Anomalies Explained · 3 of 12
The i of quantum mechanics is two ticks of an eight-tick clock: on the states that half a cycle reverses, two ticks square to minus one.
Film · Where i comes from · 4 min
This page’s experiment, computed: every frame of the film comes from the same model as the figures.
The mathematics is standard (the first hard question names it). What Recognition Science adds, reading the world as a record of events, is the source of the square root of minus one. The record’s registers hold real values, one register in each of its cells; in every cell, the cell’s own two ticks on its core supply its i, made the same way in each; taken as the i of quantum mechanics, it is made, not added.
The paradox in three lines
Quantum mechanics is written with the number i, the square root of minus one, and its textbooks take that as a starting rule. A version built from real numbers alone matches every experiment with one shared source, and whether it could do all the same work stayed open for decades; since 2022, experiments on small quantum networks have excluded it whenever it combines two systems by the textbook’s usual product and its two sources share nothing.
A cell of the record holds eight real values, its register, and its clock moves them one place per tick. On the registers that four ticks reverse, every value turned into its opposite (the core), two ticks square to minus one: they are i. Only two independent real registers keep their shape under the tick, so the premise that every register is a sum of registers that keep their shape calls for numbers beyond the real ones, the complex numbers being the smallest that serve. Each cell’s own two ticks are its i on its core; on a pair of core registers taken together over the complex numbers, two ticks of the joint clock are just minus one, not an i.
Every number the record returns here is the textbook’s own. In the network test, real-number versions that combine two systems by the usual product, with sources that share nothing, reach a score of at most about 7.66, and complex quantum mechanics reaches 6√2 ≈ 8.49 (Renou et al., 2021); measured, 8.09 ± 0.01 on one chip (Chen et al., 2022) and 7.83 ± 0.03 with the sources and measurements kept apart (Wu et al., 2022). To return the complex value the record needs three owed statements: composition (that two registers held in two laboratories agree on their unit, the operation that plays i), the position read, and attachment (section 06). On the record’s picture of two registers, the measurements are evidence for that agreement; what they exclude is real-number versions of that kind: the usual product, with sources that share nothing.
01 · The question
Every count, every click and every probability a laboratory records is a real number. Yet quantum mechanics is written in complex numbers. A complex number can be drawn as an arrow; multiplying by i turns the arrow a quarter turn, and doing that twice turns it to point the opposite way, the same as multiplying by −1: i × i = −1. Schrödinger’s equation carries i, the square root of minus one, in front of its rate of change; a state is a list of complex values; the chance of an outcome is a squared size.
Elsewhere in physics complex numbers are a convenience. An engineer writes an alternating current as an arrow turning in the complex plane because the algebra is shorter, and keeps the real part at the end. Quantum mechanics has no real part to keep. Its first rule says a state is a vector in a complex space, and that rule is a postulate, found by working backwards from experiments; since 2001, reconstructions of quantum theory (Hardy; Chiribella, D’Ariano and Perinotti) derive it from axioms about what an experimenter can do. So is the i a convenience here too, or does the world need it? And if the world needs it, where does it come from?
02 · The history
“What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ψ is surely fundamentally a real function …”
Fifteen days after that letter, the Annalen der Physik received the fourth paper of Schrödinger’s wave mechanics, with i in a pair of equations for how a wave changes in time, one of which is the form used today. For forces that change in time he could not take the i out, and the paper’s last paragraph says so.
The longer history of i, back to the sixteenth century, is told on the page for the eight-tick register. Here is the part about quantum mechanics and real numbers.
1947
Paul Dirac
In the third edition of his textbook, argues from light. Add two polarization states with real weights and you reach only a one-parameter family of states; light has a two-parameter family, every elliptical polarization, so the weights must be complex.
The Principles of Quantum Mechanics, third edition, chapter I, §5
1960
Ernst Stueckelberg
Writes quantum theory with real numbers only. In place of i he needs an operation J with J² = −1 that commutes with every observable, and shows it is needed for an uncertainty principle. With that rule his theory is ordinary quantum mechanics in real notation; the real version that the network test examines drops the rule. “The problem was to show students, why the imaginary unit enters quantum theory.”
Helvetica Physica Acta 33, 727
1990
William Wootters
Points to what the real version lacks: in complex quantum theory, the statistics of measurements made on each of two systems, with their correlations, fix the state of the pair.
“Local accessibility of quantum states”
2009
McKague, Mosca and Gisin
Show that a real version reproduces every experiment in which the parties share one source.
Physical Review Letters 102, 020505
2013
Aleksandrova, Borish and Wootters
Build a real version in which one shared two-level system with real values, the “ubit”, plays the part of i for every system.
Physical Review A 87, 052106
2021
Renou and colleagues
Find a test with two sources that tells the two apart when those sources share nothing (section 06).
Nature 600, 625
2022
Chen; Li; Wu and colleagues
Three experiments pass the real ceilings of their tests; Figure 7 shows the two that use Renou’s score.
Physical Review Letters 128 and 129
2025–26
Hoffreumon and Woods; Barrios Hita and colleagues; Weilenmann, Gisin and Sekatski
Real-number versions reproduce the quantum predictions, by combining systems with a different rule from the usual product, or by letting the two sources share a suitable state: in the network test, one bit of entanglement between them is needed and enough to reproduce the complex predictions (Weilenmann, Gisin and Sekatski, who also bound how much the sources must share to explain the scores measured so far), and when the independence of the sources is defined by what can be observed, every correlation of the complex theory in a finite network is reached (Hoffreumon and Woods).
arXiv:2504.02808; Physical Review Letters 136, 240202 (2026); Physical Review Letters 135, 180201 (2025); arXiv:2603.19208; arXiv:2610.03644
Now
This preprint
Starts from a register of real values and finds i in two ticks of its clock (sections 03 to 05).
03 · The register
Recognition Science, the physics the institute works on, starts here. Some things can be told apart, comparing two readings needs both, and reading an old record is a new event (the three facts). Telling two things apart writes an entry, a posting, and no posting is ever erased. Each cell of the record has three on/off axes, one for each dimension of space. Three on/off choices give 2 × 2 × 2 = 8 states, and visiting all eight one change at a time takes eight ticks (the count); the paper cites the count of eight from earlier work and does not derive it.
The register of a cell is what it holds: one real value at each of its eight positions (the textbook’s nearest term for the register is its state, a list of amplitudes; on the core, two positions together carry one complex amplitude, section 05). One tick of the clock moves every value one position along: after a tick, position t holds what position t + 1 held. Eight ticks bring every value home.
(S f)(t) = f(t + 1), S⁸ = 1
A register that the tick only multiplies, S v = λ v, keeps its shape from tick to tick: every value is scaled by the same number λ, its multiplier. Such a register is a stationary state of the clock, a steady one (the textbook’s eigenvector of the tick). The question is now concrete. Does the register have a full set of stationary states, enough that every register is a sum of them?
04 · The hunt
Suppose one tick multiplies a nonzero register by a real number λ. Eight ticks then multiply it by λ⁸, and eight ticks bring it home, so λ⁸ = 1. A real number whose eighth power is 1 is 1 or −1. With λ = 1 every value equals its neighbour: the constant register. With λ = −1 the values alternate in sign: the alternating register. Either way the register repeats every two positions, so two of its values fix it, and such registers fill only two of the eight dimensions the register has, one for each of its eight positions. No full set of real stationary states exists. Proved in the paper · Theorem 1
Over the complex numbers the hunt succeeds. Let ω = eiπ/4, an eighth of a full turn. The eight registers dk(t) = ωkt, for k = 0, 1, …, 7, are multiplied by ωk at each tick, and every register is a sum of them, each with its own complex weight. The multiplier of d₂ is ω² = i.
So six of the register’s eight dimensions hold no real stationary state. The tick turns them, in three planes, by an eighth, a quarter and three eighths of a full turn, and a turn in a plane is multiplication by a complex number.
Premise That a register’s states are complex follows from Theorem 1 once every state is required to be a sum of stationary states of the clock: a register is exhausted by how it stands from tick to tick, with no remainder the tick does not account for. The paper names this as a premise and says it is close to the conclusion, since it asks for numbers that contain the tick’s multipliers.
Drop the premise and Theorem 1 rules out no real description of this kind: one that keeps the six dimensions as real planes and their turns, with no stationary state for them, is consistent with it. What the record supplies beyond the premise is the result of section 05, which needs none.
Figure 2 · Break it
Figure 3 · The eight turns
05 · Two ticks
The core is the set of registers that four ticks reverse: the value at each position is minus the value opposite it, f(t + 4) = −f(t). The register in the clock at the top of this page is one. Its first four values fix a core register, so the core has four dimensions, and a tick takes core registers to core registers.
So on the core four ticks are minus one, S⁴ = −1, which is what makes it the core, and two ticks are an operation on the core whose square is minus one: what mathematicians call a complex structure and the paper calls a unit. Proved in the paper · Theorem 3
i := S² on the core, i² = −1
That is the quarter turn in the clock at the top of this page: two ticks carry a core register to a second one at right angles to it and just as long, and two more carry it to its reverse. Is this the same i as the complex numbers of the hunt? Written in the complex stationary states of section 04, a core register is fixed by its d₁ and d₅ parts, and two ticks multiply those parts by the number i. Proved in the paper · Proposition 4 This proposition and Theorem 3 need no premise.
With this i the core’s four real dimensions become two complex ones, and each complex value is carried by two positions two ticks apart: z₂ = f(2) + i f(0) and z₃ = f(3) + i f(1), the coordinates along the register that holds 1 at position 2 and −1 at position 6, and the one that holds 1 at position 3 and −1 at position 7. The tick commutes with this i, so it acts on the complex values linearly, but its two multipliers there are ω and −ω, so the tick itself is not a number. The eighth turn ζ = (1 + S²)/√2 is a number on the core: it squares to i, and eight of them make a full turn, ζ⁸ = 1. Proved in the paper · Theorem 3
The eighth turn is worth a second look. Take a core register that holds a at position 0, b at position 2, their opposites at positions 4 and 6 and zero elsewhere: one complex value, z₂ = b + i a. On its two halves the eighth turn acts as (a, b) ↦ ((a + b)/√2, (b − a)/√2), and on z₂ itself it is a common turn by an eighth (every complex value of the register turned by the same angle). That map has the form of the one a balanced beam splitter applies to the values in its two arms; the likeness is one of form only, since here the two values are the halves of one complex value and the map is a turn of the register, not a device. The paper uses the form to derive the squared read from five premises (section 06 lists them). Among them: a read’s total is blind to a common turn, and the owed position read gives the form of a read, one function of each value, added up.
Figure 4 · Your register
Two ticks are not the only operation on the core that squares to minus one and commutes with the tick. (S + S³)/√2, one tick plus three ticks divided by √2, does too (commuting: it gives the same result applied before a tick or after it), and so do the negatives of both: four in all, and no more. Two things mark out two ticks and six ticks (which are minus two ticks) among the four: they are whole numbers of ticks, and they move the value at each position to one position, so the complex values they make each sit on just two positions; (S + S³)/√2 splits the value at each position over two positions. Between two ticks and six, the choice is the direction in which the clock is read. Proved in the paper · Remark 5 Taking the unit that is a whole number of ticks as quantum mechanics’ i is the record’s working identification.
Figure 5 · The other units
Working statement A register is held while nothing is being posted about what it holds. That a held register lies in the core, each value answered by its opposite half a cycle later, is the record’s closing read at one register: a working statement of the record, not proved in the paper. In plain words: the record takes a register left alone to lie in the core, and the paper does not prove it.
06 · Two registers
Can an experiment tell real values from complex ones? The best-known test of quantum correlations, Bell’s test in the form Clauser, Horne, Shimony and Holt gave it, cannot: two parties sharing one source reach the same ceiling, 2√2, with real values as with complex ones. In 2021 Marc-Olivier Renou and colleagues found a test that can, on one condition. It has three parties and two sources. One source links the first party to the middle one, the other links the middle party to the last, and the middle party measures its two halves together.
If the two sources share nothing, registers composed as real blocks (real values combined by the usual product of two systems) reach a score of at most about 7.66 in that test, a bound found numerically and not known to be tight; composed over the complex numbers they reach 6√2 ≈ 8.49. The measured values are 8.09 ± 0.01 on one chip, with nothing kept apart, and 7.83 ± 0.03 with the sources and the measurements kept apart (Figure 7). So real blocks whose sources share nothing are not how two laboratories compose. The condition matters: real blocks whose sources are allowed to share a suitable state reproduce the complex values (Weilenmann, Gisin and Sekatski, 2025; Hoffreumon and Woods, in two papers of 2026).
On a pair of core registers the clock ticks both, so two ticks act as i on each at once, and that operation squares to plus one: it is not a unit. On the half of the pair’s states that two ticks reverse, it is simply minus one. One register’s own two ticks, with the other left alone, is a unit for the pair: it squares to minus one. Taking the first register’s or the second’s gives the same unit exactly on that half, and there the pair is the product of the two over one complex unit. Proved in the paper · Proposition 29, the complex product inside the real one
On the record, the suitable state that real blocks’ sources would have to share lies in the half of the pair’s product that two ticks reverse, both registers on the same sense of the two-tick turn (Figure 6). The network measurements are therefore evidence that two registers held in two laboratories agree on their unit; with each register’s unit taken as its own two ticks, as the record takes it, that puts their joint states in the half that two ticks reverse. They do not by themselves tell the complex product from real blocks that share that state, and they do not measure the owed statement (below).
This page’s own remark, beyond the paper On the other half the two registers’ units are opposite, and that half is a product over one complex unit too, with the second register’s unit taken as its six ticks (on the core, six ticks are minus two ticks), so every score reachable on one half is reachable on the other: what makes the shared state the same sense of the two-tick turn is taking each unit as the register’s own two ticks. Keeping a pair there also needs each laboratory’s acts to commute with its own register’s two ticks: an act that flips the signs at positions 1 and 5 moves into the other half the whole part of a state held at that register’s odd positions.
Owed Composition: that two registers held by two recognizers (in the textbook, two separate laboratories) are two blocks of one cube, one set of on/off axes, each recognizer acting on its own block, and that their joint states lie in the half of the product that two ticks of the clock reverse, which is their product over one complex unit, and that the read is the one read of the whole cube. The paper does not prove it. The network tests bear on it as evidence: they exclude real blocks whose sources share nothing.
Returning the complex value 6√2 also needs the squared read. It rests on five premises. Two are principal: the position read, owed, that a read weighs each position of a register by the value held there and by nothing else; and that a read’s total is blind to a common turn of the register (every complex value of the register turned by the same angle). Three are mild: reads add over positions, an empty position is silent, and a value larger in size never reads less. Outcomes that are not sets of positions take the textbook’s squared weights, which the paper assumes and does not derive. Returning the value needs attachment too, owed: which posting of the record a laboratory click is, and that one is posted.
Figure 6 · Two registers
Figure 7 · The measurements
| What | Value | Source |
|---|---|---|
| Ceiling, real blocks whose sources share nothing | 7.66 | Renou et al., Nature 600, 625 (2021): 7.6605 to four places, found numerically, not known to be tight |
| Ceiling, complex values | 6√2 ≈ 8.49 | Renou et al. (2021) |
| Measured, superconducting qubits on one chip, nothing kept apart | 8.09 ± 0.01 | Chen et al., Phys. Rev. Lett. 128, 040403 (2022): 43 standard deviations above 7.66 |
| Measured, photons, with the sources and the measurements kept apart (all relevant events too far apart for a light-speed signal); detection loophole still open | 7.83 ± 0.03 | Wu et al., Phys. Rev. Lett. 129, 140401 (2022): 5.30 standard deviations above the real bound, as the authors report (the rounded figures shown give about 5.7) |
Each point is a published measurement, drawn with its error bar at the scale of the axis; the shaded band lies between the two ceilings. What would let real blocks reach these values: two sources that share one bit of entanglement (Weilenmann, Gisin and Sekatski, 2025).
07 · What it means
Recognition Science needed this standard mathematics because its registers hold real values. A register of real values has no i unless something makes one, and the textbook writes quantum mechanics with i. The clock makes one on every cell’s core, and the record takes it as the i of quantum mechanics: part of its own working, not a convenience of the description.
01 · One register
On one register, the record takes the textbook’s imaginary unit to be two ticks of its clock, not an invented number laid over a real world. On the core its square is minus one because half a cycle reverses a core register, and it is the number i itself.
02 · Two registers
Every cell makes its own i the same way, from its own two ticks, but that does not give two cells one i: on a pair of core registers composed over the complex numbers, two ticks of the joint clock are minus one. Two laboratories share one i only if their registers compose over one complex unit, each laboratory’s own two ticks, which is owed (section 06).
03 · The record
Quantum mechanics is not a separate set of laws for small things. It is what the law of the record does between two postings: histories that leave no record add, and what is posted is counted. Nothing is outside the record, and the premise of section 04 is that statement read at the clock.
08 · Hard questions
09 · Not claimed
10 · Read it
The Five Postulates of Quantum Mechanics on the Record: Theorems, Premises, and What Is Owed
Jonathan Washburn · Recognition Physics Research Institute
This paradox is number 3 of the twelve in its Section 8. Theorems 1 and 3 and Proposition 4 are in Section 3; the network test is in Section 7.
Sources