On 1/5/2025 3:21 PM, Jesse Mazer wrote:
On Sun, Jan 5, 2025 at 5:35 PM Bruce Kellett <[email protected]>
wrote:
On Mon, Jan 6, 2025 at 9:14 AM Jesse Mazer <[email protected]>
wrote:
On Sun, Jan 5, 2025 at 12:44 AM Bruce Kellett
<[email protected]> wrote:
On Sun, Jan 5, 2025 at 7:46 AM John Clark
<[email protected]> wrote:
*About a month ago Sean Carroll uploaded a very good
video explaining the Many Worlds theory, but it's over
an hour long so I know there's about as much chance of
a dilettante such as yourself of actually watching it
is there is of you reading a post of mine if it's
longer than about 100 words. *
*
*
*The Many Worlds of Quantum Mechanics | Dr. Sean
Carroll
<https://www.youtube.com/watch?v=FTmxIUz21bo&t=8s> *
I watched this video, but it is not as comprehensive as
Carroll's book "Something Deeply Hidden".
However, something came up in the question period that
might warrant a comment. Talking about the Born rule,
Carroll justifies it by saying that if you measure the
spin of 1000 unpolarized particles, you get 2^1000
different UP-DOWN sequences. However, the vast majority of
these sequences will show proportions of UP vs DOWN close
to the Born rule prediction of 50/50. In the limit, if
such a limit makes sense, the proportion of sequences that
show marked deviations from the Born Rule proportions will
form a set of measure zero, and can be ignored.
That is just the law of large numbers at work, and is all
very well if the amplitudes are such that the Born
probabilities are equal to 0.5. But it is easy to rotate
your S-G magnets so that the Born probabilities are quite
different, say, 0.9-Up to 0.1-DOWN. Now take 1000 trials
again. According to Everett, you necessarily get the same
2^1000 sequences of UP-DOWN that you had before. The law
of large numbers will then tell you that the majority of
these will have approximately a 50/50 UP/DOWN split, which
is grossly in violation of the Born rule result of a 90/10
split. In other words, MWI. or Everettian QM. has a
problem reproducing the Born rule. It works in the simple
case of equal probabilities, but fails miserably once one
departs substantially from equal probabilities.
Bruce
David Z Albert mentions that if you define a measurement
operator that just tells you the *fraction* of spin-up vs.
spin-down in a large sequence of identical measurements, then
even without any collapse assumption, in the limit as #
measurements goes to infinity the wavefunction will approach
an eigenstate of this operator that matches the probability
that would be predicted by the Born rule. See his comments on
p. 238 of The Cosmos of Science at
https://books.google.com/books?id=_HgF3wfADJIC&lpg=PP1&pg=PA238#v=onepage&q&f=false
<https://books.google.com/books?id=_HgF3wfADJIC&lpg=PP1&pg=PA238#v=onepage&q&f=false>
"Then, even though there will actually be no matter of fact
about what h takes the outcomes of any of those measurements
to be, nonetheless, as the number of those measurements which
have already been carried out goes to infinity, the state of
the world will approach (not as a merely probabilistic limit,
but as a well-defined mathematical epsilon-and-delta-type
limit) a state in which the reports of h about the statistical
frequency of any particular outcome of those measurements will
be perfectly definite, and also perfectly in accord with the
standard quantum mechanical predictions about what the
frequency out to be."
But then Albert goes on to say that there are all sorts of reasons
why this simple theory cannot be the answer to the origin of the
Born rule. I have pointed out one of the most cogent of these. If
you perform similar measurements on N identically prepared systems
(say z-spin measurements on systems prepared in an x-spin-left
state), then according to Everett, you get all 2^N possible
sequences of UP/DOWN spins. This exhausts the possibilities for
the outcome of N trials, and, significantly, you must get exactly
the same 2^N sequences whatever the amplitudes of the initial
superposition might be. So you get these 2^N sequences if the
amplitudes are equal, and also if the amplitudes are in the ratio
0.9/0.1. This behaviour is incompatible with the Born rule, and
hence with ordinary quantum mechanics.
You do get all these sequences but this tells us nothing about what
their relative probabilities/frequencies are. I assume as an extension
of his analysis, if we did repeated experiments where on each trial we
performed exactly N measurements and this was repeated over many
trials (approaching infinity), then you could define a measurement
operator that would tell you the fraction with any specific N-sequence
(for example, for N=3 there would be an operator giving the fraction
of trials with result 000, likewise other operators for 001 and 010
and 011 and 100 and 101 and 110 and 111). If you had a setup where the
relative probability of these sequences was not uniform according to
the Born rule,
But there's the catch. The relative probability of those sequences
corresponds to p=0.5 no matter what p is. In order that they
instantiate the true value of p there must be an axiom that requires
they satisfy the Born rule.
then if the number of trials with that setup goes to infinity, it will
presumably likewise be true that the state approaches the eigenstate
of this operator with the frequency predicted by the Born rule,
without ever actually invoking the Born rule.
That's only the case for p=0.5, which as I read it Albert gratuitously
generalizes.
Albert would presumably say that this still doesn't resolve the
measurement problem because it doesn't give an outcome on any
particular trial, only a sort of aggregate over many trials, but this
is different from the criticism you are making. Even if we do use the
Born rule in the above scenario, it's still true that each of the
specific outcomes that are possible for a given trial with N
measurements (eg the outcomes 000, 001, 010, 011, 100, 101, 110, and
111) will occur in the long term, but that doesn't mean they are
equiprobable.
They are not, and that's ASSUMING the Born rule.
Brent
Jesse
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