Wei's question reminded me that I never responded to this.  I was not
able to find the reference that I was looking for so my answer will be
a bit sketchy.

Mitchell Porter, <[EMAIL PROTECTED]>, writes:
> Okay, can we agree on the following:
>
> I. The universal wavefunction (UWF) will have some representation 
> of the form
>
> sum(i) c_i |i>,                               (1)
>
> where the |i>s are a set of basis states.

One thing to keep in mind is that this may be an integral over an infinite
number of base states.  That is actually how Everett wrote it.  This is
useful especially when analyzing the question of why the probabilities
work as they do.  In your web article you mention that some people have
considered the discrete case (like when there are only two basis states)
and thought that the solution is that there should be multiple copies
of some of the basis states ("worlds").

I don't like this solution because it does not generalize well if there
are an infinite number of basis states.  Instead in that case what you
need is to have a weighting function or norm over the infinite number
of states, where the function represents the probability of falling
into that state.  This function is in fact the square of the amplitude,
but there is no simple (and misleading, IMO) interpretation in terms of
multiple instances of some "worlds".

> II. A complete description of each of the many worlds must be 
> found somewhere in (1), for the many-world interpretation 
> to live up to its name.

You need more than (1), presumably; at a minimum you need the Schrodinger
equation which controls how the state function evolves based on the
physical situation.  It is controversial whether you also need to invoke
a norm like I mentioned earlier, or whether that can be deduced from the
other parts of the theory.

> III. For the concept of 'relative state' to make sense, we
> must be able to attach at least two labels to the basis
> states, so that the UWF is written as
>
> sum(i,j) c_ij |i;j>                   (2)
>
> Then we can say that the relative state of subsystem j,
> given that subsystem i is in state X, is
>
> sum(j) c_Xj |j>

This seems like a good description for what we have in a typical
measurement interaction such as Everett is discussing.

Let me skip ahead a bit to the main point:

> I think what *is* true is that you can start with a pure-state
> density matrix for a coupled system, and the density matrix
> of a subsystem will evolve from a pure state into a mixture.
>
> Back at the level of wavefunctions, decoherence does not
> explain how observables come to have definite values. All
> decoherence does is reduce the correlation between subsystems;
> a decoherent state is still a superposition, it's just a
> superposition with very little entanglement.

But, do you agree that in practice, systems such as the kind we see
around us will decohere and evolve into a state composed of multiple
relative states which have little entanglement?  I'm confused by your
phrase "reduce the correlation between subsystems".  Above in III you
used the term "subsystems" to refer to physically separate systems which
are interacting, like a quantum particle and a macroscopic device which
measures its state.  But I don't think you are using it to mean the same
thing here.

What I hope, because then I would both understand you and agree, is that
"subsystems" here means separate branches of the wave function, separate
relative states.  Decoherence reduces the correlation between that branch
of the wave function where the particle is spin up and that branch where
it is spin down.  The wave function is still a superposition of the two,
but there is little entanglement between the two states.

If this is what you mean, then the next step is pretty straightforward.
You look at the state of the universe relative to each of these decoherent
base states, and consider that these relative states are effectivelly
independent, hence they are different worlds in the MWI.

Therefore the recipe for knowing when and how worlds split is to look
at when a new set of basis vectors can be found which are mutually
decoherent.  This can in principle be derived solely from the Schrodinger
equation.  Therefore the "topology" of the universe, the manner in which
it can be said to split into many worlds, does in fact follow from this
basic equation which is at the core of QM.

The issue of probablity is another matter, but if we restrict ourselves
solely to the question of when and how worlds split, I believe the MWI
does answer this question pretty clearly.

You earlier asked about a simple harmonic oscillator and asked when it
would split.  The answer, based on this reasoning, is that it does not
split, because in such a simple system there is no decoherence which
occurs.  If you had a large number of oscillators, though, coupled in
some complicated way, then you might well see decoherence and therefore
splitting.

How much of this do you agree with?

Hal

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