straycat said:
What I am interested in is how to set up a workable theory founded on the APP -- that is, a theory that makes the same predictions as QM. In your paper, you make the point that when we assume the probability measure to be prescribed by the Born rule, then we are in fact making an independent assumption. What I would like to do is to point out yet another assumption that is made by standard MWI.
That's what many people try to do: find a "natural" counting scheme that ends up being equivalent to the Born rule. What I tried to argue is that if ever you find such a scheme, it will be for a sufficiently specific value of "natural" :-), that you could call it an extra postulate.
Nevertheless (we also had some e-mail exchange about that) I think that even _before_ you consider counting stuff, there is a more fundamental issue to be put forward: namely why having to choose ONE branch/term/state at all ?
The only thing you get out of unitary QM is that after "measurement" the observer's physical construction (body) ends up in entangled states with the rest of the universe. I think it is already an extra postulate to say that the observer will only be aware of ONE of these terms in a probabilistic way.
Of course, EVEN after this is postulated, then you can find different ways to assign probabilities to each of the possibilities, and I wanted simply to indicate that at least 2 radically different ways of this assignment were possible: Born rule and the APP. The logical existence of at least two assignment schemes then means that not one can be logically DEDUCED: you will again have to postulate which one, independently. Now you can find minimalistic ("natural") postulates which do this, or you can bluntly say: by postulate, it's the Born rule.
As is rather well known, requiring "non-contextuality" is already sufficient (using Gleason's theorem).
But, before you can require "non-contextuality", you have already to say that you ARE going to assign probabilities to individual terms in the wave function (otherwise there is nothing you can require non-contextuality of !).
So all this means that unitary QM, all by itself, does not generate anything like a natural probabilistic quantum theory, as was the original goal of Everett.
So here's my line of reasoning. It seems clear to me that when a world "splits" into N branches, this is understood to mean that the *observer* state evolves from one physical state into N physically distinct states. When an observer couples to a system under observation (say, a spin 1/2 particle), then we start out by enumerating the number of eigenstates of the particle (say, N = 2). We then go on (in the standard MWI) to assume that the observer-state is represented by the same number of eigenstates -- eg, N = 2. That is, we assume that there is a one-to-one correspondence between the number of eigenstates of the observer and the number of eigenstates of the system under observation. So this is the assumption that I wish to challenge.
I think I see more or less what you're aiming at, which is a kind of "phase space" argument.
But first I'd like to point out that we do not start by enumerating the number of eigenstates of the particle, but the number of eigenspaces with distinct measurement outcomes (the distinct eigenvalues). The original system's state can then be constructed by a unique sum of one single eigenvector out of each eigenspace, say v_n, to make up the whole state, and the observer will entangle with each of these individual eigenvectors.
psi_sys = sum_n a_n v_n (n runs over the different eigenspaces of the system).
Now, you seem to think somehow that you might have "more" states of the observer associated with one v_n than with another, giving it more weight. But I don't think that that is tenable.
Indeed, BEFORE the measurement, the observer was in a specific state psi_obs0. This can be a very complex state, even unknown (part of a mixture), but it is in a certain state, even if we don't know which one. Now, due to linearity of the time evolution operator, with ONE v_n, this psi_obs0 will evolve into ONE specific psi_obsn ; there's no choice, it is fixed by the evolution operator corresponding to the measurement action. So I don't see how you can have "more" psi_obsn than psi_obsm. There's exactly one for each.
However, you could say that psi_obs0 is drawn from an ensemble, and that they populate differently the phase space around psi_obsn than around psi_obsm if we let them all evolve through the time evolution operator.
But that also has a serious problem: this would then mean that the probabilities of outcomes of measurement are determined, not by the state of the system, but by the mixture of states of the observer before measurement. That cannot generate a Born rule which only depends on the system states.
It seems to me that if we modify the MWI via the APP, and we *further* modify it by considering the notion that the number of observer-states and the number of observed-states are *not* necessarily in one-to-one correspondence, then we might be able to come up with a formulation that does, in fact, make the same predictions as standard QM.
Well, I see obstacles, as I outlined above. That doesn't mean that it cannot be done, but I see difficulties on how to approach things that way.
Nevertheless, in all this, no matter how "natural" it may sound in the end, I'm pretty convinced that EXTRA postulates are somehow necessary and that it doesn't follow logically from unitary QM.
That's what I wanted to show in my paper.
cheers,
Patrick.
) IMHO. I think everybody agrees that we have to have the Born rule to make QM work, so the point is not to argue that the Born rule must be correct.
