A Quantum Clock Is Not a Decoherence Law

The latest result in this project is a correction. An argument I had been using to connect quantum clocks to gravitational decoherence asks one mathematical statement to do two different jobs. It identifies how a system evolves, then treats that as a prediction of how an interference pattern fades. The second claim needs its own calculation.

I have written this up in a short note, From Modular Flow to Fringe Visibility: A Missing Dynamical Step, also available as a five-page PDF. It is a working paper, not peer-reviewed. The result is narrower than a new theory of gravity: some familiar, exactly solvable quantum systems expose a missing implication in this programme's proposed argument.

That is worth sharing. It changes what I can responsibly claim, and it makes the next research question more precise.

Moving a fringe is not fading it

An interferometer splits a quantum system between two paths and recombines them. By changing a phase setting, you obtain a pattern of bright and dark outcomes. Two changes to that pattern are easy to confuse.

A relative phase moves the pattern. Loss of visibility makes the difference between bright and dark smaller. Looking at one detector setting cannot reliably tell these apart: a count can fall because the fringe moved, even if its contrast stayed perfect.

For balanced paths, a simple description of the measured probability is

Here $\phi$ is the phase you scan, $c$ is the apparatus contrast, and $z(t)$ carries both the phase and the coherence associated with the two paths. Its angle shifts the pattern; its magnitude controls the visibility. The normalized visibility of this scan is $c|z|$. The oscillation in probability about one-half has half that amplitude.

To calculate $z$, you need to know what happens along each path and what information is inaccessible at readout. In ordinary quantum mechanics, a record in the environment can reduce the interference visible to the experimenter. This is standard decoherence physics, not a new mechanism proposed here.

The distinction matters because an energy scale divided by Planck's constant has the units of a rate. That does not tell you whether it produces a phase rotation, an oscillation, or persistent loss of contrast.

An example that comes back

The simplest check is almost embarrassingly small. Let a two-state environment evolve in exactly the same way whichever path is taken. Its state can change with time, and carry information about elapsed time, while the interferometer retains full visibility. Evolution alone has left no record distinguishing the paths.

That example might seem too easy, so the note also treats an environment that does respond differently to the paths: a harmonic oscillator pushed in opposite directions. Both conditional Hamiltonians have a lower energy bound. The overlap decreases as the oscillator states separate, then returns when they meet again.

Starting in the oscillator vacuum, the exact visibility is

Here $\omega$ is the oscillator's angular frequency and $g$ sets the strength of the path-dependent push, also in frequency units. For $g/\omega=1/4$, the visibility is approximately $0.607$ after half a period and exactly one after a full period. Matrix calculations with increasing oscillator cutoffs reproduce the analytic result.

There really is reduced interference during part of the cycle. There is no positive, constant exponential decay rate describing the whole evolution.

This oscillator is not a gravitational clock, and it is not a counterexample to every possible gravitational decoherence mechanism. A continuum of modes, inaccessible records, or a justified coarse-grained description may yield effectively irreversible decay. The example shows why those physical ingredients cannot be skipped. It does not show that they cannot exist.

The bound points the other way

A second correction concerns quantum speed limits. The Margolus–Levitin result puts a lower bound on the time an isolated system needs to reach an orthogonal state, given its average energy above the ground state. Equivalently, it caps how fast that transition can happen.

It does not require an interference experiment to lose visibility at a minimum rate.

Earlier QGC writing used this argument to support a coefficient window $C\in[2/\pi,1]$ in the proposed law $\Gamma=C E_G/\hbar$. That justification is withdrawn. The interval can still be examined as a chosen experimental hypothesis, but it is not a bound established by the speed-limit theorem. Nor does the theorem identify its energy variable with a gravitational density-difference energy automatically.

The note checks a related issue with thermal correlations: detailed balance constrains a relationship between positive and negative frequencies, but does not uniquely fix the noise spectrum. A predicted spectral feature needs a specified physical coupling too.

What changed—and what did not

The QGC programme explores a connection between quantum information, spacetime geometry, and an observer's notion of time. In the mathematics, a state together with an algebra of observables determines a particular flow, called modular flow. Identifying that flow with physical time is a proposed physical interpretation, not a prediction of a detector's counts.

The programme's gravitational decoherence rate remains a hypothesis. Showing that a modular generator represents physical time would be important, but would not, on its own, finish the interference calculation.

The contribution here is an audit of that logical step. The individual quantum-mechanical examples and speed-limit theorem are established material; I am not claiming their discovery. Applying them to the programme removes an overclaim and separates two research obligations that had been bundled together.

No experiment in this work falsifies or confirms QGC. Passing numerical checks does not change that. The broader audit includes conditional experimental-design calculations, but their assumed contrast and background bounds are not measurements of a real apparatus, so I am not announcing an experimentally demonstrated sensitivity or run time.

The next useful calculation is specific: choose a gravitational model, specify its prepared state and accessible interference measurement, evolve it through recombination, and calculate the counts. If an exponential law emerges, identify the regime and coefficient. If an extra noise or coarse-graining assumption is needed, say so explicitly.

Research notes, one result at a time

I want these updates to become a readable record of the work: one question, one calculation or correction, links to the details, and a clear statement of what remains open. Some entries will add a result. Others will retire an argument that did not hold up.

For this entry, the block we can keep is simple: a quantum clock's evolution and an interferometer's visibility are different observables. A proposed connection between them has to survive an actual measurement calculation.

Paper and reproducibility

• Read the technical note or download the PDF.
• Computational audit and scope.
• Tests and reproducible audit script: from the repository root, run python3 VERIFIED/tests/test_visibility_audit.py and python3 TOOLS/run-visibility-audit.py.

Prepared with AI assistance for drafting, algebra review, and code checks. This is not independent peer review. Corrections to this note will be dated and linked here.