On 10/16/2025 3:41 AM, Alan Grayson wrote:
I only wrote "... the Born rule says..." and "... the Born rule predicts..." If you don't understand how a mathematical formula can "say" or "predict" I can't help you.On Wednesday, October 15, 2025 at 7:52:07 PM UTC-6 Alan Grayson wrote: On Tuesday, October 14, 2025 at 10:24:42 PM UTC-6 Brent Meeker wrote: 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? AGWhere 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? AGSure. 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? AGYou 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 I didn't mean to imply that Born's rule is violated. But what you need to do IMO, is show how Born's rule is applied to your assumed events as seen without collapse in some world of the MWI. Otherwise, you just have a set of claims without any proof of their validity. AGYou say Born's rule will do this or that, but you don't say exactly HOW it will do this or that. AG
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