M1: The worlds split over a short period of time starting as soon as the atom enters the field. They don’t split very far. In the Copenhagen interpretation no measurement occurs (EDIT: until they hit the screen) (you can tell because later on we’ll “unmeasure” this alleged measurement, which would be impossible if the wavefunction had collapsed.). EDIT: The worlds don’t split very far while the atoms are in the field (they only split in one direction). When they hit the screen they suddenly split a lot further. In Copenhagen this would constitute a measurement.
M2: The atoms become unaligned again. EDIT: You can’t necessarily tell this apart from 50⁄50 aligned unaligned. If I could be bothered I’d calculate the density matrices. The situations are distinguishable iff the density matrices are different. My intuition says they’re the same.
M3: Two blobs. No interference. EDIT: Because the spins are different.
M4: The detector answers soon after the atom enters the field. It will of course end up agreeing with the light formed by the atom hitting the screen. EDIT: In particular the acceleration would vary with time exactly as if the atom were classical.
M1: The atom has a wavefunction, a function from {up,down} x R^3 to C. We can view this as two spacial wavefunctions (i.e. from R^3 to C) one for spin up, one for spin down. Before entering the field the up and down wave functions are the same, and localised in space (i.e. all the amplitude is in a small lump around a point). This lump of amplitude moves along until it reaches the field. At this point the two wavefunctions cease to be the same (the spin of the particle becomes entangled with its position). The one associated with spin up translates upward, its velocity increasing just like a particle in a classical field. Similarly the spin down wavefunction moves downward. The time that the worlds take to split is the time until the two lumps (corresponding to up and down) cease to overlap spatially. I don’t view the “worlds” as ontologically fundamental, only the wavefunction, so the previous sentence is close to tautology. If we allow the wavefunction to have thin tails off to infinity then the lumps never truly split, but they do still mostly separate.
Since the two lumps haven’t separated very far (say at most a few cms, or however big the S-G apparatus is) and the atom hasn’t entangled with anything else, it will be easy to remove the entanglement between spin and position by reversing the field. This is what I mean by “the worlds aren’t very far apart”. To formalise the notion of distance I suppose one could take the root of the sum of the squares of the displacements of every particle in the universe between the two worlds. So in this case the distance between worlds would be the literal distance between the two lumps of amplitude.
Once they hit the screen we suddenly have that lots of particle’s positions differ between the two worlds, and so the distance between them becomes very great. This is decoherence.
M2: In the experiment given, any single particle is essentially returned to exactly the same superposition of states it was in before it entered the fields. Exactly like neither of the fields was ever there. (Also, I hold that I can still use density matrices to deal with my subjective uncertainty, even about single particles).
At this point the two wavefunctions cease to be the same (the spin of the particle becomes entangled with its position).
So it seems that you define the degree of separation of worlds as the spatial overlap of the spin-up and spin-down components of the wave function, probably the inner product of the two normalized terms. I did not follow your musings on “displacements of every particle in the universe between the two worlds”, however.
M2: In the experiment given, any single particle is essentially returned to exactly the same superposition of states it was in before it entered the fields.
Are you saying that the screen with the two holes has no effect whatsoever? Just wondering.
(Also, I hold that I can still use density matrices to deal with my subjective uncertainty, even about single particles).
You sure can, but it seems unnecessary for a pure state. Or maybe I misunderstand what you mean by “subjective uncertainty”.
M1: The worlds split over a short period of time starting as soon as the atom enters the field. They don’t split very far. In the Copenhagen interpretation no measurement occurs (EDIT: until they hit the screen) (you can tell because later on we’ll “unmeasure” this alleged measurement, which would be impossible if the wavefunction had collapsed.). EDIT: The worlds don’t split very far while the atoms are in the field (they only split in one direction). When they hit the screen they suddenly split a lot further. In Copenhagen this would constitute a measurement.
M2: The atoms become unaligned again. EDIT: You can’t necessarily tell this apart from 50⁄50 aligned unaligned. If I could be bothered I’d calculate the density matrices. The situations are distinguishable iff the density matrices are different. My intuition says they’re the same.
M3: Two blobs. No interference. EDIT: Because the spins are different.
M4: The detector answers soon after the atom enters the field. It will of course end up agreeing with the light formed by the atom hitting the screen. EDIT: In particular the acceleration would vary with time exactly as if the atom were classical.
Neither statement is clear to me. How short a time? How do you define the distance between worlds?
Re M2: Assume you are dealing with a single atom, so it’s a pure state, no need for density matrices. What would be your answer?
M1: The atom has a wavefunction, a function from {up,down} x R^3 to C. We can view this as two spacial wavefunctions (i.e. from R^3 to C) one for spin up, one for spin down. Before entering the field the up and down wave functions are the same, and localised in space (i.e. all the amplitude is in a small lump around a point). This lump of amplitude moves along until it reaches the field. At this point the two wavefunctions cease to be the same (the spin of the particle becomes entangled with its position). The one associated with spin up translates upward, its velocity increasing just like a particle in a classical field. Similarly the spin down wavefunction moves downward. The time that the worlds take to split is the time until the two lumps (corresponding to up and down) cease to overlap spatially. I don’t view the “worlds” as ontologically fundamental, only the wavefunction, so the previous sentence is close to tautology. If we allow the wavefunction to have thin tails off to infinity then the lumps never truly split, but they do still mostly separate.
Since the two lumps haven’t separated very far (say at most a few cms, or however big the S-G apparatus is) and the atom hasn’t entangled with anything else, it will be easy to remove the entanglement between spin and position by reversing the field. This is what I mean by “the worlds aren’t very far apart”. To formalise the notion of distance I suppose one could take the root of the sum of the squares of the displacements of every particle in the universe between the two worlds. So in this case the distance between worlds would be the literal distance between the two lumps of amplitude.
Once they hit the screen we suddenly have that lots of particle’s positions differ between the two worlds, and so the distance between them becomes very great. This is decoherence.
M2: In the experiment given, any single particle is essentially returned to exactly the same superposition of states it was in before it entered the fields. Exactly like neither of the fields was ever there. (Also, I hold that I can still use density matrices to deal with my subjective uncertainty, even about single particles).
So it seems that you define the degree of separation of worlds as the spatial overlap of the spin-up and spin-down components of the wave function, probably the inner product of the two normalized terms. I did not follow your musings on “displacements of every particle in the universe between the two worlds”, however.
Are you saying that the screen with the two holes has no effect whatsoever? Just wondering.
You sure can, but it seems unnecessary for a pure state. Or maybe I misunderstand what you mean by “subjective uncertainty”.