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.
