But this doesn't mean there are actually three possible ways the universe could be, etc. That's just a trick for visualizing the path integral. All the amplitude flows actually happen, they are not possibilities.

Corroborated by this peer-reviewed article, published two days ago in Physical Review Letters:

Is a System’s Wave Function in One-to-One Correspondence with Its Elements of Reality?

Although quantum mechanics is one of our most successful physical theories, there has been a long-standing debate about the interpretation of the wave function—the central object of the theory. Two prominent views are that (i) it corresponds to an element of reality, i.e., an objective attribute that exists before measurement, and (ii) it is a subjective state of knowledge about some underlying reality.

A recent result [M. F. Pusey, J. Barrett, and T. Rudolph] has placed the subjective interpretation into doubt, showing that it would contradict certain physically plausible assumptions, in particular, that multiple systems can be prepared such that their elements of reality are uncorrelated. Here we show, based only on the assumption that measurement settings can be chosen freely, that a system’s wave function is in one-to-one correspondence with its elements of reality. This also eliminates the possibility that it can be interpreted subjectively.

Link: Physical Review Letters 108, 11. April 2012.

So an initially factorizable distribution, evolved into an "entangled system"—a joint amplitude distribution that is not viewable as a product of distinct factors over subspaces.

Actually, that's still factorizable. You just need to change basis to the diagonal. Effectively, you're treating them as one particle. The other dimension is their relative motion, and 'being a particle' corresponds to one value in that dimension.

The ability to make this transformation is why we can speak of protons and neutrons and atoms and quasiparticles. And bricks.

I found some unanswered questions, and despite they are 1 and 4 years old, I'll try to answer them, because someone might have the same question now (just as I did, in another comment).

(Psy-Kosh) I'm still confused as to why the individual blobs would tend to be more factorable. Why would they factorize easily post decoherence?

Intuitively, factorizing is translating a description to a set of shorter, independent descriptions. I will give a mathematical, non-physical, analogy. Imagine that you have a knowledge "X=3 and Y=5". You can translate it into two shorter pieces of knowledge: "X=3", "Y=5". Together they mean the same thing as the original knowledge. But it allows you speak about X while ignoring Y. (Quantum analogy: It allows to to speak about one particle, while ignoring the rest of the universe.)

More complex example: "either (X=2 and Y=5) or (X=2 and Y=6) or (X=3 and Y=5) or (X=3 and Y=6)". Fortunately, this can be factorized into "either X=2 or X=3", "either Y=5 or Y=6". Two independent knowledges. Now assume that you only care about X and ignore Y, and your colleague only cares about Y and ignores X. Later, your colleague discovers that in fact Y=5. Is this information useful for you? Absolutely not.

Another example: "either (X=2 and Y=5) or (X=3 and Y=6)". This knowledge cannot be factorized. If your colleague later discovers that in fact Y=5, it helps you know that X=2.

Now imagine parallel universes, in an old-fashioned sci-fi meaning, not quantum mechanical meaning. You have information "either (X=2 and Y=5 and we live in universe U1) or (X=3 and Y=6 and we live in universe U2)". This information, as it is, is not factorable. But if you know which universe you live in, the rest of it becomes factorable.

Quantum analogy: replace "we live in universe U1" with "my brain (which also consists of elementary particles) is in state B1". If your brain can execute the algorithm "state B1 believes that X=2 and Y=5; state B2 believes that X=3 and Y=6", then you are thinking about the particles as if the situation is easy to factorize. And it's not only about state of your brain, but also about the state of the whole universe.

(DavidAgain) I'm not clear on why the amplitude involving more particles means that they're further apart in configuration space.

Imagine a line of length 1; the distance between its ends is 1. Imagine a square of size 1; the distance between its opposite corners is 1.4. Imagine a cube of size 1; the distance between its opposite edges is 1.7. Add more dimensions, and the distance of the opposite edges increases.

Or more precisely, imagine an N-dimensional cube for very large N (something like number of elementary particles in the situation). The distance between two edges becomes greater if they differ in more coordinates. Point (0, 0, 0) is closer to point (0, 0, 1) than to point (1, 1, 1). If you can't imagine it, just calculate the distance by formula distance((x0, y0, z0), (x1, y1, z1)) = sqrt((x1 - x0)^2 + (y1 - y0)^2 + (z1 - z0)^2). The more coordinates differ, the greater the resulting distance.

If instead we'd started out with a big light-gray square—meaning that both particles had amplitude-factors widely spread—then the second law of thermodynamics would prohibit the combined system from developing into a tight dark-gray diagonal line.

In the model we suppose that the particles are attracted to each other. So what exactly does this sentence mean -- that if both particles have wide amplitude spread, they somehow stop attracting each other?

I would expect that even if we start with a system "a positive article can be anywhere, a negative particle may be anywhere", it would develop towards "positive and negative particles anywhere, but probably close to each other". Wouldn't that make a dark-gray line?

Please help.

EDIT: Found this.