Pierz wrote:
On Tuesday, August 11, 2015 at 6:08:31 PM UTC+10, Bruce wrote:
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.
That's the first objection one hears to MWI, but Deutsch's version of
MWI makes that argument redundant. It's not that there is one universe
per branch - that leads to the absurdity you describe. Rather there is
an infinity of identical universes which differentiates into 'stacks' of
differing size infinities. Deutsch explicitly repudiates the Identity of
Indiscernibles, and I'm personally fine with that. The Identity of
Indiscernibles seems to make sense logically, but it's a philosophical
idea that predates QM by a few centuries. Leibniz could not have
foreseen quantum logic either! I'm much more troubled by the basis problem.
One could take that line. But it smacks of desperation to me, because
there is nothing to support that in the formalism of QM. However, you
are quite right in that it sinks on the basis problem. It is all very
well to postulate an infinity of universes -- but according to which
basis? Do you have an uncountable infinity of uncountable infinities of
universes? Rather a high price to pay for a probabilistic theory.
And even then, you have not addressed the circularity problem that
attempted derivations of the Born rule encounter. Unless you have an
independent account of probabilities, you can't even talk about "worlds"
in any differentiated sense. In any case, the identity of indiscernibles
is well supported by QM itself -- the statistics of identical particles
depend on that principle.
Bruce
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