On 11/26/2019 7:51 PM, Bruce Kellett wrote:
A standard objection to the many-worlds interpretation of quantum
mechanics concerns energy conservation. When the universe splits on
some quantum event and a new branch(world) is created, where does the
energy come from?
Sean Carroll tackles this question on page 173 of his new book. But I
am not convinced that he gives a convincing answer. Basically, he says
that since the universe as a whole evolves according to the
Schrödinger equation, this unitary evolution conserves energy. He goes
on:
"Not all worlds are created equal. Think about the wave function. When
it describes multiple branched worlds, we can calculate the total
amount of energy by adding up the amount of energy in each world,
times the weight (the amplitude squared) for that world. When one
world divides in two, the energy in each world is basically the same
as it previously was in the single world (as far as anyone living in
it is concerned), but their contributions to the total energy of the
wave function of the universe have divided in half, since their
amplitudes have decreased. Each world got a bit thinner, although its
inhabitants can't tell any difference."
I see some problems here. One is that the total number of branches in
the branching wave function is continually increasing, and the number
of branches is not well defined -- indefinite even if not actually
infinite.
For radioactive decay the split is a continuum of "events".
So the energy in each branch is effectively zero, unless we
renormalize or something on each split. The second worry is that taken
at fact value, multiplying the energy by the weight of each branch on
a split would mean that if we have a Stern-Gerlach measurement of
spin, or a photon on a half silvered mirror, the weights of each of
the two new branches is one half, so the energy of the photon that is
reflected off my half-silvered mirror should be one-half the energy of
the incident photon. The other half of the energy has gone to the
photon (in another world) that was transmitted. This is not what is
seen, and contradicts the assertion that energy is conserved in each
branch.
Is it? If you measured the momentum change of the mirror due to the
photon, you would find it was either zero (transmitted) or h/f
(reflected). Of course you would also have welcher weg information.
If new branches are continually forming out of any branch, there is no
way the energy could be conserved without it being obvious to the
observer of the photon incident on the half-silvered mirror. (Or is
any other quantum interaction.) As any world branches, energy cannot
be conserved without it being obvious along any decohered history.
Carroll given another example; "I have, say, a bowling ball, with a
certain mass and potential energy. But then someone in the next room
observes a quantum spin and branches the wave function.
The other room? How about Alpha Centari? Or another galaxy? In fact
the lesson the C60 buckyball experiment is that there doesn't have to be
anyone measuring anything. All that's needed is decoherence into the
environment and you and the rest of the universe have split.
Now there are two bowling balls, each of which has the energy of the
previous one. No?" He answers: "That ignores the amplitudes of the
branches. The contribution of the bowling ball to the energy of the
universe isn't just the mass and the potential energy of the ball;
it's that, times the weight of its branch of the wave function. After
the splitting it looks like you have two bowling balls, but together
they contribute exactly as much to the energy of the wave function as
the single bowling ball did before."
Clearly, when the split is due to a quantum event in another room,
you are not aware of the split and of the sudden reduction of the
mass-energy of everything around you. So you could get away with that
by a simple renormalization. But if you are observing the atom in the
S-G magnet, how does this approach avoid the conclusion that you would
have to see its energy halve?
How you would see it. The act of observing it splits you too.
Brent
I do not think that locutions about the energy of the wave function of
the universe being conserved, and branches decreasing in energy by
their Born weights, are actually going to avoid the problem of
accounting for energy conservation, as observed in each continuing
branch.
It seems to me that the best one can do is say that energy is
conserved in each branch, even over splitting. That is, after all,
what is observed. Consequently, the energy of the overall wave
function is not conserved. This might cause some problems for the
insistence on unitary evolution of the wave function as a
whole...........
Bruce
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