On Sun, Jan 5, 2025 at 6:36 PM Bruce Kellett <bhkellet...@gmail.com>
wrote:
On Mon, Jan 6, 2025 at 10:21 AM Jesse Mazer <laserma...@gmail.com>
wrote:
On Sun, Jan 5, 2025 at 5:35 PM Bruce Kellett
<bhkellet...@gmail.com> wrote:
On Mon, Jan 6, 2025 at 9:14 AM Jesse Mazer
<laserma...@gmail.com> wrote:
On Sun, Jan 5, 2025 at 12:44 AM Bruce Kellett
<bhkellet...@gmail.com> wrote:
On Sun, Jan 5, 2025 at 7:46 AM John Clark
<johnkcl...@gmail.com> 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, 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.
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.
The trouble that I have pointed to is that if every possible
outcome occurs for each measurement, then the sequences are all
present whatever the amplitudes in the wavefunction. so the
sequences 000...,001...,010...,100...,011...,... etc are all
equiprobable, whatever the wave function.
How do you get from "all present" to "equiprobable"? If you flip a
pair of coins enough times you will surely get all the sequences HH,
HT, TH and TT, but you could easily be using weighted coins that don't
have 50/50 chances of landing heads and tails, in which case those 4
sequences won't occur equally often.
Thus, the operator that Albert talks about does not give the
relative probabilities for each sequence.
But he's talking very generally about any sort of operator that gives
relative frequencies of different results of quantum experiments, so
do you disagree that it's plausible this could be generalized to an
operator that gives frequencies of different multi-measurement
sequences in the way I described? Making N measurements in a row can
itself be considered a type of repeatable experiment that has 2^N
possible outcomes each time you perform it. And each of those 2^N
outcomes would be assigned some probability by the Born rule, so you
should be able to design an operator that gives you the frequency of
any one of those 2^N outcomes, which may not be equiprobable depending
on the experimental setup.
