Born rule?

[Alex A]

i recently re-read Eliezer's many world post here: http://lesswrong.com/lw/q8/many_worlds_one_best_guess/

where he says, and i quote:

If we try to get probabilities by counting the number of distinct observers, then there is no obvious reason why the integrated squared modulus of the wavefunction should correlate with statistical experimental results.  There is no known reason for the Born probabilities, and it even seems that, a priori, we would expect a 50/50 probability of any binary quantum experiment going both ways, if we just counted observers.

anyone want to meet and try and give me some more informal context for this statement?

[doug moen]

You quote Eliezer as saying "there is no known reason for the Born probabilities".

However, there are a number of people who claim to derive the Born rule from first principles. Including people like David Deutsch. There's a nice survey and discussion of some of this work here:

 http://www.preposterousuniverse.com/blog/2014/07/24/why-probability-in-quantum-mechanics-is-given-by-the-wave-function-squared/

I'm definitely not an expert and I can't vouch for any of these claims.

Doug Moen.

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[Jess Riedel]

Alex: The time evolution of the wavefunction is well understood, and the wavefunction appears to develop branches which look like distinct macroscopic outcomes. Eliezer is saying that, if we assume nothing else, then there is nothing that tells us what probabilities we should assign to those possible outcomes.  In other words, the mathematical description of the outcomes encoded in the wavefunction is logically distinct from the mapping from those outcomes to real-number probabilities.  Without such a way of assigning probabilities, the theory would have no predictive power.  Furthermore, the naive "equal probability" assignment (i.e., all outcomes have equal probability) does not agree with experimental observations.  Therefore, in order for quantum mechanics to be a useful theory for making predictions, we need a new additional principle, beyond the wavefunction's dynamical evolution, for calculating probabilities.

Doug: All derivations of the Born probabilities require the assertion of new desiderata, e.g., various symmetries.  There's nothing wrong with this, since all theories require assertions, and ultimately any assertion that leads to the Born probabilities can be justified since we observe Born probabilities in experiment.  But Eliezer is still correct in pointing out that something else must be asserted.  Since there are many possible new assertions that would could imagine, and they all lead to only a single prediction (the experimentally observed probabilities), choosing between them is largely a matter of taste and elegance.

Incidentally, in my opinion the decision-theory derivations are among the least attractive.  Adrian Kent offers a good critique here:

http://arxiv.org/abs/0905.0624

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