What Everett says in his thesis is that if the measure is additive between orthogonal states, it’s the norm squared. Therefore we should use the norm squared of observers when deciding in how to weight their observations.
But this is a weird argument, not at all the usual sort of argument used to pin down probabilities—the archetypal probability arguments rely on things like ignorance and symmetry. Everett just says “Well, if we put a measure on observers that doesn’t have weird cross-state interactions, it’s the norm squared.” But understanding why humans described by the Schrodinger equation wouldn’t see weird cross-state probability flows still requires additional thought (that’s a bit hard in the non-Hamiltonian-eigenstate observer + environment basis Everett uses for convenience).
But I think that that’s an argument you can make in terms of things like ignorance and symmetry, so I do think the problem is somewhat solved. But it’s not necessarily easy to understand or widespread, and the intervening decades have had more than a little muddying of the waters from all sides, from non-physicist philosophers to David Deutsch.
I don’t totally understand the Liouville’s theorem argument, but I think it’s aimed at a more subtle point about choosing the common-sense measure for the underlying Hilbert space.
What Everett says in his thesis is that if the measure is additive between orthogonal states, it’s the norm squared. Therefore we should use the norm squared of observers when deciding in how to weight their observations.
But this is a weird argument, not at all the usual sort of argument used to pin down probabilities—the archetypal probability arguments rely on things like ignorance and symmetry. Everett just says “Well, if we put a measure on observers that doesn’t have weird cross-state interactions, it’s the norm squared.” But understanding why humans described by the Schrodinger equation wouldn’t see weird cross-state probability flows still requires additional thought (that’s a bit hard in the non-Hamiltonian-eigenstate observer + environment basis Everett uses for convenience).
But I think that that’s an argument you can make in terms of things like ignorance and symmetry, so I do think the problem is somewhat solved. But it’s not necessarily easy to understand or widespread, and the intervening decades have had more than a little muddying of the waters from all sides, from non-physicist philosophers to David Deutsch.
I don’t totally understand the Liouville’s theorem argument, but I think it’s aimed at a more subtle point about choosing the common-sense measure for the underlying Hilbert space.