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. 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. 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. 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? 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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