- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

From: Hal Finney <hal.domain.name.hidden>

Date: Mon, 16 Feb 1998 19:20:55 -0800

Mitchell Porter sent this only to me, by accident, and asked me to forward

it to the list:

*> From: Mitchell Porter <mitch.domain.name.hidden>
*

*> Message-Id: <199802121223.WAA09131.domain.name.hidden>
*

*> Subject: Re: many worlds interpretation
*

*> To: hal.domain.name.hidden (Hal Finney)
*

*> Date: Thu, 12 Feb 1998 22:23:51 +1000 (EST)
*

*> In-Reply-To: <199802120602.WAA09202.domain.name.hidden> from "Hal Finney" at Feb 11, 98 10:02:13 pm
*

*>
*

*> > Mitchell Porter, <mitch.domain.name.hidden>, 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
*

Received on Mon Feb 16 1998 - 21:12:25 PST

Date: Mon, 16 Feb 1998 19:20:55 -0800

Mitchell Porter sent this only to me, by accident, and asked me to forward

it to the list:

Received on Mon Feb 16 1998 - 21:12:25 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:06 PST
*