Mitchell Porter sent this only to me, by accident, and asked me to forward
it to the list:

> From: Mitchell Porter <[EMAIL PROTECTED]>
> Message-Id: <[EMAIL PROTECTED]>
> Subject: Re: many worlds interpretation
> To: [EMAIL PROTECTED] (Hal Finney)
> Date: Thu, 12 Feb 1998 22:23:51 +1000 (EST)
> In-Reply-To: <[EMAIL PROTECTED]> from "Hal Finney" at Feb 
>11, 98 10:02:13 pm
>
> > 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".
>
> So in that case, what does this weighting function (which is derived
> from the UWF, rather than already present in it) actually denote?
> You refer to probabilities. If all the basis states are held to exist,
> then what probabilities are there to talk about?
>
> It's like saying "Mars exists, and Earth exists, and the probability
> of Mars is 3/4." It doesn't make any sense.
>
> > > 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.
>
> Sure, the Schroedinger equation is part of the overall theory (unless
> we try to do 'timeless' quantum cosmology, in which case there's
> just a Hamiltonian constraint, H|psi> =0). So I should have said
> "the universal wavefunction, at any time, will have a representation...",
> and II then becomes the claim that the states of the many worlds
> must be found within the instantaneous states of the UWF.
>
> > > 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.
>
> It is, I think, the exact definition of 'relative state' (although
> again, those sums could be integrals).
>
> > 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 meant was this: suppose the universal Hilbert space has the form
> H_A X H_B X H_C X ..., where H_A, H_B, H_C,... are themselves Hilbert spaces.
> (A, B, C,... are the subsystems.) Then the UWF is of the form
>
> PSI = sum(i_A,i_B,i_C,...) c_(i_A,i_B,...) |i_A> |i_B> |i_C> ...
>
> where i_A is a state from H_A, etc.
>
> Suppose we want to look at the correlation between A and B given
> that the UWF is in state PSI. We project PSI into H_A x H_B, and obtain
> the state
>
> sum(i_A,i_B) d_(i_A,i_B) |i_A> |i_B>
>
> where 
>
> d_(i_A,i_B) = sum(i_C,i_D,...) c_(i_A,i_B,i_C,i_D,...)
>
> The correlation between A and B is a function of the magnitude of
> the d's. For example, if the d's are all large for i_A = i_B, and
> small for everything else, then A and B are highly correlated,
> since (roughly speaking) knowing the A-state allows you to
> predict the B-state with high probability. Or, phrasing it in
> terms of relative states, the relative state of B, given a
> certain A-state, will be strongly peaked at a particular |i_B>.
>
> Everett quantitatively defines correlation early on in his
> thesis, in the section on information. I will confess that I
> haven't sat down and compared his definitions to those of the
> decoherence theorist Wojciech Zurek. But it is certainly my
> impression that a decoherent state is one in which the correlation
> between two systems, in the sense of 'correlation' defined by
> Everett, is very low.
>
> You ask
>
> > 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?
>
> Do you mean decohere "internally", or decohere from us, or from
> the rest of the universe...?
>
> > 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.
>
> There is a difference between "branches of the wavefunction" and
> "relative states" if we are still talking about a total wavefunction.
> Consider the product system A X B. Branches of the wavefunction
> of the total system will be elements of the Hilbert space H_A X H_B,
> but the relative states will be elements of H_A or of H_B.
>
> Let's say that A is a spin-1/2 particle and B is everything else,
> and that the universe is in a state in which A has decohered from B.
> The state of the universe will be of the form
>
> sum(i) (c_1i |up>|i_B> + c_2i |down>|i_B>)
>
> where the |i_B>'s form a basis for H_B. For this state to be
> decoherent, it must be the case that the relative state of A,
> given a B-state of |i_B>, is one in which c_1 (the coefficient
> of |up>) and c_2 (the coefficient of |down>) are of similar 
> magnitude - otherwise there would be a correlation
> between the A-state and the B-state.
>
> Now, when you refer to "that branch of the wavefunction where
> the spin is up", what do you mean? Do you mean |up>, which is
> an element of H_A, or do you mean sum(i) c_1i |up>|i_B>,
> which is an element of H_A X H_B?
>
> In any case, under neither interpretation do I see a way to
> talk about "the correlation between that branch of the wave function 
> where the particle is spin up and that branch where it is spin down".  
> Correlation (or entanglement) is a relationship between different 
> subsystems, not between different states of the same subsystem.
>
> > 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.
>
> But the universe as a whole doesn't have relative states. Only
> subsystems have relative states.
>
> > 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?
>
> Well, we seem to have different ideas about the meaning of decoherence.
> If I were to translate your prescription for knowing when worlds are
> splitting into terms I could agree with, it would have to be, the
> world splits when some subsystem decoheres from everything else.
> The problem with this is that decoherence is a matter of degree (the
> 'correlation'), whereas splitting is surely and all-or-nothing
> process - otherwise the notion of distinct worlds is meaningless.
>
> I'm glad we still have some degree of mutual understanding. I just
> hope you can make something of my formalisms above - if we can
> reach an agreement there, we'll be getting somewhere.
>
> -mitch
> http://www.thehub.com.au/~mitch

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