Eli: It seems like it would be much better to use the original name “relative state” rather than “many worlds”. The word “many” suggests that they can be counted. However, in standard QM we are usually talking about particles whizzing around in the continuum, which gives us an infinite-dimensional Hilbert space. If we restrict ourselves to Hilbert spaces of finite dimension, for example the states of some spins, then naively counting worlds remains bogus, because the number of “worlds” (i.e. entries of the state vector) with nonzero amplitude depends entirely on choice of basis. I suppose in a finite dimensional Hilbert space we could make a sensible definition of world counting as follows: the answer to how many worlds am I in is the rank of my reduced density matrix. However, this seems far removed from the main point of the “MWI”. Furthermore, it appears that the term many worlds does actually lead people astray in practice. In the posts many people keep referring to counting the worlds in which something happens in order to assess probability. This is wrong. The probabilities arise from squaring amplitudes, not from counting. If the probabilities arose from counting then in a finite dimensional Hilbert space, all the probabilities would be rational numbers. Standard QM does not have this property.
Eli: It seems like it would be much better to use the original name “relative state” rather than “many worlds”. The word “many” suggests that they can be counted. However, in standard QM we are usually talking about particles whizzing around in the continuum, which gives us an infinite-dimensional Hilbert space. If we restrict ourselves to Hilbert spaces of finite dimension, for example the states of some spins, then naively counting worlds remains bogus, because the number of “worlds” (i.e. entries of the state vector) with nonzero amplitude depends entirely on choice of basis. I suppose in a finite dimensional Hilbert space we could make a sensible definition of world counting as follows: the answer to how many worlds am I in is the rank of my reduced density matrix. However, this seems far removed from the main point of the “MWI”. Furthermore, it appears that the term many worlds does actually lead people astray in practice. In the posts many people keep referring to counting the worlds in which something happens in order to assess probability. This is wrong. The probabilities arise from squaring amplitudes, not from counting. If the probabilities arose from counting then in a finite dimensional Hilbert space, all the probabilities would be rational numbers. Standard QM does not have this property.