From: Hal Finney (hal@finney.org)
Date: Fri Jul 04 2003 - 12:52:12 MDT
Robin writes:
> Damien's critic is right; reproducing the Born probabilities is a serious
> problem with many worlds. In a stochastic reduction theory, you can just
> posit the probabilities; you haven't explained them, but at least they
> don't contradict anything. In many worlds, you have a straightforward way
> to calculate probabilities, namely counting worlds, that gives a
> *different* answer, which is a much more serious problem. You can deal
> with this problem by positing an infinity of "minds" per "world", which
> then split during measurements due to some unknown process. Or you can
> state decision theory axioms that declare that we do not care about
> counting worlds. Neither of these is very satisfactory in my opinion.
I think the decision theory approach is relatively promising. I don't
know that the axioms really declare that we don't care about counting
worlds. In the end, it's true, we conclude that rationally we should
act as though the standard QM "Born" probabilities hold and ignore
the simple mechanism of counting worlds. That is a deduction from the
axioms and is therefore implicit in them. But the axioms are intended
to present a plausible definition of what constitutes rational behavior.
They don't start right off declaring that we ignore world counts.
A simpler approach just occured to me. Let's consider a photon which is
emitted in a polarized state and encounters a polarizer tilted to give
a 10% chance of passing. As Damien describes, if each photon splits
the world into two, then the fraction of worlds that see the 10/90
probabilities becomes vanishingly small.
But does it really split like this? In focusing on just the absorption
aspect of the experiment, we are ignoring many other quantum mechanical
variables. That's appropriate for experimental purposes, but not,
perhaps, for philosophy.
The photon emission itself is a quantum mechanical process. It was
emitted by an excited electron dropping to a lower orbital. This quantum
transition is more correctly modeled as a continuous process. The excited
atom emits a constant, steady-state photon wave, and at some point the
wave function collapses when we measure the photon as a particle.
Therefore even a single emission and polarization-measurement of a photon
does not actually split the universe into two parts; it splits it into
an infinite number of parts. Even if it turns out that photon emission
is not truly continuous, that it can only happen at multiples of the
Planck time or whatever, that's still a truly enormous number of parts.
And the point is that among this humongous split, 10% of the universes
will see the photon pass through the polarizer, and 90% will see it
be absorbed. Given this reality, a simple counting rule does in fact
reproduce the Born probabilities. It was only because we abstracted away
the enormous additional complexity of the world in order to focus on the
polarizer interaction that we thought the universe split into two just
two branches.
Now, this is of course just one specific example. I don't know if it can
be generalized, or if we could create quantum systems which literally
have exactly two states, with no additional parameters or dimensions
that can fatten things up and give us the Born rules. But at least I
think this shows that these simple counting arguments are more suspect
than they seem.
Hal
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