How Many Worlds?

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New Comment

64 comments, sorted by Click to highlight new comments since: Today at 3:41 AM

How many ink blots are in this picture ?

For "many worlds", imagine that evolving in time, with fuzzy borders.

The Many Worlds Interpretation doesn't imply a countable number of worlds that suddenly branch, it's more like a fuzzy continuum; talking of branching worlds and timelines is just a high-level abstraction that makes things easier to discuss.

How many ink blots are in this picture?

This is a *really* good analogy to explain what a wrong question is.

Thanks for answering!

I guess I was confused by this:

What about the Ebborians? The Ebborians, you recall, have brains like flat sheets of conducting polymer, and when they reproduce, the brain-sheet splits down its thickness. In the beginning, there is definitely one brain; in the end, there is definitely two brains; in between, there is a continuous decrease of causal influence and synchronization. When does one Ebborian become two?

Those who insist on an objective population count in a decoherent universe, must confront exactly analogous people-splitting problems in classical physics!

Heck, you could simulate quantum physics the way we currently think it works, and ask exactly the same question! At the beginning there is one blob, at the end there are two blobs, in this universe we have constructed. So when does the consciousness split, if you think there's an objective answer to that?

When you mix regions of stability and chaos, eventually things settle down into relatively discrete zones... along some directions... while being smeared out all over the place in others.

Edited to add: Are these downvotes from people who know quantum mechanics or dynamics in phase space, or is my comment just making people confused again (a bad thing to be sure), or what? I can probably fix it, but it'd be best to know what about it needs fixing.

No idea why whoever downvoted you did so, but here’s why I think I felt your comment was not useful to me, or much less useful of what it could have been if you happen to know what you’re talking about (I don’t so I can’t tell):

Your statement states a fact without any explanation, examples or pointers to such. If you had said something like “When you mix regions of stability and chaos, things never settle down into discrete zones... it’s all smeared out all over the place.” — then the effect of reading it would pretty much have been the same unless I already knew about the subject enough not to need your comment.

Imagine someone not having *any* education in astronomy saying something like “I thought the sun and stars turn around the Earth”, and you commenting “Actually, the Earth spins around itself, and it turns around the sun, while the other stars pretty much go every which way.” Unless the first person knew you were a good astronomer, they don’t really learn anything. And even if they did believe you knew you to be an expert on what you were talking about, they might learn it as a rote fact, but won’t really understand much.

So... your comment was an attempt at “done easy”, or was it “tough”?

(It occurs to me that the line above would be normally interpreted as snarky. My intent was half friendly joke, half “if you have that opinion about Chaos theory, what did you try to achieve in your earlier comment?” I just don’t know how to express that in written English...)

So the key idea is that of "Hilbert space," which is a way to describe the universe named after a guy called Hilbert.

So for example if I flip a fair quantum coin, it's 0.5 heads and 0.5 tails. "Heads" and "tails" here are actually *dimensions,* like x and y, in Hilbert space, and the universe is at the point (0.5, 0.5). If the coin wasn't fair, then the universe could be at the point (0.6, 0.4) or even (0.999, 0.001). The number of dimensions didn't change, because there's still just heads and tails, but the point that represents our universe changed.

When you look at the coin, the universe collapses to two possible points: (0,1) and (1,0). The coin is either heads or tails. This corresponds to two "worlds." It doesn't matter whether, previously, your description was fair or not - the coin is still either heads or tails, so there are two worlds. Though I suppose if your previous description was (1,0) - definitely heads - you wouldn't assign any probability to it being tails, so there would only one "world".

Of course, it can get much more complicated. If you roll a quantum d20 instead of flipping a coin, you have to assign a point with 20 coordinates: (0.05,0.05,0.05,0.05,0.05,0.05, 0.05,0.05,0.05,0.05,0.05,0.05, 0.05,0.05,0.05,0.05,0.05,0.05,0.05,0.05). And if you throw a dart at a continuous dartboard, you have to assign a value to an infinite number of points! But don't worry - that's just the same as a function, like x^2 or sin(x). But if you throw a dart at a dartboard, does that mean you just split off an infinite number of worlds? If you flip a coin, and *then* throw the dart, is that 2*infinity = infinity?

So basically, when there are lots of possible outcomes the idea of "worlds" becomes not so useful.

When you look at the coin, the universe collapses to two possible points: (0,1) and (1,0).

Although this gives the correct answer as far as the number of worlds is concerned, it sounds strange within the MWI (which was supposedly assumed in the original question).

I think you can't do MWI justice without introducing the observer in the game: apart from the coin which lives in a two-dimensional universe, there is the observer whose mental state lives in a (at least) three-dimensional universe. The dimensions of the observer's mind are "think the coin landed heads (TH)", "think the coin landed tails (TT)" and "don't know (DN)". Together we have six dimensions, all combinations of coin and observer states:

- coin:heads, observer:TH
- coin:heads, observer:TT
- coin:heads, observer:DN
- coin:tails, observer:TH
- coin:tails, observer:TT
- coin:tails, observer:DN

In this space there are three planes defined by the observer's mental states: for example, the plane TT consists of vectors that have all coordinates except the second and fifth equal to zero. The observer's consciousness has a peculiar property of seeing only projections to these planes. Those projections are what is called *worlds*.

In the beginning, the observer doesn't know and the coin is 50% heads and 50% tails; this means the state vector of our model "universe" is (0, 0, 0.707, 0, 0, 0.707). (Have I mentioned that the probabilities aren't in fact the coordinates but their squares? Anyway, this is a technicality we don't really need now, but we should be consistent. The state vector must have lenght precisely 1.) At this moment, the projections to the aforementioned planes are

- DN: (0, 0, 0.707, 0, 0, 0.707)
- TH: (0, 0, 0, 0, 0, 0)
- TT: (0, 0, 0, 0, 0, 0)

In a sense, three "worlds" already exist, but since two of them have zero length, they can be disregarded.

Now the observer measures (looks at) the coin. Measurements are mysterious processes which, over some time, get the observer into correlation with the coin. The universe state vector becomes (0.707, 0, 0, 0, 0.707, 0) and the projections are now

- DN: (0, 0, 0, 0, 0, 0)
- TH: (0.707, 0, 0, 0, 0, 0)
- TT: (0, 0, 0, 0, 0.707, 0)

Now we have two non-zero projections and can speak about two worlds. But remember, there is still only one six-dimensional Hilbert space with one universe state vector. It is believed that under normal circumstances no processes can put the state vector back to the state where there are less non-zero projections than before. But in principle it could happen and if it does, the worlds would merge again.

Congratulations, you just earned yourself one "click." I've never really gotten quantum physics, not that I've tried much. But your description as a Hilbert space makes a lot of sense to me. It also helps me understand why "decomposing the wavefunction" is important or even necessary as a concept.

I'm surprised that this has not been said, so I'll present the way I think about branching, though it will be a bit heavy on the mathematics and I apologize for that. Perhaps someone else can pare it down a bit. Also, I am not a physicist, I am a mathematician, so my model is probably more optimized for making me feel like quantum mechanics describes a world in the abstract, and less optimized for describing the specific world we live in.

In the Schrodinger's cat experiment, we have a vast number of elementary objects, which are essentially all wave functions. If we consider the reduced density matrices, the set of all possible reduced density matrices is... well naively I might guess that it is the n^2-fold product of the unit interval, where n is the dimension of the matrix, but it's also possible that the space of reduced density matrices is some other lie group (if it turns out the space is NOT a lie group, this interpretation is in serious trouble!). Either way, there is a Haar measure on it; which is to say we can, in some sense, have a continuous space of all the elements of that group. Now conceptually I'll consider each of the coefficients of these matrices as sort of like a probability. Now I construct one universe for every member of the direct product (maybe direct sum?) of the group, indexed by the set of wave functions in my experiment, and I call this set of universes the "branches which causally descend from my circumstance" because that sentence makes me feel warm and fuzzy. In each of those universes, each wave function expresses itself as though it collapsed in one direction if the coefficients of the matrix indexed by that wave function are greater than the reduced density matrix of that wave function, and the other if they are less. The fact that I don't know what should happen if the coefficients are equal bothers me, but this isn't really a good expression of the "directions" that a matrix can "collapse in" so I will guess that there is a better formulation that a physicist could make that resolves this issue, and if I am convinced there isn't, I'll start reading up on quantum physics for the purpose of sleeping better at night.

The naive picture that I had that I tried to comb out into a real model here is that there are a bunch of continuous probabilities (intervals [0,1]) which resolve as either 0 or 1. So the number of universes coming out should be indexed by intervals [0,1] for every probability in the situation, with that universe coming out with a 0 if the value in the index is less than the probability and a 1 if it is greater. I suppose, since the big deal here is measure, that you could arbitrarily assign the equality case here to either side and it would never make a difference.

I'll reiterate at this point that I'm a mathematician, not a physicist. This is what's gone on in my head as an explanation for what it REALLY MEANS to have many worlds. I would love to hear a physicist's perspective on why this is all complete nonsense.

How many universes "branch off" from a "quantum event", and in how many of them is the cat dead vs alive, and what about non-50/50 scenarios, and please answer so that a physics dummy can maybe kind of understand?

(Is it just 1 with the live cat and 1 with the dead one?)