The Diósi–Penrose Rate as a Thermal Fluctuation

A famous number without a derivation

In 1987, Lajos Diósi proposed — and Roger Penrose independently refined — a candidate decoherence rate for a mass in spatial superposition:

$\Gamma_{\mathrm{dec}} \;=\; \frac{E_G}{\hbar},\qquad E_G \;=\; \frac{G M^2}{d}.$

The energy scale $E_G$ is the Newtonian self-energy of the difference between the two branch mass densities. It is a classical, uncontroversial quantity. The rate is more audacious: it says gravity's own characteristic energy, divided by $\hbar$, tells you how quickly a macroscopic superposition decoheres.

For a one-microgram particle separated by one millimetre, this is a decoherence time near a nanosecond. For a dust grain in two places, it is faster than any experiment can resolve. The number is at the heart of the whole gravitational-decoherence programme.

Nothing in quantized linearized gravity actually derives it. A companion paper, Quantized Linearized Gravity Does Not Supply the Diósi–Penrose Decoherence Rate, spells out why: for a static, inertial source with equal-mass branches, the free graviton bath has no zero-frequency noise that couples to the mass-density difference at any temperature, and relational clock readouts depend on mass and time only through the combination $GMt$ — the wrong structure for a Diósi–Penrose rate. The rate reproduces exactly, however, if you postulate a source-independent potential noise with the right spatial correlation. The question this essay reports on is whether that noise can be understood as something physical, rather than a hypothesis grafted on from outside.

The thermal move

The quantum-geometric programme reads Einstein's equation as an equation of state — a statement that spacetime's geometry is an equilibrium variable set by a thermodynamic condition on entropy. If that is right, the coarse-grained geometry fluctuates in the way any equilibrium variable fluctuates: with a stiffness and a temperature.

The paper, Diósi–Penrose Noise from Relaxing Emergent Geometry (also as a PDF), computes those two numbers from first principles — within the framework's own axioms — and then asks whether the resulting noise matches the Diósi–Penrose kernel.

Stiffness, from entropy. Tile a region of space with little entanglement-equilibrium balls of some coarse-graining size $\ell$. Each ball has a generalized entropy; a small perturbation of the Newtonian potential changes it by a quadratic amount. Summing those area deficits, and applying the canonical fluctuation formula at the balls' modular temperature, gives — at linear order — a Gibbs weight

$\exp\!\bigl[-\beta\, E_{\mathrm{Newton}}[\Phi]\bigr].$

Here $E_{\mathrm{Newton}}$ is the ordinary Newtonian energy functional. The Poisson equation is its maximum; the attraction between masses is the ordinary one. The effective temperature works out to

$k_B T_{\mathrm{eff}} \;=\; \frac{5\,\hbar c}{2\pi\,\ell}.$

That one line is already striking: it ties a thermodynamic temperature of geometry to the coarse-graining scale, through nothing but $\hbar$ and $c$.

The equal-time correlator of the potential that comes out of this calculation is exactly Diósi's kernel. The right stiffness is there.

Relaxation, from modular mixing. A stiffness alone is not enough. General relativity propagates fluctuations at the speed of light; a Diósi–Penrose rate needs them to relax. The paper's fourth and sharpest assumption is that the coarse geometry relaxes at the modular mixing rate of the field that carries the noise, independently in successive cells.

That assumption is less arbitrary than it sounds. Modular flow is the natural dynamics attached to any local algebra and KMS state in algebraic quantum field theory; the mixing rate is set by the Planckian scale $2\pi k_B T / \hbar$. Demanding that one and the same KMS state describe both the static fluctuation and its decay forces a single rate coefficient. For a scalar field of operator dimension one — a Newtonian potential — the answer is

$\boxed{\;C \;=\; \frac{1}{\pi}\;}$

a factor $\pi$ below the "natural" Diósi–Penrose value $C=1$. For a one-microgram mass separated by one millimetre this moves the predicted decoherence time from $1.6$ nanoseconds up to about five nanoseconds.

Three fingerprints that could kill it

A model is only as serious as the ways it can fail. The thermal relaxation form predicts three measurable departures from the parameter-free Diósi–Penrose law:

1. Rate coefficient $C=1/\pi$. Any absolute measurement of the decoherence time at a known mass and separation reads off $C$. The two values differ by exactly $\pi$: a scan that finds $C=1$ falsifies this mechanism; one that finds the slower rate corroborates it (and leaves pure Diósi–Penrose with a mystery).

2. A Yukawa-regularized kernel, not a Gaussian one. Standard treatments smear the mass density with a Gaussian of size $R_0$. Here the kernel instead picks up a Yukawa cutoff $(1 - e^{-r/R_0})/r$ with $R_0 = \ell/5$. The two regularizations behave differently at short distances and at high photon energies.

3. A finite noise temperature. Collapse models traditionally feature white noise — the same at all frequencies. The thermal model's noise has a temperature $k_B T_{\mathrm{eff}} = \hbar c /(2\pi R_0)$, and emission above that scale is suppressed by the ordinary Bose factor $e^{-\hbar \omega / k_B T_{\mathrm{eff}}}$.

Together these say the model is not parameter-free Diósi–Penrose with a different cover. It is a specific thermal member of the same family, with a definite coefficient and a built-in high-frequency cutoff.

What the hard X-ray searches actually see

The parameter-free Diósi–Penrose model is already excluded. The Gran Sasso and MAJORANA experiments watched for spontaneous photon emission from idle germanium and quartz; the measured null signal rules out the smearing length $R_0$ staying below about $5 \times 10^{-11}$ metres. That looks like a knockout.

It is not a knockout for the thermal form. The Bose factor exponentially suppresses emission above the effective temperature, which for the surviving range of $R_0$ corresponds to soft X-rays (sub-keV). Recomputing the published bounds with the thermal spectrum: Gran Sasso allows $R_0$ down to $2 \times 10^{-13}$ metres; MAJORANA down to $4 \times 10^{-12}$ metres; the recent XENONnT X-ray constraint, which reaches keV energies, becomes the binding one, allowing $R_0$ down to $3 \times 10^{-11}$ metres. Heating bounds also weaken (by a factor of several). The hard-photon search that killed Diósi–Penrose simply does not see the thermal version.

Equivalently, the model picks a coarse-graining scale $\ell \gtrsim 1.3 \times 10^{-10}$ metres and predicts the surviving noise sits in the soft-X-ray band. The decisive test is now a dedicated thermal-template reanalysis of existing xenon or germanium data below a few keV. Hard-photon searches and low-frequency force-noise bounds have already said what they can say.

Where to be careful

The paper is honest that this is a model, not a derivation from first principles.

• The stiffness follows at linear order from three stated assumptions (an additive tiling of generalized entropy; Einstein-constrained fluctuations; the canonical reservoir formula). None is beyond doubt.
• The relaxation requires a fourth assumption — the modular mixing postulate — that general relativity does not supply.
• The dimension-one input $\Delta_\Phi = 1$ is motivated by the Newtonian potential's kernel shape, but it is not forced.

Every one of these can turn out wrong, and the model would fall with them. What has changed is the shape of the task. Instead of a free rate stapled on from outside, there is now a thermodynamic mechanism with one coefficient, three distinctive predictions, and an experimental lever — a sub-keV reanalysis of existing data — that it can be measured against. That is more than Diósi–Penrose had, and more than any derivation at linear order in $G$ has yet supplied.

The theory remains, as the honest ledger says, conjectural at the operator level. The novelty here is that the postulate it adds has started to look more like a hypothesis about the thermodynamics of geometry than like an arbitrary choice of noise.

Update (October 2026). A follow-up shows why the relaxation assumption cannot come from gravity in the vacuum at all, and what that commits the model to: Why Gravity Alone Cannot Decohere a Superposition.

Related: The Quantum-Geometric Correspondence; The BMV experiment and the cube law. Working paper; not peer-reviewed.