Put a small mass in two places at once and ask how long the superposition survives. Diósi and Penrose proposed an answer built from gravity alone: the superposition decays at a rate set by the gravitational self-energy of the difference between the two positions, divided by Planck's constant,
For a microgram separated by a millimetre this is a few nanoseconds. The formula is simple and testable, and it sits at the centre of this research programme. The question I keep returning to is whether gravity actually produces it, or whether it has to be added by hand.
This post reports a sharper answer than I had before. It is a negative result, and I think it is the most useful thing the project has found in a while.
What a decay rate needs
A superposition decoheres when its two branches leave different records in their surroundings. A steady rate of decay, rather than a one-off loss of contrast, needs something more specific: an environment that keeps fluctuating at arbitrarily low frequencies, with a white-noise floor at zero frequency.
That is ordinary physics. A dust grain in a warm gas decoheres at a steady rate because the gas keeps supplying slow, random kicks. In the language of the fluctuation–dissipation theorem, a zero-frequency floor requires the environment to have friction.
The companion paper, Quantized Linearized Gravity Does Not Supply the Diósi–Penrose Decoherence Rate, checked this for weak, quantized gravity. For a static mass whose two branches have the same mass, the gravitational field leaves a record whose size is first order in Newton's constant, but that record stops growing. Coherence drops a little and then holds. There is no rate.
The general argument
That calculation was done at lowest order. The obvious hope was that something beyond it, a strong-gravity or non-perturbative effect, could supply the missing noise. It cannot, and the reason turns out to be general.
A white-noise floor needs fluctuations with zero energy but non-zero momentum: slow wiggles with a definite wavelength. In a gas these are everywhere (diffusing particles, sound waves that have damped out). But take any quantum vacuum whose energy is positive and that looks the same everywhere and in every inertial frame. In such a vacuum every excitation carries at least as much energy as its momentum times the speed of light. There are no zero-energy excitations with a wavelength, and so no white-noise floor, at any order in the coupling and whatever the interactions.
This holds for weak gravity, strong gravity, and any well-behaved quantum field theory. It also closes the hope that the thermal-looking "modular temperature" of a region of spacetime could help. That temperature is a property of the vacuum itself, and the vacuum obeys the argument. A numerical check over every kind of allowed spectrum confirms it: the noise floor always vanishes. Spectra that do produce a floor all belong to a medium.
So what would make the rate real?
If experiment ever sees the Diósi–Penrose rate, one of two things must be true:
- Spacetime behaves like a medium. At small enough scales the coarse-grained geometry has a rest frame and fluctuations that relax back to equilibrium, the way a warm material does. Then the zero-energy fluctuations exist, and the rate follows.
- Quantum mechanics is modified. The evolution of a superposition is not unitary: some collapse rule acts on it directly. This is Penrose's own picture.
Neither can come out of standard, unitary, Lorentz-invariant quantum gravity. The rate is therefore an independent law of nature, to be stated and tested, not derived from what we already have.
Why I chose the medium
The thermal model already in this programme is a medium in exactly this sense. Its coefficient, , comes from treating the coarse geometry as a system in thermal equilibrium, with one consistent temperature for its fluctuations and its relaxation. A collapse rule has no temperature. It cannot fix , and it heats matter without limit. So the medium is the reading in which the theory makes a prediction, and it keeps the second law intact. That is now the stated form of the theory.
A medium with a rest frame sounds alarming: doesn't it break relativity? I checked the obvious tests.
- Atomic clocks and preferred-frame tests of gravity. The medium's pull on matter at rest is just ordinary Newtonian gravity. Its rest frame shows up only through relaxation that takes about seconds. The resulting effects come out between ten and twenty orders of magnitude below current limits.
- Gravitational waves. If the medium also damped gravitational waves, they would die out within the width of an atom. LIGO and Virgo see them, so the medium may relax only the Newtonian part of gravity, never the waves. This is a real constraint, and the model now states it explicitly.
- Energy. The medium very slightly heats matter, so energy is not conserved for matter alone. The way the theory derives Einstein's equations turns out to allow this: it holds volumes fixed, which gives the trace-free form of the equations, in which a small energy imbalance shows up as a slow drift of the cosmological constant. Over the whole history of the universe, the drift is about one part in a hundred million of the observed dark energy. The same estimate shows the medium cannot be an ordinary hot gas: one at the medium's temperature would carry about times the observed dark-energy density. Its temperature has to be the vacuum's own.
What survives as a visible signature is a tiny daily modulation, about one part in a million, in a decoherence rate as the Earth turns relative to the cosmic rest frame. Nobody can measure that yet, but no collapse model predicts it.
The test that decides
The cheapest decisive test does not need a new experiment. Collapse-type models make matter emit faint X-rays. A white-noise collapse emits a flat spectrum. The thermal medium emits a spectrum that cuts off exponentially near a keV, so the ratio of counts between a low and a high energy band is between 3 and 500 rather than 1. Existing data from underground xenon and germanium detectors can be reanalysed for exactly that shape. A thermal cutoff would favour the medium. A flat excess would favour collapse and rule out this theory's coefficient. No excess pushes the medium's grain below about an atom's size.
Where this leaves things
The honest summary is shorter than the work behind it. Gravity, as we know how to quantize it, does not decohere superpositions at the Diósi–Penrose rate, and no improved calculation within standard physics will change that. If the rate exists, spacetime is a medium or quantum mechanics is incomplete. This project now bets on the medium, says what that commits it to, and points at data that can prove it wrong.
Working research notes; not peer-reviewed. The published papers are listed on the research page.