Where the quantum world ends decoherence and the classical limit

Quantum mechanics was built for atoms, and its founders drew a line around it. Niels Bohr held that a measuring instrument must be described in the plain terms of everyday physics, while the thing it measures follows the quantum rules. Werner Heisenberg called the line between them the cut. In 1932 John von Neumann showed that the cut can be moved anywhere along the chain from the atom to the pointer of the instrument without changing a single prediction. The rules said where the line could go. They never said where it is.

In 1935 Erwin Schrödinger carried the chain up to a cat in a box (the cat’s own page). In 1970 H. Dieter Zeh pointed to what everyday bodies never escape: they are never alone, their surroundings touch them all the time, and each touch can carry away which way they went. Wojciech Zurek (1981, 1982) showed how the surroundings pick out the states that last, and Erich Joos and Zeh (1985) worked out how fast the rest are lost: for a dust grain in air, the two paths stop showing bands almost at once. Physicists call this decoherence, and it moved the question from how big a body is to what touches it.

Then laboratories pushed the tests outward, to heavier bodies and to deliberate contacts, in the experiments below. None has found a size at which the bands stop by themselves.

Between two postings about it, a body is held: nothing has been written about which path it took. The record cannot tell the two paths apart, and on the paper’s premise that a step is linear it adds their values, which can cancel: Feynman’s rule, add the amplitudes of alternatives that nothing tells apart.

At each place on the screen the value from one path, a, and the value from the other, b, arrive together. The paper proves that the read is the square (Theorem 16, with Corollary 17 for complex values; the Born rule has its own page): a read of a register weighs the squared size of what is held at each place. Taken to the screen, which is part of what is owed, it says that a place shows the squared size of what arrives there,

|a + b|² = |a|² + |b|² + 2 |a| |b| cos φ

where φ is how far out of step the two values arrive. The last term is the bands: in step it adds, and half a turn out of step it subtracts, down to an empty gap when the two sizes are equal.

The premises, exactly

That the two paths’ values add is the step between records, which is linear by premise: the paper takes linearity as part of what a step is (§4). “The read is the square” (Theorem 16) is proved for reads of a register’s positions (a register is what one cell of the record holds, one value at each of its places), with Corollary 17 for complex values, from five premises. Two are principal: that a read weighs each place by the value held there and nothing else (the position read, which is owed), and that the total read is blind to a turn of the whole register together (a premise). The other three are rules of a read: reads of separate places add, an empty place reads nothing, and a value of larger size never reads less. The paper calls the last two mild and shows that neither can be dropped. That what each place on the screen shows is a read of a register’s positions (each at least one complex value, held at two positions) is part of attachment, which is owed.

Now let the body touch something on the way that keeps a mark telling the paths apart wholly: a gas molecule that bounced off one path one way and off the other path another way. The two paths now end at different places of the body and the gas molecule taken together, and reads add over separate places (a premise of the square theorem, Theorem 16). If the screen’s places are such reads, which is owed, the read of a place on the screen becomes |a|² + |b|²: the last term is gone, and so are the bands. Nothing had to be posted about the gas molecule. A mark that tells the paths apart wholly takes the bands out of every read of the body’s positions alone; one that tells them apart in part takes part of them (section 03).

Bands on a screen can also blur with nothing told apart: when the instrument shakes, or what the body passes is itself in a mix, each body’s bands land shifted by an amount nobody records, and their sum is flatter. That blur writes no mark and can in principle be undone; physicists usually distinguish it from decoherence.

While a mark that tells the paths apart wholly is only held, the bands are out of every read of the body’s positions alone, but a read of the marker that is blind to the path can still sort the landed dots into two sets, one showing bands and the other their complement: that is what the delayed-choice eraser does (the double slit). Once the mark’s value is posted, that sorting brings out no bands, and a posting is never erased: the bands are gone for good.

A gas molecule that bounces off the body may bounce a little differently depending on which path the body took. Then it holds a mark, and the better its two possible states can be told apart, the more of the bands it takes. The paper proves what a contact of its form does to a state, the two parts it leaves and their weights (Theorem 25, “a contact leaves the state or applies the crossing”); the algebra is the textbook’s own.

A contact is built from a change called the crossing, X, which keeps lengths and undoes itself (X² = 1), and has a strength θ, an angle. With i a square root of −1 (the imaginary unit, used throughout quantum mechanics), the paper’s form is

C(θ) = cos θ + i sin θ X

Where the crossing moves a state ψ wholly off where it stood, the contact makes it a sum of two parts at right angles: one in which nothing was posted, ψ itself exactly as it was, with squared length cos²θ, and one with the crossing applied, Xψ, with squared length sin²θ. Contacts through one and the same crossing add their angles: C(θ)C(θ′) = C(θ + θ′).

For a stand-in channel with two states, a gas molecule simplified to two outcomes, this works out simply. Let the contact touch the body with opposite signs on its two paths. Then the channel ends in one of two states, one after each path, and the bands keep

contrast left = |cos 2θ|

of their contrast: the size of the overlap of the channel’s two states. At θ = 45° the two states are at right angles and one contact takes all of the bands; weaker contacts take less. Past 45° the two states draw together again, and at 90° a whole crossing flips the channel alike after either path, tells nothing apart, and only moves the bands by half a spacing. The number is the textbook’s own, computed here with squared weights for a contact of the form in the contact theorem (Theorem 25) under the conditions below; the paper does not print it. The trade-off it shows, that the better one contact tells the paths apart the less of the bands it leaves, is the textbook’s duality between path knowledge and contrast (Wootters and Zurek, 1979; Greenberger and Yasin, 1988; Englert, 1996).

The conditions, exactly

The contact theorem (Theorem 25) has a second case. Where the crossing keeps the state where it stood, nothing is moved: the contact turns the state’s two parts of definite sign in opposite senses and changes the length of neither, and a state whose sign is already definite is left alone at any strength, up to a common turn.

At a register, on Recognition Science’s working statement, the crossing is a sign on the register times a flip F on the channel that touched it, here the gas molecule. Which case holds for the register itself is not settled; on Recognition Science’s working reading, one recognition per tick (a working statement), it is the second. Wherever the gas molecule stands in a state e that the flip moves wholly off itself, the first case holds on the gas molecule either way: with squared length cos²θ it stands where it stood and the register is exactly as it was; with squared length sin²θ it is flipped and the register carries the sign.

The contrast |cos 2θ| also needs each path to carry a definite sign on the register (the sign the crossing applies there), +1 on one and −1 on the other, which the working reading allows and the first case does not. The gas molecule then ends in cos θ e + i sin θ Fe after one path and cos θ e − i sin θ Fe after the other, and these overlap by cos²θ − sin²θ = cos 2θ. Taking a laboratory body’s two paths as such a register is part of attachment, below.

What the theorem does not give

It does not give the event: that one part is the one posted and the other is not, and what the posting then does to the body’s history. The paper states what is owed in these words, where a posting is an entry and the ledger is the record:

(O1) Attachment. State which postings of the ledger a laboratory system, its apparatus and its outcome are: which counted event takes an entry from a held history, how often, and what it does to the history.

Note what the two parts are. In the figures’ model “as it stood” and “crossed” carry the same weights on both paths, so posting which of them occurred would tell nothing about the path: sorted by it, the bands come back, shifted by half a spacing in the crossed set. What ends the bands for good is a posting of what the channel holds about the path, a further event; which events do that is attachment, owed.

A molecule in a good vacuum can cross an interferometer untouched; a dust grain in air meets contacts constantly. Each gas molecule that bounces off it is a separate contact holding its own mark, whole or small, and the shares of the bands they leave multiply: after n contacts of strength θ the contrast left is |cos 2θ|ⁿ. Many nearly harmless touches add up to the loss of the bands, as the figure below shows for up to a thousand of them. How much one touch tells depends on how far apart the two paths are, compared with the wavelength of what touches the body: light much longer than that separation carries only a small part of a mark.

A bigger body is a bigger target: it meets more gas molecules, scatters more light and, when warm, gives off more light of its own. Its size changes the count of contacts, and in standard decoherence theory, as in this figure, the contacts are what take the bands. The mass dial at the top of the page leaves the computed contrast unchanged because it moves the mass and not the count: the square and contact theorems (Theorems 16 and 25) have no mass in them, and the figure counts contacts alone. For massive bodies Recognition Science holds one open candidate for what the counted event is (section 05); it gives no number, so the figure does not draw it.

The loss of contrast from two of these kinds of contact has been measured in molecule interferometers: collisions with gas molecules (section 05) and light given off by hot molecules (Hackermüller and colleagues, 2004). In both, the loss of contrast agrees with standard decoherence theory; for the collisions the calculation has no adjustable parameter.

Suppose the line were a size: let the bands fade with mass alone, with no contact at all. Then a body’s bands would be fixed by what it is, and keeping it better isolated could never bring them back. In the laboratory it does: with less gas on the way the bands are deeper, and the loss of contrast follows the count of collisions (Hornberger and colleagues, 2003). That rules out a line at a size alone for bodies like the ones measured. The serious proposals for a line, collapse models, make the loss depend on mass, separation and time; no sign of such a loss has been found, and searches for the X-rays such models predict have ruled out some of their versions (Donadi and colleagues, 2021); Recognition Science’s own candidate for massive bodies is of that kind too, though it ends the bands at a counted event (the list below). Try it:

What holds Recognition Science’s line in place, each with its status:

• The square and the adding. The square theorem (Theorem 16), proved for reads of a register’s positions, under the premises named in section 02, one of them owed (the position read); that the screen’s places are such reads is part of attachment, owed; the step that adds values is linear by premise.
• The contact. The contact theorem (Theorem 25), proved; the crossing’s form at a register is a working statement, and |cos 2θ| needs the working reading (section 03).
• Only a posting resolves (the paper calls it “only a contact resolves”; in the paper a contact is a posting that crosses from outside into a register, not any touch). A body never turns definite on its own: its histories add until a posting tells them apart, and no rate of ending belongs to the body alone. A working statement, not a theorem.
• One open candidate for massive bodies. For a massive body Recognition Science holds one candidate for the counted event, a working statement and not derived, in the form Roger Penrose proposed (see also Lajos Diósi): two histories of one body that are already apart would stop being one history when the gravitational self-energy of the difference between their two mass distributions has built up one unit of action. Its unit is open, so it gives no time for any body, and the paper does not say how it fits the working statement above.
  The candidate, in full

  The strain that would end the two histories is the record’s own, which Recognition Science identifies with the gravitational field. The rule is a threshold in action, set by how the body’s mass is spread in its two histories and how far apart they are, not by size alone; what would end the bands is again something counted. The paper counts that event as the record’s own, and gives as the reason for the rule that a record that is one and is counted can hold two strains only while they differ by less than one of its own entries.

  Under it an isolated massive body would lose its bands with nothing outside touching it, at a time set by how its mass is spread, how far apart its two histories are, and the unit, which is open: the form physicists call gravity-related wave-function collapse, Penrose’s “objective reduction”. As worded, that sits uneasily with two clauses of the working statement above, that a body never turns definite on its own and that no rate of ending belongs to the body alone, and the paper does not say how the candidate and those clauses fit.

  Penrose’s own rule takes the unit to be the reduced Planck constant, giving a time of about that constant divided by the self-energy; the paper leaves the unit open. Even with its unit open, experiments bound it: any unit small enough to end the bands of the bodies already seen to interfere, during their flight, is ruled out (the paper computes no such bound). The parameter-free Diósi–Penrose model, in Diósi’s stochastic form, is ruled out by the absence of the radiation it predicts (Donadi and colleagues, 2021); the candidate specifies no such dynamics, so whether that test applies to it is not said.

• Attachment, owed. Which events in a laboratory are postings, and that one outcome is posted (section 03).

Isn’t this just decoherence?

For everything measured, yes: the mathematics (C(θ) = exp(iθX) is the textbook’s coupling of a two-valued pointer), the contrasts and the absence of a size line are the decoherence account’s own (Zeh, Joos, Zurek). Decoherence explains why the bands leave every read of the body alone but not why one outcome occurs; what Recognition Science adds is an event, a posting, at which one outcome is written and never erased. The textbook’s measurement postulate and collapse models add such an event too, and which events are postings is owed (attachment), so, like the textbook’s cut, Recognition Science does not yet say where that event happens.

Is any of this different from ordinary quantum mechanics?

Not in any number: Recognition Science computes nothing for a laboratory body that the textbook does not, so none of these experiments can tell the two apart; like the textbook, it needs an act the linear step does not make, and when that act happens is owed. The paper names one place among its twelve paradoxes where Recognition Science says something the unitary step alone does not: a posting is never undone, so a device that has kept a record of a result could not be made to interfere, however well isolated. What counts as keeping a record in a laboratory is owed, so, in the paper’s words, this is not yet a test. The paper does not count its working candidate for massive bodies in that sentence; it lists it separately, as a place to look. It too would end the bands of an isolated body with nothing outside touching it, which the unitary step alone never does; only its time waits on the unit, which is open (section 05).

What is not proved?

Two owed statements: attachment (section 03), which includes when a posting happens and so whether a lost band is gone for good, and the position read (section 02); the square is proved only for reads of a register’s positions. Beyond them, Recognition Science derives no strength θ for any laboratory contact and so no decoherence rate (the rates are standard decoherence theory’s), the crossing’s form and the reading that |cos 2θ| needs are working statements, and the candidate for massive bodies sits uneasily with “only a posting resolves” (section 05).

• It names no size, mass or time at which interference ends.
• It gives no collapse rate for any body: its one candidate for massive bodies, of Penrose’s form, is not established, has its unit open and gives no number.
• It gives no number of its own for any laboratory body.

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. Where the quantum world ends is the fifth of its twelve paradoxes (§8). The results this page uses are Theorem 16 (the read is the square, §5) with Corollary 17 (the read of a complex value is its squared modulus), and Theorem 25 (a contact leaves the state or applies the crossing, §6.2). The working statement about the crossing is in §6.2; “only a contact resolves” and the candidate for massive bodies are in §6.3.

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

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