Anomalies Explained. Twelve paradoxes of quantum mechanics come apart the same way.

One particle at a time, sent at two slits, builds bands on a screen, as if each one took both paths. Mark which slit it took, and the bands vanish. In the eraser the mark rides on a twin particle; “erase” it later, even after the particle itself has landed, and the bands come back in sorted subsets of the dots (Kim and colleagues, Physical Review Letters, 2000).

Two paths that nothing tells apart are one history in two parts. What each path brings to a place on the screen, its value (the textbook’s amplitude), adds to the other’s, and values can cancel: where the two arrive out of step there are gaps between the bands. A full mark, a value held by something else, tells the two parts apart: they now sit at different positions of the screen and the marker taken together, so the screen counts each on its own. Counts never cancel, so the bands go, and nothing about the mark has to be written down for that. A landed dot is written down for good, as an entry in the record (a posting), so a later choice at the marker cannot touch it; the choice only sorts dots already there. This uses a premise of the paper, that values add while nothing is written down, and its proof, from five premises with the owed O2 (line 3) among them, that reads are squares; that a landing is such a read is O1.

The textbook’s values: with a full mark, no bands among the unsorted dots, however long the delay, and, sorted by a read of the marker blind to the path, bands and their exact opposite in the two halves, adding back to the hump; for any mark, the bands’ strength V and how well the path can be told, D, obey V² + D² ≤ 1 (Englert, 1996), a textbook computation the paper recomputes and its theorems do not produce; measured with single photons, the exit each photon took, not a mark, telling the path: V² + D² = 0.97 ± 0.03 (Jacques et al., Physical Review Letters, 2008). Owed: O1, O2, and O3 for the two detectors.

Quantum mechanics gives each outcome a number, its amplitude, and the chance of the outcome is that number’s size squared. Why the square, and not the size itself or its fourth power?

A read taken position by position, whose total a common turn of all the register’s complex values cannot change, is a constant times the sum of squares, given three mild premises. With equal halves, the clock’s one-eighth turn leaves one half of a complex value empty, as a beam splitter with equal values in its arms leaves one output dark, so the read must double when a value grows by √2 (proved in the paper: “The read is the square”). On a system of three or more complex values whose squared sizes add to one, such as registers taken together, a read that totals one and meets four further conditions gives each position its squared size (“Unit total forces the weights”).

Each probability is the squared size of its amplitude, and because the read is a square, three paths whose values add interfere only in pairs, so for ideal paths the third-order term is zero (Sorkin, 1994; measured below 1% of the pair terms by Sinha and colleagues, Science, 2010, where the textbook’s own looped paths also add a small term). Owed: O2, O1.

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, published 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: the record takes them as its 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. 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.

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 (a bound found numerically, not known to be tight), 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). The record returns the complex value if two registers agree on their unit, which is O3; the measurements are evidence for that, and what they exclude is real blocks whose sources share nothing. Owed: O3, O2, O1.

The law that moves a quantum state is linear: a sum of possibilities stays a sum, and the law never picks one. Couple a cat to one atom that may or may not decay, and the law leaves the whole box as a sum: atom decayed and cat dead, plus atom whole and cat alive. Every look finds the cat alive or dead.

The order is the other way round. Records come first and are definite: an entry is one entry, never a sum of entries. Adding possibilities is what the law does between two entries (for laws with finitely many levels and bounded rates, the form of that step is the theorem “The step in real time”). A measurement is an entry. A contact of the paper’s form loses nothing and gains nothing (the theorem “A contact leaves the state or applies the crossing”): where its crossing moves a state wholly off where it stood, the state becomes a sum of a part left exactly as it was, weight cos²θ, and a part with one crossing, weight sin²θ; where the crossing keeps the state where it stood, no part leaves it: its two parts of definite sign turn in opposite senses, and neither changes length. A cat is in contact with its surroundings at every tick; that one part is posted is O1.

The weights are squares, and no laboratory body yet gets a number of its own; the working candidate for massive bodies has an open unit. A held register is written only at a contact. Owed: O1, of which this paradox is the plainest statement, and O2.

Molecules of about two thousand atoms, sent through an interferometer, show bands, as if each took more than one path at once (Fein and colleagues, 2019). No dust grain has been seen to. Where is the line between the quantum world and ours, and what draws it?

The line is not a size. A body’s two paths keep all their bands for exactly as long as nothing tells them apart. Once something it touched holds a mark, a clue that tells the paths apart wholly, the bands are gone from every read of the body alone (for a register, the square and contact theorems, Theorems 16 and 25, with O2 among the square’s premises; for a laboratory body, O1 too, below); once that mark is posted, written into the record, they are gone for good, because a posting is never erased. A large warm body is told apart constantly: every gas molecule that bounces off it and every photon it gives off carries some of a mark away.

The record gives no number of its own for any laboratory body, and the measured loss of contrast follows the count of collisions (Hornberger and colleagues, 2003). Owed: O1, with a working candidate for massive bodies whose unit is open, and O2.

A friend measures a photon inside a sealed laboratory and has a result. Wigner, outside, describes the whole laboratory as a sum of both results, and in principle he can make the two interfere. Which description is right?

A written result is a posting, one entry in the record, and a posting is never undone. Making the friend’s two results interfere needs them to be one history with no record. Both cannot be true of the same friend: a friend about whose result nothing has been posted is a held register, and Wigner can make the friend’s two possible results interfere; a friend who has written it is on the record, and he cannot. The answer follows from the definition of the record, that a posting is never erased, and from no theorem of the paper; two theorems serve it: “A contact leaves the state or applies the crossing”, which lets Wigner run a contact backward while nothing about its mark has been posted, and “Three ceilings”, which place the measured score.

Every friend tested so far was one Wigner could reverse or make interfere; the record counts such a friend as having written nothing, and until O1 says what writing is, that cannot be checked apart from the reversal. For those, correlations of the textbook’s kind come back, above Brukner’s bound of 2, which holds if every result is a fact for everyone, settings are freely chosen and nothing passes between the sides: 2.416 ± 0.075 with photons as the friends (Proietti et al., 2019), and, with a photon’s path as the friend, further bounds broken (Bong et al., 2020). For a friend who has written the result down, the record’s answer is no bands, the number of the textbook’s rule for a measurement; the textbook’s step alone would allow them back, and no experiment has reached such a friend. Owed: O1 (which posting a friend’s result is), O3 and O2.

Two stations, far apart, each choose one of two settings at random and record a result, +1 or −1. If every result were written in advance, carried by the particles like answers on a card, one sum of the four correlations, S, could not pass 2 on average (a correlation is how often the two results agree minus how often they differ). Experiments pass 2.

An act is what a station does to get its +1 or −1. Acts that only read values already written cannot pass 2, so passing 2 rules out answers held in advance, each fixed by what the pair carried and its own setting. The record’s account of what is left: some act writes its result. There is one record, and in its account the two stations are two blocks of one cube, sharing one state. On the record no signal travels: each station’s acts touch only its own block, and acts whose order makes no difference do not move each other’s weights. The correlation lives in one shared state, which is not local in Bell’s sense; that two laboratories share one cube is O3.

At most 2√2, with no signal from either station to the other. Measured in a loophole-free test (each setting chosen at random, too late for a signal at light speed to carry it to the other station, and every heralded pair counted): 2.42 ± 0.20 (Hensen et al. 2015). Owed: O3, O2, O1.

Two stations far apart, A and B, each choose one of two settings and get a result, +1 or −1. A correlation is how often the two results agree minus how often they differ: +1 if they always agree, −1 if they always differ. Add three of the four correlations and subtract the fourth. In quantum mechanics that score, S, stops at 2√2 ≈ 2.83. Yet the rule that seems to govern distant stations, that neither station’s choice can change the odds of the other’s results, allows S = 4. Why does it stop at 2√2?

On the record, the ceiling comes from three conditions: both stations act on one shared state; each act has two outcomes, +1 and −1, and is its own undoing: done twice, it gives back the state it started from, so asked twice it agrees with itself (the textbook’s observable with values +1 and −1), and every act at one station commutes with every act at the other, which for acts means that on every state they give the same odds in either order; and each read (the textbook’s probability, once the total is set to one) is a square: the squared length of the part of the state that matches that outcome. With these, S is at most 2√2, the combined length of a unit square’s two diagonals, and suitable acts on one cube of on/off switches reach it exactly; one table of probabilities that merely does not signal reaches 4. The ceiling, that one cube reaches it, and the table that reaches 4 are proved again in the paper’s Section 7, after Tsirelson (1980) and Popescu and Rohrlich (1994), in the theorems “Three ceilings” and “One cube reaches the middle ceiling”.

The ceiling is 2√2 = 2.828427…, and the closest of the measurements the paper cites, 2.82759 ± 0.00051 (Poh et al. 2015), sits 0.00084 under the ceiling, 1.6 times its quoted uncertainty; no measurement has passed it. Within the textbook the bound is Tsirelson’s theorem; why nature stops there is the open question. Not resolved on the record until O3 is a theorem: this paradox is the composition law. Owed: O3, O2, O1.

Some quantum measurements can be made together: those whose order makes no difference, which physicists say commute. Nine measurements, each giving +1 or −1, can be set in a square so that the three in each row, and the three in each column, can be made together. If each carried its answer in advance, the same whichever row or column it is measured in, the three rows’ products would multiply to the same number as the three columns’, since both multiply all nine answers once. Quantum mechanics says the rows give +1, +1, +1 and the columns +1, +1, −1.

A measurement is a read of an act: it writes the act’s result as a new posting; it is not a look at an entry already there. Two acts can be read together exactly when they commute (proved in the paper). The square’s nine acts work on two of the three on/off axes of a register of the record, and each row’s and each column’s product is exact, on every state. If the answers were entries already on the record, one for each act whatever it is read with, they would form a table of nine signs, and no table has both the rows’ products and the columns’. It is a count.

On the square’s test the nine acts score 6 and any table of nine signs written in advance at most 4, by the textbook’s computation, which the paper recomputes and its theorems do not produce. Two trapped ions measured 5.46 ± 0.04 on one state, with violations of like size on the other states they prepared (Kirchmair et al. 2009). Owed: O1, O2, and O3 where two ions carry the two axes.

Look at one object at three times, each look giving +1 or −1: a current running one way round a loop, or the other. Give each pair of times its own runs and score the pair: how often its two looks agree, minus how often they differ. If the object is in one definite state between looks, and a careful look leaves it as it was, the scores for times 1 and 2 and for times 2 and 3, minus the score for times 1 and 3, K, cannot pass 1. Is a large object like that?

Not while nothing writes it. What is definite is what has been written. Between two looks, an object that writes nothing holds values that add, as waves add, not one settled state; and a look is itself a writing: a posting, an event written into the record. Reading one yes-or-no look first moves the chances of another by an exact amount: never more than one half, and zero whatever the state exactly when the order of the two looks makes no difference. Any turn other than none or a half turn makes the order matter, and a turn is what carries the score past 1; a register only shuffled between looks, and read, never passes 1.

The 3/2 at a turn of π/3 is the textbook’s computation with squared weights; the paper recomputes it, and its theorems do not produce it. The bound 1 comes from counting over triples of values that careful looks leave undisturbed, the three-look form of Leggett and Garg’s inequality. Measured on a superconducting loop: a middle look whose result nobody inspected moved the last look’s average beyond what a definite, undisturbed current allows, by about 84 standard deviations (Knee et al. 2016). The test refutes definite values and undisturbing looks together and cannot say which fails; the record gives up both. Owed: O1, O2.

A particle meets a barrier it has too little energy to get over, and sometimes it turns up on the far side anyway: it tunnels. How long was it inside? The ways of defining that time give different answers, and some of them level off as the barrier gets wider (Thomas Hartman, Journal of Applied Physics, 1962).

Time is a count of postings, the entries written in reality’s record. While nothing is posted about a particle it is held, and a value held inside a barrier posts nothing: no entry marks it going in or coming out, so the crossing has no count of its own. That is not a time of zero: the record’s count runs on while the particle is held, but no entry divides that count, so no part of it is a time inside. Every measured “tunnelling time” is the reading of some apparatus making its own contact with the particle; what a contact of the paper’s form does to a held state, in the two cases the paper treats, is its contact theorem.

The record gives no number of its own here. Two clocks, two answers: at most 1.8 attoseconds for electrons leaving hydrogen atoms, by an attoclock (Sainadh et al., Nature, 2019), and 0.61 ± 0.07 milliseconds for rubidium-87 atoms crossing a barrier 1.3 micrometres thick, by a spin clock (Ramos et al., Nature, 2020). Owed: O1.

Particles of half-integer spin, such as electrons, protons and neutrons, exclude one another: no two of the same kind share a state. Particles of whole-number spin, such as photons, can crowd together into one (Wolfgang Pauli, Physical Review, 1940). Spin is about how a particle behaves when it is turned. Why should it decide how two of them behave together?

The record’s clock has eight ticks (from earlier work). Its core is the set of states whose sign four ticks, half the cycle, reverse, and eight restore. One full turn reverses the sign of a half-integer spin’s state, and two restore it; that the two reversals are one is not proved. Given that a swap of two identical records (the textbook’s identical particles) reverses the sign of their joint state, exclusion follows at once: where the two would be in the same state, the joint state equals minus itself, so it is zero.

Only signs come back, −1 and +1; given the odd sign, exclusion is exact, as in the textbook, with no violation, and none has been found. The puzzle is not resolved on the record; in relativistic field theory the link is a theorem (Pauli, 1940, for free fields), and what is missing is an elementary reason for it. Owed: O4, together with which turn of a body four ticks are.