The Insufficiency of Born's Rule

[Alan Grayson]

FWIW, I've always thought Born's Rule refers to the Frequentist Theory of Probabilility (FTP). Moreover, I don't recall my professors explicitly stating this, but it was, AFAICT, the unstated assumption, likely inherited from slit experiments and the need for a multitude of trials to establish the interference pattern. And I've always assumed the FTP means that for a given probability value, and some assumed tolerance (the epsilon number in the concept of limits), there exists a certain number of trials for which there's a convergence to that value. But Born's Rule doesn't tell us what that number is (which will be different for each context of probability). AG

 

[Russell Standish]

Born's rule is a = <ψ|A|ψ>, which is basically the mean of some distribution.

σ² = <ψ|A²|ψ>-<ψ|A|ψ>² is the standard deviation.

The normal law of large numbers
(https://en.wikipedia.org/wiki/Law_of_large_numbers) has

P(|<X>-a|≥ε) ≤ σ²/(nε²)

or in English, the probability the average measured value <X> over n trials differs from the Born rule prediction goes to zero as n→∞ as 1/n.

So no, the Born rule is not insufficient.


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