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.
