Why Is the Quantum World So Strange? Four Odd Rules, One Reason.

Take three polarizing lenses, the kind in good sunglasses. A polarizing lens has a direction. It passes light that swings along that direction and blocks light that swings across it.

Hold one in front of a lamp and about half the light gets through. Put a second lens behind it and turn it until the lamp goes dark. Crossed, the two lenses block everything.

Now slide a third lens in between, turned 45 degrees from each. Light comes back. With ideal lenses, one eighth of the lamp's light gets through all three.¹ You added a blocker and got more light.

Richard Feynman shared the 1965 Nobel Prize in Physics for his work on quantum electrodynamics. In his 1964 lectures at Cornell he told his audience: "I think I can safely say that nobody understands quantum mechanics."² The same physics describes how every chip and every laser works.

So why is it so strange? The lenses show two of four odd rules, and all four share one source.

The short answer

The quantum world is strange only next to the picture of a finished book. In that picture every answer is already written on its page, and looking only reads the page. Reality's record isn't a finished book. It's written, one step at a time, and writing is an act. Next to a finished book four things look odd, and each has its own reason:

1. The numbers turn. The numbers that describe a state have a direction, because each step of the record turns them. They're complex numbers.
2. The odds are squares. The chance of an outcome is the square of the size of its amplitude. Five plain rules for reading a record leave no other law.
3. Plans fail. A plan is a set of answers written down beforehand. In a certain two-lab game a plan scores at most 2, which wins 75 rounds in 100. A shared record moved by the ledger's own flips, not only read, reaches exactly 2√2, which wins 85.4 in 100. Real labs beat 75.
4. A look writes. A look is a new entry, written after the thing it looks at. It changes the thing it looks at when that thing has no settled answer yet, and it never changes the past.

Why this matters: "strange" measures the distance from the picture you start with. Next to a finished book, quantum physics looks like four separate miracles. Next to a record that is written, each rule has a reason you can follow, with its conditions named beside it, and one precise question is left over.

The record, and why nothing in it is erased

This post stands on one idea from Part 1 of the series The Universe Writes in Pen, What You Are. Reality keeps a permanent record of every event, and nothing in it is ever erased. We call it the ledger, and each event in it is an entry.

Here's the reason. Physics already agrees on the part it covers: its basic laws never destroy information. In quantum mechanics that's called unitarity. The present fixes the past exactly, and running the laws backward recovers every earlier state. Our work adds a second reason. The record only grows: each new entry is added with every earlier entry kept, and writing a new entry never alters an earlier one.³ I draw the record as a book with a line for each entry. The book is a drawing, and the record is real.

The one reason: a record that is written

Start with the opposite picture. In a finished book every answer is already on its page. A look turns to the page and reads it, and reading changes nothing. If the world were a finished book, odds would only measure our ignorance of the pages, two friends sharing the book couldn't beat the best plan, and nothing in it would turn, so ordinary numbers would do. That picture is a drawing of an idea most of us carry, not of anything in nature.

The ledger isn't a finished book. It's written one step at a time. Each step turns the patterns it carries (Fact 1). Reading the record has one law (Fact 2). Acts that move a shared record can do what no plan can (Fact 3), and a look is one more act, so it adds a line (Fact 4). The four don't come from one proof. Each has its own reason and conditions, below. What they share is where they come from: a record that is written.

Fact 1: The numbers turn

The numbers that describe a state have a direction, and each step of the record turns them. A number with a length and a direction is a complex number, and quantum physics is written in them from its first page. Here's where they come from in the ledger, with eight seats and eight clock hands.

The ring. The ledger stores the state of a system, its complete description, in a register: a ring of eight slots, one for each corner of a cube. Why eight? Space has three dimensions, and each dimension is one yes-or-no direction with two sides, so there are 2 × 2 × 2 = 8 combinations, the eight corners of a cube. (Why three: lasting structure needs a connection that no smooth motion can undo, and linking is that connection. Two loops can link, like two rings of a chain, only in three dimensions. In two they can't wind around each other, and in four or more they slip apart.) A complete pass that visits every corner takes at least eight steps, and eight is enough. (A step is one step in the writing of the record.)⁴ In each step, every slot's content moves one place along the ring.

Each slot holds a number called its amplitude, which I'll draw as a clock hand with a length and a direction.

Steady patterns. Some patterns come out of a step unchanged except that every hand is multiplied by the same number, which stretches, flips or turns every hand the same way. Call them steady. Eight steps bring every hand home, so that number, multiplied by itself eight times, has to give 1.

Ordinary numbers first. An ordinary number is a hand that points right (positive) or left (negative). Two patterns are steady: hands all pointing right, which a step leaves alone (multiplier 1), and hands alternating right, left, right, left, which a step flips (multiplier −1). No other ordinary number gives 1 when multiplied by itself eight times.

Now let the hands point anywhere. Set them to 3, 6, 9 and 12 o'clock, and again 3, 6, 9 and 12, so that one trip around the table is two trips around the clock. Pass every hand one seat along, and the ring reads 12, 3, 6, 9, 12, 3, 6, 9. Compare seat by seat: every hand has turned a quarter turn, all the same way. The step moved the pattern and turned it. The number that turns a hand a quarter turn is called i. (Two quarter turns make a half turn, which flips a hand, so i × i = −1.) This pattern's multiplier is i, and multiplied by itself eight times it gives 1: two full turns.

In all there are eight steady patterns, one for each way of turning a hand a whole number of eighths of a circle per step, 0 to 7 eighths. Two of them, 0 and 4 eighths, keep the hands on the left-right line. The other six send hands off it. Only numbers that can turn have room for them: complex numbers.⁵

What this rests on. A state lives on the ring, a step moves every slot one place along it, and the state is described by steady patterns: it can be built from them. Those are the conditions. With ordinary numbers a ring of eight slots has two steady patterns, and building every state takes eight. With numbers that turn there are eight. So, on those conditions, the amplitudes are complex numbers.

Outside work reaches complex numbers by another road. In 2021 Marc-Olivier Renou and colleagues showed that real-number quantum physics, with the standard rule for combining separate systems, gives a different answer from complex quantum physics for a certain network of independent sources, and in 2022 two experiments found results it can't produce.⁶ That argument isn't ours, and it doesn't use the ring.

Next, and why: each amplitude has a length. What does the length do? It sets the odds.

Fact 2: The odds are squares

The chance of an outcome is the square of the size of its amplitude. The words first, then the reason.

Take a photon, one packet of light, at a lens. Two of its eight slots are in use, pass and stop. Draw its state as an arrow of length 1 on a flat sheet. The lens has two lines on the sheet, at right angles: the pass line and the stop line. Drop a line from the arrow's tip straight onto each one. Those are the arrow's two shadows, and a shadow's length is the size of that answer's amplitude. Multiply a size by itself and you get its share. A chance is a number from 0 to 1 that says how often an answer turns up, and the rule is that an answer's chance is its share. Max Born gave the rule in 1926, and textbooks take it as a postulate.

Turn the arrow 45 degrees from the pass line. Each shadow is 0.71 long, the square root of a half, so its share is exactly 1/2. Half the photons pass: that's the half at the middle lens and again at the last lens in the opening. Turn the arrow 60 degrees and the pass shadow is 0.5 long, so the chance is 0.5 × 0.5 = 1/4: one photon in four passes.⁷ The arrow is a drawing of the real state, and the numbers it carries are exact.

Why the square, and not some other power? Five plain rules for reading a record leave one law:

1. The chances of all the slots total 1.
2. No chance is negative.
3. An empty slot has chance 0.
4. The chance of "this slot or that one" is their two chances added.
5. A slot of a given size has the same chance in every state, whoever its neighbors are and wherever it sits.

Four of the five are what it takes to be a chance at all. The fifth, the same-size rule, is the physical premise: the kind of assumption Andrew Gleason's 1957 result starts from. No angle, no smooth curve and no calibration goes in, and nothing else fits.⁸

Each rule does work. Take away the same-size rule and a fourth-power law survives the other four. Under it, two labs holding detectors prepared to agree can send messages: one splits its own outcome in two, and the other's odds jump from 1/2 to 2/3, with no wire between them. Under the square, the other lab's odds stay at 1/2. Is the Universe Rolling Dice? works the five rules through.

Next, and why: the odds are exact. Are the results already written inside the particles, with the odds only measuring our ignorance? A game decides it.

Fact 3: Plans fail

Answers written down beforehand can't win a certain game more than 75 times in 100. Nature beats that, and quantum physics tops out at 85.4.

Start with a pair of gloves. Seal a left glove and a right glove in two boxes and mail one to a friend in another city. You open yours and find a left glove, so your friend's box holds a right one. Nothing travelled between the boxes: the answer was packed in advance. Nobody calls that spooky. It's gloves. A plan is that: every answer fixed before anyone looks, so a look only reads it.

Now a game that tests for plans. Two friends sit in separate rooms and can't talk. A referee hands each a random bit, 0 or 1, and each answers 0 or 1. They win if their answers match, unless both were handed a 1. Then they win by answering differently.

Whatever plan they make beforehand, they win at most three rounds in four. No plan wins all four pairs of bits: winning the three that need matching answers makes all four answers the same, which loses the pair that needs different ones. Physicists score the game with a number S, and the win rate is 1/2 + S/8. No plan scores above S = 2, which is 75 in 100. That limit is Bell's inequality, in the form Clauser, Horne, Shimony and Holt gave it in 1969.⁹

A shared record that's only read is a plan. In a real test each friend holds one particle of an entangled pair: one record with two ends. If both friends only look at that record, every answer was in it already, and the limit stays at 2.

A shared record moved by the ledger's own flips beats every plan. Let each friend's question move the record, with acts built from the ledger's own basic move: flipping one yes-or-no mark (full to empty, or empty to full), with its sign. Each friend works on at least two marks: each friend's two questions are built from two different flips. Reads never get in each other's way. Two of one friend's moves do: in one order and then the other, they leave different records, so no single page holds both answers beforehand. The score is exactly 2√2, about 2.83, a win rate of 85.4 in 100. That's the number quantum physics has, and Boris Tsirelson proved in 1980 that it's quantum physics's ceiling.¹⁰

Two things sit under that. The result starts from a pair that already shares one record. How two separately made particles come to share one is its own question, and Entanglement Isn't Spooky. It's Shared. says where it stands. And the ceiling of 2√2 rests on the two labs being two parties, each acting on its own side. A rule that forbids only messages allows scores up to 4, so "no messages" isn't what stops the score at 2.83.

Real labs beat plans. In 2015 a team at Delft ran the game with electron spins in two labs 1.3 kilometres apart. Over 245 rounds they scored S = 2.42, give or take 0.20: about 80 wins in 100. Vienna and NIST reported loophole-free photon tests the same year, and the 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser and Anton Zeilinger for experiments like these.¹¹ What these tests rule out is answers set beforehand and carried along by the particles. They don't by themselves rule out every hidden determinism, and they don't test the ledger against quantum physics, because the ledger gives the quantum number. Nothing is sent between the labs: each friend's answers, taken alone, look like fair coin flips whatever the other does.

Next, and why: the move that beat the plan did more than read. So does a look: it writes.

Fact 4: A look writes

A look is a new entry, written after the thing it looks at. It changes the thing it looks at when that thing has no settled answer yet, and it never changes the past.

Start with the past. At noon you write a note: buy milk. At 12:05 you look at it. Take the ledger as it stood at noon: every entry through noon, and none after. Your 12:05 look isn't in it, because it hadn't happened yet. So the look isn't an edit to the noon entry. It's a new line below it.¹²

A look needs no eyes and no mind. It's a contact: two things meeting. The smallest event there is takes two things and one yes or no: one thing told apart from another. Telling them apart takes information about both, and the least information there can be is one yes or no. You're doing it right now: reading this tells one letter from another. A lens does it to a photon. So does a camera, a detector, or a stray particle that bumps another. And in the ledger, writing is an act: no flow that just drifts along on its own, at any field strength and for any length of time, is the same as a record write.¹³

Back to the lenses. Suppose each photon carried a fixed answer to every lens's question, and each lens only read it. A photon that gets through all three lenses would also get through the first and the last with the middle lens taken out. But those two, crossed, pass nothing. So no photon would get through all three. About one in eight does, so the lenses aren't just reading answers that were already there.

Here's what the middle lens does. A photon that has passed the first lens lies along that lens's axis. For the middle lens, turned 45 degrees, it has no settled answer: its arrow sits halfway between the pass line and the stop line. The look cuts the arrow into its two shadows, each 0.71 long. The ledger writes one entry, pass or stop. If it's pass, the photon now lies along the middle lens's axis: before the look it lay along the first lens's axis, so the look changed it. The last lens sees a photon 45 degrees off its own axis and passes half of them. Of 800 photons that start, about 400 pass the first lens, 200 the second and 100 the third.

What picks the entry? That last step isn't derived yet: what picks which entry the ledger writes on a given run, pass or stop. Here's what we have beside it. The odds are exact (Fact 2). Experiments rule out a plan in the particle (Fact 3) and show one result per run, at those odds. Nothing already written is erased. And before the entry, when an ideal detector is fed a state with a nonzero part for each of two outcomes, its complete output holds both: everything it kept, taken together, lies in neither outcome alone. That's proved. For any time limit, one detector built from the ledger's own parts exists that finishes inside it and whose output still holds both outcomes.¹⁴ No collapse step is used.

Is the Universe Rolling Dice? works through that. Does Looking at Something Change It? covers settled and unsettled answers, clashing looks and why no look is perfect. Can We Ever Know Everything? shows why the record is never a finished book: every look adds an entry, so no list of everything is ever finished. Is There a Theory of Everything? puts the ring of eight and the squared odds on its scorecard.

What's still open

One step isn't derived.

• What picks the entry. Experiments show one entry per run, at the squared odds. That rule says what gets written. It doesn't say what, in an actual detector at an actual moment, with the energy it takes and the time it lasts, settles whether this run's entry reads pass or stop. Textbooks add a rule called collapse by hand, and collapse isn't derived here either. More clicks can't settle it, because no list of results, however long, can tell chance from a rule that picks the entries. It's a derivation to be done.

Three conditions stand under the four facts, each named where it was used: that a state lives on the ring, a step moves every slot one place along it, and the state is described by its steady patterns (Fact 1); the same-size rule (Fact 2); and that the two labs are two parties (Fact 3).

The whole trip, in one breath

Reality's record is written one step at a time, and a finished book is the wrong picture of it. Each step turns the patterns it carries, and with ordinary numbers a ring of eight slots holds only two of its eight steady patterns, so the numbers that describe a state turn. Five plain rules for reading a record leave one law for the odds, the square. A shared record that's only read is a plan and scores at most 2, which wins 75 in 100. One moved by the ledger's own flips scores exactly 2√2, which wins 85.4, and real labs beat 75. A look is a new entry, written after the thing it looks at. It never changes the past, and it changes the thing when the thing has no settled answer yet. One step is left: what picks the entry.

That's how reality is built: not a book already written, but a record written one step at a time, with every look one more line.