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.
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.