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.
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.
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.