Born rule

In plain terms

Quantum mechanics predicts probabilities rather than certainties, so if you run the same experiment a thousand times you get stable statistics, and the recipe for those statistics is the Born rule. Each possible outcome is assigned a quantity called an amplitude, and the probability of the outcome is the square of the amplitude's size. The recipe works perfectly, but the textbook never says where it comes from and simply lays it down as an axiom, so you're left wondering why the square rather than the fourth power or the plain size.

That silence should be troubling, because everywhere else in physics probabilities come from somewhere, whether from counting cases, from ignorance of fine detail, or from the relative sizes of possibilities. Quantum mechanics is the only place where probability is written in by hand at the ground floor.

CSD's answer is that the recipe is geometry. The space of all quantum states divides into regions, one for each possible outcome of a measurement, and the probability of an outcome is the fraction of the space that its region occupies. The square in the textbook rule is simply what those fractions turn out to be when you measure the regions with the one measure the symmetry of the space allows.

In CSD

This is the programme's central result, the one everything else supports. The outcome regions divide the space of states, the Fubini-Study measure says how large each piece is, and the ratio of a region's size to the whole equals the textbook weight exactly, so there's no approximation and no restriction to special cases.

Three features carry the weight of the claim. The result is proved at every dimension, so it isn't an accident of the simplest system. It never uses Gleason's theorem, the standard route to the Born rule, so the corpus holds two genuinely independent derivations rather than one argument in two costumes. And symmetry forces the measure doing the work, which is what keeps the reasoning from being circular.

Much turns on this. If the result failed, CSD would be a reinterpretation of quantum mechanics, and because it holds CSD can claim a reconstruction, which is the whole stake.

Mathematically

For a preparation vector psi on CP^(N-1), the Fubini-Study measure of the barycentric outcome region attached to basis vector e_k equals the squared modulus of the inner product of e_k with psi. The corpus proves this coordinate by coordinate at general N, using neither Gleason nor Busch. It's unconditional in the sense that decides circularity, since nothing about the measure or the weights is assumed, while the headline form carries a genericity hypothesis of no vanishing amplitude. The companion unconditional forms retire that hypothesis, as a vanishing amplitude simply gives a cell of measure zero and the identity still holds.

The route runs through the moment map. Outcome regions pull back to regions of the probability simplex, where the Fubini-Study measure becomes flat by Duistermaat- Heckman, so the curved-space volume ratio collapses to an ordinary Lebesgue computation. A companion apex statement covers the dropped vertex, so all N coordinates of a generic preparation are covered and not just a chart that misses one.

The name

Max Born (1882-1970) was born in Breslau, now Wroclaw. He studied at Breslau, Heidelberg, Zurich and Gottingen, and returned to the Gottingen chair in 1921. During the 1920s his institute trained or hosted Heisenberg, Pauli, Jordan, Fermi, Oppenheimer and Teller, and for a decade it was the centre of the quantum world.

The probability rule has one of the best publication stories in physics. Born's 1926 paper on collisions stated, in the body of the text, that the probability was the amplitude itself, and the correction, that it's the square of the amplitude, went in as a footnote added at proof stage. Few footnotes have carried so much.

Born was dismissed in 1933 under the Nazi civil-service laws, so he went to Cambridge and then to the Tait Chair in Edinburgh. The Nobel Prize came in 1954, twenty-eight years after the footnote, and he felt the delay keenly. Olivia Newton- John was his granddaughter.

Related entries

• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Born volume ratioIn every finite dimension the Born weight of an outcome equals a ratio of volumes in the space of quantum states, for every outcome and every state, with no genericity condition attached.
• Typicality volumeAlmost every way the experiment could have started gives the quantum statistics, and that isn't an assumption about probability, as it's a measured volume.
• Outcome regionAn outcome is a region, and the apparatus alone draws its boundary, so the preparation decides where the trajectory lands and doesn't get to draw the map.
• Is the Born rule derived or assumed in CSD?It's derived, as the volume of a region of the underlying state space measured with the one ruler its symmetry allows, and the assumptions that remain are named posits, listed openly rather than hidden in the machinery.
• Fisher-Rao metricThe natural way to measure distance between quantum states agrees with the statistician's way to measure distance between the probability distributions they produce, exactly and with no constant left to fix.