Pierz wrote:
Thanks Bruce, that actually makes a lot of sense ... and kind of
completely trashes my previous understanding! It also makes QM weirder,
and even makes me doubt MWI, which reading Deutsch had convinced me was
the true account.
MWI is popular, but it is not without its problems.
As an aside, quantum probabilities as given by the Born Rule do not
come from branch counting -- the probability is not a measure
of the underlying universe count.
Then where do they come from? Without that notion of a measure, MWI
seems not to be telling us all that much.
That just the question! Deutsch and Wallace have devoted a lot of energy
trying to derive the Born rule from MWI, plus something like rational
decision theory. These attempts have been widely criticized, and I agree
with the criticism that the approach is basically circular: they get the
Born rule only by assuming that small amplitudes correspond to small
probabilities -- which is just another way of expressing the Born rule.
.......
Sorry can you clarify why branch counting is ruled out on this, ah,
basis? Do you mean that there are an uncountable infinity of possible
values and therefore you can't count anything? Actually this was going
to be another part of my original question - surely "worlds" have to at
least be countable, even if infinite? How much sense does a "continuum
of worlds" make?
If space and time are continuous, there are an infinite number of
possible eigenvalues for position and momentum. So you need to consider
an infinite number of branches for most real-valued probabilities. And
the reals form an uncountable infinity.
I think there are some technical arguments against branch counting that
I can't call to mind at the moment, but I think simple arguments
suffice. Consider the two-valued spin case. We only ever have at most
two components to the wave function, whatever the basis. If the
coefficients are such that one result has probability 1/3, and the other
2/3, branch counting would require that there be two branches giving one
result with only one for the other result. Where did that extra branch
come from? It is not in the formalism. And since it is necessarily
identical to one of the other branches, the identity of indiscernibles
would say that we only have two distinct branches. In which case, all
probabilities would equal 0.5, whatever the measurement angle. And this
is absurd.
Bruce
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