The short answer is yes, Many Worlds is falsifiable. For example, right now
there are experiments underway in an attempt to prove that the GRW theory
of objective quantum wave collapse makes predictions that Many Worlds does
not, if they are successful it will prove that Everett was dead wrong, it's
as simple as that.  GRW claims that Schrodinger's equation is incomplete
and that another very complex term needs to be added to it because it's the
only way they could think of to get rid of all those worlds that for some
reason they dislike, there was simply no other reason to add that extra
term. With this new term Schrodinger's equation is no longer completely
deterministic because a random element is added such that the larger the
wave function is (the more particles it has) the more likely the quantum
wave function will objectively collapse. They carefully tuned their very
complex extra term inserted into Schrödinger's equation in just such a way
that, because an individual electron is so small the probability of you
being able to observe one objectively collapse is almost but not quite
zero; but the probability of you NOT observing something as large as a
baseball NOT collapsing is also almost, but not quite, zero. Despite heroic
efforts. up to the present day nobody has found a speck of experimental
evidence in support of the GRW theory of objective quantum wave collapse,
and until and unless they do Many Worlds must be the preferred theory
according to Occam's razor because it makes fewer assumptions, it has no
need to complicate matters by adding that extra term to Schrodinger's
equation.

But GRW is not the only or even the most popular competitor to Many Worlds,
that honor would have to go to the Copenhagen interpretation, and there is
certainly no way to falsify that, but back in 1986 in his book "The Ghost
in the Atom" David Deutsch proposed another way to falsify Everett's Many
Worlds; the experiment would be difficult to perform but Deutsch argues
that is not Many Worlds fault, the reason it's so difficult is that the
conventional view says conscious observers obey different laws of physics,
Many Worlds says they do not, so to test who's right we need a mind that
uses quantum properties.

In Deutsch's experiment, to prove or disprove the existence of many worlds
other than this one, a conscious quantum computer shoots electrons at a
metal plate that has 2 small slits in it. It does this one at a time. The
quantum computer has detectors near each slit so it knows which slit the
various electrons went through. The quantum mind now signs a document for
each and every electron saying it has observed the electron and knows which
slit it went through. It is very important that the document does NOT say
which slit the electron went through, it only says that it went through one
and only one slit and the mind has knowledge of which one. Now just before
the electron hits the plate the mind uses quantum erasure to completely
destroy the memory of what slits the electrons went through, but all other
memories including all the documents remain undamaged. After the document
is signed the electron continues on its way and hits the photographic
plate. Then after thousands of electrons have been observed and all
which-way information has been erased, develop the photographic plate and
look at it. If you see interference bands then the Many World
interpretation is correct. If you do not see interference bands then there
are no worlds but this one and the conventional interpretation is correct.

Deutsch is saying that in the Copenhagen interpretation when the results of
a measurement enters the consciousness of an observer the wave function
collapses, in effect all the universes except one disappear without a trace
so you get no interference. In the Many Worlds model all the other worlds
will converge back into one universe when the electrons hit the
photographic film because the two universes will no longer be different
(even though they had different histories), but their influence will still
be felt. In the merged universe you'll see indications that the electron
went through slot X only and indications that it went through slot Y only,
and that's what causes interference.

I know that what I said in the above is a fair representation of what
Deutsch was saying because some years ago I wrote to him about this and he
said it was an accurate paraphrase.

It must be admitted that like every theory Many Worlds makes predictions
that cannot be tested, but a theory is not judged on the basis of what
predictions it makes that have neither been confirmed nor falsified
experimentally,  instead they are judged by how well they conform to
experiments that HAVE been performed, and in Many Worlds  case it conforms to
every physics experiment that has ever been made up to the present day. Yes
Everett's idea produces a lot of worlds, but Occam does NOT say the best
theory is the one that produces the simplest outcome, the best theory is
the one that makes the fewest assumptions and still agrees with
experimental observations.

John K Clark    See what's on my new list at  Extropolis
<https://groups.google.com/g/extropolis>
fwm

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