Ergodocity and unitary evolution

A. Neumaier

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How is ergodicity relevant here?
This is not what MWI does. In MWI it is the worlds that form an ensemble. Thus we have at least one world in which all throws show sixes - in fact many more since throwing dice is not the only measurable thing that happens in the world. Similarly, for any other sequence of throws that we may think of, there are plenty of worlds realizing just this. But we are actually in just one of these worlds - all others are (from our point of view) imagined, like in your case 1.

MWI ''works'' by saying that the ensemble probabilities are such that Born's rule holds. This means that if we look at enough worlds we find that most (in the sense of ensemble probaility) of the worlds [the ordinary ones] have random throw sequences, whereas an infinite minority of worlds [the exceptional ones] has throw sequences with a nontrivial pattern or correlations. Moreover, there are an even larger minority of mixed worlds where at times the dice behave random, and at other times the dice behave systematic.

Obviously we are in one of the ordinary worlds since we experience random throw sequences. But if the other worlds are as real as ours, the inhabitants of the infinitely many exceptional worlds or mixed worlds observe very different laws, and the same MWI ''explains'' their laws. Thus MWI explains every possible set of laws or unlawfulness or intermittent lawfulness. For me, this implies that MWI explains nothing.

The situation were different if MWI were ergodic. in this case, the ensemble statistics would be aritrarily closely replicated in time in each individual world, i.e., the observations of the inhabitants in each world would satisfy the same statistics, and there would be only ordinary worlds.

Without ergodicity, positing that we are in an ordinary world is equivalent to positing that the Born rule holds in our world - i.e., we assume what we would want to derive. Since we experience only one world, it is no good to argue with typicality - since this typicality is precisely what we want to infer!
 

A. Neumaier

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ergodicity allows us to interpret probability measure-theoretically, on an ensemble, and also as a fraction of time, on a single system.
Yes, and this is important, since each single cup of tea has the properties derived from the ensemble.

Already Gibbs (who invented towards the end of the 19th century the notion of a thermodynamic ensemble) noticed (and stated explicitly) that the ensembles are fictitious repetitions, used just to be able to apply statistical intuition - although he gets deterministic results applicable to each instance.

Note that the formal developments in equilibrium thermodynamics never use the notion of repetition, so it doesn't matter in this case whether the ensemble is real or fictitious. Gibbs was well aware of that.

But if the ensemble is used to describe stochastic experiments that extend over times significant at human scales, the repetitions count. In this case it makes a big difference whether these repetitions are actual and hence affect the statistics computed from the experiments, or if they are imagined, and hence have no impact at all.
 

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