Wednesday, September 16, 2009

Measure of Consciousness versus Probability

In the last post, Meaning of Probability in an MWI, it was explained that in a deterministic Many-Worlds model, with known initial conditions, that which plays the role for of a probability for practical purposes is the ratio

(the measure (amount) of consciousness which sees a given outcome)
/ (the total measure summed over outcomes)

I call that the effective probability of the outcome.

Although the effective probability is quite similar to what we normally think of as a probability in terms of its practical uses, there are also important differences, which will be explored here.

The most important differences stem from the fact that measure of consciousness need not be a conserved quantity. By definition, probabilities sum to 1, but that is not all there is to it. In a traditional, single-world model, a transfer of probability indicates causality, while the total measure remains constant over time. This is not necessarily so in a MW model.

For example, suppose there are two branches, A and B. A has 10 observers at all times. B starts off with 5 observers at T0, which increases to 10 observers at T1 and to 20 observers at T2. All observers have the same measure, and observe which branch they are in.

So the effective probability of A starts off at 2/3 at T0, while the effective probability of B is 1/3. At T1, A and B have effective probabilities of 1/2 each. At T2, the effective probability of A is 1/3 and that of B is 2/3.

There are two important effects here. First, the effective probability of B increased with time. In a single-world situation, that would mean that a system which was actually in A was more likely to change over to B as time passes. But in this MW model, there is no transfer of systems, just changes in B itself.

This means that probability changes that would require nonlocality in a single-world model don't necessarily mean nonlocality in a MW model. If A is localized at X1, and B is localized at X2 which is a light-year away, there need not be a year's delay before the effective probability of B suddenly increases.

In a single-world local hidden variable model, probability must be locally conserved, so that the change of probability in a region is equal to the transitions into and out of adjacent regions only. This need not be so in an MW model.

The second important effect of nonconservation of measure in a MW model is that total measure changes as a function of time. Observers can measure, not only what branch they are on, but also what time it is. They will be more likely to observe times with higher measure than with lower measure, just as with any other kind of observation.

A good example of this is a model proposed by Michael Weissman - a modification of physics designed to make world-counting yield the Born Rule. His scheme involved sudden splitting of existing worlds into proposed new degrees of freedom, with a higher rate of such splitting events for higher amplitude worlds. The problem with it is that if new worlds are constantly being produced, then the number of observers would be growing exponentially. The probability of future observations, as far into the future as possible, would be much greater than that of our current observations. Thus, the scheme must be false unless we are highly atypical observers, which is highly unlikely.

It is important to realize that since changes in measure mean changes in the number of observers, decreases in measure are undesirable. This will be discussed further in the next post.

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