A Rate Is Just the Zero-Frequency Part of a Spectrum

Modular Spectroscopy — Quantum-Geometric Correspondence series

The Problem With Being Right About the Rate

A massive body in spatial superposition decoheres through its own gravitational field. Diósi and Penrose fixed the rate by the branch-pair self-energy, $\Gamma \sim E_G/\hbar$ with $E_G = GM^2/d$. Continuous spontaneous localisation reaches a comparable scale. The Quantum-Geometric Correspondence derives the same $E_G/\hbar$ at first order in $G$. For a microgram separated by a millimetre they all say about $6\times10^{8}\ \mathrm{s}^{-1}$.

This agreement is usually presented as reassuring. It is actually the central experimental difficulty. Measuring that rate — already at the frontier of levitated optomechanics — would tell you something real and important: that the decoherence is gravitational and intrinsic rather than environmental. It would not tell you which mechanism did it. Every candidate predicts the same number, so the number cannot discriminate between them.

Unless you stop treating it as a number.

What a Static Experiment Throws Away

A decoherence rate is the zero-frequency content of a noise spectrum. The which-path environment has a spectral density $S(\omega)$, and a static interference experiment reads only its floor, $S(0)$. Everything the spectrum does at nonzero frequency is discarded before the measurement begins.

So the question becomes: does the mechanism predict a frequency dependence, and does it differ from the structureless white noise of a collapse model by enough to resolve?

It does, and the reason is structural rather than lucky.

Why the Principle Fixes the Whole Spectrum

The programme rests on identifying the modular Hamiltonian of an observer's algebra with the physical Hamiltonian, $K = 2\pi H_{\mathrm{phys}}$ — the sharp form of the Connes–Rovelli thermal-time hypothesis. This paper does not try to derive that relation. It posits it as a principle, the Modular–Physical Correspondence, in the way the equality of inertial and gravitational mass was posited rather than derived, and then follows the consequences.

The consequence here is immediate, and it is stronger than one might expect.

That relation carries the Bisognano–Wichmann normalisation: the flow it generates is thermal at the universal inverse temperature $\beta_{\mathrm{mod}} = 2\pi$ — the same temperature behind the Unruh effect. If physical time is modular flow at that temperature, then the environment carrying which-path information is in a KMS (thermal) state at that temperature. And once you know a bath is thermal at a known temperature, the fluctuation–dissipation theorem hands you its entire noise spectrum. No freedom left:

This is a white plateau of height $S(0) = \eta/\pi$ — which reproduces the static rate $E_G/\hbar$ exactly, as it must — crossing over to a linear rise at a knee:

One posited principle, and the whole spectrum falls out. That is the kind of rigidity worth testing.

Two Features, Honestly Separated

The spectrum has two testable features, and it matters a great deal not to conflate them.

The floor is uncoolable. It carries the modular temperature, not the laboratory temperature. Cooling your apparatus does not reduce it. It is the irreducible decoherence that survives perfect isolation — and it would be the first evidence of a noise floor no engineering can remove.

But this feature is shared with the entire Diósi–Penrose and collapse class. So is the $M^2/d$ scaling of the rate itself. These separate intrinsic gravitational decoherence from environmental decoherence. They do not separate this mechanism from its competitors, and claiming otherwise would be the easy overclaim to make here.

The knee is the discriminating feature. Its frequency is set by the universal modular temperature, it scales as $M^2/d$ with no free parameter, and — this is the useful part — it is independent of the open rate coefficient $C$. That coefficient, bounded to $[2/\pi,\,1]$ but not pinned, is the programme's main residual ambiguity. It enters $S(\omega)$ only as an overall multiplicative factor, so it scales the floor and leaves the knee frequency untouched.

An experiment measuring the knee is therefore reading a prediction with no adjustable parameter at all. That is rare, and it is what makes this worth building.

The Measurement

The protocol is noise spectroscopy: sweep a which-path modulation across roughly 0.1–100 Hz for femtogram-to-nanogram masses at micron-to-millimetre separations, and watch how the decoherence responds.

The discriminator is clean:

Two independent handles. The response above the knee separates a thermal modular bath from white collapse noise. And even a collapse model retrofitted with a cutoff to fake a knee cannot make that cutoff scale as $M^2/d$ — the cutoff is a property of the model, not of the mass and separation you happen to load into the trap.

Sweep the mass, and either the knee moves the way the principle says or it does not.

The Honest Bottleneck

Worth stating plainly, because it determines whether this is a real proposal or a thought experiment: the bottleneck is coherence and isolation, not modulation.

Modulating a which-path degree of freedom in the 0.1–100 Hz band is not hard. Holding a femtogram-to-nanogram superposition coherent and isolated long enough to sweep that band is the entire difficulty, and it is precisely where levitated optomechanics is currently pushing. This proposal does not ask for a new capability; it asks for the capability the field is already chasing, used differently.

What a Measurement Would Settle

The relation $K = 2\pi H_{\mathrm{phys}}$ is posited, not proven. Proving it for interacting gravitons amounts to constructing quantum gravity's modular theory, which is open.

The spectrum above is a derived consequence of it. That is the logical structure worth being clear about: the principle is an assumption, the knee is a theorem given the assumption, and the measurement tests the assumption through the theorem.

If the knee is there, at the frequency the principle demands, tracking $M^2/d$ as the mass is swept — then the thermal-time hypothesis has been measured rather than argued about, and the modular identity graduates from something to hope for into something to explain.

If the noise is flat, the principle is wrong, and the agreement on the rate was a coincidence rather than a clue.

Either way, the experiment stops being a test of whether gravitational decoherence exists and starts being a test of what it is.

This is the modular-spectroscopy paper of the Quantum-Geometric Correspondence series, deriving the full decoherence noise spectrum from the Modular–Physical Correspondence and identifying the C-independent knee that discriminates it from Diósi–Penrose collapse. The full paper carries the derivation, the protocol, and an explicit tiering of what is mechanism-specific versus shared with the broader first-order-in-G class.