On 10/14/2025 12:20 PM, Alan Grayson wrote:
On Monday, October 13, 2025 at 10:28:30 PM UTC-6 Brent Meeker wrote:
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
Sorry, I don't quite understand your example? What has this to-do with
collapse of the wf and the MWI? Where is collapse implied or not? How
is Born's rule applied when the wf is discrete? AG
You wrote, "...claiming that without collapse,/Born's rule would fail in
some world of the MWI/....Can you prove it using mathematics?" So I
showed that in MWI, which is without collapse, 6 out of 16
experimenters will observe p=0.5 even in a case in which the Born rule
says the likelihood of p=0.5 is 0.049. Of course your challenge was
confused since it is not Born's rule that fails. Born's rule is well
supported by thousands if not millions of experiments. Rather it is
that MWI fails...unless it includes a weighting to enforce the Born
rule. But as Bruce points out there is no mechanism for this. If the
experiment is done to measure the probability (with no assumption of the
Born rule) then there are 16 possible sequences of four measurements and
6 of them give p=0.5 and 6/16=0.375, making p=0.5 the most likely of the
four outcomes. What this has to do with collapse of the wave function
is just that the Born rule predicts the probabilities of what it will
collapse to. So (assuming MWI) there are still 6 of the 16 who see 2h
and 2t but somehow those 6 experimenters have only a small weight of
some kind. Their existence is kind of wispy and not-robust.
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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