trentmkelly/LessWrong-43k huggingface dataset:
No Individual Particles
by Eliezer Yudkowsky — 2008-04-18T04:40:19.000Z

Followup to: Can You Prove Two Particles Are Identical?, Feynman Paths

Even babies think that objects have individual identities. If you show an infant a ball rolling behind a screen, and then a moment later, two balls roll out, the infant looks longer at the expectation-violating event. Long before we're old enough to talk, we have a parietal cortex that does spatial modeling: that models individual animals running or rocks flying through 3D space.

And this is just not the way the universe works. The difference is experimentally knowable, and known. Grasping this fact, being able to see it at a glance, is one of the fundamental bridges to cross in understanding quantum mechanics.

If you shouldn't start off by talking to your students about wave/particle duality, where should a quantum explanation start? I would suggest taking, as your first goal in teaching, explaining how quantum physics implies that a simple experimental test can show that two electrons are entirely indistinguishable —not just indistinguishable according to known measurements of mass and electrical charge.

To grasp on a gut level how this is possible, it is necessary to move from thinking in billiard balls to thinking in configuration spaces; and then you have truly entered into the true and quantum realm.

In previous posts such as Joint Configurations and The Quantum Arena, we've seen that the physics of our universe takes place in a multi-particle configuration space.

The illusion of individual particles arises from approximate factorizability of a multi-particle distribution, as shown at left for a classical configuration space.

If the probability distribution over this 2D configuration space of two classical 1D particles, looks like a rectangular plaid pattern, then it will factorize into a distribution over A times a distribution over B.

In classical physics, the particles A and B are the fundamental things, and the configuration space is just an isomorphic way of looking at them.

In quantum physics, the configuration space is the fundamental thing, and you get the appearance of an individual particle when the amplitude distribution factorizes enough to let you look at a subspace of the configuration space, and see a factor of the amplitude distribution—a factor that might look something like this:

This isn't an amplitude distribution, mind ...
Feynman Paths
by Eliezer Yudkowsky — 2008-04-17T06:32:28.000Z

Previously in series: The Quantum Arena

At this point I would like to introduce another key idea in quantum mechanics. Unfortunately, this idea was introduced so well in chapter 2 of QED: The Strange Theory of Light and Matter by Richard Feynman, that my mind goes blank when trying to imagine how to introduce it any other way. As a compromise with just stealing his entire book, I stole one diagram—a diagram of how a mirror really works.

In elementary school, you learn that the angle of incidence equals the angle of reflection. But actually, saith Feynman, each part of the mirror reflects at all angles.

So why is it that, way up at the human level, the mirror seems to reflect with the angle of incidence equal to the angle of reflection?

Because in quantum mechanics, amplitude that flows to identical configurations (particles of the same species in the same places) is added together, regardless of how the amplitude got there.

To find the amplitude for a photon to go from S to P, you've got to add up the amplitudes for all the different ways the photon could get there—by bouncing off the mirror at A, bouncing off the mirror at B...

The rule of the Feynman "path integral" is that each of the paths from S to P contributes an amplitude of constant magnitude but varying phase, and the phase varies with the total time along the path. It's as if the photon is a tiny spinning clock—the hand of the clock stays the same length, but it turns around at a constant rate for each unit of time.

Feynman graphs the time for the photon to go from S to P via A, B, C, ... Observe: the total time changes less between "the path via F" and "the path via G", then the total time changes between "the path via A" and "the path via B". So the phase of the complex amplitude changes less, too.

And when you add up all the ways the photon can go from S to P, you find that most of the amplitude comes from the middle part of the mirror—the contributions from other parts of the mirror tend to mostly cancel each other out, as shown at the bottom of Feynman's figure.

There is no answer to the question "Which part of the mirror did the photon really come from?" Amplitude is flowing from all of these configurations. But if we were to ignore all the parts of the mirror except the middle, we would calculate essentially the same amoun...
Redistributing fairness
by KatjaGrace — 2008-04-17T05:39:00.000Z

From Kwame Anthony Appiah’s fascinating longer article on fairness in politics, via Greg Mankiw:

In the 1970s, the Nobel Prize-winning economist Thomas Schelling used to put some questions to his students at Harvard when he wanted to show how people’s ethical preferences on public policy can be turned around. Suppose, he said, that you were designing a tax code and wanted to provide a credit — a rebate, in effect — for couples with children. (I’m simplifying a bit.) In a progressive tax system such as ours, we try to ease the burden on the less well off, so it might make sense to adjust the child credit accordingly. Would it be fair, do you think, to give poor parents a bigger credit than rich parents? Schelling’s students were inclined to think so. If the credit was going to vary with income, it seemed fair to award struggling families the bigger tax break. It would certainly be unfair, they agreed, for richer families to get a bigger one.

Then Schelling asked his students to think about things in a different way. Instead of giving families with children a credit, you’d impose a surcharge on couples with no children. Now then: Would it be fair to make the childless rich pay a bigger surcharge than the childless poor? Schelling’s students thought so.

But — hang on a sec — a bonus for those who have a child amounts to a penalty for those who don’t have one. (Saying that those with children should be taxed less than the childless is another way of saying that the childless should be taxed more than those with children.) So when poor parents receive a smaller credit than rich ones, that is, in effect, the same as the childless poor paying a smaller surcharge than the childless rich. To many, the first deal sounds unfair and the second sounds fair — but they’re the very same tax scheme.

That’s a little disturbing, isn’t it?

Why do people respond this way? There’s no real paradox. The above questions seem to have elicited from the subjects a confusion of aims, in combination with a strong conceptually unpolished [IF rich THEN confiscate money] reflex.

Assume (very) hypothetically that a bonus or penalty should be applied. If it is as an incentive it should apply to rich and the poor equally, unless there is some reason to incentivise one economic class over the other ...
The Quantum Arena
by Eliezer Yudkowsky — 2008-04-15T19:00:15.000Z

Previously in series: Classical Configuration Spaces

Yesterday, we looked at configuration spaces in classical physics. In classical physics, configuration spaces are a useful, but optional, point of view.

Today we look at quantum physics, which inherently takes place inside a configuration space, and cannot be taken out.

For a start, as you might guess, in quantum physics we deal with distributions of complex amplitudes, rather than probability distributions made up of positive real numbers. At left, I've used up 3 dimensions drawing a complex distribution over the position of one particle, A.

You may recall that yesterday, 3 dimensions let us display the position of two 1-dimensional particles plus the system evolution over time. Today, it's taking us 3 dimensions just to visualize an amplitude distribution over the position of one 1-dimensional particle at a single moment in time. Which is why we did classical configuration spaces first.

To clarify the meaning of the above diagram, the left-to-right direction is the position of A.

The up-and-down direction, and the invisible third dimension that leaps out of the paper, are devoted to the complex amplitudes. Since a complex amplitude has a real and imaginary part, they use up 2 of our 3 dimensions.

Richard Feynman said to just imagine the complex amplitudes as little 2-dimensional arrows. This is as good a representation as any; little 2D arrows behave just the same way complex numbers do. (You add little arrows by starting at the origin, and moving along each arrow in sequence. You multiply little arrows by adding the angles and multiplying the lengths. This is isomorphic to the complex field.) So we can think of each position of the A particle as having a little arrow associated to it.

As you can see, the position of A bulges in two places—a big bulge to the left, and a smaller bulge at right. Way up at the level of classical observation, there would be a large probability (integrating over the squared modulus) of finding A somewhere to the left, and a smaller probability of finding it at the small bulge to the right.

Drawing a neat little graph of the A+B system would involve having a complex amplitude for each joint position of the A and B particles, which you could visualize as a hypersurface in 4 dimensions. I'd draw it for you, bu...
Classical Configuration Spaces
by Eliezer Yudkowsky — 2008-04-15T08:40:56.000Z

Previously in series: Distinct Configurations

Once upon a time, there was a student who went to a math lecture. When the lecture was over, he approached one of the other students, and said, "I couldn't follow that at all. The professor was talking about rotating 8-dimensional objects! How am I supposed to visualize something rotating in 8 dimensions?"
"Easy," replied the other student, "you visualize it rotating in N dimensions, then let N go to 8."
—old joke

Quantum configuration space isn't quite like classical configuration space. But in this case, considering that 8 dimensions is peanuts in quantum physics, even I concede that you ought to start with classical configuration space first.

(I apologize for the homemade diagrams, but this blog post already used up all available time...)

In classical physics, a configuration space is a way of visualizing the state of an entire system as a single point in a higher-dimensional space.

Suppose that a system is composed of two particles, A and B, each on the same 1-dimensional line. (We'll call the two directions on the line, "forward" and "back".)

Then we can view the state of the complete system A+B as a single point in 2-dimensional space.

If you look at state 1, for example, it describes a state of the system where B is far forward and A is far back. We can view state 1 as being embodied either in two 1-dimensional positions (the representation on the right), or view it as one 2-dimensional position (the representation on the left).

To help grasp the idea of viewing a system as a point, this alternate graph shows A and B on the same line.

When A and B are far apart, they both move toward each other. However, B moves slower than A. Also, B wants to be closer to A than A wants to be close to B, so as B gets too close, A runs away...

(At least that's what I had in mind while trying to draw the system evolution.)

The system evolution can be shown as a discrete series of states: Time=1, Time=2, Time=3... But in configuration space, I can draw the system evolution as a smooth trajectory.

If I had the time (to learn to use the appropriate software), I'd be drawing neat-o 3D diagrams at this point. Like the diagram at right, only with, like, actual graphics.

You may have previously heard t...
Can You Prove Two Particles Are Identical?
by Eliezer Yudkowsky — 2008-04-14T07:06:34.000Z

This post is part of the Quantum Physics Sequence.
Followup to: Where Philosophy Meets Science, Joint Configurations

Behold, I present you with two electrons. They have the same mass. They have the same charge. In every way that we've tested them so far, they seem to behave the same way.

But is there any way we can know that the two electrons are really, truly, entirely indistinguishable?

The one who is wise in philosophy but not in physics will snort dismissal, saying, "Of course not. You haven't found an experiment yet that distinguishes these two electrons. But who knows, you might find a new experiment tomorrow that does."

Just because your current model of reality files all observed electrons in the same mental bucket, doesn't mean that tomorrow's physics will do the same. That's mixing up the map with the territory. Right?

It took a while to discover atomic isotopes. Maybe someday we'll discover electron isotopes whose masses are different in the 20th decimal place. In fact, for all we know, the electron has a tiny little tag on it, too small for your current microscopes to see, reading 'This is electron #7,234,982,023,348...' So that you could in principle toss this one electron into a bathtub full of electrons, and then fish it out again later. Maybe there's some way to know in principle, maybe not—but for now, surely, this is one of those things that science just doesn't know.

That's what you would think, if you were wise in philosophy but not in physics.

But what kind of universe could you possibly live in, where a simple experiment can tell you whether it's possible in principle to tell two things apart?

Maybe aliens gave you a tiny little device with two tiny little boxes, and a tiny little light that goes on when you put two identical things into the boxes?

But how do you know that's what the device really does? Maybe the device was just built with measuring instruments that go to the 10th decimal place but not any further.

Imagine that we take this problem to an analytic philosopher named Bob, and Bob says:

"Well, for one thing, you can't even get absolute proof that the two particles actually exist, as opposed to being some kind of hallucination created in you by the Dark Lords of the Matrix. We call it 'the problem of induction'."

Yes, we've heard of the problem of in...
Where Philosophy Meets Science
by Eliezer Yudkowsky — 2008-04-12T21:21:33.000Z

Looking back on early quantum physics—not for purposes of admonishing the major figures, or to claim that we could have done better if we’d been born into that era, but in order to try and learn a moral, and do better next time—looking back on the dark ages of quantum physics, I say, I would nominate as the “most basic” error…

… not that they tried to reverse course on the last three thousand years of science suggesting that mind was complex within physics rather than fundamental in physics. This is Science, and we do have revolutions here. Every now and then you’ve got to reverse a trend. The future is always absurd and never unlawful.

I would nominate, as the basic error not to repeat next time, that the early scientists forgot that they themselves were made out of particles.

I mean, I’m sure that most of them knew it in theory.

And yet they didn’t notice that putting a sensor to detect a passing electron, or even knowing about the electron’s history, was an example of “particles in different places.” So they didn’t notice that a quantum theory of distinct configurations already explained the experimental result, without any need to invoke consciousness.

In the ancestral environment, humans were often faced with the adaptively relevant task of predicting other humans. For which purpose you thought of your fellow humans as having thoughts, knowing things and feeling things, rather than thinking of them as being made up of particles. In fact, many hunter-gatherer tribes may not even have known that particles existed. It’s much more intuitive—it feels simpler—to think about someone “knowing” something, than to think about their brain’s particles occupying a different state. It’s easier to phrase your explanations in terms of what people know; it feels more natural; it leaps more readily to mind.

Just as, once upon a time, it was easier to imagine Thor throwing lightning bolts, than to imagine Maxwell’s Equations—even though Maxwell’s Equations can be described by a computer program vastly smaller than the program for an intelligent agent like Thor.

So the ancient physicists found it natural to think, “I know where the photon was… what difference could that make?” Not, “My b...
Distinct Configurations
by Eliezer Yudkowsky — 2008-04-12T04:42:36.000Z

The experiment in the previous essay carried two key lessons:

First, we saw that because amplitude flows can cancel out, and because our magic measure of squared modulus is not linear, the identity of configurations is nailed down—you can’t reorganize configurations the way you can regroup possible worlds. Which configurations are the same, and which are distinct, has experimental consequences; it is an observable fact.

Second, we saw that configurations are about multiple particles. If there are two photons entering the apparatus, that doesn’t mean there are two initial configurations. Instead the initial configuration’s identity is “two photons coming in.” (Ideally, each configuration we talk about would include every particle in the experiment—including the particles making up the mirrors and detectors. And in the real universe, every configuration is about all the particles… everywhere.)

What makes for distinct configurations is not distinct particles. Each configuration is about every particle. What makes configurations distinct is particles occupying different positions—at least one particle in a different state.

To take one important demonstration…

Figure 1 is the same experiment as Figure 2 in Configurations and Amplitude, with one important change: Between A and C has been placed a sensitive thingy, S. The key attribute of S is that if a photon goes past S, then S ends up in a slightly different state.

Let’s say that the two possible states of S are Yes and No. The sensitive thingy S starts out in state No, and ends up in state Yes if a photon goes past.

Then the initial configuration is:

“photon heading toward A; and S in state No,” (−1+0i)

Next, the action of the half-silvered mirror at A. In the previous version of this experiment, without the sensitive thingy, the two resultant configurations were “A to B” with amplitude −i and “A to C” with amplitude −1. Now, though, a new element has been introduced into the system, and all configurations are about all particles, and so every configuration mentions the new element. So the amplitude flows from the initial configuration are to:

“photon from A to B; and S in state No,” (0−i)

“photon from A to C; and S in state Yes,” (−

... ,,,,,,,,
Joint Configurations
by Eliezer Yudkowsky — 2008-04-11T05:00:58.000Z

The key to understanding configurations, and hence the key to understanding quantum mechanics, is realizing on a truly gut level that configurations are about more than one particle.

Continuing from the previous essay, Figure 1 shows an altered version of the experiment where we send in two photons toward D at the same time, from the sources B and C.

The starting configuration then is:

“a photon going from B to D,

and a photon going from C to D.”

Again, let’s say the starting configuration has amplitude (−1+0i).

And remember, the rule of the half-silvered mirror (at D) is that a right-angle deflection multiplies by i, and a straight line multiplies by 1.

So the amplitude flows from the starting configuration, separately considering the four cases of deflection/non-deflection of each photon, are:

The “B to D” photon is deflected and the “C to D” photon is deflected. This amplitude flows to the configuration “a photon going from D to E, and a photon going from D to F.” The amplitude flowing is (−1+0i)×i×i=(1+0i).
The “B to D” photon is deflected and the “C to D” photon goes straight. This amplitude flows to the configuration “two photons going from D to E.” The amplitude flowing is (−1+0i)×i×1=(0−i).
The “B to D” photon goes straight and the “C to D” photon is deflected. This amplitude flows to the configuration “two photons going from D to F.” The amplitude flowing is (−1+0i)×1×i=(0−i).
The “B to D” photon goes straight and the “C to D” photon goes straight. This amplitude flows to the configuration “a photon going from D to F, and a photon going from D to E.” The amplitude flowing is (−1+0i)×1×1=(−1+0i).

Now—and this is a very important and fundamental idea in quantum mechanics—the amplitudes in cases 1 and 4 are flowing to the same configuration. Whether the B photon and C photon both go straight, or both are deflected, the resulting configuration is one photon going toward E and another photon going toward F.

So we add up the two incoming amplitude flows from case 1 and case 4, and get a total amplitude of (1+0i)+(−1+0i)=0.

When we wave our magic squared-modulus-ratio r...
Configurations and Amplitude
by Eliezer Yudkowsky — 2008-04-11T03:14:21.281Z

So the universe isn’t made of little billiard balls, and it isn’t made of crests and troughs in a pool of aether… Then what is the stuff that stuff is made of?

In Figure 1, we see, at A, a half-silvered mirror, and two photon detectors, Detector 1 and Detector 2.

Early scientists, when they ran experiments like this, became confused about what the results meant. They would send a photon toward the half-silvered mirror, and half the time they would see Detector 1 click, and the other half of the time they would see Detector 2 click.

The early scientists—you’re going to laugh at this—thought that the silver mirror deflected the photon half the time, and let it through half the time.

Ha, ha! As if the half-silvered mirror did different things on different occasions! I want you to let go of this idea, because if you cling to what early scientists thought, you will become extremely confused. The half-silvered mirror obeys the same rule every time.

If you were going to write a computer program that was this experiment— not a computer program that predicted the result of the experiment, but a computer program that resembled the underlying reality—it might look sort of like this:

At the start of the program (the start of the experiment, the start of time) there’s a certain mathematical entity, called a configuration. You can think of this configuration as corresponding to “there is one photon heading from the photon source toward the half-silvered mirror,” or just “a photon heading toward A.”

A configuration can store a single complex value—“complex” as in the complex numbers (a+bi), with i defined as √−1. At the start of the program, there’s already a complex number stored in the configuration “a photon heading toward A.” The exact value doesn’t matter so long as it’s not zero. We’ll let the configuration “a photon heading toward A” have a value of (−1+0i).

All this is a fact within the territory, not a description of anyone’s knowledge. A configuration isn’t a proposition or a possible way the world could be. A configuration is a variable in the program—you can think of it as a kind of memory location whose index is “a photon heading toward A”—a...