Does Looking at Something Change It?

Check your bike tire with a pressure gauge. Press the gauge onto the valve and a little air hisses out. The check took some of the thing it was checking.

Dip a thermometer into a small cup of espresso. The thermometer starts out cold, so it pulls some heat out of the coffee. You've measured a coffee that's slightly cooler than the one you were holding.

Anyone who has measured anything knows the feeling: looking costs something. A better gauge costs less. So here's the question physics has to answer. Is the cost just clumsiness, something a better instrument can shrink to nothing? Or is there an exact rule underneath, one that holds for any look at anything, even a perfect one?

There's an exact rule, and it comes in three parts. Two more results go with them: what a look is, and how close to perfect any real instrument can come. The puff of air and the lost heat are the clumsy part of the cost. The rule is about the cost that stays when the instrument is perfect.

This post stands on one idea from Part 1 of this series, What You Are. Nothing that happens is ever erased. Every event leaves a complete trace, and the full record of what happened is part of reality now. The past still exists. We call that record the ledger, and each event in it is an entry. I draw it as a book with one line for each entry. The book is a drawing, and the record is real.

Physics already agrees on the part it covers. Its basic equations never destroy information about the past. In quantum mechanics that's called unitarity: the present fixes the past exactly, and running the equations backward recovers every earlier state. Textbook quantum mechanics sets measurement aside as an exception. In the ledger, a measurement is one more entry, and it erases nothing. Our claim goes further: the ledger is reality's own record, and our institute's work proves that it keeps every entry. Writing a new entry never alters an earlier one.¹ (Another post follows the no-erasing rule into your trash can: The Universe Has No Delete Key.)

Part 1 also gave three plain facts, and the third is where today starts: checking a note is a new event. It adds a moment after the note. It doesn't change the note. Step 1 shows why.

The short answer

Looking never changes the past. Looking changes the thing it looks at only when it asks something the thing has no settled answer to. And when two looks clash, the damage between them is exact and never more than a quarter of the whole.

Here's the whole answer in six steps. The rest of the post takes them one at a time and shows how we know each one.

1. A look never changes the past. A look is a new entry, written after the thing it looks at. Nothing in the ledger reaches back and edits an earlier line.
2. A look is a contact that writes a record. It needs no eyes and no mind. While the contact lasts, the page is in pencil. Once the contact is released, it's pen.
3. A look at a settled answer changes nothing. If the thing already has a definite answer to your question, the look reads it and leaves it as it was.
4. A look at an unsettled answer makes the answer. Some questions have no yes or no yet. The look has to write one, and the thing is changed.
5. Two looks that clash pay an exact price. Some pairs of looks can't be answered together. They spoil each other by an exact amount, never more than a quarter of the whole. Reading a settled mark and flipping that same mark pay exactly a quarter.
6. No look is perfect. A recorder that obeys the ledger's conservation rules can't record exactly. Its error is at least 1/(2(r² + 3)) for a recorder with r charged units, and that floor is never zero.

Why this matters: a look needs no mind. Any contact that writes a record is a look: a camera, a detector, a bump from another particle. The exact rules below hold for every one of them the same way. And the past is safe. No look, by anyone or anything, ever edits a line that's already written.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got, and why the next step comes next.

Step 1: A look never changes the past

A look is a new entry, written after the thing it looks at. Here's how we know, with a sticky note.

At noon you write a note: buy milk. At 12:05 you look at it.

Now take the ledger as it stood at noon: every entry up to noon, and none after. Is your 12:05 look in it? No. It hadn't happened yet, and a record of everything through noon can't hold something that happens at 12:05. If somebody had written down a description of your look before noon, the description would be in there. But a description of a look is one entry, and the look is another.

That's the third plain fact from Part 1 in its exact form: a record of events through an earlier time doesn't contain the later event of examining it. The examination is an additional event, after the cutoff.²

So your look isn't an edit to the noon entries. It's a new line below them. The noon entry still says buy milk, in the same ink as before.

That holds for every look at everything: a glance at a photograph of your seventh birthday party, a read of last week's lab notebook, a detector click about a particle prepared at noon. The look lands after what it looks at. (How the past gets read is the business of Part 2, Time.)

What a look can do, in the steps ahead, is change how the thing goes on. What it can't do is reach back and change what was written about the thing before.

Zoom out. Step 1 is done. A look is a new entry, written after the thing it looks at, and nothing in the ledger reaches back to edit an earlier line. Next, and why: we haven't said what a look is. If it's an entry, something writes it. Step 2 is about that something, and it turns out not to be a mind.

Step 2: A look is a contact that writes a record

A look asks a thing a yes-or-no question and writes the answer in the ledger. What does the asking and writing is a contact: two things meeting. It needs no eyes and no mind.

Part 1 said it this way: when two things bump into each other, the bump is the entry. That's because the smallest event there is, one thing told apart from another, takes two things and one yes or no, and you're doing it right now: reading this tells one letter from another. A photon lands on a detector. Light lands on a retina. One particle bumps another. The contact is the look, and it's a look whether or not anyone is watching.

A natural question: a look is when I look, isn't it? Doesn't it take somebody watching?

No. A person watching is one more contact. Everything in this post holds for a person, a camera and a stray particle in exactly the same way, and none of it depends on anyone watching.

So what does writing an entry take? Three exact facts. I'll draw them as pencil and pen. The pencil and pen are a drawing of the real mechanism, and the three facts are the mechanism itself.

1. Writing takes contact. A record changes only while a contact acts on it, and no faster than the contact pushes.
2. A contact that writes can unwrite. For every state a contact writes into, there's a mirror state it erases from, at the same rate. The mirror state is easy to build: keep the part of the state that's already recorded and flip the sign of the rest. It has the same recorded weight, meaning the same amount of the state sits in the record, and the contact moves that weight the other way. So no contact writes one way for every input. While a contact is on, the page is in pencil.
3. Release makes pen. Once the recorder lets go, free motion leaves whatever is recorded exactly as it is, for every later time.

The graph follows the recorded weight. With no contact, nothing moves. With the contact on, a state it writes into starts gaining recorded weight, and its mirror state, starting from the same weight, starts losing it at the same rate. When the contact lets go, both stay where they ended.

Pencil isn't the ledger erasing a line. A line isn't posted until the contact is released, and once it's posted it stays. Nothing posted is ever erased.

One more fact, about the push itself: writing is never passive. In the ledger, 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. Writing takes an active step. So a look is never a passive glance. It's an act, and it adds an entry.

Zoom out. Step 2 is done. A look is a contact that writes an entry. It needs no mind. While the contact is on, the page is in pencil, and a contact that writes in one place erases in its mirror. At release it becomes pen. And writing is always an act. Next, and why: what does a look do to the thing it looks at? That depends on whether the thing already has an answer to the question. The easy case comes first.

Step 3: A look at a settled answer changes nothing

If the thing already has a definite answer to your question, the look reads that answer and leaves it exactly as it was.

Here's what that means in the ledger. The ledger writes in rows of marks, and each mark is full or empty. I'll write full as 1 and empty as 0. The simplest look is a read: it asks a yes-or-no question about which marks are full. Is mark 1 full? Are marks 2 and 3 both empty? A read reports a fact about where the marks are, and that is all it does. It doesn't move a mark. It doesn't send anything from one arrangement of marks to another. It stamps each arrangement with a yes or a no.

Take a row where mark 1 is full and marks 2 and 3 are empty. Read it: full. Read it again: full. A thousand reads, a thousand fulls. The answer was settled before the first look and it's settled the same way after the last. Nothing about the answer moved.

The look still writes an entry, and writing is an act (Step 2). What it doesn't touch is the answer. The row reads the same before and after, and the ledger has one more line.

What about the gauge in the hook? The puff of air came from the gauge's contact with the valve, and a contact is an act. Step 3 is about the look's question and its answer, and about those alone: a read of a settled answer leaves the answer where it was.

Reads also never get in each other's way. Ask is mark 1 full?, then is mark 2 full?, then are marks 1 and 3 equal? Ask them in any order, or all at once, and you get the same answers every time. On that row: yes, no, no. Every collection of reads can be answered together.

Hold on to one thing: settled always means settled for a question. A thing can be settled for one question and unsettled for another. Step 4 starts with an unsettled one. Step 5 ends with a settled one that a second question unsettles.

Zoom out. Step 3 is done. A read reports a yes-or-no fact about where the marks are and moves nothing. A settled answer is found and left alone, and reads never get in each other's way. Next, and why: the harder case. What if the thing has no answer yet to the question you ask? Is there such a case at all? There is, and Step 4 shows one, with a picture.

Step 4: A look at an unsettled answer makes the answer

If the thing has no yes or no yet to your question, the look can't read one. It writes one, and the thing is changed. To see it, I need a picture of one mark.

The picture. I'll draw the state of one mark as an arrow of length 1 on a flat sheet of paper, the same drawing as in Entanglement Isn't Spooky. It's Shared. The arrow is a drawing of the real state. The numbers it carries are exact. Two lines cross on the sheet: the empty line, running right, and the full line, running up. An arrow lying along the empty line is a mark that's empty, for certain. An arrow along the full line is a mark that's full, for certain. Those are Step 3's settled marks. One is in the left panel below.

Every look has its own pair of lines on the sheet, at right angles. The read of a mark uses the empty line and the full line. A look cuts the arrow into its shadows on its two lines: drop a line from the tip of the arrow, at right angles, onto each. For a settled mark, nothing is cut away. The whole arrow is its own shadow on one line, and there's nothing on the other.

Now an unsettled mark: an arrow between the lines. Say its shadow on the empty line is 0.8 long and its shadow on the full line is 0.6 long. (Check: 0.8 × 0.8 + 0.6 × 0.6 = 0.64 + 0.36 = 1, so the arrow's length is 1.) A shadow's length is its size. Its share is its size times itself: 0.64 and 0.36. The chance of an answer is its share, so a read of this mark says empty 64 times in 100 and full 36 times in 100. Why the share? The odds have to be nonnegative, give a certain answer odds of 1, and add up across answers that exclude each other. For a ledger of two or more marks, the share is the only rule that does all three. Physics has had that result since Gleason proved it in 1957, and it calls the share rule the Born rule.³ (Is the Universe Rolling Dice? works it through.)

Where an unsettled mark comes from. Take a particle sitting in one cell of the ledger, with a hop to the next cell. Let the mark we're watching be is the particle in the next cell? Full means yes, empty means no. At the start it's empty. Now wait a moment, however short. Part of the particle is already in the next cell. That's exact, and it's what a hop does in quantum mechanics: it turns the state continuously away from the first cell, so after any short time the next cell holds a small weight, not zero. At every small moment after the start, a part of its weight sits in the neighboring cell, and no yes-or-no answer equals that part: it's a number strictly between 0 and 1. The arrow has turned away from the empty line, toward the full line, and it isn't on either.

The particle has no answer to which cell? It has a weight in between. At every one of those moments there's no yes or no to find.

What the look does. The look cuts the arrow into its two shadows. The ledger writes one entry, and the state becomes that one piece, scaled back up to length 1. That's the right panel. If the entry is empty (64 times in 100), the arrow now lies along the empty line. If it's full (36 in 100), it lies along the full line. Either way the arrow has moved. Before the look it was between the lines. After the entry it's on one. The thing is changed, because the look had no answer to read, and made one.

The same shows up one level higher, in a measuring instrument. Take an ideal one, or a close copy built from the ledger's own parts, and feed it a thing that has a share above zero for each of two answers. Its complete output, everything it kept, taken together, isn't inside either outcome alone. It holds both, until the ledger writes an entry. Physicists know this as the measurement problem: an instrument that obeys the laws of quantum mechanics ends up holding both outcomes, not one.⁴

Which piece does the ledger write? That's the one open step in this post. Not yet derived: how the ledger attaches an entry to an actual event, and so which piece it takes. Derived: the odds of each piece, exactly, and that the instrument's complete output holds both outcomes. The odds are the Born rule of quantum mechanics, and experiments have never found a departure from it. Is the Universe Rolling Dice? is about exactly that step.

What recording does to a cell. So far, one look at a time. Here's an exact fact about a thing that's looked at and recorded again and again: on average, its state gets sharper as the entries pile up, and its circulation fades. Take one cell of the ledger. It holds two clocks. Each is full or empty and has one of four settings, so each has 2 × 4 = 8 ways to be, and the pair has 8 × 8 = 64. After each entry written about it, the cell is in the state that agrees with everything recorded so far. How blurry is that state? Call it the cell's blur. Zero means one exact state, and more means more spread out.

• The record leaves nowhere to hide. There are 64 sharp reads of the cell that don't overlap and together cover all of it. No part of the cell is dark to the record. (Physicists call a part where no record can tell one state from another a dark subspace.)
• Entries sharpen it. On average no entry makes it blurrier, and after k entries the blur is at most C/(k + 1), where C is one fixed number, set by the ledger's own law, that works for every starting state. Ten times as many entries cut the bound to about a tenth. Keep recording and the blur goes to zero. Physicists call this purification.⁵

Recording also wipes out the circulation inside the cell. A cell can circulate, going round one way or the other. This isn't the flow the ledger writes down at each step. It's hidden in the cell's state: no string of the ledger's ordinary reads can tell one way from the other. At the start, a look built for it says which way, for certain. After k entries, the average of what that look reads is at most √(2C/(k + 1)), with the same C. It fades to nothing. A cell that began circulating one way and a cell that began the other way end up the same cell. Which way it went isn't in the cell any more. It survives only in the phases between the alternative records the ledger can write: in how those records line up with each other.

Zoom out. Step 4 is done. When the thing has no answer yet, the look writes one and the thing is changed: the arrow moves onto a line. A cell that's recorded again and again has an average blur of at most C/(k + 1) after k entries, and loses its circulation. Next, and why: so far each look came alone. What happens when a thing is asked two different questions? Sometimes the questions get along. Sometimes they don't, and the clash has an exact price.

Step 5: Two looks that clash pay an exact price

Two looks can be answered together exactly when neither spoils the other. When they do spoil each other, the damage has an exact size, and it's never more than a quarter of the whole.

Looks that get along. Reads get along (Step 3), and so does any pair of looks where it makes no difference which you ask first. Such a pair cuts the arrow into four pieces, one for each pair of answers, and the pieces don't overlap. The chance of a pair of answers is the share of its piece, the same rule as for one look. It runs the other way too. If two looks have a clean joint answer on every state, it makes no difference which you ask first. So "can be answered together" and "don't spoil each other" are one and the same thing.

A look that clashes with a read. The ledger's own basic move is the flip: turn a mark over, 1 to 0 or 0 to 1. A flip can be asked about as a look too. The flip look asks what the flip would do to the arrow. Does it leave the arrow as it is, or turn it exactly round? On the sheet, flipping is a mirror across the diagonal, and the flip look's two lines are the two diagonals. An arrow on one diagonal is left alone by the flip. Call that answer plus. An arrow on the other diagonal is turned exactly round. Call that answer minus.

Now take a mark that's settled: empty, for certain. A read finds empty every time. But that arrow sits exactly halfway between the two diagonals, so for the flip look it's as unsettled as an arrow can be.

Ask the flip look first. It cuts the arrow into a plus piece and a minus piece. Each is about 0.71 long (the square root of a half), so each has share 1/2. Now read each piece. The read cuts the plus piece into an empty piece and a full piece, and does the same to the minus piece. Four pieces, each with share 1/4.

Look at the two pieces that lead to empty. One came by the plus route and one by the minus route. They aren't two different pieces. They're the very same arrow, half a unit long, lying along the empty line. The two routes to full are opposite arrows: one up, one down.

The overlap of two arrows is their two lengths multiplied, times how much they point the same way: 1 for the same way, 0 at right angles, −1 for exactly opposite. The two routes to empty: 1/2 × 1/2 × 1 = 1/4. The two routes to full: 1/2 × 1/2 × (−1) = −1/4.

That overlap is the price of the clash. Here's what it costs. Read the mark with no flip look first, and empty comes up with share 1: certain. Ask the flip look first, then read, and empty has two routes, each with share 1/4, so together 1/2. A certain answer has become a coin toss. The missing half is exactly twice the overlap:

1 = 1/4 + 1/4 + 2 × 1/4

When two routes lead to the same answer, the share that answer has when read alone is the two routes' shares plus twice their overlap. That's just how arrows add. So the overlap isn't a bookkeeping trick. The first look changed the odds of the second by exactly twice the overlap. Those odds come from the law of the entry in Step 4: the ledger writes one piece, and the state becomes that piece. Experiments have seen this case. Send a spin that is certainly up along one axis through an analyzer along a perpendicular axis. Whichever beam comes out, a second analyzer along the first axis finds the spin up half the time, not every time. That's the sequential Stern–Gerlach arrangement of textbook quantum mechanics, with the read as the first-axis analyzer and the flip look as the perpendicular one. Polarizing filters show it with light: two crossed filters pass nothing, and a third slipped between them at 45 degrees lets some light through.

The price in general. Nothing here was special to one mark. For any two looks on any thing, ask one, then the other. Take one answer of the second look, together with each of the first look's two answers. The commutator of the two looks is the difference between doing them in one order and in the other. The two pieces overlap by exactly one eighth of a number built from it, with a sign that goes with the answer you took.⁶ To build the number, pass the commutator back through the first look and see how far the result points along the thing's own arrow. The commutator is zero exactly when the order doesn't matter, and then the price is zero too. (Physicists know clashing looks as noncommuting, and the uncertainty principle is the best-known case. Here the clash has an exact price.) The 8 is 2 × 2 × 2: each of the three cuts in the overlap, the first look's two answers and the second look's one, carries a factor of a half. The footnote multiplies it out.

For a read and a flip on one mark, the formula comes down to something you can do in your head: the price is a quarter of the mark's imbalance, its share of empty minus its share of full.

The last row is Step 3 again. A mark on the plus line is settled for the flip look, so the flip look finds its answer and leaves it alone, and there's nothing to pay.

The ceiling. However two looks clash, on whatever thing, the price never goes above a quarter of the whole. The reason is short. Each look can be turned into a move that undoes itself and keeps every arrow's length, like the flip. Done in one order and then in the other, two such moves land an arrow in two places at most twice its length apart. Divide by 8, and the most you can get is 2/8, which is 1/4. That quarter falls out of the arithmetic of two looks.

And a quarter is reached when two looks clash as hard as looks can, where one order gives exactly the opposite arrow from the other, and the thing is settled for the look you ask second. Reading a mark and flipping that same mark is exactly such a pair. Lay out every arrangement of the marks as a corner of a cube (three marks give an ordinary cube with eight corners). The read of any one mark and the flip of that mark clash this way, and on a thing sitting at one corner they pay exactly 1/4.⁷ So the quarter isn't a loose bound. It's the most one look can ever cost another, and the ledger pays it in full.

Since the odds shift by exactly twice the price, the ceiling has a plain face. No look can move another look's odds by more than a half. A certain answer can fall to a coin toss. It can't fall further.

Zoom out. Step 5 is done. Two looks can be answered together exactly when they don't spoil each other. When they clash, the price is the overlap of their pieces, it never exceeds a quarter, and a read and a flip on a settled mark pay all of it: a certain answer falls to a coin toss. Next, and why: we've treated the look as an ideal act. A real look is made by a recorder, a machine with a limited number of parts. How exactly can a recorder write an answer down?

Step 6: No look is perfect

Every real look is made by something. A recorder is a machine that takes a look and writes the answer down: a detector with a memory. The ledger puts an exact floor under how well any recorder can do, and the floor isn't zero.

The recorder. Build one from the ledger's own parts:

• It has r charged units: cells of the ledger that carry its two conserved charges, energy and occupation (how many cells are occupied).
• It has any amount of neutral memory, a finite system that carries no charge. The whole recorder starts in one fixed state, the same whatever it's about to record.
• It records by an exchange with the source that keeps the total energy and the total occupation exactly the same.
• It reports with a sharp pointer: a plain yes or no.

Take the source to be a pair of the ledger's cells, and ask the recorder to report one yes-or-no fact about it: is the first one occupied and the second one not? Say that, for every state the source can be in, the recorder's chance of answering yes is within e of the true chance. Then

e ≥ 1/(2(r² + 3)).

That's the floor.

The floor drops as r grows, so a bigger recorder has room to do better. Turn the floor around: an error of e needs r² ≥ 1/(2e) − 3. To get the error down to 1 in 100 takes at least 7 charged units. For 1 in a million it takes at least 708. But the floor only falls. It never reaches zero. No recorder with a finite number of charged units records exactly.

The proof uses no noise: no friction, no heat, no shaky hand. The reason is conservation alone. The two conserved quantities force a conserved count, and a recorder with r charged units can only spread that count over so many values. Physics has known the shape of this limit since Wigner in 1952 and Araki and Yanase in 1960: a conservation law limits how well a finite apparatus can measure a quantity that doesn't commute with the conserved one. The floor here is a bound of that kind, with an exact number on it.⁸

That's what "no look is perfect" means. The look as an idea can be exact: a pair of lines on a sheet. The machine that makes it can't be.

Zoom out. Step 6 is done. A recorder that conserves energy and occupation has an error of at least 1/(2(r² + 3)), so exact recording is impossible. Next, and why: that's all six steps. What's left is what they add up to.

The whole trip, in one breath

A look is a new entry, written after the thing it looks at, so no look ever changes the past. It's a contact that writes a record, with no eyes or mind needed: pencil while the contact is on, pen once it's released. If the thing already has an answer to the question, the look reads it and leaves it alone. If it doesn't, the look writes one and the thing is changed, and a cell that's recorded again and again gets sharper on average and loses its circulation. Two looks that clash overlap by an exact amount that never exceeds a quarter, and a read and a flip on a settled mark pay the full quarter, taking a certain answer down to a coin toss and no further. And no recorder that obeys the ledger's conservation laws is ever exact.

That's what looking is in reality: every look is a new line in the book, and the book never changes a line that's already written.