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I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
That's what Wikipedia claims (here), and their reference for that is this 1968 article by James Hartle. I checked it out some time ago and he's clearly also assuming that the Hilbert space of a physical system is the tensor product of the Hilbert spaces of its subsystems. That's a very strong assumption. I don't have all the details figured out, but it seems to me that this assumption is essentially equivalent to assuming that the Born rule holds. The weak assumption that you mentioned is probably just the piece that needs to be added to make them completely equivalent.I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
I still don't think that QM is anything more than a set of rules that tells us how to calculate probabilities of possibilities, but I think that using Occam's razor as an argument against the MWI makes about as much sense as using it against special relativity because it includes more than one inertial frame. If (the Dirac-von Neumann version of) quantum mechanics actually describes reality (which is hard to dismiss based only on Occam, considering that no other theory does a better job), this reality clearly must include many worlds. Even if some other version of QM is an accurate description of the world, then why would we consider it "simpler"? I don't think e.g. Bohm or a realist intepretation of the path integrals formulation is any simpler. The worlds are only "points of view" in the linear and deterministic evolution of a single point in a vector space, and I think the people who try to use Occam against the MWI have completely failed to understand this point....an internal sentient observer would be perfectly able to describe his own history and experience using, say, Copenhagen and wavefunction collapse, and treat everything about splitting of himself as Occam razor violating nonsense.
I think every human is always in a superposition in most decompositions into subsystems, but the only decomposition that mattters to that human is the one that describes the universe as consisting of his memory and everything else, and in that decomposition, his memory states keep developing correlations with eigenstates of whatever he observes. Conscious experience is the development of such correlations. An important detail here is that the correlations form so quickly that the human won't ever notice that he failed to experience the time of decoherence.I wonder if it's possible to set up an experiment that creates a superposition of a human being in different states.
Good point. I agree.This all depends on what you mean by "world". There's a definite branching of sentient beings and of classical worlds. There isn't any branching of the whole state of the universe in the big Hilbert space - that one simply evolves according to Schrodinger's equation.
It's a way to interpret QM as a description of what actually happens, instead of as nothing more than a set of rules that tells us how to calculate probabilities of possibilities. That's exactly what interpretations are about.What's the advantage of the MWI?
(What's the aim of any interpretation in general?)
What do you consider conventional?To me it seems MWI makes an even more abstract mess than what we had before with "conventional" thinking.
Hi Gerenuk,Actually I do favour interpreting the set of rules of QM as to make the picture either more intuitive or so that one can grasp QM effects better in the mind or make the picture more well defined so that one has never doubt about what the result of a question might be. I cannot see MWI achieving either of these two. I also believe a good interpretation will give as a more correct and complete version of QM.
No there aren't - as far as I know.Thanks a lot for the link. These kind of interpretation are actually really my favourite, but I also haven't studied them yet. So I've collected information and books and will study them soon.
My first impression was that the pilot wave is still a bit awkward.
Are there any less-known interpretations that are similar to that?
We don't need one, but it would be nice to have one, if it really does describe what actually happens.Why do we need an interpretation and what should it achieve?
I do too.Hmm, then I personally prefer the set of rules without the complicating extension like MWI around it.
A correct interpretation would certainly do that, but I doubt that there is such a thing.I also believe a good interpretation will give as a more correct and complete version of QM.
I don't think there's a universally accepted definition of the Copenhagen interpretation, but most people would say that it asserts that the laws of QM do not apply to measuring devices(!) even though it applies to the components they're made of(!!), and that measurements "collapse" wave functions into eigenstates. The problem with this is that it's complete rubbish that no one has ever believed is true. The first assumption introduces an obvious inconsistency into the theory, and the second implies that we have not one, but two rules that specify how systems change with time. That makes another inconsistency possible.Conventional I consider the Copenhagen interpretation I suppose.
The main epistemological point of MWI is that it is essentially the only interpretation that is not an extension.Hmm, then I personally prefer the set of rules without the complicating extension like MWI around it.
Perhaps I'm missing your point, but how could the Hilbert space of a physical system NOT be the tensor product of its subsystems? That seems axiomatic to me.That's what Wikipedia claims (here), and their reference for that is this 1968 article by James Hartle. I checked it out some time ago and he's clearly also assuming that the Hilbert space of a physical system is the tensor product of the Hilbert spaces of its subsystems. That's a very strong assumption. I don't have all the details figured out, but it seems to me that this assumption is essentially equivalent to assuming that the Born rule holds. The weak assumption that you mentioned is probably just the piece that needs to be added to make them completely equivalent.
The observer is never able to experience a splitting of himself, because he's always in a state of definite memory. In places where MWI says that the observer is split, the observer instead observes wavefunction collapse. So, from the point of view of the observer, those "other" states of him are unobservable and do not exist. From his point of view, the Occam-minimal, positivist interpretation is Copenhagen and not MWI. Even if MWI is the proper description of the totality of the universe.using Occam's razor as an argument against the MWI makes about as much sense as using it against special relativity because it includes more than one inertial frame. If (the Dirac-von Neumann version of) quantum mechanics actually describes reality (which is hard to dismiss based only on Occam, considering that no other theory does a better job), this reality clearly must include many worlds.
There's more than one way to use two Hilbert spaces to construct a third. We use the tensor product because we want to make sure that the probability of obtaining two specific results in two independent measurements on two non-interacting systems is the product of the two probabilities assigned by the Born rule. See this post for a few more details about this, and this one for more about the tensor product in general.Perhaps I'm missing your point, but how could the Hilbert space of a physical system NOT be the tensor product of its subsystems? That seems axiomatic to me.
I think I will too when I have learned it. I have bought the books already. 1, 2. I just need to get through them. It looks like it will take a long time. I'm going to finish another book (3) before I get deep into these two.I prefer something more like the C*-algebra picture.
OK, that's a statement I haven't heard before. How does the C*-algebra formulation deal with subsystems, and how is it relevant? Does it imply that something I said is wrong?While the ket picture is useful for some calculations, it obscures what's happening when we want to restrict to subsystems or whatever.
There's more than one way to use two Hilbert spaces to construct a third. We use the tensor product because we want to make sure that the probability of obtaining two specific results in two independent measurements on two non-interacting systems is the product of the two probabilities assigned by the Born rule. See this post for a few more details about this, and this one for more about the tensor product in general.
The subsystems aren't represented by orthogonal subspaces. For example, if you take the tensor product of a 2-dimensional and a 3-dimensional Hilbert space, the result is 6-dimensional, not 5-dimensional. The choice to use the tensor product is definitely non-trivial.In order for the tensor product construction to work, all we need is for the two Hilbert spaces to be orthogonal, which is automatically true in all interpretations of QM as long as two systems are non-overlapping.
I think this is a motivated question. I posed the same in post 43, where I gave my view.What is your opinion? Why do we need an interpretation and what should it achieve?
(I repeat that my opinion is that it should make QM either easier or extend it)
Let's play with idea of the many observer view rather than many world view? (or just think of MWI, but where there is a physical basis for each world, which is an subjective view)“Definitely not. Then intensity is not important. Even if we have Frequent event (90%) and Rare event (10% probability), and we make 100 tries, then all combinations are possible, like FFFFFFFFFFFF… (100 Fs), and RRRRRRRRR (100Rs which is also rare). All 2^100 branches must exist! There are 2^100 observers observing all these branches”
“Lets make that experiment. I bet we get about 85-95Fs and 5-15Rs. What is a prediction of MWI?”
“Hmmmm…. Everything is possible…”
I am blocked at this point.
According to my way of reasoning, these two ideas does not even mix consistently.The same as the axioms for the statistical interpretation (Link), plus the additional assumptions that it makes sense to consider the Hilbert space of the universe (even though it includes yourself), and that a state vector in that Hilbert space is a representation of all the properties of a physical system (the omnium). (The statistical interpretation doesn't assume that, and it never includes the observer in the Hilbert space).
The problem with such an additional assumption is that it destroys all the beauty of pure MWI without that assumption. This is because such an assumption raises questions that cannot be answered within MWI:I think that in the MWI, the Born rule can be derived from the weaker assumption that measuring an observable of a system that is in an eigenstate will yield the corresponding eigenvalue with certainty.
the Born rule… personally, I think for MWI it must be interpreted differently.
We can and should interpret it differently, but I don't know if there's a way to interpret it the way you're suggesting. There are several different ways to derive the Born rule's assignment of probabilities from the assumption that the Hilbert space of the omnium can be decomposed into a tensor product of Hilbert spaces of subsystems (and some minor technical assumption such as the one Count Iblis mentioned). We can also prove the converse. This suggests that in the MWI, we should think of the Born rule as the assumption that these decompositions are allowed. The actual probability assignment should be thought of as a result derived from that axiom. Just don't forget that the this decomposition axiom is very non-trivial....like, total number of observers observing X divided by the total number of observers in some subbranch on a given basic...
Such as?There are several different ways to derive the Born rule's assignment of probabilities from the assumption that the Hilbert space of the omnium can be decomposed into a tensor product of Hilbert spaces of subsystems (and some minor technical assumption such as the one Count Iblis mentioned).