In a series of papers, Simon Saunders (arxiv:2103.01366 and arxiv:2103.03966) offers an extensive argument for the Everettian interpretation of quantum mechanics. The paper on structure (arxiv:2103.01366) contains the following paragraph:
"Everett called them *branches*. It is not hard to see that the connection between amplitudes and Born-rule probability is retained for multiple experiments. The amplitude of each branch, at the end of N experiments, as determined by the unitary evolution alone (together with the initial state), equals the square root of the Born-rule probability for that sequence of outcomes )just multiply together the probabilities for the results taken sequentially). Now consider the superposition of all those branches with the same relative frequency for the "+" outcome; not quite so obviously, the amplitude of this superposition is highly sensitive to the discrepancy, if any, between that relative frequency and the Born rule quantity for the "+" outcomes, the quantity *p*. Let the discrepancy be *eps*; then the amplitude falls off exponentially as exp( -* N eps*^2/*kappa), *where *kappa = 4p(1 - p)* and *N* is, as before, the number of trials. It is the first of a number of quantum Bernoulli theorems, the quantum analogues of the laws of large numbers: the amplitudes of branches with the "wrong" relative frequencies fall off exponentially quickly in the number of trials, in comparison with the amplitudes of Born-rule compliant branches." This argument was criticized by Adrian Kent (arxiv:0905.0624), but the argument persists, largely because it is central to the Everettians' case that their theory is consistent with the Born rule. While Kent's criticism still stands, he has made much the same mistake as Saunders. This is to assume that in Everett's picture, each trial is effectively a Bernoulli trial, with a probability of success p. This is not the case. Since every outcome occurs on every Evettian trial, the process cannot be seen as a Bernoulli trial. The crucial point of Saunders' argument hinges on the normal approximation to the Binomial (Bernoulli) distribution as the number of trials becomes large. Since the distribution from Everettian trials is not binomial, this approximation does not hold, and the whole argument collapses. The essence of a Bernoulli trial is that one outcome occurs with probability p (a 'success'), and other outcomes do not occur. This is in obvious conflict with Everett's approach in which every outcome occurs on every trial. Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAFxXSLQTG%2BinTsdK1zc2VjrGZEWm2VUj%3D%2BjP1wVXkAGiYhr6mw%40mail.gmail.com.
