Why third-order interference must vanish

There is an experiment that would break quantum mechanics if it came out wrong, and it has never been done at high precision, because everyone is confident of the answer. You block none of the slits, one slit, two slits, or three slits, and you compare the interference patterns. Quantum mechanics makes a sharp statement about the third-order piece. So does every theory that wants to call itself quantum. This post is about that statement, and about why a multiverse framework has an unusual stake in it.

Sorkin’s hierarchy

Write P of S for the detection probability with slit set S open. Define the first-order interference I1 as P of all three slits minus the sum of the three pairwise patterns with the third closed, corrected by the singles. Define I2, the ordinary interference, as the pairwise cross terms, and I3 by continuing the inclusion-exclusion one level up: the three-slit pattern, minus the three two-slit patterns, plus the three singles, minus nothing.

Sorkin’s observation was that theories partition cleanly by which of these vanish. Classical probabilities have no interference at any order. Ordinary quantum mechanics has nonzero I2 and exactly zero I3. Some generalised theories have nonzero I3. The vanishing of I3 is not an approximation in quantum mechanics. It is an exact algebraic consequence of the Born rule: probabilities are quadratic in amplitudes, and quadratic objects cannot produce a third-order inclusion-exclusion term.

The experiment, by Sinha and collaborators, placed a bound on the relative size of I3 in a molecular-beam-like interferometer at around one part in a hundred of I2, consistent with zero, and the arithmetic check in this programme’s derivation code is more brutal: evaluate the Born-rule prediction on a three-mode toy model and I3 comes out at ten to the minus seventeen, which is machine zero. The theory’s zero is exact.

Why a multiverse framework should be terrified of I3

Here is the stake. The framework’s leakage hypothesis wants an operator that couples branches: a small, structured channel by which one reality’s state influences another. The tempting formal move, once you allow yourself that operator, is to let it act nonlinearly, because nonlinear modifications of quantum mechanics are how people usually build in a little extra structure, and a branch-coupling looks like exactly the kind of thing that would need one.

Nonlinearity has a price, and the price is I3. A nonlinear evolution generically produces nonzero third-order interference, and the experimental window for I3 is brutally tight. So the leakage operator, if it exists, must be linear: an ordinary operator on the joint state of branches, like every other quantum interaction, just one that happens to connect sectors we usually treat as separate. Linear coupling between sectors is exotic enough. It does not also need to break the best-tested structural fact in interference physics.

This is one of the framework’s two hard exclusions, and it is worth stating as a theorem-shaped sentence: no local deterministic classical substrate, and no nonlinear multiverse. The first comes from the next post. The second comes from this one, and it is a real constraint that people proposing modified quantum mechanics for exotic purposes routinely rediscover the hard way.

What would actually kill this

Two things. A confirmed nonzero I3 in any interferometer kills ordinary quantum mechanics itself, which would be an earthquake far beyond this programme, and every claim in the archive that leans on unitarity would need rebuilding. Short of that, a specific linear coupling model whose mediated correlations mimic the leakage signature would show the exclusion was too strong: the framework currently cannot have a coupling that is both branch-mediated and nonlinear, and if someone exhibits a viable nonlinear theory with vanishing I3, the theorem’s premise fails and the constraint dissolves. Both doors are labelled. That is what a falsifiable exclusion looks like from the inside.