Schrödinger’s cat and the measurement problem

Quantum mechanics is taught with two laws of change. In 1932 John von Neumann set them side by side: left alone, a state changes smoothly and linearly, by Schrödinger’s equation; measured, it jumps to one outcome. He showed that the line between the thing measured and the thing measuring can be drawn anywhere along the chain from the atom to the pointer of the apparatus, and the numbers come out the same. The rules say a measurement happens. They never say what one is.

In the summer of 1935, after Einstein, Podolsky and Rosen had published their paper on what the quantum state leaves out, Einstein and Schrödinger wrote to each other about what the linear law does to large things. On 8 August Einstein offered a charge of gunpowder that can go off by itself: after a year, he wrote, the quantum state would describe a blend of exploded and not yet exploded, though in reality nothing lies between the two. Schrödinger answered with a cat, and that November he printed it, in a long paper on the present situation in quantum mechanics:

He meant it as an absurdity. His next paragraph says that such a case “keeps us from accepting, so naively, a ‘blurred model’ as a picture of reality”: the law, read as a picture of what is there, leaves the cat smeared, and no one has ever seen a smeared cat. Nine decades on, the textbook still has no account of the moment a sum becomes one result; the readings that do without such a moment, Bohm’s of 1952 and Everett’s of 1957, change what is taken to be there. What physics has learned since is why the sum is so hard to see.

Between two entries there are histories that leave no entry. The record cannot tell them apart, so it does not count them one by one: they add as values, and values can cancel each other. What is written is counted, and counts never cancel.

How such a history steps is a theorem. Given its premises, and for the laws it covers, its answer is plain: left to itself, a state only turns, each part at its own rate. Nothing is lost, nothing is gained.

Call T the law that carries the record from one entry to the next. The paper asks for one rule that steps a history across a stretch with no entry. The rule acts on a sum of histories term by term (the paper takes this linearity as part of what a step is) and meets four demands:

• it uses the law and nothing else, in the same way for laws of every size;
• it loses nothing and gains nothing, because there is no record for anything to go to or come from;
• it regroups: any number of stretches read as one are the same as that many steps;
• it changes continuously with the law’s factors.

Then, for the laws the theorem covers (section 05), the rule is

step = exp(−icH),   H = −log T

for one real number c, the unit of the tick (the theorem “The step in real time”, Theorem 8, proved in the paper). In words: each stationary state, a state the law only shrinks, by the same factor each time it acts, turns instead, at a rate set by how fast the law would shrink it; one the law leaves at rest stays at rest. That c is not zero, so that histories turn at all, is not proved in the paper. Through one identification, a working statement, light is the long wave of the law; c is then the speed of light in cells per recognition, and the law of motion adopts c = 1 as its unit. That is an adopted unit, not a measured value. This is the textbook’s unitary step, the one Schrödinger’s equation generates, with i the square root of −1. One thing differs from the textbook: the energy operator H is not a second object supplied beside the law. It is minus the logarithm of the law between records, and here that law is the record’s own.

Given those premises, and for the laws the theorem covers, the textbook’s step is the record’s own step between two entries, and it does exactly what the puzzle says: it carries the parts of a held history forward as one sum. What it never does is write. The puzzle asks how a sum turns into one result. In Recognition Science a result is never a sum: it is an entry, written at a contact.

The paper proves what a contact does, in the theorem “A contact leaves the state or applies the crossing” (Theorem 25). A register is what one cell of the record holds: one real value at each of its eight places. The crossing X keeps lengths and undoes itself: X² = 1. A contact of strength θ, an angle (at 0° it does nothing; at 90° it is a whole crossing), is taken to be the operator (a premise)

C(θ) = cos θ + i sin θ X

The theorem treats two cases of what a crossing does to where a state ψ stood: it moves it wholly off, or it keeps it there. Where it moves it, the contact turns ψ into a sum of exactly two parts at right angles, relative to where it stood; the crossing picks the split:

C(θ)ψ = cos θ · ψ + i sin θ · Xψ

In the first part no crossing was applied, and it is ψ itself, exactly as it was, with squared length cos²θ. In the second the one crossing was applied, with squared length sin²θ.

Where the crossing keeps the state where it stood, nothing is moved off it: no part leaves, and no weight changes. On the place where the state stood, the crossing then only multiplies, by +1 or by −1, so the state is the sum of its +1 part and its −1 part (either can be empty). The contact turns the values of these two parts opposite ways and changes the length of neither:

C(θ)ψ = e^iθ Π₊ψ + e^−iθ Π₋ψ

Here Π_±ψ are the parts of ψ that the crossing multiplies by +1 and by −1. A contact of this kind acts on that one two-valued sign and nothing else: it keeps the weight of each sign and turns the two apart, and a state whose sign is already definite is left alone by a contact of any strength, up to a common turn. Two more statements are proved in the same theorem. Repeated contacts through one and the same crossing, with nothing else acting between them, add up: C(θ)C(θ′) = C(θ + θ′). And a state crossed an even number of times has not been crossed; one crossed an odd number of times has been crossed once.

On the record’s working statement, at a register the crossing is a sign on the register times a flip on the channel, the thing that touched it, and on the record’s working reading a held register stays held (section 05). Wherever the channel stands in a state the flip moves wholly off itself, the split of the first case holds anyway, on the channel: with squared length cos²θ the channel stands where it stood and the register is exactly as it was; with squared length sin²θ the channel is flipped and the register carries the sign.

What the theorem does not give

The theorem does not give the event, in either case: that one part is the one written and the other is not, and what the entry then does to the history. That is owed (attachment, below). That a part’s chance is its squared length, as the dial’s tally takes it, is the textbook’s rule, which the paper proves only for some parts (section 05). The paper states what is owed in these words, where a posting is an entry and the ledger is the record:

An atom held in a vacuum can go a long while untouched. A cat cannot. Every air molecule that bounces off it and every photon it gives off is a contact, a place where an entry can be written: a large warm body is touched constantly.

The figure below runs one way contacts can pile up, each read before the next; on the record’s working statement and working reading, the state each contact moves off where it stood is that of the thing that touched; a held register stays held. A real body has other ways. Contacts through one and the same crossing, with nothing written and nothing else acting between them, add their angles (section 03). Contacts with different molecules are different crossings. With nothing written, each one of strength strictly between 0° and 90° whose flip moves its molecule leaves the surroundings holding more about which part the body is in. The two parts then interfere less: the textbook’s decoherence. Which of these a real body undergoes, which touches tell a live cat from a dead one, and which event writes, are owed (attachment).

The laboratory pushes the line between the quantum world and ours by keeping a body’s surroundings from telling its parts apart (the timeline in section 01). In one experiment with fullerene molecules, the loss of interference followed the count of collisions with gas molecules (Hornberger and colleagues, 2003). In Recognition Science, only a contact resolves a held history: its parts add until an entry tells them apart, and stop at the first one that does, and no rate of resolution belongs to the body alone. That is a working statement, not a theorem.

Each result above holds under conditions, and each condition has a name.

• The step rests on four premises, and on linearity. The four demands of section 02 are adopted as premises, and that a step acts on a sum term by term is part of the paper’s definition of a step. Deriving the four from the record’s starting point, that occurrences can be told apart, is owed by the foundation of the record, not by quantum mechanics separately. The theorem is stated for laws on a finite number of dimensions with a full set of stationary states at right angles, each shrunk by a factor above zero and at most one, so their rates are bounded. The textbook writes the energies of an atom, and so of a cat, with no ceiling; for them the step is not proved in this paper.
• Which case holds for a register is not settled. That the crossing is a sign on the register times a flip on the channel is a working statement. Whether it moves a held register off its held states or keeps them, reversing the register’s own tick, depends on one count the record has not settled: the number of the register’s own moves in a window of the clock. The record’s working reading is the second case: a held register stays held, and a contact reads the sign of its moves and nothing else. On that working statement, and under either reading, the split falls on the thing that touched the register, wherever the flip moves that thing wholly off where it stood (section 03).
• That states are complex rests on one more premise, close to the conclusion: that every state is a sum of states the clock’s tick only multiplies (§3 of the paper). The complex unit itself, two ticks of the clock, needs no premise (“Two ticks are i”, Theorem 3, proved in the paper); that it is the textbook’s i is a working statement, and through it, from the step on, the paper takes its spaces complex, as the textbook does. That a held register lies in its core, where each complex value is carried by two places two ticks apart, is a working statement. The unit has its own page: Why complex numbers.
• The square rests on five premises, and is proved only for some parts. For a register, the read of a set of its positions is, up to one common factor, the sum of the squared values held there, given five premises about a read. Two are principal: that a read weighs each position by the value held there and nothing else (the position read, owed), and that the total read is blind to the eighth turn of a register in its core, which turns all its complex values together (such a turn is not a tick, which only moves values from one position to the next). Three are mild: reads of separate places add, an empty place reads nothing, and a value larger in size never reads less. The outcomes a held register offers are the sets of positions whose read is unchanged when all its complex values turn together; the least of them are the complex values themselves, each weighed by its squared length. One position alone is half a complex value: its read is a term of the total, not an outcome. For other parts the square is a premise, the textbook’s rule; what an entry does to the state is owed (attachment). The square has its own page: The Born rule.

The figure below breaks the second demand (“Counts show beats” and “Weights and turns are one complex rate”, Propositions 10 and 11, proved in the paper). Two possibilities that the law shrinks at different rates (the textbook’s two energies), once added, show up as a beat, a count that rises and falls; once unrecorded histories can lose something, it dies away. This is not the surroundings telling the parts apart: the possibilities themselves fade. At the dial’s far end the step is the law itself, the record being counted: a weight only shrinks, and nothing in it repeats.

Isn’t this just decoherence, or known mathematics?

The mathematics is known: energy as minus the logarithm of a positive law is how lattice physics reads a transfer matrix; the turn to exp(−icH) is the Wick rotation, given rigorous form by Osterwalder and Schrader (1973, 1975), and the paper’s argument that four demands leave only that turn ends with a step of Cauchy’s (1821); C(θ) = exp(iθX) is the textbook coupling of a two-valued pointer; and Hardy, Chiribella, D’Ariano and Perinotti, and Gleason derive the formalism and the square from other premises. The paper derives less than they do: it leaves the square for other parts a premise and the product for two laboratories owed. What it adds is a restatement of its premises, of the same strength and not a derivation: said of the record, they become statements a record makes of itself, each carried by an identification, and for the step, with linearity and continuity, they leave nothing free but a unit (section 05). Decoherence, and Feynman’s rule that alternatives which nothing, even in principle, tells apart add their amplitudes, give the same numbers and, like the record, do not say which part occurs; the record adds that an entry, once written, is one entry and permanent, and it owes the rest (attachment).

Is any of this different from ordinary quantum mechanics?

No number the record gives for this paradox differs from the textbook’s, so none of the experiments on this page can tell the record from the textbook. What differs is what is taken as given: the textbook starts from a state that turns and must somehow be measured; the record starts from definite entries and proves the form of the turning, up to one unit, under named premises, for finitely many levels with bounded rates. At the measurement itself it uses the textbook’s weights and state afterwards, and owes the event.

What is not proved?

Owed, for this paradox: attachment, which event writes the entry, that one part is written and what the entry does to the history (quoted in section 03), and the position read. The premises, the theorem’s limits and the working statements are each named where they enter (sections 02 to 05). For a massive body the record holds one more working statement, a candidate for the counted event: the record’s own strain, which physics calls the gravitational field, with a rule in the form Penrose proposed (see also Diósi). It concerns which event writes an entry, not a change to the step between entries, and its unit is open, so it gives no time for any body.

• No time and no size at which a cat, or any laboratory body, is resolved, and no collapse rate, a rate at which a held state would turn into one result by itself.
• No statement of which outcome occurs at a given contact.
• No special part for any kind of system: a contact is any touch between the register and what is outside it.
• No number of its own for a cat or any laboratory body (the third line at the top).

Jonathan Washburn. “The Five Postulates of Quantum Mechanics on the Record: Theorems, Premises, and What Is Owed.” Preprint, Recognition Physics Research Institute, posted 14 November 2026. Not peer reviewed.

The paper takes the five rules quantum mechanics is taught from and says, rule by rule, what is proved on the record, what is a premise and what is owed. The cat is the fourth of its twelve paradoxes (§8). The results this page uses are “The step in real time” (Theorem 8, §4); “Counts show beats” and “Weights and turns are one complex rate” (Propositions 10 and 11, §4); “The read is the square” and “The read of a complex value is its squared modulus” (Theorem 16 and Corollary 17, §5); and “A contact leaves the state or applies the crossing” (Theorem 25, §6); section 04 follows its fifth paradox, “Where the quantum world ends”.

Read the preprint (PDF), the institute’s library

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