---
title: 'Diósi–Penrose Model: Gravity-Driven Collapse'
url: https://www.emergentmind.com/topics/diosi-penrose-model
type: topic
---

# Diósi–Penrose Model: Gravity-Driven Collapse

Searching arXiv for recent 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 [1001.4857]. 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 [1401.0176] [1701.00343] [2010.03971] [2601.17384]. 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 [1408.6460] [2111.13490] [2406.18494].

## 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 [1001.4857]. 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 [1001.4857]. 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 \(|\mathbf g_1-\mathbf g_2|^2\) [1701.00343] [1701.00353].

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 [1001.4857]. 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 [1401.0176]. 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 [1401.0176].

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 \(|c_1|^2|c_2|^2E_{G12}\), where \(E_{G12}\) is precisely the Diósi–Penrose energy [1701.00343]. In this reading, the DP energy is the energetic cost of forcing branches with different preferred geometries to share one classical spacetime [1701.00343]. 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 [1701.01765].

## 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
\[
\frac{d}{d t}\hat \rho(t)=-\frac i \hbar[\hat H_N,\hat \rho(t)]+\mathcal D[\hat \rho(t)],
\]
with dissipator
\[
\mathcal D[\hat \rho(t)]=-\frac{4\pi G}{\hbar}\int d r\,\int d r'\,\frac{1}{|r-r'|}\,[\hat \mu(r'),[\hat \mu(r),\hat \rho(t)]]
\]
[2406.18494]. The collapse-driving operator is the smeared mass density \(\hat\mu(r)\), typically regularized as
\[
\hat \mu(r)=\sum_{i=1}^N \frac{m_i}{(2\pi R_{\text{eff},i}^2)^{3/2}} e^{-\frac{(r-\hat \x_i)^2}{2R_{\text{eff},i}^2}},
\qquad
R_{\text{eff},i}=\sqrt{R_0^2+R_i^2}
\]
[2406.18494]. The cutoff \(R_0\) is necessary because the unsmeared Newton kernel is divergent for pointlike constituents [1408.6460] [2406.18494].

For rigid bodies, the center-of-mass dynamics becomes translation covariant,
\[
\frac{d \hat \rho_\text{\tiny CM}(t)}{d t} =-\frac{i}{\hbar}[\hat H_\text{\tiny CM},\hat \rho_\text{\tiny CM}(t)] +\int d\p\, F(\p)\left(e^{i\p\cdot\hat \x}\hat \rho_\text{\tiny CM}(t)e^{-i\p\cdot\hat \x}-\hat \rho_\text{\tiny CM}(t)\right),
\]
with a form factor exhibiting the amplification mechanism through a double sum over constituents [2406.18494]. When free motion can be neglected, spatial coherences decay approximately as
\[
\langle \x|\hat \rho_\text{\tiny CM}(t)|\y\rangle \simeq \langle \x|\hat \rho_\text{\tiny CM}(0)|\y\rangle \exp\left[-t/\tau(\x-\y)\right],
\qquad
\tau(d)=\frac{\hbar}{\Delta E(d)},
\]
where \(\Delta E(d)\) is the DP gravitational mismatch energy for branch separation \(d\) [2406.18494].

Another important mathematical lineage is the Schrödinger–Newton self-gravity model,
\[
-\frac{\hbar^2}{2m}\Delta \psi + U\psi = E\psi,
\qquad
\Delta U = 4\pi G m^2 |\psi|^2,
\]
which is often taken as a stationary self-localization model associated with the Diósi–Penrose idea [2010.03971]. 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 [2010.03971]. A relativistic embedding was constructed through coupled Dirac and Maxwell-like equations,
\[
i\gamma^\mu(\partial_\mu-iA_\mu)\psi+\psi=0,
\qquad
\Box A^\mu=\bar\psi\gamma^\mu\psi,
\qquad
\partial_\mu A^\mu=0,
\]
with the nonrelativistic limit reducing exactly to the spherical Schrödinger–Newton system [2010.03971]. 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 [2010.03971].

A more recent reformulation recasts the DP stochastic equation as a quantum filter. In that approach, the Newton kernel
\[
g(x,y)=\frac{G}{|x-y|}
\]
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 [2601.17384]. Continuous measurement of an output quadrature yields a quantum Kushner–Stratonovich equation whose pure-state unraveling reproduces the DP stochastic law [2601.17384]. 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 [2601.17384].

## 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 \(1/Q^2\) [1408.6460]. This is why a coarse-graining or cutoff parameter \(R_0\) must be introduced [1408.6460] [2406.18494]. 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 \(10^{-7}\,\mathrm m\), introduced to avoid severe heating [1408.6460]. Experimental constraints now impose a lower bound \(R_0\gtrsim 4\times 10^{-10}\,\mathrm m\), excluding the natural parameter-free version tied to nuclear-wavefunction scales in matter [2111.13490] [2406.18494].

The localization scale of self-gravitating stationary states is likewise a persistent issue. In the nonrelativistic Schrödinger–Newton picture, the characteristic width is
\[
a_0=\frac{\hbar^2}{Gm^3},
\]
which becomes
\[
a_0=l_P\left(\frac{m_P}{m}\right)^3
\]
when written in Planck units [2103.00994]. For ordinary particles this is enormous compared with microscopic localization scales. In the relativistic Dirac–graviMaxwell embedding, the characteristic ground-state width is
\[
\ell_{\rm gs}\sim \frac{\hbar^2}{Gm^3}=\frac{L}{\alpha},
\]
with \(L=\hbar/(mc)\) and \(\alpha=(m/M_{\rm Pl})^2\), so for a nucleon the width is macroscopically huge on particle scales [2010.03971]. This leads to the conclusion that a nucleon cannot be localized sufficiently by self-gravity alone to model realistic wavefunction reduction [2010.03971].

A separate quasi-relativistic correction based on the Grave de Peralta parametrization modifies the standard width to
\[
a=a_0\sqrt{1-\left(\frac{m}{m_P}\right)^4},
\]
showing that the self-localization width collapses to zero as \(m\to m_P\) and becomes undefined for larger masses [2103.00994]. That result is interpreted not as literal zero-size collapse, but as a breakdown of the nonrelativistic self-gravity picture near the Planck scale [2103.00994].

At the mesoscopic and macroscopic level, geometry and dimensionality matter strongly. In the center-of-mass DP energy for composite objects,
\[
\Delta E(d)=8\pi Gm^2\sum_{i=1}^N\sum_{j=1}^Nf(r_{ij},R_0,d),
\]
the double sum over constituent pairs makes collapse highly sensitive to packing, dimensionality, and branch separation [2406.18494]. For a \(D\)-dimensional crystal in the asymptotic regime analyzed there,
\[
\Delta E \propto N^{3/2}\quad (D=2),
\qquad
\Delta E \propto N^{5/3}\quad (D=3),
\]
so three-dimensional bodies collapse much more effectively than thin two-dimensional plates [2406.18494].

## 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 [1701.00353]. In that analysis, a nucleus is modeled with Gaussian density
\[
\rho(\mathbf x) = \frac{m}{(\sqrt{2\pi}\,\sigma)^3} \exp\!\left(-\frac{\mathbf x^2}{2\sigma^2}\right),
\]
and the DP energy for a displacement \(\Delta s\) becomes
\[
E_{DP} = \frac{G m^2}{\sqrt{\pi}\,\sigma}\, f_\sigma\!\left(\frac{\Delta s}{\sigma}\right),
\]
with \(f_\sigma(x)\approx x^2/12\) for \(x\ll1\) and saturation for \(x>4\) [1701.00353]. The spatial variation \(\sigma\) is set by lattice vibrations and is typically of order \(0.1\,\text{\AA}\) at room temperature [1701.00353].

This yields a clear regime structure. For displacements much smaller than \(\sigma\), the microscopic nuclear contribution dominates, and the parameter-free DP model predicts substantially faster decay than the smeared-density Diósi version [1701.00353]. For displacements exceeding roughly ten lattice constants, the nuclear-granularity contribution can be neglected and both approaches effectively agree [1701.00353].

That framework was extended to electrical components such as capacitors, resistors, wires, and piezo actuators, and then applied to a single-photon detector [1701.00353]. In the specific detector analyzed there, the predicted superposition lifetime was about \(93\,\mathrm{ms}\) in the parameter-free model and about \(28\,\mathrm{s}\) in the smeared-density model, while coupling the detector to a piezo-displaced mass reduced the lifetime to about \(0.4\,\mu\mathrm s\) [1701.00353]. 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 [1408.6460]. 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 [1001.4857] [1401.0176].

The cutoff problem is persistent. The unsmeared model is divergent, while the introduction of \(R_0\) is mathematically necessary but theoretically awkward [1408.6460] [2406.18494]. 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 [1408.6460]. 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 \(10^{11}\,\mathrm{amu}\) when the natural cutoff is retained [1408.6460].

Another criticism targets the standard timescale itself. A focused analysis of superpositions of energy eigenstates argued that the DP collapse time
\[
T \approx \frac{c^4 \hbar R}{G(\Delta E)^2}
\]
can be too long to prevent probability-density oscillations with period \(\hbar/\Delta E\) from becoming shorter than the Planck time when \(\Delta E\) exceeds the Planck energy, creating a tension with discrete-spacetime assumptions [1001.4857]. 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 [1001.4857].

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 \(O(\alpha^2)\), but it simultaneously shows that self-gravity alone is far too weak to localize ordinary particles realistically [2010.03971]. 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 [2103.00994]. These results suggest that simple self-gravity is not by itself a satisfactory microscopic collapse mechanism [2010.03971] [2103.00994].

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 [2601.17384]. 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 [2601.17384].

## 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,
\[
\partial_tW_{\hat\rho(t)} = \{W_{\hat H},W_{\hat \rho(t)}\}_\star + D\,\Delta_{\bm p}W_{\hat \rho(t)},
\qquad
D=\frac{G\hbar m^2}{3\sqrt{\pi} R_0^3},
\]
which implies unbounded heating [2502.03173]. 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 [2502.03173]. 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 [2606.06259] [2502.03173].

The dissipative DP extension modifies the Lindblad operators by adding a momentum-dependent friction term,
\[
\hat L_k=me^{ik\cdot \hat x}-\frac{\hbar \beta}{8}\left\{k\cdot \hat p,e^{ik\cdot \hat x}\right\},
\]
introducing a dissipation parameter \(\beta\) [2606.06259] [2502.03173]. 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 [2502.03173]. 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 \(\beta^3\) [2606.06259]. This establishes that curing the heating problem does not simply convert DP into ordinary thermal Brownian motion [2606.06259].

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
\[
R_0 > 0.54 \times 10^{-10}\,\mathrm m
\]
at \(95\%\) probability [2111.13490]. This is about three orders of magnitude stronger than previous bounds and excludes the natural parameter-free Penrose version in which \(R_0\) is identified with the actual nuclear wavefunction size in the crystal [2111.13490]. The current experimental lower bound is also summarized in recent phenomenological work as \(R_0\gtrsim 4\times10^{-10}\,\mathrm m\) [2406.18494].

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
\[
R_0\lesssim 10^{-4}\,\mathrm m
\]
[2406.18494]. That same study emphasizes strong dependence on geometry and dimensionality: a directly visible 2D plate can fail to collapse within \(0.01\,\mathrm s\) for any \(R_0\), whereas 3D objects of similar lateral size collapse much more effectively [2406.18494].

A final development points toward non-Markovian variants. In a string-theoretic scenario, a DP-like spatial kernel \(1/k^2\) emerges, but with temporally colored rather than white noise, strongly suppressing high-frequency spontaneous-radiation constraints while retaining low-frequency collapse behavior [2603.24429]. 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 [2603.24429].

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 [2010.03971] [1408.6460] [2111.13490] [2406.18494] [2502.03173].

Source: https://www.emergentmind.com/topics/diosi-penrose-model