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 , with i defined as . 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 .

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 numbers *amplitudes*.

There are two other configurations, which we’ll call “a photon going from *A* to Detector 1” and “a photon going from *A* to 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 *A* toward 1” and “a photon going from A toward 2” using the value of “a photon going toward *A*,” and the rule that describes the half-silvered mirror at *A*.

Roughly speaking, 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.” 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 , into the half-silvered mirror at *A*, and this transmits an amplitude of to “a photon going from *A* toward 1,” and also transmits an amplitude of to “a photon going from *A* toward 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 “*A* to 1” and “*A* to 2” respectively. (Actually, the values inherited should be multiplied by another complex factor, corresponding to the distance from *A* to 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:

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)

*and optionally*

Configuration “Detector 1 gets a photon”: (0−i)

Configuration “Detector 2 gets a photon”: (−1+0i).

This 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 *complicated* reasons 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 no *simple*measuring 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 *do* have a magical measuring tool that can tell us the *squared modulus* of a configuration’s amplitude. If the original complex amplitude is , we can get the positive real number . 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 *ratios* of 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 *complicated* experiments (to be seen shortly), we can tell that the original complex numbers had a ratio of *i* to 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 *not* something we are going to consider yet. Walk before you run. It is not possible to understand what happens *all the way up* at 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. *Reality* is 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, *B* and *C* are full mirrors, and *A* and *D* are half-mirrors. The line from *D *to *E* is dashed for reasons that will become apparent, but amplitude is flowing from *D* to *E* under 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 .

We proceed to compute the amplitude for the configurations “a photon going from *A* to *B*” and “a photon going from *A* to *C*”:

“a photon going from Ato B” = a photon heading toward A” =

Similarly,

“a photon going from Ato C” = 1 a photon heading toward A” =

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:

“a photon going from B to D”= “a photon going from A to B”= ,

“a photon going from C to D”= “a photon going from A to C”=

“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)

Amplitude of flows to “a photon going from D to E.”

Amplitude of flows to “a photon going fromDtoF. ”

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

Amplitude of flows to “a photon going from D to F.”

Amplitude of flows to “a photon going from D to E.”

Therefore:

- The
*total*amplitude flowing to configuration “a photon going from*D*to*E*” is . - The total amplitude flowing to configuration “a photon going from
*D*to*F*” is .

(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 *no* photons detected at *E*. Every photon seems to end up at *F*. The ratio of squared moduli between “*D* to *E*” and “*D* to *F*” is 0 to 4. That’s why the line from *D* to *E* is 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 *got* to 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 *another* experiment that disproves your *new* alternative hypothesis, it’s Figure 3.

Here, we’ve left the whole experimental setup the same, and just put a little blocking object between *B* and *D*. This ensures that the amplitude of “a photon going from *B *to *D*” is 0.

Once you eliminate the amplitude contributions from that configuration, you end up with totals of in “a photon going from *D* to *F*, ” and in “a photon going from *D* to *E*.”

The squared moduli of and 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 *C* and *D*. 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 *possibly* happen 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 *i* and −*i*, amplitude flows can cancel each other out. Amplitude flowing from configuration *X *into configuration *Y* can be canceled out by an equal and opposite amplitude flowing from configuration *Z* into configuration *Y*. In fact, that’s exactly what happens in this experiment.

In probability theory, when something can either happen one way or another, *X* or ¬*X*, then . And all probabilities are positive. So if you establish that the probability of *Z* happening given *X* is , and the probability of *X* happening is , then the total probability of *Z* happening is *at least* no matter *what* goes on in the case of ¬*X*. There’s no such thing as negative probability, less-than-impossible credence, or credibility, so *degrees of belief *can’t cancel each other out like amplitudes do.

Not to mention that probability is in the mind to begin with; and we are talking *about* the territory, the program-that-is-reality, not talking *about* human cognition or states of partial knowledge.

By the same token, configurations are not *propositions*, not *statements*, not *ways the world could conceivably be*. Configurations are not semantic constructs. Adjectives like *probable* do not apply to them; they are not beliefs or sentences or possible worlds. They are not *true* or *false* but simply *real*.

In the experiment of Figure 2, do not be tempted to think anything like: “The photon goes to either *B* or *C*, but it *could* have gone the other way, and this possibility interferes with its ability to go to *E*…”

It makes no sense to think of something that “could have happened but didn’t” exerting an effect on the world. We can *imagine* things 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, and *that* is 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 event* of a car hitting you—this event which could have happened to you, but in fact didn’t—is directly exerting a *causal* effect 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 *is* a 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 *should* have 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 real* configurations 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 of *experimental results* for later essays.

In the real world, amplitude is a continuous distribution over a continuous *space* of 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 by *i*.” 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.]

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.

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

For a rather silly reason, I wrote something about:

Please ignore the "lowest known layer" part. I accidentally committed a mind projection fallacy while writing that comment.botogol:

I have the impression Eliezer writes blog entries in much the same way I read Wikipedia: Slowly working from A to B in a grandiose excess of detours... =)

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?

Eliezer, I realise there's still a way to go, but I just wanted to let you know that this is already much more useful than any conversion I've had about QM with anyone in the past. Thank you.

Eadwacer, I might be wrong, but I'd assumed both operations are always performed.

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

isthe 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

feellike 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.

aren't you writing this on the wrong blog?As far as I know Robin doesn't actually have a separate economics blog and he seems to drop any economics topic that interests him into this one, so neither Eliezer nor Robin always stick closely to the "bias" theme. Does it really matter?

Jess: You mean that physics, as we understand it, absolutely requires that there will exist complex phase differences such that when divided by 2*Pi, the result will be irrational?

Oh well then, bang goes that idea. :)

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... (read more)

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... (read more)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

verydifferent 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

verydifferent 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)

Just so everyone's on the same page, continuum atheism doesn't entail disbelief in all irrationals. The hypoteneuse of a right triangle, pi and e are all in the countable set of computable reals.

If a photon hits two full mirrors at right angles, then its amplitude is changed by i*i = -1. Does it matter whether the second mirror turns the photon back towards its source, or causes the photon to continue in the direction it was going originally? Do you get -1 in both cases?

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... (read more)

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

lighterpencil, 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... (read more)

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.

i = (1 + 0i) flows to "A photon going from D to F" o Amplitude of (0 + -i)1 = (0 + -i) flows to "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.

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 -- ... (read more)

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... (read more)

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..."...

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 ... (read more)

I thought that was a really good, logical, simple explanation. Looking forward to reading the next episode.

Thanks!

This is very cool. I know that's just in my head, but now I just want a half-silvered mirror to test this with my kids.

An xkcd take on this material

I collapsed laughing. Your results may vary.

"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... (read more)

"Adjectives like probable and possible do not apply to them; they are not beliefs or sentences or possible worlds. They are not true or false but simply real."

Based on all the "i"s in the equations I think you meant to say "complex" =p

Is it possible in reality to fire a single photon?

(Post modified)

after the computer program above calculates the amplitude (the same every time we run the program), can we incorporate in the program additional steps to simulate our magical measurement tool (the detector)?

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:

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.

ratiosbetween amplitudes are “real” in that sense, because if you multiply the amplitude of everything by, say, exp(2πi/3), nothing actually changes.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.

I... Er... What. Where did the whole 'amplitude' thing come from? I mean, it looks a lot like they are vectors in the complex plane, but why are they two dimensional? Why not three? Or one? I just don't get the idea of what amplitude is supposed to describe.

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?

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?

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?

Get a blog and stop commenting here.

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 m... (read more)

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... (read more)

Could you answer the question with a yes or no? That's really all that it takes.

This is true but irrelevant because the site you are citing contrasts general relativity to Newtonian physics, not the model Miles Mathis is claiming which issues completely different predictions.

Please stop trying to continue this conversation.

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.

Please go away.

I would be shocked if Eliezer did anything to straighten us out if he ever looked into the matter.

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

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

Dude, I

ama mod. I don't like slinging the banhammer around as a first resort, but you're annoying.You didn't finish.

Clarification: an amplitude is the value of a configuration?

so { a photon going from

AtoB= (-1 + 0i) } is a configuration and { (-1 + 0i) } is an amplitude?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 ... (read more)

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

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?

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... (read more)

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... (read more)

Why doesn't the block between B and D absorb the photon a third of the time, since it should have the same modulus as the detectors? What's so special about things that tell us that they've been hit by a photon?