> First off, how is it that the MWI does not imply
>quantum immortality?
MWI is just quantum mechanics without the wavefunction collapse postulate.
This then implies that after a measurement your wavefuntion will be in a
superposition of the states corresponding to definite outcomes. But we
cannot just consider suicide experiments and then say that just because
branches of the wavefuntion exist in which I survive, I'll find myself there
with 100% probability. The fact that probabilities are conserved follows
from unitary time evolution. If a state evolves into a linear combination of
states in which I'm dead and alive then the probabilities of all these
states add up to 1. The probability of finding myself to be alive at all
after the experiment is then less than the probability of me finding myself
about to perform the suicide experiment.
The probability of me finding myself to be alive after n suicide experiments
decays exponentially with n. Therefore I should not expect to find myself
having survived many suicide experiments. Note that contrary to what you
often read in the popular accounts of the multiverse, the multiverse does
not split when we make observations. The most natural state for the entire
multiverse is just an eigenstate of the Hamiltonian. The energy can be taken
to be zero, therefore the wavefunction of the multiverse satisfies the
equation:
H|psi> = 0
And |psi> is thus stationary. We can say that the multiverse is timeless.
Now, we find ourselves to be in some sector of the multiverse that is
clearly not stationary. For any person, you can consider an operator (let's
call it A) that will answer the question: Is this person present in our
universe? This operator has two eigenvalues: 0 (the person is not present),
1 (the person is present). You can then split the Hilbert space into two
sectors that are mapped to zero and 1 respectively (the two eigenspaces of
A). If the person is present, you can ask more questions about the person.
For each question there is an operator. The simultaneous eigenstates of all
these operators forms a basis of the sector where the person is present. The
basis vectors specify everything there is to know about the person (and
about the rest of the universe).
Let's denote these basis states by |b1>, |b2>, ....|bn>
The probability for the person to be alive at all is:
P_alive = Sum over k of |<psi|bk>|^2
Suppose that you are in state |b1>. The a priori probablity for this is P1 =
|<psi|b1>|^2. Since the multiverse does not evolve in time:
<psi|exp(-i/hbar H t) = <psi|
So:
<psi|b1> = <psi|exp(-i/hbar H t)|b1>
exp(-i/hbar H t)|b1> will be some linear combination of the |bk>'s and
states from the sector in which the person is dead:
exp(-i/hbar H t)|b1> = sum over k of ck |bk> + |dead>
where |dead> satisfies A|dead> = 0
The best way to think about exp(-i/hbar H t)|b1> is that it is just a
unitary mapping that happens to give the subjective time an observer will
experience in the multiverse which in reality is static. If |b1> happens to
describe the person about to enter a suicide machine, then
|<dead|exp(-i/hbar H t)|b1>|^2 will converge to 1 quite fast. One minus this
probablity is the sum of the |ck|^2 and gives the probability that the
person finds himself alive in a state in which he remebers stepping in the
suicide machine a time t ago. So, that probability tends to zero very fast
as t goes to infinity.
--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to everything-list.domain.name.hidden
To unsubscribe from this group, send email to everything-list-unsubscribe.domain.name.hidden
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en
-~----------~----~----~----~------~----~------~--~---
Received on Tue Apr 15 2008 - 20:22:32 PDT