysis, a nucleus is modeled with Gaussian density

$\Delta E_G$8

and the DP energy for a displacement $\Delta E_G$9 becomes

$$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$0

with $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$1 for $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$2 and saturation for $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$3 (Quandt-Wiese, 2017). The spatial variation $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$4 is set by lattice vibrations and is typically of order $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$5 at room temperature (Quandt-Wiese, 2017).

This yields a clear regime structure. For displacements much smaller than $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$6, the microscopic nuclear contribution dominates, and the parameter-free DP model predicts substantially faster decay than the smeared-density Diósi version (Quandt-Wiese, 2017). For displacements exceeding roughly ten lattice constants, the nuclear-granularity contribution can be neglected and both approaches effectively agree (Quandt-Wiese, 2017).

That framework was extended to electrical components such as capacitors, resistors, wires, and piezo actuators, and then applied to a single-photon detector (Quandt-Wiese, 2017). In the specific detector analyzed there, the predicted superposition lifetime was about $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$7 in the parameter-free model and about $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$8 in the smeared-density model, while coupling the detector to a piezo-displaced mass reduced the lifetime to about $$\Delta E_G = \frac{1}{G}\int \left(\Phi_2-\Phi_1\right)^2\, d^3x,$$9 (Quandt-Wiese, 2017). These results underscore that DP phenomenology depends not only on total mass, but on how the branch-dependent mass distribution is realized mechanically.

5. Critiques, limitations, and theoretical tensions

Several lines of criticism recur across the literature. A basic conceptual objection is that Diósi’s stochastic dynamics is not actually derived from a deeper gravitational theory; gravity enters only through the choice of a Newtonian spatial noise correlator, so the model remains phenomenological in much the same sense as GRW or CSL (Bahrami et al., 2014). Penrose’s original criterion is conceptually suggestive but also heuristic: the relation between spacetime ambiguity and the particular Newtonian self-energy formula is not derived from full general relativity (Gao, 2010, Oosterkamp et al., 2013).

The cutoff problem is persistent. The unsmeared model is divergent, while the introduction of $\Phi_1,\Phi_2$0 is mathematically necessary but theoretically awkward (Bahrami et al., 2014, Figurato et al., 2024). With a nucleon-Compton-scale cutoff, the original DP dynamics predicts an unrealistically large heating rate; with a much larger cutoff, the overheating problem is suppressed, but the cutoff becomes ad hoc (Bahrami et al., 2014). Even with dissipation, one analysis concluded that physically acceptable behavior requires either an artificially large cutoff or restriction of applicability to sufficiently massive systems, roughly above $\Phi_1,\Phi_2$1 when the natural cutoff is retained (Bahrami et al., 2014).

Another criticism targets the standard timescale itself. A focused analysis of superpositions of energy eigenstates argued that the DP collapse time

$\Phi_1,\Phi_2$2

can be too long to prevent probability-density oscillations with period $\Phi_1,\Phi_2$3 from becoming shorter than the Planck time when $\Phi_1,\Phi_2$4 exceeds the Planck energy, creating a tension with discrete-spacetime assumptions (Gao, 2010). The critique does not reject gravity-induced collapse as such, but suggests that the standard DP criterion may be quantitatively inadequate if Planck-scale discreteness is taken seriously (Gao, 2010).

There are also structural limitations of the self-gravity program. The relativistic Dirac–graviMaxwell embedding demonstrates mathematically that active gravitational mass, inertial mass, and the bare Dirac mass coincide up to $\Phi_1,\Phi_2$5, but it simultaneously shows that self-gravity alone is far too weak to localize ordinary particles realistically (Kassandrov et al., 2020). The quasi-relativistic correction that drives the self-localization width to zero near the Planck mass indicates breakdown of the semiclassical nonrelativistic picture in that regime (Poveda et al., 2021). These results suggest that simple self-gravity is not by itself a satisfactory microscopic collapse mechanism (Kassandrov et al., 2020, Poveda et al., 2021).

A further conceptual ambiguity concerns whether the DP dynamics is a true collapse theory or a decoherence-like effective description. Quantum-filtering and open-system formulations preserve the same master equation while changing the interpretation of the stochasticity (Gough et al., 24 Jan 2026). This suggests that at least part of the DP structure can be re-read as an unraveling of an open-system dynamics rather than a fundamental modification of quantum theory (Gough et al., 24 Jan 2026).

6. Thermodynamics, dissipation, and experimental status

A major modern theme is the thermodynamic consistency of gravity-related collapse. In the original frictionless DP model, the Wigner equation reduces to pure Hamiltonian flow plus momentum diffusion,

$\Phi_1,\Phi_2$6

which implies unbounded heating (Artini et al., 5 Feb 2025). Entropy-production analysis for a harmonic oscillator shows that the model is thermodynamically consistent only if interpreted as coupling to an infinite-temperature noise field (Artini et al., 5 Feb 2025). That result has been sharpened further by phase-space studies of dissipative extensions, which confirm that the standard model’s heating pathology is genuine and that dissipation is necessary for physical viability (Melo et al., 4 Jun 2026, Artini et al., 5 Feb 2025).

The dissipative DP extension modifies the Lindblad operators by adding a momentum-dependent friction term,

$\Phi_1,\Phi_2$7

introducing a dissipation parameter $\Phi_1,\Phi_2$8 (Melo et al., 4 Jun 2026, Artini et al., 5 Feb 2025). In the weak-dissipation regime, the effective phase-space equation becomes a Klein–Kramers-type equation with both diffusion and friction, yielding a finite thermal stationary state and non-negative entropy production (Artini et al., 5 Feb 2025). Beyond that regime, however, the dynamics is intrinsically non-Gaussian. In a more complete analysis, the dissipative model does not thermalize to a Gibbs state but rather relaxes to a non-equilibrium steady state with heavy tails, and the asymptotic non-Gaussianity scales as $\Phi_1,\Phi_2$9 (Melo et al., 4 Jun 2026). This establishes that curing the heating problem does not simply convert DP into ordinary thermal Brownian motion (Melo et al., 4 Jun 2026).

Experimentally, the most stringent present constraint comes from spontaneous radiation searches. Since DP-induced momentum diffusion causes Brownian motion of charged matter and therefore radiation, a dedicated underground HPGe experiment at Gran Sasso constrained the nuclear smearing length to

$$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$0

at $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$1 probability (Donadi et al., 2021). This is about three orders of magnitude stronger than previous bounds and excludes the natural parameter-free Penrose version in which $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$2 is identified with the actual nuclear wavefunction size in the crystal (Donadi et al., 2021). The current experimental lower bound is also summarized in recent phenomenological work as $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$3 (Figurato et al., 2024).

The remaining viable window is therefore bounded below by experiment and, more loosely, above by the requirement that collapse be effective enough to ensure macroscopic classicality. A recent analysis of a visually macroscopic graphene plate found that not all macroscopic systems collapse fast enough under the strictest classicality criterion; relaxing that criterion yields a plausible upper bound

$$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$4

(Figurato et al., 2024). That same study emphasizes strong dependence on geometry and dimensionality: a directly visible 2D plate can fail to collapse within $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$5 for any $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$6, whereas 3D objects of similar lateral size collapse much more effectively (Figurato et al., 2024).

A final development points toward non-Markovian variants. In a string-theoretic scenario, a DP-like spatial kernel $$E_{DP} = \frac{G}{2} \int d^3x\, d^3y\, \frac{\big(\rho_2(\mathbf x)-\rho_1(\mathbf x)\big)\big(\rho_2(\mathbf y)-\rho_1(\mathbf y)\big)}{|\mathbf x-\mathbf y|},$$7 emerges, but with temporally colored rather than white noise, strongly suppressing high-frequency spontaneous-radiation constraints while retaining low-frequency collapse behavior (Itzhaki, 25 Mar 2026). This suggests that some empirical objections to the standard white-noise DP model are model-structure dependent rather than generic to all gravity-related collapse proposals (Itzhaki, 25 Mar 2026).

In sum, the Diósi–Penrose model is best understood not as a single equation but as a research program organized around one central idea: mass-density superpositions carry a gravity-related instability scale. Its enduring significance lies in making that idea mathematically concrete and experimentally falsifiable. Its unresolved status lies in the fact that every concrete realization so far—stochastic, semiclassical, self-gravitating, dissipative, relativistic, or filtering-based—solves some problems while exposing others (Kassandrov et al., 2020, Bahrami et al., 2014, Donadi et al., 2021, Figurato et al., 2024, Artini et al., 5 Feb 2025).