What confuses me about the actual verification of these experiments is that they require perfect timing and distancing. How exactly do you make two photons hit a half-silvered mirror at exactly the same time? It seems that if you were off only slightly then the universe would necessarily have to keep track of both as individuals. In practice you're always going to be off slightly so you would think by your explanation above that this would in fact change the result of the experiment, placing a 25% probability on all four cases. Why doesn't it?

Jordan, the three answers to your question are:

1) Each photon is actually spread out in configuration space - I'll talk about this later - so an infinitesimal error in timing only creates an infinitesimal probability of both detectors going off, rather than a discontinuous jump.

2) Physicists have gotten good at doing things with bloody precise timing, so they can run experiments like this.

3) I didn't actually perform the experiment.

"The probability of two events equals the probability of the first event plus the probability of the second event."

Mutually exclusive events.

It is interesting that you insist that beliefs ought to be represented by classical probability. Given that we can construct multiple kinds of probability theory, on what grounds should we prefer one over the other to represent what 'belief' ought to be?

If the photon sources aren't in a super position I think you have to represent the system with a vector of complex numbers. IANAP either.

Gray, fixed.

Jesus Christ, the complex plane. I half-remember that.

Eliezer, this may come as a shock, but I suspect there exists at least some minority of individuals beyond just me who will find the consistent use of complex numbers at all to be the the most migraine-inducing part of this. You might also find that even those of us who supposedly know how to do some computation on the complex plane are likely to have little to no intuitive grasp of complex numbers. Emphasis on computation over understanding in mathematics teaching, while pervasive, does not tend to serve students well. I could be the only one for whom this applies, but I wouldn't bet heavily on it.

Thankfully, there's already a couple of "intuitive explanations" of complex numbers on the 'net. And I dug up the links. And the same site has a few articles about generalizations of the Pythagorean Theorem, on a related note. Basically, anyone who is having any trouble with the mathematical side of this is likely to find it a bit of a help. It's also a lot like overcoming bias in its explanatory approach, so there's that.

Imaginary/complex numbers:

http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

http://betterexplained.com/articles/intuitive-arithmetic-with-complex-numbers/

General math:

http://betterexplained.com/articles/category/math/

I should probably also take this opportunity to swear up and down that I'm not trying to generate ad revenue for that site, but you'd have to take my word on it. I might also add that I'm quite certain I still don't understand complex numbers meaningfully, but that's a separate thing.

Given that we can construct multiple kinds of probability theory, on what grounds should we prefer one over the other to represent what 'belief' ought to be?

Make sure you understand Cox's theorem, then exhibit for me two kinds of probability theory, then I will reply.

Here's what bugs me: Those two photons aren't going to be exactly the same, in terms of say frequency or maybe the angle they make against the table. So how close do they have to be for the configurations to merge? Or is that a Wrong Question? Perhaps if we left the photon emmiter the same but changed the detector to one that could tell the difference in angle, then the experimental results would change? What if we use a angle-detecting photon detector but program it to dump the angle into /dev/null?

Regarding Larry's question about how close the photons have to be before they merge --

The solution to that problem comes from the fact that Eliezer's experiment is (necessarily) simplifying things. I'm sure he'll get to this in a later post so you might be better off waiting for a better explanation (or reading Feynman's QED: The Strange Theory of Light and Matter, which I think is a fantastically clear explanation of this stuff.) But if you're willing to put up with a poor explanation just to get it quicker...

In reality, you don't have just one initial amplitude of a photon at exactly time T. To get the full solution, you have to add in the amplitude of the photon arriving a little earlier, or a little later, and with a little smaller or a little larger wavelength, and even travelling faster or slower or not in a straight line, and possibly interacting with some stray electron along the way, and so on for a ridiculously intractable set of complications. Each variation or interaction shows up as a small multiplier to your initial amplitude.

But fortunately, most of these interactions cancel out over long distances or long times, just like the case of the two photons hitting opposite detectors, so in this experiment you can treat the photon as just having arrived at a certain time and you'll get very close to the right answer.

Or in other words--easier to visualize but perhaps misleading--the amplitude of the photons is "smeared out" a little in space and time, so it's not too hard to get them to overlap enough for the experiment to work.

--Jeff

Jeff, I would not call your last paragraph misleading. It is quantum mechanics--as opposed to quantum field theory--and I think QED is a gratuitously complicated way to answer the question. Perhaps it's a way to get from classical notions of particles to quantum mechanics, but it seems to me to go against the spirit of trying to understand QM on its own terms, rather than as a modification of classical particles.

I found your explanations of Bayesian probability enlightening, and I've tried to read several explanations before. Your recent posts on quantum mechanics, much less so. Unlike the probability posts, I find these ones very hard to follow. Every time I hit a block of 'The "B to D" photon is deflected and the "C to D" photon is deflected.' statements, my eyes glaze over and I lose you.

Dan, Emmett,

I hear you. Unfortunately, I can't put as much work into this as I did for the intuitive explanation of Bayes's Theorem, plus the subject matter is inherently more complicated.

Feynman's QED uses little arrows in 2D space instead of complex numbers (the two are equivalent). And if I had the time and space, I'd draw different visual diagrams for each configuration, and show the amplitude flowing from one to another...

But QM is also inherently more complicated than Bayes's Theorem, and takes more effort; plus I'm trying to explain it in less time... I'm not figuring that all readers will be able to follow, I'm afraid, just hoping that some of them will be.

If the problem is not that QM is confusing but that you can't follow what is being said at all, you probably want to be reading Richard Feynman's QED instead.

Feynman's QED uses little arrows in 2D space instead of complex numbers (the two are equivalent).

this is quite offtopic, but there are important analogies [1] [2] between Feynman diagrams, topology and computer science. being aware of this may amke the topic easier to understand.

So you calculate events for photons that happen in parallel (e.g. one photon being deflected while another is deflected as well) the same way you would for when they occur in series (e.g. a photon being deflected, and then being deflected again)? It seems that in both cases you are multiplying the original configuration by -1 (i.e. i * i).

FWIW, my first instinct when I saw the diagram was to assume 2 starting configurations, one for each photon, though I guess the point of this post was that I can't do stuff like that. In fact, when I did the math that way, I came up with the photons hitting different detectors twice as much as when they both hit the same one.

I think I'll be picking up the Feynman book.....

Richard: Cox's theorem is an example of a particular kind of result in math, where you have some particular object in mind to represent something, and you come up with very plausible, very general axioms that you want this representation to satisfy, and then prove this object is unique in satisfying these. There are equivalent results for entropy in information theory. The problem with these results, they are almost always based on hindsight, so a lot of the times you sneak in an axiom that only SEEMS plausible in hindsight. For instance, Cox's theorem states that plausibility is a real number. Why should it be a real number?

Grey Area asked, "For instance, Cox's theorem states that plausibility is a real number. Why should it be a real number?" For that matter, why should the plausibility of the negation of a statement depend only on the plausibility of the statement? Mightn't the statement itself be relevant?

Gray Area,

In reply to "why a real number question", we might want to weaken the theory to the point were only equalities and inequalities can be stated. There are two weaker desiderata one might hold. Let (A|X) be the plausibility of A given X.

Transitivity: if (A|X) > (B|X) and (B|X) > (C|X), then (A|X) > (C|X)

Universal Comparability: one of the following must hold (A|X) > (B|X) (A|X) = (B|X) (A|X) < (B|X)

If you keep both, you might as well use a real number -- doing so will capture all of the desired behavior. If you throw out Transitivity, I have a series of wagers I'd like to make with you. If you throw out Universal Comparability, then you get lattice theories in which propositions are vertexes and permitted comparisons are edges.

On the other hand, you might find just a single real number too restrictive, so you use more than one. Then you get something like Dempster-Shafer theory.

In short, there are alternatives.

As for why should the plausibility of the negation of a statement depend only on the plausibility of the statement, the answer (I believe) is that we are considering only the sorts of propositions in which the Law of the Excluded Middle holds. So if we are using only a single real number to capture plausibility, we need f{(A|X),(!A|X)} = (truth|X) = constant, and we have no freedom to let (!A|X) depend on the details of A.

Well, an alternative way to suggest probability is the Right Way is stuff like dutch book arguments, or more generally, building up decision theory, and epistemic probabilities get generated "along the way"

The arguments I like are of the form that each step is basically along the lines of "if you don't follow this rule, you'll be vulnerable to that kind of stupid behavrior, where stupid behavior basically means 'wasting resources without making progress toward fulfilling your goals, whatever they are'"

Frankly, I also like those arguments because mathematicaly, they're cleaner. Each step gets you something of the final result, and generally doesn't require anything more demanding than basic linear algeabra.

It's nice to know Cox's Theorem is there, but it's not what, to me at least, would be a simple clean derivation.

Psy-Kosh,

Curiously, I have just the opposite orientation -- I like the fact that probability theory can be derived as an extension of logic totally independently of decision theory. Cox's Theorem also does a good job on the "punishing stupid behavior" front. If someone demonstrate a system that disagrees with Bayesian probability theory, when you find a Bayesian solution you can go back to the desiderata and say which one is being violated. But on the math front, I got nothing -- there's no getting around the fact that functional equations are tougher than linear algebra.

Cyan: I certainly admit that the ease of the math may be part of my reaction. Maybe if I was far more familiar with the theory of functional equations I'd find Cox's theorem more elegant than I do.

(I've read that if one makes a minor tweak to Cox's theorem, just letting go of the real number criteria and letting confidences be complex numbers, the same line of derivation more or less hands you quantum amplitudes. I haven't seen that derivation though, but if that's correct, it makes Cox's theorem even more appealing. QM for "free"! :))

The vulnerabiliy ones though actively motivate the criteria rather than a list of reasonable sounding properties an extention to boolean logic ought to have. The basic criteria, I guess, could be describes as "will you end up in a situation in which you'd knowingly willingly waste resources without in any way benefiting your goals?"

So each step is basically a "Mathematical Karma is going to get you and take away your pennies if you don't follow this rule." :)

But yeah, the bit about only reasonable extention of vanilla logic does make the Cox thing a bit more appealing. On the other hand, that may actively dissuade some from the Bayesian perspective. Specifically, constructivists, intuitionists in particular, for instance, may be hesitant of anything too dependant on law of excluded middle in the abstract. (This isn't an abstract hypothetical. I've basically ended up in a friendly argument a while back with someone that more or less had them rejecting the notion of using probability as a measure of belief/subjective uncertainty because it was an extention of boolean logic, and the guy didn't like law of excluded middle and basically was, near as I can make out, an intuitionist. (Some of the mathematical concepts he was bringing up seem to imply that))

Personally, I just really like the whole "math karma" flavor of each step of a vulnerability argument. Just a different flavor than most mathematical derivations for, well, anything, that I've seen. Not to mention, in some formulations, getting decision theory all at once with it.

I am having a bit of trouble with this series. I can see that you are explaining that reality consists of states with "amplitude" numbers assigned to each state.

1. You seem to assign arbitrary numbers to the initial states and an arbitrary amplitude change rules to mirrors. Why is this in any way applicable to objective reality? Or are these numbers non-arbitrary? Or am I just missing something elementary?
2. Why states of photons or detectors are complex numbers and mirror is a function?
3. How does time factor into all of this?

I am sorry. I should have read the rest of the series BEFORE starting to ask questions about this particular article. Please disregard my previous post.

Thanks for helping me remove the classic hallucination to now understand the double slit phenomenon.

I thought I understood quantum mechanics. I have studied it for years. Passed exams. Got a degree.

This is the first time I have ever heard of the HOM experiment, and it is causing a crisis in my mind, since it does not match what I know about quantum mechanics.

They really should at least tell us of this experiment and its results when they start teaching us. Even after these years I was still under the impression that quantum mechanics was about unbreakable limitations on the measurement of data.

This experiment proves that the universe itself does not work in an intuitive way. All the people who are trying to outwit quantum mechanics by measuring position and energy at the same time are not just attempting something impossible, what they are trying doesn't even make sense.

AAAAAAAAAAAGH!

I found the calculation of the amplitude flows for cases 1 to 4 confusing at first. I think the part I missed is that the reflection of a photon at a half silvered mirror multiplies the configuration (ie the state of both photons) by i. So in case 1 we get each photon multiplied by i twice, so the amplitude at both detectors is 1.