The Measurement Problem That Won't Die

No better experiment will settle it. The numbers are already right. The open question is what the mathematics means: whether the wave it uses is a real thing, whether that wave ever becomes one definite outcome, and when. That question is the measurement problem.

There are several ways out — several accounts of what the recipe describes. Each keeps the predictions and pays somewhere else: in extra worlds, in a thinner account of what exists, or in locality. None is free.

Three things before you start. One: pick the way out you would like to be true — the essay asks you this first. Two: read each section; each one prices one option. Three: watch the meter and the ledger — the meter tracks the four options, the ledger records the bill. Both update as you read.

The problem comes from two rules that sit in every textbook. The first is the Schrödinger equation. Give it a state and it fixes the state at every later moment — smoothly, deterministically, with no chance anywhere. Its key property is linearity: a sum of states evolves into the sum of what each would have become on its own. Feed in a superposition and the equation carries it forward intact.

The second rule applies when you measure. On measurement the state jumps to one definite outcome, chosen at random, with probabilities set by the square of the wave. This jump is the only place chance enters the theory.

The two rules do not fit together. The first is deterministic and applies to everything. The second is probabilistic and applies when you "measure." The second should be a special case of the first, but it cannot be, because one is deterministic and the other is not. Here are the two rules on one system:

One more principle turns the tension into a contradiction. A property has a definite value only when the state is an eigenstate of that property — the eigenstate–eigenvalue link. In a superposition there is no fact of the matter about the value at all. So the linear equation, left to run, does not produce a blurred outcome or an unknown one — it produces no outcome at all.

Follow it up the chain. A device measures a superposed electron; by linearity the pointer ends up reading "hard" and "soft," with no fact about which. Put a person in front of the pointer — call her Martha — and the same linearity puts her belief into superposition: she believes "hard" and believes "soft," but there is no fact about what she thinks.

Schrödinger made the same point with a cat: the superposition scaled up to something alive, so the wave has the cat dead and alive with no fact about which. Bell stated the choice this forces: either the wave is not everything, or it is not right. Each way out in this essay takes one side of that choice.

Where does the jump happen — at the detector, at the cat, at Martha? The mathematics does not say. You can slide the boundary between the "quantum" part, described by the wave, and the "classical" part, described in plain definite terms, but the predictions barely change wherever you put it. Bell's complaint was that the boundary can be moved and the theory never says where it belongs. Tolerating that ambiguity permanently, at the most fundamental level, was for him the real departure from classical physics. Climb the chain and slide it yourself:

Start with what the wave function is. Essay 1 ended on this point: a system of particles is not described by one small wave each, but by a single wave over all the ways the whole system could be arranged. That space — configuration space — has three dimensions for every particle. For two particles it is six-dimensional; for a speck of dust, the dimensions are astronomically many.

Is that wave a physical thing or a bookkeeping device? The case that it is physical is direct. Interference changes where individual particles land — not just the averages — and it does so depending on real conditions, such as whether both slits are open, regardless of what anyone knows. That cannot be an effect of bookkeeping. It has to be the behaviour of something physical. The PBR theorem makes the conclusion exact: two different wave functions cannot describe the same physical state.

The wave also does not reduce to parts. Take two particles far apart. Each has its own summary, but the summaries leave out how the two are correlated — the joint wave carries more than the sum of its pieces. Pushed to the limit, only the whole universe has a wave function of its own. The theory's basic object is global, and it is defined on configuration space rather than on ordinary space. Here it is for two particles:

That is the second of the three questions. The first was outcomes: why does a measurement give one definite result? The second is ontology: what exists, and where? The third is locality, and it comes later. No way out answers all three cleanly. From here, a ledger tracks what each option buys and what it pays.

Each serious option keeps the experimental success but changes something else — what counts as a theory, how many outcomes occur, what exists, or the equation itself. They are not variants of one another. Each is a full position on what is real, and each pays.

The first is the one working physicists use: take the boundary as given, apply the collapse rule when you measure, and do not ask what "measure" means. It is the first of Bell's six worlds, the pragmatic one — when the cut gets awkward, fold more of the apparatus into the quantum description. It works for all practical purposes, and for most people that is the end of it.

Its price is that it is not a physical theory. It names no ontology and no dynamics, and it uses "observer" and "measurement" without defining them. Bohr's reply to Einstein was supposed to settle the matter; Bell could make little sense of it. The rule delivers predictions but gives no account of the world.

The second option keeps the wave and nothing else. If the equation never collapses, then after a measurement both outcomes are real: the world divides, and each branch holds one definite result and one version of you who saw it. This is Everett's position — the wave is everything and it is right, and the cat is dead in one branch and alive in another, and the branches no longer interfere.

Its price is probability. If every outcome happens with certainty, what does a 70% chance mean? The difficulty splits in two. The Everettians answer the practical problem — how to bet when both outcomes are coming — at best with a rule about caring more for higher-amplitude branches. They do not answer the epistemic problem: if every branch is real, nothing explains why betting on the odds has worked.

There is an ontology cost too. If reality is only the wave, the ordinary world of located things is hard to find inside it, and the branches themselves are only approximately defined — never sharp.

The third option adds something to the wave: particles, with definite positions at every moment. The wave never collapses; it guides the particles, and that is its only job. Nothing is ever indefinite — the electron goes through one slit, and the cat is alive or dead outright. What people call the "Schrödinger-cat state" is a state of the wave. The cat itself was never in doubt. This is de Broglie and Bohm's theory, and the probabilities follow from a typicality argument once you add the hypothesis that the particle positions start out distributed as $|\psi|^2$, the square of the wave.

Bell admired the theory and put its neglect down to fashion rather than physics — worth teaching, he said, as an antidote to the idea that vagueness is forced on us. He also named its price. To coordinate distant particles the guiding wave has to act across any distance at once: the theory is openly nonlocal. Only position is ever really definite, a preferred frame of rest returns, unobservably, and the arrangement keeps this machinery permanently undetectable. The positions are definite, but the price is nonlocality.

The fourth option changes the equation. Keep the wave as everything, but let it jump on its own from time to time — a spontaneous collapse, at random, with no reference to measurement. Ghirardi, Rimini and Weber give the jumps two new constants: a rate so low that a single particle jumps about once in a hundred million years, and a width for the collapse. A single particle would almost never be caught. A cat contains an astronomical number of entangled particles, so one jump anywhere drags the whole body onto a single outcome almost at once. The cat cannot stay both.

Bell wrote the theory out approvingly in his own hand — a small change, he said, that is enough to make the theory rational. It has prices: the two new constants, an ontology that must still be added by hand (sparse "flashes" in space, or a smeared matter density), and the tails of the collapse curve, which never quite reach zero. It also has a feature the others lack.

Because it changes the physics, GRW makes different predictions from the recipe — a universe running very slightly warmer than it should, a faint spontaneous radiation from ordinary matter. Experiments can look for those deviations and bound them. So GRW is different in kind. It changes the physics, it can be tested, and that makes it more vulnerable than the others, and more serious.

A standard reply says decoherence solves all of this. When a system touches its surroundings — air, light, a stray particle — it becomes entangled with them, and the interference between its branches disappears from anything you could easily measure. The system stops looking quantum. Decoherence is real, and it is why you never see a coffee cup in two places.

What decoherence does not do is select an outcome. The interference is not destroyed — it is exported into correlations with the whole environment, still there in principle and hopeless to gather back. No observer is required: in a real experiment, the "observer" that kills the interference can be a single proton, and the whole process runs on the plain Schrödinger equation. By the equation's own account the system has become more entangled — and that is exactly when it starts to look classical.

All the branches are still there, and nothing has selected one. The evidence that measurements have outcomes is not a decoherence experiment. It is that we see them. Decoherence hides the superposition. It does not collapse it, and it does not deliver a result. Turn the coupling up and watch:

The measurement problem survives decoherence. That is one reason it has not died.

One cost appears on nearly every line of the ledger, and it comes from an experiment with two particles. Prepare them together and send them far apart. Measure the same property on both and the results always match — or always oppose — at any distance. Bell's example was Bertlmann's socks: see that one sock is pink and you know at once the other is not. No mystery there; the socks were always that way.

Einstein's argument was exactly that. If a measurement here cannot disturb its partner far away, each particle must have carried its answers all along, fixed in advance. The argument does not assume the outcomes were predetermined; it concludes they were, from locality plus the perfect matching. Bell insisted on this because it is usually reported backwards: locality is the premise, and the predetermination is forced, since any looseness would spoil the matching.

Bell then asked whether answers fixed in advance, locally, could reproduce the full quantum predictions at every setting. They cannot. At intermediate settings the local accounting gives one answer, quantum mechanics gives another, and experiment agrees with quantum mechanics. A purely local common-cause story is not enough — some nonlocal dependence enters. He then removed every extra assumption — no determinism, no spin, no "particles," just clicks in boxes with the settings entered too late for any signal to reach the other side — and the conclusion did not change. Play the correlations and try to fake them with a local rule:

The result needs two clarifications. It is a real dependence between distant events — a fact about nature itself, whatever theory describes it. And it cannot be used to send a message: the statistics on your side do not shift with what is done on the far side, so no signal passes.

Bell listed four remaining positions. Quantum mechanics is wrong in these cases — the experiments say it is not. The settings were never free: the choice of what to measure was itself fixed in advance, along with everything else, so the argument's premise fails. Something travels faster than light, and Lorentz invariance is in trouble. Or Bohr was right and there is no reality below the everyday scale — which leaves the deepest theory vague about what "everyday" means. Each way out earlier in this essay has to meet this menu somewhere.