The Born rule: why probability is a square.

From the record’s starting fact (part 01), earlier work derives three dimensions of space, so one cell of the record has three on/off axes and 2 × 2 × 2 = 8 positions, and a clock of eight ticks (the count; the Overview). Number the register’s positions 0 to 7; each holds a real value. One tick of the clock shifts the register by one place: afterwards each position holds the value that sat one place further along (position 7 takes the value from 0), and eight ticks bring the register back.

Take the registers that four ticks reverse: the value at each position is minus the value opposite it. On them, two ticks are a quarter turn. Do it twice and every value is reversed, so two ticks square to −1: they are the number i, made of ticks (proved in the paper: “Two ticks are i”, Theorem 3, and “The clock’s i is the number i”, Proposition 4).

Half a quarter turn is an eighth turn: add the register to itself two ticks on, and divide by √2. On a register that holds a at position 0 and b at position 2, with their opposites at 4 and 6, the eighth turn leaves

(a, b) ⟶ ( (a + b)/√2 , (b − a)/√2 )

at positions 0 and 2 (Theorem 3). It is built from two ticks, an addition and a division by √2, and it has the form of the balanced two-port of optics: a beam splitter applies the same map to the values in its two arms. Here the beam splitter is the half-silvered mirror of part 01 used with two ways in: values come in by its two arms and leave by its two outputs, and with equal values in, at the right phase, one output, the dark port, stays empty. The likeness is one of form. Here the two values are the two halves of one complex value, b + ia, and the map is a common turn of the register, not a device.

Seen on that complex value, the eighth turn multiplies it by one number of the clock’s own making, (1 + i)/√2, which turns it by an eighth of a circle. It does the same to every complex value of the register: this is the textbook’s change of global phase, which changes nothing the textbook measures. For the reads of part 04 it leaves the read of each complex value unchanged (“The read of a complex value is its squared modulus”, Corollary 17); that a read’s total does not move under it is a premise. Seen on the two positions, it is the map above, and with equal halves it leaves one half of the complex value empty: position 2, with its opposite at 6. The eighth turn is not a tick. A tick only moves values from one position to another, so the total of a read taken position by position and added up is the same after it.

A read gives a weight to every set of positions of a register. The paper asks five things of it (its B1 to B5, the definition “Reads of the register”, Definition 14). Each is a premise of the theorem “The read is the square” (Theorem 16): the first and the last are its two principal premises, the middle three are mild, and the first is also owed, on the paper’s list as O2:

1. B1 · Position by position. The read of one position uses the value held there and nothing else: one rule φ turns each value into its term of the total. Owed
2. B2 · Additive. The read of a set of positions is the sum of the reads of its positions. Mild premise
3. B3 · An empty position is silent. A position that holds 0 reads 0. Mild premise
4. B4 · More is not less. A value at least as large in size never reads less, so a value and its opposite read the same. Mild premise
5. B5 · Blind to a common turn. On the registers that four ticks reverse, the total read does not change under the eighth turn, which turns every complex value together (why: What it means, 02). Principal premise

Now take equal halves, a = b. Before the turn, positions 0, 2, 4 and 6 hold a, a, −a, −a (the other four hold 0); after it they hold √2a, 0, −√2a, 0. The turn sends everything to positions 0 and 4 and leaves 2 and 6 empty, the form of a beam splitter’s dark port. A value and its opposite read the same (the fourth item), so the total before is four reads of a, and the total after is two reads of √2a plus two empty positions, which read nothing. Blindness says the totals are equal:

φ(√2 a) = 2 φ(a).

A value √2 times larger reads exactly twice as much. Of all the powers |x|^p, only p = 2 does that: the magnitude grows by √2 ≈ 1.414 and the fourth power by 4. Among all reads the doubling alone does not force the square; the whole equation below does, with the rule that more is not less. (The figure at the top runs the same arithmetic on a beam splitter’s two arms, a and b; the register holds each half twice, once with its sign reversed, so its totals are exactly twice the figure’s.)

With any two values the same blindness gives φ(a) + φ(b) = φ((a + b)/√2) + φ((a − b)/√2). The doubling clears the √2 and leaves the parallelogram law, φ(a + b) + φ(a − b) = 2φ(a) + 2φ(b). A rule that obeys it and never reads less for a value larger in size is a constant times the square (“The parallelogram law forces the square”, Lemma 15). So the read of any set of positions, of any register, is that constant times the sum of the squared values there (Theorem 16). The constant is free until the total is set to one; then the read of a set is its share of the sum of squares.

Complex values read as their squared size

On the registers that four ticks reverse, each complex value z is carried by two positions two ticks apart, with their negatives opposite them; the read of the two positions is the constant times |z|², the squared size, and of all four, twice that. Multiplying the register by any unit complex number leaves that read unchanged. The read of one position alone does change: with the constant set to 1, the register with 1 at position 0 and −1 at position 4 reads 1 at position 0 before the eighth turn and 1/2 after (Corollary 17). So the read is still taken position by position, and what a common turn cannot move is its total over each pair: the squared size of that pair’s complex value, the form in which the textbook states the rule.

A single position is half of a complex value, and its read is a term of the total, not an outcome. Were it posted alone, the record would hold a count that a common turn moves (1 to 1/2 above). Between records, a state left alone turns by the step (the second rule in part 01). A stationary state, one that the law between records only multiplies by a number, turns all together at a rate of its own (that histories turn at all is not proved in the paper). The record’s working statement takes the step’s i to be the clock’s two ticks, and with it that turn is a common turn of the register. So the count would rise and fall at twice the state’s own rate, while the record’s counts show only differences of rates (“Counts show beats”, Proposition 10).

So the outcomes a held register offers (one about which nothing has been posted) are sets of positions whose read no common turn moves, and the least of them are its complex values, each carried by two positions two ticks apart. The position read gives the form of a read, one function of each value, added up; it does not say that each term can be posted by itself. On these registers, what is owed is that the read of one complex value is the sum of one function of each of its two halves. That a laboratory’s outcomes are such reads, and which posting a click is, belong to attachment, the paper’s owed statement O1.

A unit total forces the same weights

A second theorem starts from the other end, and it is about a different object: not the two complex values of one register that four ticks reverse, but a system with several outcomes, each carrying a complex value, such as registers taken together. Take such a system with three or more positions, each holding one complex value (the paper states it for eight; the proof uses three), whose squared sizes add to one, and ask five things of a read: the reads add to one, none is negative, an empty position reads nothing, two positions holding values of the same size read the same whatever the other positions hold, and reads add over sets of positions. Then each position reads exactly its squared size (the theorem “Unit total forces the weights”, Theorem 18).

It drops the clock, the rule that more is not less, and every assumption of continuity. It takes the sum of squared sizes as the measure of the whole, as the read above gives on the registers that four ticks reverse, and carries the square from the whole to each part: had fourth powers added to one instead, the same five items would give fourth powers. The argument is of the kind Everett gave in 1957 (part 02), and the proof weighs three positions holding √x, √y and √(1 − x − y), with the rest empty. Three positions are needed: with two, the weight sin²(πx/2) of a squared size x meets every item and is not x (“Three positions are needed”, Remark 19). On the two complex values of the registers that four ticks reverse, the square comes from the first theorem instead.

Both principal premises carry weight: the paper shows it with two reads that break one each (“Each premise carries weight”, Proposition 20). The magnitude, |x|, meets every item of part 04 but blindness: the eighth turn changes its total. The fourth power, rescaled so the reads add to one, meets every item of the unit-total theorem but the position read: the weight of one position depends on what the others hold, and a split somewhere else moves it.

The two mild premises about a single value cannot be dropped either (Section 5). Without silence at an empty position, a constant plus the square, k + cx² with k ≠ 0 and c ≥ 0, meets the other four. Without the rule that more is not less, take an additive function A that is linear over the numbers p + q√2 (p and q rational) and not continuous: its square meets the other four and is not a multiple of x².

The fourth power’s failure, a split elsewhere moving the weight here, is the paper’s reason the position read is physical, not a convenience, and it is a reason about outcomes, so about whole complex values (What it means, 03).

Here the theorems stop. They are about reads of a register’s positions: that the outcomes of a laboratory apparatus are reads of such positions is owed (attachment, the paper’s O1). The position read is owed too (O2): the paper gives the reason above, not yet its derivation from the record, and inside one complex value, where it asks that the read be a sum over the two halves, the paper gives no reason beyond the form. Blindness to a common turn is a premise, not a property of the tick (part 03). For reads along other perpendicular splittings of a space the square is a hypothesis of the paper; Gleason’s theorem (part 02) derives it there, in three or more dimensions, from non-negative weights that add over every such splitting.

The read is the square, so three paths whose values add interfere only in pairs. In the textbook’s terms, the intensity at a point of the screen is the squared size of the summed amplitudes, and |A + B + C|² is the three single squares plus one cross term for each pair. So the pattern with three slits open is fixed by the patterns of the pairs and of the single slits, and the combination

I_ABC − I_AB − I_BC − I_CA + I_A + I_B + I_C = 0

holds exactly if the wave with several slits open is the sum of the single slits’ waves, where I is the intensity at a point of the screen with the named slits open. With real slits that is very nearly so: light that loops from one open slit to another adds a small term of the textbook’s own, which a test must subtract (Sawant and colleagues, 2014).

Sorkin wrote this combination down in 1994; it is now usually called the third-order interference term. Each pair term, such as I_AB − I_A − I_B, is the interference two slits add to their singles. With any other power the third-order term is not always zero, so measuring it tests the square, given that the paths’ values add.

Sinha and colleagues sent photons through three slits, from an attenuated laser and as heralded single photons, and bounded the third-order term below 1% of the pair terms (2010). Kauten and colleagues, in a five-path interferometer (ten sets of three paths), measured it as a fraction of the pair terms at (0.0 ± 1.6) × 10⁻³ with heralded single photons, and at (0.0 ± 3.1) × 10⁻⁵ with classical light from a laser, where it tests the square in the energy of a light wave, after correcting for the nonlinearity of their detectors (2017; in their numbering, which counts the pair terms as first order, it is the second-order term). For paths whose values add, the square gives exactly zero.

What would break it is shared with the textbook: for paths whose values add, a three-path term that the pairs do not account for would break the theorem “The read is the square” (Theorem 16), and a weight at one place moved by a split at another would break the position read. Each would break the textbook too.