Branch counting feels like it makes sense because it feels like the particular branch shouldn’t matter, i.e. that there’s a permutation symmetry between branches under which the information available to the agent remains invariant.
But you have to actually check that the symmetry is there, which of course, it isn’t. The symmetry that is there is the ESP one, and it provides the correct result. Now I’ll admit that it would be more satisfying to have the ESP explicitly spelled out as a transformation group under which the information available to the agent remains invariant.
You can’t actually presume that… The relevant quantum concept is the “spectrum” of an observable. These are the possible values that a property can take (eigenvalues of the corresponding operator). An observable can have a finite number of allowed eigenvalues (e.g. spin of a particle), a countably infinite number (e.g. energy levels of an oscillator), or it can have a continuous spectrum, e.g. position of a free particle. But the latter case causes problems for the usual quantum axioms, which involve a Hilbert space with a countably infinite number of dimensions—there aren’t enough dimensions to represent an uncountable number of distinct position eigenstates. You have to add extra structure to include them, and concrete applications always involve integrals over continua of these generalized eigenstates, so one might reasonably suppose that the “ontological basis” with respect to which branching is defined is something countable. In fact, I don’t remember ever seeing a many-worlds ontological interpretation of the generalized eigenstates or the formalism that deals with them (e.g. rigged Hilbert space).
In any case, the counterpart of branch counting for a continuum is simply integration. If you really did have uncountably many branches, you would just need a measure. The really difficult case may actually be when you have a countably infinite number of branches, because there’s no uniform measure in that case (I suppose you could use literal infinitesimals, the equivalent of “1/alephzero”).
I explored my skepticism about this paper in a brief dialogue with Claude…
Branch counting feels like it makes sense because it feels like the particular branch shouldn’t matter, i.e. that there’s a permutation symmetry between branches under which the information available to the agent remains invariant.
But you have to actually check that the symmetry is there, which of course, it isn’t. The symmetry that is there is the ESP one, and it provides the correct result. Now I’ll admit that it would be more satisfying to have the ESP explicitly spelled out as a transformation group under which the information available to the agent remains invariant.
Branch counting stops making sense when there are uncountably many branches, and there are (presumably).
You can’t actually presume that… The relevant quantum concept is the “spectrum” of an observable. These are the possible values that a property can take (eigenvalues of the corresponding operator). An observable can have a finite number of allowed eigenvalues (e.g. spin of a particle), a countably infinite number (e.g. energy levels of an oscillator), or it can have a continuous spectrum, e.g. position of a free particle. But the latter case causes problems for the usual quantum axioms, which involve a Hilbert space with a countably infinite number of dimensions—there aren’t enough dimensions to represent an uncountable number of distinct position eigenstates. You have to add extra structure to include them, and concrete applications always involve integrals over continua of these generalized eigenstates, so one might reasonably suppose that the “ontological basis” with respect to which branching is defined is something countable. In fact, I don’t remember ever seeing a many-worlds ontological interpretation of the generalized eigenstates or the formalism that deals with them (e.g. rigged Hilbert space).
In any case, the counterpart of branch counting for a continuum is simply integration. If you really did have uncountably many branches, you would just need a measure. The really difficult case may actually be when you have a countably infinite number of branches, because there’s no uniform measure in that case (I suppose you could use literal infinitesimals, the equivalent of “1/alephzero”).