On 14/12/2017 8:23 am, smitra wrote:
On 12-12-2017 23:13, Bruce Kellett wrote:
On 13/12/2017 2:12 am, smitra wrote:
On 12-12-2017 12:33, Bruce Kellett wrote:
On 12/12/2017 9:46 pm, smitra wrote:
Yes, it's only an estimation but it yields a good order of
magnitude estimate for the center of mass. What the calculation
shows is that quantum superpositions do exists at the macroscopic
level and these can then be amplified by chaotic dynamics. Of
course, it then becomes incoherent, but in the MWI that's besides
the point.
MWI splitting depends on coherence, so it is certainly not beside the
point for the coin toss.
It doesn't depend on coherence. Why would it matter if the state of
the coin gets entangled with a zillion other environmental degrees
of freedom? The dynamics according to unitary time evolution leads
toa superposition, no matter how many degrees of freedom are
involved in the entanglement.
You are missing the point. Splitting according to the Schrödinger
equation does depend on coherence. The decoherence that entangles the
coin with a zillion other environmental degrees of freedom occurs
after the splitting. Given decoherence, the process is irreversible
FAPP, which means that there is no practical way, by design or chance,
that a decohered state can recohere. Sure, in the many worlds of MWI
the superposition, if it once existed, is still intact. But if no such
superposition ever existed, then it can't be created from non-coherent
interactions.
So Schrödinger's cat was once a coherent state of a cat in a box, and
the splitting occurs with the decay of a nucleus; decoherent
entanglement then leads to the splitting of worlds FAPP. But given an
arbitrary coin, it is already non-coherently entangled with many
environmental degrees of freedom, but there is no state that can lead
to {heads>+|tails>} in a unitary manner, so there is no state that can
then evolve into a splitting and decoherence into worlds distinguished
by either |heads> or |tails>. If you think that there is, write out
the schematic sequence of states evolving under the SE that leads to
this result.
So, let's examine this more closely. We start with a state that is a
superposition of branches that each undergo classical evolution, so
I'm not now appealing to the arguments in the paper by Albrecht. Then
we end up with a state of the form:
Sum over j [|heads(j)>|env(heads,j)> +
Sum over j |tails(j)>|env(tails,j)>]
where j enumerates microstates of the coin and the state of the
environment in each sector where the coin is heads or tails depends on
the microstate of the coin. The coin states are not normalized, the
norms are chosen to yield the correct probabilities. E.g. if tails
has zero probability then all the |tail(j)> are given a norm of zero.
So, the above expression is completely general.
Now, I'm part of the environment, so we can write:
|env(X,j)> = Sum over k of |me(X,Y,j,k)>|env'(X,Y,j,k)>
where X is heads or tails, Y, denotes my macrostate that I'll define
below, j denotes the microstate of the coin and k sums over
microstates corresponding to macrostate Y.
This splitting of my state into a macrostate Y and microstate k is in
principle arbitrary, we can choose this splitting such that Y keep
tracts of what I'm aware of and then k takes into account all the
other degrees of freedom in my body and brain that I'm not aware of.
E.g. if I were a digital computer then Y would correspond to some
bitstring defined by the computational state of the computer and we
then sum over all the possible microstates that correspond to some
fixed macrostate.
Then when I'm not yet aware of the result of the coin throw, we have:
|me(X,Y,j,k)> = |Awareness(Y)>|body(X,Y,j,k)>
where Y contains all the information that I am aware of, and that
doesn't include the result if the coin throw nor information contained
only at the microscopic scale. So, Awareness (Y) for a given Y defines
who I am, where I am , and what I'm experiencing.
It should be clear that despite decoherence, |Awareness(Y)> will
factor out of the global superposition as it doesn't depend on X, k
and j.
So, given what I know, I cannot tell in which branch I am. Therefore
unless all the branches corresponding to one outcome have zero norm,
I will have copies in both branches.
There is just no way that all the information describing what I'm
aware of at some moment is going to constrain how someone else can
throw a coin to such a degree that the outcome will be fixed.
So the end point of your analysis is that you can't see how it could be
possible that you are not right? The outcome is fixed -- and the same --
in all worlds in which you participate!
But it is quite easy to see what is going on. Since it is all
independent of your awareness, we can factor that out of the summations.
And you still do not get a coherent superposition from a decoherent
mixture. What goes on in the world is independent of whether or not the
rock on the floor is aware of it or not. You are not in a position any
different from that of said rock. Look at the expansion for
Schrödinger's cat:
|nucleus>|box>|cat>you>|environment> -->
{|decayed>|poison spilt>|cat dead>|You>|environment records dead cat>
+ |undecayed>|poison intact>|cat alive>|You>|environment records live
cat>}
You can see that your state does not change (is irrelevant) until you
interact with the environment in such a way that your consciousness
becomes entangled with the state of the cat. This is not mysterious, and
it is not changed just because you claim there is some magic associated
with consciousness that makes a difference. You are just reverting to
Copenhagen or many minds where consciousness is necessary for the
understanding of QM.
Bruce
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