I'm extremely interested, and the question isn't meant as anything but an inquiry. Finding a gap in one's knowledge is one of the more exciting aspects of life.
Amplitudes are complex numbers, out there in the real world (as far as we understand it). This is quite different from probabilities, who are real numbers between 0 and 1 who exist only as a mental constructs. Simply put, while amplitudes share some mathematical properties with probabilities, they are not probabilities.
Now that we're done turning colloquial words into misleading jargon, we can talk about the theory itself. Namely Everett's many-worlds and decoherence.
We could play the "where's the electron" game, but I'd rather play "where's the photon" instead —it's simpler. So you throw a photon through a half-sieved mirror, to be detected by one of to judiciously placed detectors. Oh, and have one detector linked to a kitten murdering system for good measure.
If you repeat the experiment often enough, you will witness a kitten death half the time, with absolutely no way to predict the outcome in advance. The results are the same if you put the kitten in a box, and open that box after the fact. So it certainly looks like the universe is not deterministic.
The equations on the other hand are definitely deterministic. Future amplitude distributions are perfectly predicted by past amplitude distributions —which by the way you can't fully observe, but that's another issue entirely. So, if you look at the amplitude distribution, you'll see that once the photon hit the mirror, there will be a blob of amplitude for both cases: passing through and being reflected. Going further, there will be a blob of amplitude for each of the detectors being hit. Finally, there will be a blob of amplitude for the living kitten, and another one for the dead kitten. Oh, and the equations also says that the blobs quickly cease to interact —that's decoherence.
Basically, what the equations say is that the universe splits itself in two, generating one version with the dead kitten, and one version with the live kitten. The equations also say that the inhabitants of either version don't get to see the other one (they've ceased to interact).
Now the only question left is why we experimentally find ourselves to be in one version and not in another. But never forget that every time you run that Schrödinger experiment, a cat will die. If not in your universe, then the other.
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The Copenhagen interpretation, which would have the blob of amplitude corresponding to the other universe just collapse into nothingness (that is, set to zero), has no basis in the equations which by now are backed up by mountains of evidence. It is an additional hypothesis layered on top of the equations, conveniently formulated in a way that wouldn't falsify any experiment. On top of that, it violates a number of long standing principles, such as locality.
You could also call those amplitudes "probabilities", but that's just a word trick. It doesn't explain anything.
My understanding of (real) probabilities is that they're really about our ignorance of a system. E.g., if I'm flipping a coin but put very little spin on it, it's easy for a human to predict what side will show. As we put more spin on it, the probability of guessing correctly goes down, but that's about perceptual and cognitive limitations.
My (layman's) understanding of quantum mechanics, though, is that there are no hidden variables. In which case, I couldn't understand why anybody would call the quantum stuff "probability". It's a relief to know that they're fundamentally different.
But the fact remains that blindly following the equations leads you to Many-Words. You have to modify the results to get to any sort of collapse hypotheses. Also, simply postulating that what you don't see doesn't exist doesn't help. http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/
Pretend that you're talking to someone who doesn't see many-worlds as an obvious corollary of the current QM theories. What experiments do you propose to indicate the simultaneous existence of e.g. a universe in which the photon chose Slit A and another in which the photon chose Slit B?
Your appeal to BitII fails because that concerns e.g. conservation laws that are regularly observed to be true, and which therefore can be assumed true when a particular situation makes observation impossible. Many-worlds has never been shown by experiment, so it is not eligible for such treatment.
We already agree that there is no experimental difference between collapse/Copehagen interpretation, and the Many Worlds interpretation. In the name of what are you demanding experimental evidence for one interpretation, and not the other?
> Infinite uncountable infinities of universes is not parsimonious.
You need to remember that those universes are in no way postulated. They are derived —from the equations. So you can't use that impressive infinity to claim that the Many World interpretation would somehow have a higher Kolmogorov complexity than Copenhagen.
Many Worlds just takes the equations and run with them. It's Copenhagen that makes additional assumptions by manipulating the results of the equations: They're not real, or there's a collapse… Either way, that's an additional hypothesis on top of those equations (which by the way have massive amounts of experimental evidence behind them, and are accepted by everyone as the current best guess). That additional hypothesis is not very parsimonious, don't you think?
(You have to remember how Occam's razor really works. From the Wikipedia, "The principle states that among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected." As a simple matter of fact, Many Worlds makes strictly fewer assumptions than Copenhagen. Parsimony is not measured by the size of the universe predicted by the theory. Parsimony is measured by the number of core assumptions. Not the same thing at all.)
To get a feel of how utterly ridiculous collapse postulates are (in 20/20 hindsight, I don't want to make fun of physicists), I must quote Eliezer Yudkowsky: http://lesswrong.com/lw/q6/collapse_postulates/
If collapse actually worked the way its adherents say it does, it would be:
1. The only non-linear evolution in all of quantum mechanics.
2. The only non-unitary evolution in all of quantum mechanics.
3. The only non-differentiable (in fact, discontinuous) phenomenon in all of quantum mechanics.
4. The only phenomenon in all of quantum mechanics that is non-local in the configuration space.
5. The only phenomenon in all of physics that violates CPT symmetry.
6. The only phenomenon in all of physics that violates Liouville's Theorem (has a many-to-one mapping from initial conditions to outcomes).
7. The only phenomenon in all of physics that is acausal / non-deterministic / inherently random.
8. The only phenomenon in all of physics that is non-local in spacetime and propagates an influence faster than light.
WHAT DOES THE GOD-DAMNED COLLAPSE POSTULATE HAVE TO DO FOR PHYSICISTS TO REJECT IT? KILL A GOD-DAMNED PUPPY?
If I were to stipulate that the Copenhagen interpretation(s) is (are) silly philosophical daydreaming, could you do the same for many-worlds?
Funny how different people perceive his writings. To me, most of what he writes feels obvious —at least in retrospect. "Obvious" doesn't feel very mystic to me. I do understand however how many people would be distrustful of his casual writing style. He tends to sound like a lowly blogger, not like a respectable academic. (Personally, I don't care for those status signals.)
> I'll read that eventually, but I won't be surprised if it turns out to be like much of the rest of LW
Hmm, if you're already familiar with this material, then don't bother just yet. Start with the first few chapters of E.T. Jayne's Probability Theory: the Logic of Science. That's more basic, less crazy sounding, and more generally applicable.
Now I don't exactly know how physicists deal with quantum mechanics. I asked one, and he didn't even bother with any interpretation, sticking with the observable consequences of the equations (a more prudent attitude than either Copenhagen or Many-World).
Something however bothers me deeply: insisting on calling amplitudes "probabilities", while they're anything but. That only makes teaching harder. Seriously, I was tempted to mass delete every occurrences of "probability" from the "Probability Amplitude" article in the Wikipedia.
> If I were to stipulate that the Copenhagen interpretation(s) is (are) silly philosophical daydreaming, could you do the same for many-worlds?
Err, it doesn't work like that. If I were to convince you, I would have learned nothing, and believe then what I believe now. I can only give you my current best guess.
Which is, many-worlds is by no means certain. At the very least we don't have a Theory of Everything, and we could miss something. Collapse postulates however are just crazy. Rejecting the existence of the amplitude you didn't observe is just as insane (no less, no more) as rejecting the existence of a photon which just passed the limits of our observable universe.
When we do find a theory of everything, I bet the equations will predict the existence of things that can't be observed —not even in theory. I just hope people won't see that as a licence to not believe in those things at all.
Other unobservable stuff may or may not be incredibly important. (Though at a first glance, QM interpretation isn't.)
Most convenient reference:
> In quantum mechanics, a probability amplitude is a complex number ...
https://en.wikipedia.org/wiki/Probability_amplitude#A_basic_...
I don't understand this part: "Now the only question left is why we experimentally find ourselves to be in one version and not in another"
Surely there are different 'we' observing in both?
From the inside however, we still have the Born probabilities to contend with: apparently, our subjective probability to observe one version or another is tied to the square root of the relevant amplitudes. In the case of a half-sieved mirror, that's one chance in two. But we can easily bias this with a stronger (or weaker) mirror.