Isn't this just the result of classifying ##C^*## algebra factors? Or do you ask specifically why MWI assumes ##I_\infty##?DarMM said:It seems odd to me that in a theory that is about the deterministic evolution of the universe, with probability only arising from local subjective viewpoints, the deterministic evolution would have the mathematical form of a probability theory.
A. Neumaier said:Yes, this is exactly what I was paraphrasing..
But that's how your arguments sound to me. The words you are using convey nothing significant, since you simply postulate a lot of unseen worlds and a probabilistic law for them to justify with a weak argument only what you know already about the world you see.stevendaryl said:Well, your "paraphrase" substituted a completely different meaning. It isn't what I meant. It isn't what I said. I didn't say that because that's not what I meant.
A. Neumaier said:But that's how your arguments sound to me. The words you are using convey nothing significant, since you simply postulate a lot of unseen worlds and a probabilistic law for them to justify with a weak argument only what you know already about the world you see.
Stephen Tashi said:Taking pains to speak only of frequencies, how do we decide if MWI predicts the Born Rule? It has to be something like "On the the most frequent branches ( i.e. the most frequent "wolds") , the frequency of events observed in a repeated experiment is approximately the frequency given by the Born Rule."
I think the question I'm asking might be more on the philosophical side after some thought. It was basically that the mathematical structure of any quantum theory is that of a C*-algebra, i.e. a generalised probability theory. In MWI, probability is not a fundamental component of the theory and yet the theory has the mathematical structure of a probability theory. I was wondering if there was any discussion of this odd mathematical structure, given the non-probabilistic nature of the theory.A. Neumaier said:Isn't this just the result of classifying ##C^*## algebra factors? Or do you ask specifically why MWI assumes ##I_\infty##?
Really? I would understand if you had said "valid argument" but to deny the existence of a coherent argument is patently ridiculous.A. Neumaier said:The real question is: In what sense does MWI predict Born's rule? I cannot see any coherent argument.
Yes, if you take the usual notion of "probability" as fundamental, but my remarks are directed at attempts that begin with the MWI sans probabilities and attempt to substitute actual frequencies for probabilities. If you take the MWI, you can have unequal weights, but these have no direct interpretation as probabilities - at least not without a lengthy proof. If you want to deal only with actual frequencies, you have to figure out a way to define your frequencies in such a manner that branches that have higher weights are "frequent". For example, you could assume there are more copies of them.stevendaryl said:Probability in quantum mechanics is definitely not a matter of "counting branches". You can have just two branches, and that doesn't mean that they have equal weight.
No. Decoherence does not require the Born Rule! It's a physical process that does not resolve a state as a proper mixture anywhere.DarMM said:Just another question, doesn't decoherence already require the Born rule, to permit tracing over the environment? Hence without the Born Rule, how do you show the state vector is of essentially Schmidt form to permit the clear branching structure without the Born Rule?
How is Tracing justified without the Born Rule?Derek P said:No. Decoherence does not require the Born Rule!
No idea. Why would you want to?DarMM said:How is Tracing justified without the Born Rule?
Well there are. What you are calling a branch is a branch of the original superposition - the state of the system under observation. It's not a world.Stephen Tashi said:Yes, if you take the usual notion of "probability" as fundamental, but my remarks are directed at attempts that begin with the MWI sans probabilities and attempt to substitute actual frequencies for probabilities. If you take the MWI, you can have unequal weights, but these have no direct interpretation as probabilities - at least not without a lengthy proof. If you want to deal only with actual frequencies, you have to figure out a way to define your frequencies in such a manner that branches that have higher weights are "frequent". For example, you could assume there are more copies of them.
Derek P said:No idea. Why would you want to?
Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule.Derek P said:No idea. Why would you want to?
DarMM said:Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule.
I'm reading Wallace's book right now, so I'll see what he says.
You're both missing something. The Born Rule and tracing are not assumed in MWI, they are derived from the unitary model. Indeed it is hard to imagine how a treatment that concludes with quantum predictions could fail to reproduce the Projection Postulate - except that it would no longer be a postulate but a theorem. But let's not be too ambitious - this thread is about the Born Rule, not the entirety of MWI's mathematical underpinnings. It can be tackled by explicitly expanding the global state as a sum of many ket products. It is a result, not an assumption. And of course once the Born Rule is established, you can hit the density matrix with it if that's what floats your boat.stevendaryl said:Well, it's the most general form of a quantum prediction. If you have a state that is described by a certain density operator ##\rho##, and you have an operator ##A##, then the expected value of ##A### is ##tr(\rho A)##.
Derek P said:You're both missing something. The Born Rule and tracing are not assumed in MWI, they are derived from the unitary model. Indeed it is hard to imagine how a treatment that concludes with quantum predictions could fail to reproduce the Projection Postulate - except that it would no longer be a postulate but a theorem.
Actually, tracing and decoherence do not involve measurement, hence tracing is strictly speaking an additional assumption independent of the Born rule in its conventional formulation. The latter is a statement about measurement results, but things not yet measured have no results, hence the conventional form of Born's rule is inapplicable.DarMM said:tracing requires the Born Rule
Okay. So let's pretend there isn't a theorem and tracing is not justified. You can (I believe) derive the Born rule by expanding the state as a sum of components. Of course you will actually be taking a trace but you won't be relying on a theorem to give you permission.stevendaryl said:He wasn't saying that tracing was assumed. He was asking how it was justified. A theorem is a good justification, if there is a theorem.
You need to change your signature to "Every serious student of QM needs a copy of Ballentine and to Look into Gleason" !bhobba said:Look into Gleason.
Thanks
Bill
No, it's state counting.DarMM said:However, if it's not branch-counting, what is it?
DarMM said:I know the proof of Gleason's theorem, but it has never genuinely helped me comprehend the MWI arguments as it comes from a very different direction, Wallace argues that his proof is a separate line of argumentation to Gleason. I think I need to read Wallace's book in full perhaps.
Derek P said:Okay. So let's pretend there isn't a theorem and tracing is not justified.
DarMM said:Tracing is required to obtain decoherence and tracing requires the Born Rule. Hence decoherence does require the Born rule. I'm reading Wallace's book right now, so I'll see what he says.
Derek P said:No, it's state counting.
Derek P said:Okay. So let's pretend there isn't a theorem
There are googols of terms in "the superposition" but only a small number of worlds. It's up to you which one you call "branches" but they are certainly very different.stevendaryl said:What do you mean by "state counting"? In the MWI, there is only one state for the entire universe, so counting states always gives you the answer 1.
Do you mean counting the number of terms in the superposition? If so, then how is that different from branch counting?
DarMM said:Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.
This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.Derek P said:You can (I believe) derive the Born rule by expanding the state as a sum of components.
Schlosshauer's book is probably the best modern discussion of the measurement problem.DarMM said:Okay, just to say, I'm going to stay away from this thread until I finish Wallace and Schlosshauer's books. I'm reading Wallace's book using Mandolesi's papers as a guide, both the 2015 and 2018 ones.
I don't think it has been answered. If you decide to moderate it please just remove the offending subthreads.bhobba said:But there is a theorem - you can't waive away Gleason. The only out with Gleason is non-contextuality - non-contextual interpretations usually have there own way of handling the measurement issue.
The measurement issue has problems - but we know things about it the early pioneers did not - there is no getting around it. Progress will not be made by - let's suppose something we know is true is not.
As I have said please please try to understand interpretations pro's and cons. I think your question has been answered. But if you or anyone wants to keep it going - go ahead. It will be shut down if it becomes simply argumentative.
Thanks
Bill
I have no idea what you mean. I would not assign probabilities to worlds. As far as I know it cannot be done, and I think it is provably impossible.A. Neumaier said:This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.
A. Neumaier said:This only gives a decomposition but assigns no meaning or frequencies/probabilities to the terms. You have to postulate that each term has a given frequency/probability. This is a postulate far worse than Born's rule since Born's rule has at least an empirical support, but assigning probabilities to worlds of which only one is observed is completely arbitrary.
Derek P said:There are googols of terms in "the superposition" but only a small number of worlds. It's up to you which one you call "branches" but they are certainly very different.
Derek P said:I have no idea what you mean. I would not assign probabilities to worlds. As far as I know it cannot be done, and I think it is provably impossible.