Wei, the relationship between computing power and the probability rule is interesting, but doesn’t do much to explain Born’s rule.
In the context of a many worlds interpretation, which I have to assume you are using since you write of splitting, it is a mistake to work with probabilities directly. Because the sum is always normalized to 1, probabilities deal (in part) with global information about the multiverse, but people easily forget that and think of them as local. The proper quantity to use is measure, which is the amount of consciousness that each type of observer has, such that effective probability is proportional to measure (by summing over the branches and normalizing). It is important to remember that total measure need not be conserved as a function of time.
So for the Ebborian example, if measure is proportional to the thickness squared, the fact that the probability of a slice can go up or down, depending purely on what happens to other slices that it otherwise would have nothing to do with, is neither surprising nor counterintuitive. The measure, of course, would not be affected by what the other slices do. It is just like saying that if the population of China were to increase, and other countries had constant population, then the effective probability that a typical person is American would decrease.
The second point is that, even supposing that quantum computers could solve hard math problems in polynomial time, your claim that intelligence would have little evolutionary value is both utterly far-fetched (quantum computers are hard to make, and nonlinear ones could be even harder) and irrelevant if we believe—as typical Everettians do—that the Born rule is not a seperate rule but must follow from the wave equation. Even supposing intelligence required the Born rule, that would just tell us that the Born rule is true—but we already know that. The question is, why would it follow from the wave equation? If the Born rule is a seperate rule, that suggests dualism or hidden variables, which bring in other possibilities for probability rules.
Actually there are already many other possibilities for probability rules. A lot of people, when trying to derive the Born rule, start out assuming that probabilities depend only on branch amplitudes. We know that seems true, but not why, so we can’t start out assuming it. For example, probabilities could have been proportional to brain size.
These issues are discussed in my eprints, e.g.
Decision Theory is a Red Herring for the Many Worlds Interpretation
http://arxiv.org/abs/0808.2415
Wei, the relationship between computing power and the probability rule is interesting, but doesn’t do much to explain Born’s rule.
In the context of a many worlds interpretation, which I have to assume you are using since you write of splitting, it is a mistake to work with probabilities directly. Because the sum is always normalized to 1, probabilities deal (in part) with global information about the multiverse, but people easily forget that and think of them as local. The proper quantity to use is measure, which is the amount of consciousness that each type of observer has, such that effective probability is proportional to measure (by summing over the branches and normalizing). It is important to remember that total measure need not be conserved as a function of time.
So for the Ebborian example, if measure is proportional to the thickness squared, the fact that the probability of a slice can go up or down, depending purely on what happens to other slices that it otherwise would have nothing to do with, is neither surprising nor counterintuitive. The measure, of course, would not be affected by what the other slices do. It is just like saying that if the population of China were to increase, and other countries had constant population, then the effective probability that a typical person is American would decrease.
The second point is that, even supposing that quantum computers could solve hard math problems in polynomial time, your claim that intelligence would have little evolutionary value is both utterly far-fetched (quantum computers are hard to make, and nonlinear ones could be even harder) and irrelevant if we believe—as typical Everettians do—that the Born rule is not a seperate rule but must follow from the wave equation. Even supposing intelligence required the Born rule, that would just tell us that the Born rule is true—but we already know that. The question is, why would it follow from the wave equation? If the Born rule is a seperate rule, that suggests dualism or hidden variables, which bring in other possibilities for probability rules.
Actually there are already many other possibilities for probability rules. A lot of people, when trying to derive the Born rule, start out assuming that probabilities depend only on branch amplitudes. We know that seems true, but not why, so we can’t start out assuming it. For example, probabilities could have been proportional to brain size.
These issues are discussed in my eprints, e.g. Decision Theory is a Red Herring for the Many Worlds Interpretation http://arxiv.org/abs/0808.2415