Configurations and Amplitude

11th Apr 2008

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New Comment

361 comments, sorted by Click to highlight new comments since: Today at 2:08 PM

Some comments are truncated due to high volume. (⌘F to expand all)

Eliezer, in case you plan to discuss Bell's-inequality-type experiments in future posts, I suggest that you use the GHZ state (not the EPR pair) to show how local realism is ruled out in QM. The GHZ state is a much cleaner result, and is not obscurred by the statistics inherent in Bell's inequality.

I think some of my readers may be overestimating the degree to which I intend to explain quantum mechanics, here. I'm not doing a textbook. I'm trying to get (reasonably smart nonphysicist) readers to the point where they're *no longer confused*, and the remaining difficulties are mere matters of math.

Still a useful suggestion though, thanks.

A configuration can store a single complex value - "complex" as in the complex numbers (a + bi).Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming you still hold that position?

I'm speaking as a nonphysicist reader, so I may well be missing something awfully obvious here. Any clarification would be appreciated.

Eliezer (and Robin) this series is very interesting and all, but.... aren't you writing this on the wrong blog?

I used to like this blog better when it was all about overcoming bias

Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming you still hold that position?

In case you didn't notice, he's talking about a complex number, not all the complex numbers.

So... I'm confused. You say:

"...the half-silvered mirror rule is "Multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle."

We appear to have defined everything needful, except the word "when".

Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?

914y

It was intended to be clear that all operations are performed and propagated throughout the entire system, I think.

311y

It's not clear to everyone; I didn't get it until reading your comment.

07y

I was confused about it too, but understood that what is meant is, that the 'half-silvered mirror rule' is a rule that does two things at once, namely x: (x*1, x*i), so it's a multi-valued operation.

as far as uncountable complex states... well, the actual complex values don't matter so much as the relative phases. Maybe best to think about it almost as a geometrical principle, of sorts.

Here I'm just speculating, but maybe relative phase (that is, angle when representing the complex value in polar notation) can only be shifted by rational amounts? That is, the relative phase between thingie 1 and thingie 2 is x*2Pi, where x must be rational?

I'm not saying this *is* the way it is, but I can certainly see, based on my, as of yet, limited knowledge, that it could be that way.

I guess, Eliezer, that I would be concerned about convincing everyone that the universe runs along like a computer, computing amplitudes locally (which seems to be the gist of your discussion). To do so would certainly make people *feel* like QM isn't confusing; it would just be wave mechanics. But this would give people a false confidence, I think, and is not how the universe appears to operate.

But this is the first post, so I'll try to confine my criticism until you've wrapped up your discussion.

Psy-Kosh, when QM is formulated rigorously (something that is rarely done, and only by mathematical physicists) the amplitudes must be able to take on any number in the complex plane, not just the rationals.

Sebastian Hagen, I believe Eliezer is explaining to us the best model physicists have for the way the world works on the (sorta) lowest level we understand, not his personal beliefs on the nature of reality. This model must include the irrationals, to be self-consistent. This does not prevent the universe from being discretized (no uncountable sets) on a more fundamental level from QM.

I was a little disturbed when you offered up the experiment that allowed us to reject the hypothesis about a half-mirror changing each time it reflects a photon or lets one through. How do we know there aren't other experiments that could discredit the amplitude hypothesis? I'm sure there's a good answer, but don't expect me to take too much on faith.

I also thought it was odd that you called configurations real, when they just seem to be a mathematical construct that describes the behavior of photons bouncing off of mirrors. Couldn't some other construc...

Eadwacer: Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?

Amplitude flows to both end configurations, every time. That is the law of the amplitude flows. It is not one or the other.

Sebastian: *Any complex number? I.e. you're invoking an uncountable infinity for explaining the lowest known layer of physics? How does that fit in with being an infinite-set atheist - assuming *...

213y

I'm sorry; I'm still a bit confused by this. "Amplitude flows to both end configurations every time," so when a single photon is fired (as in figure 1), I agree that the amplitudes of A->1 and A->2 are both 1. Does that mean both detectors click? (I was under the impression that only one detector would click.)

613y

"The detector" is not a machine for "measuring the amplitude". In fact, we have no way of "measuring the amplitude". The only tool we have "measures the ratios of the squares of the amplitudes", and that tool is this: "run the simulation a bunch of times and compute the ratio of detections at 1 to detections at 2".

Psy-Kosh: I have never heard of anyone ever successfully formulating quantum (or classical) mechanics without the full spectrum of real numbers. You can't even have simple things, like right triangles with non-integer side length, without irrational numbers to "fill in the gaps". Any finite-set formulation of QM would look *very* different from what we understand now.

Psy-Kosh: I have never heard of anyone ever successfully formulating quantum (or classical) mechanics without the full spectrum of real numbers. You can't even have simple things, like right triangles with non-integer side length, without irrational numbers to "fill in the gaps". Any finite-set formulation of QM would look *very* different from what we understand now.

Eliezer,

I've found that Jaynes's infinite set atheism is a little too extreme. It forces you to take the slow route every time when you want to explore stochastic process priors like Gaussian process and Dirichlet process priors. I reserve infinite set atheism for observables -- no infinite sets of observations allowed.

Jess: Basically, my (extremely vague) notion was that since there's a "planck time" below which little (as far as we know) can be meaningfully said, effectively all quantum operations/changes over time/whatever are integer number of some "planck" versions of themselves, or combinations theirof.

Soooooo..... maybe.... possibly... there may be some sort of "quantum of phase shift"... But I concede that it was just speculation on my part based on vague notions.

Oh well, thanks. :) (would be slick if it actually did work out that way, sounds like, from you, that may not be much of an option)

If anybody wants help with the "complex number" thing there are good tutorials at http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ and the follow-on at http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/

Okay, what happens in this situation: Take figure 2. The arrow coming in from the left? Replace it with figure 1, with its mirror relabeled E and detector 2 removed (replaced with figure 2). And lengthen the distance to detector 1 so that it's equal to the total distance to detector 2 in figure 2. And I guess call the detector 1 in figure 2 "X" for "we know you won't be getting any amplitude". Now what? Here's what I get...

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1) A photon is coming from E to A (-1,0)

A phot...

Here's what I was missing: the magnitudes of the amplitudes needs to decrease when changing from one possible state to more than one. In drawing-on-2d terms, a small amount of dark pencil must change to a large amount of *lighter* pencil, not a large amount of equally dark pencil. So here's what actually occurs (I think):

A photon is coming toward E (-1,0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from E to A (-1/sqrt(2),0)

A photon is coming from E to 1 (0,-1/sqrt(2)) A photon is coming from A to B (0,-1/2) A photon is coming from A to C...

211y

What I was about to say. It really doesn't matter yet, but it's better to get the reader used to unitarity straight away. (Though I wouldn't explicitly mention unitarity this early -- I'd just replace the rule with "Multiply by 1/sqrt(2) when the photon goes straight, and multiply by i/sqrt(2) when the photon turns at a right angle" and everything that follows from that. If the maths gets too complicated with all those denominators, just make the initial amplitude -sqrt(2) rather than -1.)

The prediction for what happens when you block the B to D path is wrong. We have three final configurations, not two as in the above.

- From the configuration "A photon going from B to D", with original amplitude (1 + 0i) o Amplitude of (1 + 0i) * 1 = (1 + 0i) flows to "Block with absorbed photon".
- From the configuration "A photon going from C to D", with original amplitude (0 + -i)
o Amplitude of (0 + -i)
*i = (1 + 0i) flows to "A photon going from D to F" o Amplitude of (0 + -i)*1 = (0 + -i) flows to "A photon

14y

I think Eliezer meant "the block absorbs the photon, and then destroys the universe". It's a magical block that, instead of absorbing photons, renders it impossible for them to pass it.
If it's possible for the block to absorb the photon, then I think you got the answer wrong. Eliezer was cheating a bit with his calculations, ignoring unitarity, which was okay because everything was growing by the same constant factor. Taking into account the block, you have to start paying attention to unitarity. (See GreedyAlgorithm and [anonymous]'s comments immediately above yours.) I think this means that the half-silvered mirrors multiply by i√2 and 1√2, and hence there's ½ probability the photon will hit the block and ¼ for each of the detectors detecting a photon.

"So... I'm confused. You say:

"...the half-silvered mirror rule is "Multiply by 1 when the photon goes straight, and multiply by i when the photon turns at a right angle."

We appear to have defined everything needful, except the word "when".

Accepting that we are just performing 'operations' on 'configurations', what decides which operation will be performed? Is it the configuration of the incoming photon? Is it some magical (i.e.quantum) property of a half of a silver?" ~ Hendrik Boom

I feel the same way as above.

714y

My understanding is fuzzy, but the sense I get is that the word "when" just attaches to the possible configurations rather than being a condition. A rephrasing of the idea might be:
"Multiply by 1 for the configuration in which the photon goes straight, and multiply by i for the configuration in which the photon turns at a right angle."
Which results in two configurations every time, rather than different configurations depending on what the photon does.

011y

'I think the problem some of us are having is reading "They would send a photon toward the half-silvered mirror ..." precludes the possibility of there being two results. I didn't think to not picture a little (billiard) ball being propelled towards a "half-silvered mirror" and it ending up at one (and only one) of the detectors.

So from what I understand:

A photon is merely our way of interpreting an amplitude wave in 3d-space. Such a 3d amplitude can be described by an x,y vector or (to simplify things), a x+yi complex number. (The complex number multiplied by i is really just a way of getting the same result as an x,y vector, due to the properties of complex numbers.)

Correct me if I am wrong so far, because I am about to get a bit fuzzy.

From what I see, the half-silvered mirror sends the amplitude wave in both directions, and is capable of reversing the phase of the amplitude -- ...

Hi, I'm the kind of guy I think this article was meant to target - I did not have an understanding of QM, but did start with enough base knowledge to follow the article without tripping over language or math in it.

I must say that I fried my brain trying to decipher what you're trying to say. From one paragraph to the next, there's a constant feeling a big hidden mental leap has been made. All of a sudden, one is left lost between notions that were introduced, but never explained.

For example. In Figure 3, from prior knowledge, I would suppose if you counted...

012y

You are. If you were to put a detector 3 there instead of an absorber, it would go off half the time, and detectors 1 and 2 would each go off a quarter of the time.

111y

Are you implying that the presence of a detector instead of an obstacle changes what the other detectors detect, or not?
The text is unclear here:
Does "half the time" mean "half the time that any detector goes off", or "half the time you shoot a photon"? I would expect that, with the obstacle in place, half the time you shoot a photon no detector would go off, because the first mirror would deflect it into an obstacle. Seeing no detector go off is distinct and observable, so I don't see any way it could be eliminated as a possibility like the other case described here where two possible timelines that lead to the same world interfere and cancel out. So I would assume Eliezer means "half the time that any detector goes off". If so, I'd like to see the text updated to be more clear about this.

14y

It means "half the time that any detector goes off", assuming that the block is a bog-standard lump of wood and not a magical construct like the measurement tool.

I have a question similar to Nate's. How does a half-silvered mirror work? More specifically, what is it about light or about half-silvered mirrors that means there are two paths for a photon out of a half-silvered mirror (compared to a full mirror, for example)? My guess at the moment is that the answer might start "light doesn't actually travel in straight lines..."...

114y

How does a half-silvered mirror work?

214y

You can't explain yourself? I followed your link. It looks like part of why half-silvered mirrors "work" for the purpose of seeing someone without them seeing you is that one side is kept brightly lit while the spying side is kept dark. I think "beam-splitter" is possibly a more accurate term for my question, which I looked up and found
(Wikipedia) Of course, this doesn't actually explain anything - why should there be a thickness of aluminum such that part of the light is reflected while the remainder is transmitted?
Would a beam-splitter still work if the silvered and non-silvered parts were much larger (i.e. a chunky block pattern)? If you fired a single photon at that would it still make sense to calculate amplitude as you do in this post (considering the two outward paths and multiplying one by i, the other by 1)? Perhaps the distance between a silvered part and a non-silvered part needs to be close to the wavelength of the photon?

013y

If the cross-section of the photon was spread out so that it hit both silvered and non-silvered parts, some would reflect, yes. But it wouldn't reflect quite like a mirror - diffraction effects would make things wonky, so people use half-silvered mirrors, which are nice.
How do they work, you ask? Did you ever take a course on wave mechanics where you calculated reflection and transmission coefficients? It's exactly like that, except now the probability is essentially what's "waving." (if you haven't, see here)

011y

To answer your question as to how a half-silvered mirror works, first it might be a good idea to discuss how a full mirror works.
Classically speaking, the silver in the mirror has electrons that can freely move around. The electromagnetic fields of the incoming light accelerate the charged electrons in the silver, inducing electric current.
The currents flowing in the silver create their own electric fields, which by Lenz's law, cancel out the electric field inside the silver, and in doing so, send an oppositely-shaped wave back out into the void (the reflected wave).
Because silver is not a perfect conductor of electricity, the topmost layer of silver does not completely cancel these fields, and so the light can actually penetrate a small distance into the metal (typically nanometers) before it's finally converted into electric current.
If the silver coating is very thin, thinner than the penetration depth, then the component of the light wave that has penetrated through the metal will escape out the other side and keep going. That is, the resistance of the thin silver is high enough that the induced current doesn't completely cancel out the electric fields of the photon.
This classical explanation is also the same as the quantum one.
They also make beamsplitters that are like you describe -- I think they call them "Polka dot beamsplitters". I don't remember what they're used for. They would work the same way, but if you have a focused laser beam, the beam spot would be so small that it would either hit a full-mirrored section or a transparent section, and not both. You would need to use a lens on both sides of the beam splitter to spread the beam out to encompass the whole beamsplitter, and then gather it back. I think as long as the polka dots are not on the same scale as the wavelength, it wouldn't cause a problem.

Don't know if anyone else ever comes back and reads here, but if so, I could use a bit of help.

I'm reading the quantum sequence, and I'm far enough in that the basics like this should be coming together. And mostly they are. But I have this nagging fact at the back of my mind that even though I can see why Figure 2 works, I can't actually explain Figure 1.

I understand that the amplitude flows to both detectors. I understand that it follows the same rules each time. I understand why each end configuration gets the values it does. But why does each detector ...

14y

The detector clicks 50% of the time because "detector makes a clicking noise" is so complex that it doesn't ever end up in the same state as "detector doesn't make a clicking noise" to interfere with it. There are multiple paths this photon can take to end up in the same configuration, because the photon moving around is simple enough that we can design an experiment to make some of the amplitude that's flowed to different configurations flow back to the same configurations – but the detector is complex enough that it separates the amplitude flows far enough that next to none of the amplitude from "detector 1 goes off" and "detector 2 goes off" will flow to the same configuration; hence they won't (noticeably) interfere with each other. And then the human either hears or doesn't hear the detector; the human is also complex enough that "human hears a clicking noise from detector 1" and "human hears a clicking noise from detector 2" aren't going to interfere; there's no way they're ending up in the same configuration afterwards.
Anything that remembers where the photon went will not observe interference from the photon going the other way, because it needs to be able to reach the same configuration from both of those configurations for any amplitude flow to interfere.

The post you're replying to is from April 2008; the next part is Joint Configurations and you can follow along by selecting "Article Navigation > by author" and clicking the right arrow, or follow the whole thing in a more organised way by following the Quantum Physics Sequence.

213y

It's funnny - but your brackets don't match.

013y

Fixed. Thx.

"we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials. The ratio of these values is the ratio of the squared moduli of the amplitudes. But the reason for this is not something we are going to consider yet."

OK, but I'd still like to see a little link or something here that takes me straight to the next article where this is properly dealt with, since this seems to be the biggest gap in understanding that the current article leaves open...

Would it be possible to actually set up this experiment at home (i.e. without an expensive physics lab)? Any particular pointers would be wonderful, even if it's just giving a common name that this setup uses. The sequence seems wonderful, but I'd prefer not to take it on faith if I can take it on empirically-demonstrated-it-myself instead :)

A friend comments:

start from figure 2, turn the half mirror at D round so it faces the other way, now E will light up instead of F. Since his explanation doesn't allow for that we've just proved his explanation is wrong.

Anyone know if that's right? EDIT: seems clear to me both detectors must light up if you do this. EDIT2: it turns out that by "turn around" he means through 180 degrees, which should surely mean no change.

113y

Can you clarify the question? Do you mean turning the mirror by a quarter-turn from its current orientation, so it's diagonal in the other direction? I compute that if you do that it should work out with 1/4 chance of E, 1/4 chance of F, and 1/2 chance of neither (if it's reflected back towards B or C it will never reach either). Exactly like the classical case, actually...

113y

Edited to clarify - I agree re quarter turn, but it turns out he means half turn. I think our thought-experiment half-silvered mirrors are unchanged by a half turn.

112y

It turns out he was referring to this error; see How accurate is the quantum physics sequence?

OnlyWhat affects the world is real. (If things can affect the world without being "real", it's hard to see what the word "real" means.) Configurations and amplitude flows are causes, and they have visible effects; they are real.

Now, for complicated reasons that we aren't going to go into today - considerations that belong on a higher level of organization than fundamental quantum mechanics, the same way that atoms are more complicated than quarks - there's no simple measuring instrument that can directly tell us the exact amplitudes of each configuration. We can't directly see the program state.

I'm not sure if you cover this in further articles... but it is worth saying:

The amplitudes of each state are not unique... there are more than one (in fact, there are infinitely many) different configurations that get you the same observable probability density, each differing by a phase factor.

012y

For that matter, amplitude of a wave...but what is waving? Where's the realism?

Eliezer, regarding the Fig.1 experiment above you're saying "The half-silvered mirror obeys the same rule every time." "This same result occurs—the same amplitudes stored in the same configurations—every time you run the program (every time you do the experiment)." OK, mathematical result is the same. However, physical results at detectors 1 & 2 are not the same: click at either of them is not predictable. There is symmetry in math vs asymmetry of physical result for any individual photon. Is there any "quantum explanation" for such physical dissimilarity?

012y

The ratio of "photon at detector 1" and "photon at detector 2" (averaged over enough trials) is 1.
Edit: This was actually written as a response to one of these comments.

In regards to the first experiment (Fig.1) "the little two-dimensional arrow for the configuration "Detector 1 gets a photon" has the same squared length as for "Detector 2 gets a photon"." This mathematical equality should have resulted in each photon arriving at detectors 1 & 2 simultaneously. But this never happens. Could anybody explain to me reason for such a discrepancy between math and reality?

In regards to the first experiment (Fig.1) "the little two-dimensional arrow for the configuration "Detector 1 gets a photon" has the same squared length as for "Detector 2 gets a photon"." This mathematical equality should have resulted in each photon arriving at detectors 1 & 2 simultaneously. But this never happens. Could anybody explain to me reason for such a discrepancy between math and reality?

[This comment is no longer endorsed by its author]

Eliezer is saying that when the ratio of the squared moduli is 1, than Detector 1 goes off half the time and Detector 2 goes off half the time. But why it should be necessarily interpreted this way? Is this another QM rule? What prevents, in this case, an alternative interpretation: a photon must split in half and arrive at both detectors at the same time?

BUT MORE IMPORTANTLY, WHAT DO YOU WANT?

Get a blog and stop commenting here.

[This comment is no longer endorsed by its author]

This brilliant young mathematician can speak to you in more familiar terms and has all the math "to back up what he says."

I looked at a few of his essays and didn't find any substantial mathematics in them, brilliant or otherwise. In the process I came across assertions that for a particle in a circular orbit, v = 2 pi r/t is false for the traditional value of pi, which in this kinematic situation must have the value 4. Oh, and modern physics is a conspiracy of the intelligence communities to prevent dangerous discoveries being made. (Hm, is modern AGI a conspiracy of the SIAI to prevent UFAI?)

Crackpot. He claims to have many pseudonyms; is "MonkeyMind" one of them?

ETA: More about Miles Mathis.

I've spent a while hanging around conspiracy theorists online, and taken the time to follow up on the sorts of people who get talked about for proposing "revolutionary" theories which are kept down by the scientific orthodoxy.

What distinguishes people in this category, of which Miles Mathis is typical, is not failure to produce testable hypotheses, but the production of hypotheses that are trivially wrong. If Miles Mathis' claims about physics were correct, to point out a single instance of failure, GPS satellites, rather than being geosynchronou...

212y

The linked site (http://www.physicsmyths.org.uk/) is a crackpot site which argues that special and general relativity are wrong. So (given what I have heard from credible authorities about the workings of GPS's) I would actually bet that the claim is false.

212y

I only briefly eyeballed the site, and didn't have that much awareness of the content. I am aware that GPSes incorporate the predictions of General Relativity for their calibration, but it did not strike me as implausible that the deviations from Newtonian physics would be within their error margins, at least up to this time of operation. I admit that I was very much premature in asserting that it was true without doing the calculations myself. Thanks for pointing that out.

012y

I'm hesitant to point this out after saying I wasn't going to engage with him anymore, but this sounds way less like something someone who's sincerely spent months chasing after the idea that Eliezer is seriously misguided for what he wrote in Configurations and Amplitudes would say than someone who was deliberately trying to crank the levers of other readers. I've wavered previously on whether to give him the benefit of the doubt, but at this point I think it's only fair to assume that Monkeymind is a troll rather than a hopelessly confused person.

Please stop trying to continue this conversation.

The way to make that happen is by NOT responding to their comments. Only downvote, don't reply. Also, downvote those who reply, irrespective of how well their comments are composed, to discourage the behavior that encourages bad conversations (the conversation that sprang from your reply is currently 30 comments strong).

012y

I regret that now, but Monkeymind has kept coming back to comment now repeatedly since I previously stopped replying to his comments, and I was hoping that negative feedback would deter him where an absence of feedback had failed to.

When you reply to one of my comments, the letterbox under my username lights up in red, and it won't go away until I click on it, which links directly to the comment.

Trying to hold a discussion with you has so far proven to be fruitless as well as frustrating, and I am not going to continue engaging with you after this, but I am going to ask you to stop coming back to this conversation over and over with new comments, because it is going to cause annoyance whether I reply to them or not.

2[anonymous]12y

The entire thread was deleted, but you can still read parts of it in the history.
Previously.

012y

I can see the banned comments, but yeah. I actually read or at least skim literally every comment and post on LW, so I've been sitting back for a while now.

0[anonymous]12y

Then someone should probably know about the privilege escalation bug that allows us plebs to read banned comments.

012y

You also see banned comments? I think the person to notify is Matt of Tricycle.

412y

The ability to see banned comments on userpages (but nowhere else) is a feature.
(Someone taking the active step of clicking on a username presumably has a specific interest in seeing the comments, and ought to be able to.)

212y

I think Alicorn understood paper-machine to be claiming to see deleted comments in-place. I also think this because going to someone's user page and seeing banned comments does not seem like the kind of activity that would count as a "privilege escalation bug".

012y

Depends on why the comments were banned. If it's because they disclosed information which shouldn't have been disclosed...

-1212y

-412y

So what's up with that? I went to a lot of work writing those posts.
Is this the sort of thing done with approval of the site owner?
They were well thot out and reasoned posts. The majority were very civil and violated no posted rules. In fact there aren't any posted rules that I am aware of. Just because my posts are annoying to some folks is not reason to delete them. NO one has to read anything.
I just don't understand the reasoning there, or here:
"A specific suggestion I have is to establish a community norm of downvoting those participating in hopeless conversations, even if their contributions are high-quality."

0[anonymous]12y

It would depend, I think, on how one resolves the conflict between Friendliness and the human value of self-determination.

But OK, if that is how you roll, I'll continue on to the Singularity board or Nick Bostrom's or elsewhere and discuss the equally debunked notions of transhumanism instead. I only hope that I don't continue to encounter the immaturity and childishness I have here.

"Skies change, not cares, for those who cross the seas." --Horace

Good luck, if you ever finally graduate into the real world of solid objects and hard cold reality, I'll be surprised.

If I ever turn into something you approve of unreservedly, I will be surprised too.

Thanks for this explanation. I've tried to read it some time ago but have not really coped with it. Now after reading again it was interesting for me to check if this is explained on some other internet sources and how exactly. So I checked one of the first top search results and here is what I saw: http://physics.stackexchange.com/questions/91695/double-slit-expirement-fundamentals-half-silvered-mirror-version

There some guy asked for an explanation of this experiment and answers are all about optical refraction and phase shifting, which honestly speaking ...

Great post, Eliezer! I have one question, though, and maybe some of the folks here can answer it as well: why do we multiply amplitude in Figure 2 by i if it either turns "left" or "right" at 90 degrees? In the complex plane, we multiply a vector by i if we want to rotate it 90 degrees clockwise, and by -i if we want to rotate it 90 degrees counterclockwise...

There is one thing that confuses me about this post, which I haven't found in any of the comments

So the final program state is:

Configuration "A photon going toward A": (-1 + 0i)Configuration "A photon going from A toward 1": (0 + -i)

Configuration "A photon going from A toward 2": (-1 + 0i)

Why does the bolded configuration still exist in the same way? Shouldn't it go back to zero once the photon has reached A, since the rest of the post seems to imply a timely order of things?

07y

silver, check out rule #3 for calculating probability amplitudes:
https://en.wikipedia.org/wiki/Probability_amplitude#The_laws_of_calculating_probabilities_of_events
It basically states that in order to calculate the amplitude of a photon going to 1 is a product of the amplitude of it going to A and then from A to 1. So we do need to remember what the amplitude was for a photon going toward A.

This comment refers to the editor's note in the ebook (footnote 1). That note says that the conventional form of a half silvered mirror multiplies by -1 (not i) when a photon turns at a right angle. The note also states that Eliezer's formulation is physically realizable, but doesn't give further explanation. This seemed confusing to me. If a simple explanation of when Eliezer's version is correct can be provided, then that would be helpful. My guess is that this might be when the mirror is the same from both sides: ex. if the experimen...

What bugs me about this article is that we have 'half silvered mirrors'. By definition they divert half and allow half through. Like the one at 'A'. But then suddenly, with the one at 'D' we get "And what D does to a photon, depends on the angle at which the photon arrives" - so not a half silvered mirror, but something else, with no explanation of how or why the angle affects the outcome.

As a layperson whose understanding changed from billiard balls to waves to probabilities I suspect there is no 'reality' that everything can be reduced to - and...

Less wrong user titotal has written a new and corrected version of this post, and I suggest that anyone wanting to learn this material should learn it from that post instead.

(In case anyone is curious, the main error in *this* post is that Eliezer describes the mirror as splitting an incoming state into two outgoing states . However, the overall magnitude of this outgoing state is , whereas the incoming state had magnitude . This means that the mirror is described by a *non-unitary operator*, meaning that it doesn't conserve probability, which is forbidden in quantum mechanics. You can fix this by instead describing the outgoing state as .

While it is permitted to do quantum mechanics without normalizing your state (you can get away with just normalizing the probabilities you compute at the end), any operators you apply to your system must still have the correct normalization factors attached to them. Otherwise, you'll get an incorrect answer. To see this, consider an initial state of , where describes the photon heading towards a beam-splitter and described the photon heading in a different direction entirely. This state is unnormalized, which...

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

wasthis experiment— not a computer program thatpredictedthe 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 towardA.”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 towardA” 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”—and it’s out there in the territory.As the complex numbers that get assigned to configurations are not positive real numbers between 0 and 1, there is no danger of confusing them with probabilities. “A photon heading toward

A” has complex value −1, which is hard to see as a degree of belief. The complex numbers are values within the program, again out there in the territory. We’ll call the complex numbersamplitudes.There are two other configurations, which we’ll call “a photon going from

Ato Detector 1” and “a photon going fromAto Detector 2.” These configurations don’t have a complex value yet; it gets assigned as the program runs.We are going to calculate the amplitudes of “a photon going from

Atoward 1” and “a photon going from A toward 2” using the value of “a photon going towardA,” and the rule that describes the half-silvered mirror atA.Roughly speaking, the half-silvered mirror rule is “multiply by 1 when the photon goes straight, and multiply by

iwhen the photon turns at a right angle.” This is the universal rule that relates the amplitude of the configuration of “a photon going in,” to the amplitude that goes to the configurations of “a photon coming out straight” or “a photon being deflected.”[1]So we pipe the amplitude of the configuration “a photon going toward

A,” which is (−1+0i), into the half-silvered mirror atA, and this transmits an amplitude of (−1+0i)×i=(0−i) to “a photon going fromAtoward 1,” and also transmits an amplitude of (−1+0i)×1=(−1+0i) to “a photon going fromAtoward 2.”In the Figure 1 experiment, these are all the configurations and all the transmitted amplitude we need to worry about, so we’re done. Or, if you want to think of “Detector 1 gets a photon” and “Detector 2 gets a photon” as separate configurations, they’d just inherit their values from “

Ato 1” and “Ato 2” respectively. (Actually, the values inherited should be multiplied by another complex factor, corresponding to the distance fromAto the detector; but we will ignore that for now, and suppose that all distances traveled in our experiments happen to correspond to a complex factor of 1.)So the final program state is:

and optionallyThis same result occurs—the same amplitudes stored in the same configurations—every time you run the program (every time you do the experiment).

Now, for

complicatedreasons that we aren’t going to go into here— considerations that belong on a higher level of organization than fundamental quantum mechanics, the same way that atoms are more complicated than quarks—there’s nosimplemeasuring instrument that can directly tell us the exact amplitudes of each configuration. We can’t directly see the program state.So how do physicists know what the amplitudes are?

We

dohave a magical measuring tool that can tell us thesquared modulusof a configuration’s amplitude. If the original complex amplitude is (a+bi), we can get the positive real number (a2+b2). Think of the Pythagorean theorem: if you imagine the complex number as a little arrow stretching out from the origin on a two-dimensional plane, then the magic tool tells us the squared length of the little arrow, but it doesn’t tell us the direction the arrow is pointing.To be more precise, the magic tool actually just tells us the

ratiosof the squared lengths of the amplitudes in some configurations. We don’t know how long the arrows are in an absolute sense, just how long they are relative to each other. But this turns out to be enough information to let us reconstruct the laws of physics—the rules of the program. And so I can talk about amplitudes, not just ratios of squared moduli.When we wave the magic tool over “Detector 1 gets a photon” and “Detector 2 gets a photon,” we discover that these configurations have the same squared modulus—the lengths of the arrows are the same. Thus speaks the magic tool. By doing more

complicatedexperiments (to be seen shortly), we can tell that the original complex numbers had a ratio ofito 1.And what is this magical measuring tool?

Well, from the perspective of everyday life—way, way, way above the quantum level and a lot more complicated—the magical measuring tool is that we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials. The ratio of these values is the ratio of the squared moduli of the amplitudes. But the reason for this is

notsomething we are going to consider yet. Walk before you run. It is not possible to understand what happensall the way upat the level of everyday life, before you understand what goes on in much simpler cases.For today’s purposes, we have a magical squared-modulus-ratio reader. And the magic tool tells us that the little two-dimensional arrow for the configuration “Detector 1 gets a photon” has the same squared length as for “Detector 2 gets a photon.” That’s all.

You may wonder, “Given that the magic tool works this way, what motivates us to use quantum theory, instead of thinking that the half-silvered mirror reflects the photon around half the time?”

Well, that’s just begging to be confused—putting yourself into a historically realistic frame of mind like that and using everyday intuitions. Did I say anything about a little billiard ball going one way or the other and possibly bouncing off a mirror? That’s not how reality works.

Realityis about complex amplitudes flowing between configurations, and the laws of the flow are stable.But if you insist on seeing a more complicated situation that billiard-ball ways of thinking can’t handle, here’s a more complicated experiment.

In Figure 2,

BandCare full mirrors, andAandDare half-mirrors. The line fromDtoEis dashed for reasons that will become apparent, but amplitude is flowing fromDtoEunder exactly the same laws.Now let’s apply the rules we learned before:

At the beginning of time “a photon heading toward

A” has amplitude (−1+0i).We proceed to compute the amplitude for the configurations “a photon going from

AtoB” and “a photon going fromAtoC”:Similarly,

The full mirrors behave (as one would expect) like half of a half-silvered mirror—a full mirror just bends things by right angles and multiplies them by i. (To state this slightly more precisely: For a full mirror, the amplitude that flows, from the configuration of a photon heading in, to the configuration of a photon heading out at a right angle, is multiplied by a factor of i.

)So:

“B to D

”and “C to D”are two different configurations—we don’t simply write “a photon at D”—because the photons are arriving at two different angles in these two different configurations. And what D does to a photon depends on the angle at which the photon arrives.Again, the rule (speaking loosely) is that when a half-silvered mirror bends light at a right angle, the amplitude that flows from the photon-going-in configuration to the photon-going-out configuration, is the amplitude of the photon-going-in configuration multiplied by

i. And when two configurations are related by a half-silvered mirror letting light straight through, the amplitude that flows from the photon-going-in configuration is multiplied by 1.So:

From the configuration “a photon going from B to D

,” with original amplitude(1+0i)From the configuration “a photon going from C to D

,” with original amplitude(0−i)Therefore:

totalamplitude flowing to configuration “a photon going fromDtoE” is (0+i)+(0−i)=(0+0i)=0.DtoF” is (1+0i)+(1+0i)=(2+0i).(You may want to try working this out yourself on pen and paper if you lost track at any point.)

But the upshot, from that super-high-level “experimental” perspective that we think of as normal life, is that we see

nophotons detected atE. Every photon seems to end up atF. The ratio of squared moduli between “DtoE” and “DtoF” is 0 to 4. That’s why the line fromDtoEis dashed, in this figure.This is not something it is possible to explain by thinking of half-silvered mirrors deflecting little incoming billiard balls half the time. You’ve

gotto think in terms of amplitude flows.If half-silvered mirrors deflected a little billiard ball half the time, in this setup, the little ball would end up at Detector 1 around half the time and Detector 2 around half the time. Which it doesn’t. So don’t think that.

You may say, “But wait a minute! I can think of another hypothesis that accounts for this result. What if, when a half-silvered mirror reflects a photon, it does something to the photon that ensures it doesn’t get reflected next time? And when it lets a photon go through straight, it does something to the photon so it gets reflected next time.”

Now really, there’s no need to go making the rules so complicated. Occam’s Razor, remember. Just stick with simple, normal amplitude flows between configurations.

But if you want

anotherexperiment that disproves yournewalternative hypothesis, it’s Figure 3.Here, we’ve left the whole experimental setup the same, and just put a little blocking object between

BandD. This ensures that the amplitude of “a photon going fromBtoD” is 0.Once you eliminate the amplitude contributions from that configuration, you end up with totals of (1+0i) in “a photon going from

DtoF, ” and (0−i) in “a photon going fromDtoE.”The squared moduli of (1+0i) and (0−i) are both 1, so the magic measuring tool should tell us that the ratio of squared moduli is 1. Way back up at the level where physicists exist, we should find that Detector 1 goes off half the time, and Detector 2 half the time.

The same thing happens if we put the block between

CandD. The amplitudes are different, but the ratio of the squared moduli is still 1, so Detector 1 goes off half the time and Detector 2 goes off half the time.This cannot

possiblyhappen with a little billiard ball that either does or doesn’t get reflected by the half-silvered mirrors.Because complex numbers can have opposite directions, like 1 and −1, or

iand −i, amplitude flows can cancel each other out. Amplitude flowing from configurationXinto configurationYcan be canceled out by an equal and opposite amplitude flowing from configurationZinto configurationY. In fact, that’s exactly what happens in this experiment.In probability theory, when something can either happen one way or another,

Xor ¬X, then P(Z)=P(Z|X)P(X)+P(Z|¬X)P(¬X). And all probabilities are positive. So if you establish that the probability ofZhappening givenXis 12, and the probability ofXhappening is 13, then the total probability ofZhappening isat least16 no matterwhatgoes on in the case of ¬X. There’s no such thing as negative probability, less-than-impossible credence, or (0+i) credibility, sodegrees of beliefcan’t cancel each other out like amplitudes do.Not to mention that probability is in the mind to begin with; and we are talking

aboutthe territory, the program-that-is-reality, not talkingabouthuman cognition or states of partial knowledge.By the same token, configurations are not

propositions, notstatements, notways the world could conceivably be. Configurations are not semantic constructs. Adjectives likeprobabledo not apply to them; they are not beliefs or sentences or possible worlds. They are nottrueorfalsebut simplyreal.In the experiment of Figure 2, do not be tempted to think anything like: “The photon goes to either

BorC, but itcouldhave gone the other way, and this possibility interferes with its ability to go toE…”It makes no sense to think of something that “could have happened but didn’t” exerting an effect on the world. We can

imaginethings that could have happened but didn’t—like thinking, “Gosh, that car almost hit me”—and our imagination can have an effect on our future behavior. But the event of imagination is a real event, that actually happens, andthatis what has the effect. It’s your imagination of the unreal event—your very real imagination, implemented within a quite physical brain—that affects your behavior.To think that the

actual eventof a car hitting you—this event which could have happened to you, but in fact didn’t—is directly exerting acausaleffect on your behavior, is mixing up the map with the territory.What affects the world is real. (If things can affect the world without being “real,” it’s hard to see what the word “real” means.) Configurations and amplitude flows are causes, and they have visible effects; they are real. Configurations are not possible worlds and amplitudes are not degrees of belief, any more than your chair is a possible world or the sky is a degree of belief.

So what

isa configuration, then?Well, you’ll be getting a clearer idea of that in later essays.

But to give you a quick idea of how the real picture differs from the simplified version we saw in this essay…

Our experimental setup only dealt with one moving particle, a single photon. Real configurations are about multiple particles. The next essay will deal with the case of more than one particle, and that should give you a much clearer idea of what a configuration is.

Each configuration we talked about

shouldhave described a joint position of all the particles in the mirrors and detectors, not just the position of one photon bopping around.In fact, the

really realconfigurations are over joint positions of all the particles in the universe, including the particles making up the experimenters. You can see why I’m saving the notion ofexperimental resultsfor later essays.In the real world, amplitude is a continuous distribution over a continuous

spaceof configurations. This essay’s “configurations” were blocky and digital, and so were our “amplitude flows.” It was as if we were talking about a photon teleporting from one place to another.If none of that made sense, don’t worry. It will be cleared up in later essays. Just wanted to give you some idea of where this was heading.

1. [

Editor’s Note:Strictly speaking, a standard half-silvered mirror would yield a rule “multiply by −1 when the photon turns at a right angle,” not “multiply byi.” The basic scenario described by the author is not physically impossible, and its use does not affect the substantive argument. However, physics students may come away confused if they compare the discussion here to textbook discussions of Mach–Zehnder interferometers. We’ve left this idiosyncrasy in the text because it eliminates any need to specify which side of the mirror is half-silvered, simplifying the experiment.]