--- On Sun, 2/8/09, russell standish <li...@hpcoders.com.au> wrote:
> He must have some model in mind which tells us how
> the "amplitude" of the branches relates to the "amplitude" of the
> original state.
The Schrodinger equation is linear and unitary. As long as it applies (in
other words, assuming the MWI, so no collapse) the norm of a "branch" remains
equal to the norm of whatever term in the original wavefunction evolved to form
that branch.
In other words, in the MWI,
|psi> = a|A> + b|B> evolves to |psi'> = a'|A'> + b'|B'>
where |a| = |a'| and |b| = |b'|
and conventionally we assume <psi|psi> = 1.
If <A|B> = 0, as for a measurement, then |a|^2 = <psi| |A><A| |psi>
Now if |A'> and |B'> are decoherent branches, the Born rule states that the
probability for branch |A> is |a|^2 = |a'|^2. |a'|^2 is the squared norm of
the branch, and is more instructive for the MWer to talk about than |a|^2.
The norm of each branch "world" is no longer 1, as the collapse interpretation
would have set it to. Conceptually, in the MWI only the wavefunction of the
entire multiverse should really be normalized to 1 (or to whatever). But for
convenience, whenever we start an experiment, we renormalize what we started
with to 1 and throw out the rest of the branches from consideration.
Incidentally, I think that could be the reason QM is linear: Maybe the real
physics is not linear, but since the amplitude of each branch is so small (the
average of the squared norms is decreasing with time as the number of branches
increases), the higher order terms quickly became negligable.
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