Attempts to make the Born rule emerge explicitly from outcome counting

1. Nov 29, 2005

straycat

Attempts to make the Born rule "emerge" explicitly from outcome counting

I would like to compare and contrast the various attempts to date to reconstruct the MWI so that it: 1) assumes outcome counting instead of the Born rule, but also 2) makes correct experimental predictions. This is an offshoot of the discussions:

"my paper on the Born rule..."

and

"Are World Counts Incoherent?"

These two threads discuss arguments for and against outcome counting (= the APP); on this thread, we can just assume the APP and take it from there.

As far as I know, the only serious attempts to do this explicitly are Graham's 1973 paper, Robin Hanson's "mangled worlds" paper, Michael Weissman's paper, and my own work-in-progress. (Many-minds might also be thrown into the mix, but I think not ...) Since Mike and Robin are both here, we should have a good discussion!

Here are some points of comparison that occur to me:

1) Is the underlying theory linear or nonlinear, and which would we expect/prefer?
2) Are there any "ontological" concerns raised by the modification?
3) Are there any experimental tests?
4) How do we explain the underlying postulates?
5) Does it involve pruning or extra branching?
6) Are the number of branches (allowed to be) finite or infinite? Discrete or continuous?

If anyone knows of any other attempts, or can think of other ways to compare/contrast these schemes, then post!

David

Last edited: Nov 29, 2005
2. Nov 29, 2005

straycat

pruning or extra branching?

In a sense, Robin's scheme employs pruning, if "mangling" and "destroying" are equated with pruning. I am not sure I understand, though, what mangling means exactly. If mangled worlds continue to exist, but they have been changed so that we can ignore them by (say) an anthropic principle argument, then we have not actually "pruned" branches in the sense that they "no longer exist from a God's eye view." This is good, b/c I'm not sure what that would mean or how that could occur. So I suppose I would say that mangled worlds does not technically prune, but it does employ a "pruning-equivalent."

In my scheme, I do not employ pruning or branching. But again, I do employ what might be considered to be a "pruning-equivalent" in the first part of the tree diagram of Fig 1 E. Basically, the idea is that a particular chunk of the tree diagram can be ignored in the approximation because it is simply improbable that its branches will ever be reached. So the basic mechanism behind my pruning equivalent is essentially 1) the APP itself, plus 2) the micro-structure of the tree diagram.

Mike's involves "anti-pruning," ie extra branching, as he states in message #71 of Patrick's thread. I still have more to read of his paper to understand how this works though!

David

3. Nov 29, 2005

straycat

alternate scheme for alternate PP?

Here's a question that is related to post 79 in Patrick's thread "my paper on the Born rule."

Let's suppose that ordinary QM were described by the rule: probability = |a|^n, instead of the Born rule: probability = |a|^2. Would it be possible to modify any of the schemes that are the topic of discussion in this thread accordingly? And what would be the underlying "explanation?"

In my scheme, the reason that 2 shows up in the exponent is that there are assumed to be two "special" paths from source to detector associated with the particle. So to change the exponent from 2 to n, we could simply postulate that there are n "special" paths. If we have n=1, then the one "special" path could be thought of as "the real path that the particle takes through spacetime." To get n=2, we postulate that from the "god's eye view" perspective of the particle through 4-d spacetime, the particle has the topology of a torus; ie, it looks like a big loop of string, which traverses the source as well as the detector, and so the two special paths correspond to the two ways around the loop.

Hopefully Robin and Mike can comment on how their schemes could be modified to yield |a|^n instead of |a|^2.

David

4. Nov 29, 2005

straycat

linear or nonlinear?

The introduction to Mike's paper contains an argument that we should expect our outcome-counting schemes to be characterized by nonlinearity. So do they?

Mike's scheme is nonlinear, as it says in the abstract.

After discussion with Mike, I've come to think that my scheme is fundamentally nonlinear as well. This is because the evolution of the observer through time is governed by general relativistic laws, and Einstein's equation is nonlinear. This is a quick and dirty argument, though -- hasn't really been vetted thoroughly yet.

Robin, is your scheme linear? You seem to imply this in your introduction with the statement: "I suggest how one can derive the prediction that observed long run measurement frequencies are given by the Born rule, while still insisting that only the standard linear evolution rule always and exactly determines evolution ..." If so, does this conflict with Mike's argument that the underlying scheme need be nonlinear? Or have I misunderstood something?

David

5. Nov 30, 2005

RobinHanson

In my sheme n depends on where the mangling cutoff is in the distribution of worlds. If the cutoff is at the median world, n = 0, while if it is at the median measure, n=2. Other positions yeild other n. But the cutoff isn't intended to be a free parameter - it is expected to be derived from the rest of quantum mechanics.