On 10/13/2025 5:04 AM, Alan Grayson wrote:
On Sunday, October 12, 2025 at 11:50:58 PM UTC-6 Brent Meeker wrote:
On 10/12/2025 10:18 PM, Alan Grayson wrote:
On Sunday, October 12, 2025 at 10:37:32 PM UTC-6 Brent Meeker wrote:
If there's no collapse then every possible sequence of
results is observed in some world and the relative counts of
UP v. DOWN in the ensemble of worlds will have a binomial
distribution. So for a large numbers of trials those worlds
in which UPs and DOWNs are roughly equal will predominate,
regardless of what the Born rule says. So in order that the
Born rule be satisfied for values other than 50/50 there must
be some kind of selective weight that enhances the number of
sequences close to the Born rule instead of every possible
sequence being of equal weight. But then that is
inconsistent with both values occuring on every trial.
Brent
Why does Born's rule depend on collapse of wf? AG
Where did I say it did?
Brent
The greatest mathematicians tried to prove Euclid's 5th postulate from
the other four, and failed; and the greatest physicists have tried to
dervive Born's rule from the postulates of QM, and failed;, except for
Brent Meeker in the latter case. You claimed it in the negative, by
claiming that without collapse, Born's rule would fail in some world
of the MWI. An assertion is just that, an assertion. Can you prove it
using mathematics? AG
Sure. Consider a sequence of n=4 Bernoulli trials. Let h be the number
of heads. Then we can make a table of the number of all possible
sequences bc with exactly h heads and with the corresponding observed
proportion h/n
h bc h/n
0 1 0.0
1 4 0.25
2 6 0.5
3 4 0.75
4 1 1.0
So each possible sequence will correspond to one of Everett's worlds.
For example hhht and hthh belong to the fourth line h=3. There are
sixteen possible sequences, so there will be sixteen worlds and a
fraction 6/16=0.3125 will exhibit a prob(h)~0.5.
But suppose it was an unfair coin, loaded so that the probability of
tails was 0.9. The possible sequences are the same, but now we can
apply the Born rule and calculate probabilities for the various
sequences, as follows:
h bc h/n prob
0 1 0.0 0.656
1 4 0.25 0.292
2 6 0.5 0.049
3 4 0.75 0.003
4 1 1.0 0.000
So most of the observers will get empirical answers that differ
drastically from the Born rule values. The six worlds that observe 0.5
will be off by a factor of 1.8. And notice the error only becomes
greater as longer test sequences are used. The number of sequences peak
more sharply around 0.5 while the the Born values peak more sharply
around 0.9.
Brent
On 10/12/2025 6:56 PM, Alan Grayson wrote:
Correct me if I'm mistaken, but as far as I know the wf has
never been observed; only the observations of the system it
represents. This being the case, in a large number of
trials. Born's rulle will be satisfied regardless of which
interpretation an observer affirms; either the MWI with no
collapse of the wf, or Copenhagen with collapse of the wf.
That is, since we can only observe the statistical results
of an experiment from a this-world perspective, and we see
that Born's rule is satisfied, so I don't see how it can be
argued that the rule fails to be satisfied if the MWI is
assumed. I think the same can be said about the other worlds
assumed by the MWI, namely, that IF we could measure their
results, the rule would likewise be satisfied.AG --
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