Different interpretations? No, different theories

[Quantumental]

I'm not entirely sure, but I'm leaning towards yes.

What about the preferred basis issue ?
Have you read demystifiers blgopost that I linked to?

 

[kith]

I'm not entirely sure, but I'm leaning towards yes.

This is also my impression. There are reasonable objections against the MWI, but I don't get why people reject Everett's basic idea because of the supposable impossibility to derive the Born rule.

What about the preferred basis issue?

In order to apply the Born rule, you have to say which part of the whole Hilbert space corresponds to your physical system of interest. So the necessary separation is introduce by hand. Once you have done this, decoherence solves the preferred basis issue.

 

[Fredrik]

What about the preferred basis issue ?
Have you read demystifiers blgopost that I linked to?

I have read it before. I haven't read the article that he links to, because its main claim is something that I never doubted:

In modern literature, one often finds the claim that the basis problem is solved by decoherence. What J-M Schwindt points out is that decoherence is not enough. Namely, decoherence solves the basis problem only if it is already known how to split the system into subsystems (typically, the measured system and the environment). But if the state in the Hilbert space is all what exists, then such a split is not unique.

However, I suspect that the following (standard) idea is misguided:

To define separate worlds of MWI, one needs a preferred basis, which is an old well-known problem of MWI.

To explain why, I'm going to have to speculate a bit. I can't prove any of this rigorously at this point, and I'm not working on it.

I think it makes more sense to postulate (as part of a definition of an MWI) something like "every 1-dimensional subspace is a world". If we do, we don't need a preferred basis to tell us which 1-dimensional subspaces are worlds, because they all are. For each decomposition of the universe into subsystems (like "the cat"+"everything else"), the Born rule selects a "preferred" basis. Instead of the statement "the 1-dimensional subspaces identified by the basis are worlds and all the other ones aren't", I propose that "the 1-dimensional subspaces identified by the basis are especially interesting worlds".

In what sense are they "interesting"? I think it might be possible to prove something like "the worlds singled out by the preferred basis are the ones where the subsystems store the maximum amount of information about each other". If this can be done, then things get pretty interesting. The inner product gives us a way to assign a numerical value to how "close" two worlds are. Worlds that are very close to the "interesting" ones will be practically indistinguishable from them. And in worlds that are far from the "interesting" ones, the subsystems are going to be bad at storing information about each other. That could mean that they don't contain any conscious observers, since consciousness involves information storage.

Please don't misinterpret this as a suggestion that consciousness plays some active role in the MWI, or that consciousness transcends the physical, or some other nonsense like that. I'm just saying that I consider worlds where consciousness is present more interesting than worlds without consciousness.

 

[stevendaryl]

I think it makes more sense to postulate (as part of a definition of an MWI) something like "every 1-dimensional subspace is a world". If we do, we don't need a preferred basis to tell us which 1-dimensional subspaces are worlds, because they all are. For each decomposition of the universe into subsystems (like "the cat"+"everything else"), the Born rule selects a "preferred" basis. Instead of the statement "the 1-dimensional subspaces identified by the basis are worlds and all the other ones aren't", I propose that "the 1-dimensional subspaces identified by the basis are especially interesting worlds".

I'm sorry, what do you mean by "1-dimensional" here? In what sense is a cat one-dimensional?

 

[Fredrik]

I'm sorry, what do you mean by "1-dimensional" here? In what sense is a cat one-dimensional?

I'm talking about subspaces of the Hilbert space of the "universe", and by "universe" I mean the physical system that the interpretation we're trying to define claims that QM describes. Penrose calls it "the omnium". Pure states are, as always, represented by 1-dimensional subspaces. (Edit: It's of course more common to represent pure states as state vectors, but for all state vectors f and all complex numbers c, cf represents the same state as f. Because of this, the 1-dimensional subspace $\mathbb Cf=\{cf|c\in\mathbb C\}$ is a better representation of the pure state than the state vector f).

 

[martinbn]

Fredrik, I think this way you get something different, not MWI.

 

[Fredrik]

Fredrik, I think this way you get something different, not MWI.

It's definitely not Everett's MWI, but I think many worlds are unavoidable once we make the assumption that the state vector of the universe represents all the properties of the universe. The argument is what I said in the quote in #16.

Is there even such a thing as "Everett's MWI" or "The MWI"? All attempts to recover the Born rule have either failed or required additional assumptions that are essentially equivalent to the Born rule.

 

[JK423]

It's definitely not Everett's MWI, but I think many worlds are unavoidable once we make the assumption that the state vector of the universe represents all the properties of the universe. The argument is what I said in the quote in #16.

Or, to be stated differently, if everything is quantum mechanical (sounds more unambiguous).

 

[stevendaryl]

It's definitely not Everett's MWI, but I think many worlds are unavoidable once we make the assumption that the state vector of the universe represents all the properties of the universe. The argument is what I said in the quote in #16.

Is there even such a thing as "Everett's MWI" or "The MWI"? All attempts to recover the Born rule have either failed or required additional assumptions that are essentially equivalent to the Born rule.

On the other hand, there is a sense in which the Born rule in a limiting sense is built into the definition of the hilbert space for quantum mechanics. Two wave functions \Psi_1(x) and \Psi_2(x) are considered equal, as members of the hilbert space, if \int |\delta \Psi|^2 dx = 0, where \delta \Psi = \Psi_2 - \Psi_1. So if you have a branch of the wave function whose norm is 0, you can ignore it.

One of the arguments I've heard for recovering the Born rule is to consider the quantum problem of recording an infinite sequence of measurements of some experiment that has amplitude \dfrac{1}{\sqrt{2}}. The argument is that the Hilbert space measure of the histories where the frequencies don't equal \dfrac{1}{2} is zero, and so they don't exist (the definition of Hilbert space mods out by things of measure zero).

The mathematics of this derivation is pretty screwy, though. To get a "wave function" that can record an infinite number of measurements, I think you need something like a nonseparable Hilbert space.

 

[JK423]

Regarding the non-circular derivation of Born's rule, Zurek has done serious work on this with the concept "envariance". He seems to have derived Born's rule non-circularly:

http://prl.aps.org/abstract/PRL/v90/i12/e120404
http://pra.aps.org/abstract/PRA/v71/i5/e052105

Are you all aware of this?

 

[Fredrik]

Or, to be stated differently, if everything is quantum mechanical (sounds more unambiguous).

Does it really sound more unambiguous? I don't think it's clear what "is quantum mechanical" means. If we want your statement to mean the same as mine, plus that the entire universe can be assigned a state vector, then I think we would need to explain the words "is quantum mechanical" by a statement like mine.

Without an explanation, your statement could be interpreted as only saying that every subsystem of the universe including the universe itself can be assigned a Hilbert space and a state vector. Since we already know that an atom can be assigned a Hilbert space and a state vector, and don't know if that state vector describes all the properties of that atom, it's not clear that "the universe has a state vector" implies that that state vector describes all the properties of the universe.

I agree that my statement isn't as precise as we'd want it to be, but I think that some ambiguity is unavoidable when we're talking about the meaning of mathematical terms in the theory.

 

[stevendaryl]

One of the arguments I've heard for recovering the Born rule is to consider the quantum problem of recording an infinite sequence of measurements of some experiment that has amplitude \dfrac{1}{\sqrt{2}}. The argument is that the Hilbert space measure of the histories where the frequencies don't equal \dfrac{1}{2} is zero, and so they don't exist (the definition of Hilbert space mods out by things of measure zero).

As I said in another post, there's a similar philosophical problem with understanding the empirical meaning of classical probability. A coin flip having probability \dfrac{1}{2} doesn't mean that when you flip the coin 100 times, you're going to get 50 heads. But you can say that the set of all possible histories in which an infinite sequence of coin flips don't yield 50% heads has measure zero. So you can understand the meaning of "probability 50%" for a single flip in terms of "probability 0" for an infinite sequence of flips.

 

[Fredrik]

Regarding the non-circular derivation of Born's rule, Zurek has done serious work on this with the concept "envariance". He seems to have derived Born's rule non-circularly:

http://prl.aps.org/abstract/PRL/v90/i12/e120404
http://pra.aps.org/abstract/PRA/v71/i5/e052105

Are you all aware of this?

Yes. He uses that the Hilbert space of a composite system is the tensor product of the Hilbert spaces of the subsystems. (I think all of these derivations do). I think the best motivation for that assumption is the argument made by Aerts and Daubechies in 1978. (pdf). I believe that all of their axioms about propositional systems can be derived from the usual Hilbert space version of QM with the Born rule.

 

[JK423]

Does it really sound more unambiguous? I don't think it's clear what "is quantum mechanical" means. If we want your statement to mean the same as mine, plus that the entire universe can be assigned a state vector, then I think we would need to explain the words "is quantum mechanical" by a statement like mine.

Without an explanation, your statement could be interpreted as only saying that every subsystem of the universe including the universe itself can be assigned a Hilbert space and a state vector. Since we already know that an atom can be assigned a Hilbert space and a state vector, and don't know if that state vector describes all the properties of that atom, it's not clear that "the universe has a state vector" implies that that state vector describes all the properties of the universe.

I agree that my statement isn't as precise as we'd want it to be, but I think that some ambiguity is unavoidable when we're talking about the meaning of mathematical terms in the theory.

Yes, ok, we agree. We just ascribe different meaning to the words. When i say "everything is quantum mechanical" i mean that quantum mechanics is "all there is", so there are no extra properties not described by QM.

 

[JK423]

Yes. He uses that the Hilbert space of a composite system is the tensor product of the Hilbert spaces of the subsystems. (I think all of these derivations do). I think the best motivation for that assumption is the argument made by Aerts and Daubechies in 1978. (pdf). I believe that all of their axioms about propositional systems can be derived from the usual Hilbert space version of QM with the Born rule.

So, since you are aware of this, are there any objections against it? Why isn't it considered the ultimate proof that Born's rule can be derived from quantum theory? For some reason you don't accept it.

 

[Fredrik]

So, since you are aware of this, are there any objections against it? Why isn't it considered the ultimate proof that Born's rule can be derived from quantum theory? For some reason you don't accept it.

What I said is the objection. The argument is circular. It's roughly like this:

QM with the Born rule → the rules for propositional systems → QM without the Born rule + tensor products → QM with the Born rule​

What Zurek did is perhaps the ultimate proof of that last implication. But the relevant implication

QM without the Born rule → QM with the Born rule​

has never been proved, and I don't think it can be done.

 

[JK423]

What I said is the objection. The argument is circular. It's roughly like this:

QM with the Born rule → the rules for propositional systems → QM without the Born rule + tensor products → QM with the Born rule​

What Zurek did is perhaps the ultimate proof of that last implication. But the relevant implication

QM without the Born rule → QM with the Born rule​

has never been proved, and I don't think it can be done.

Thank you for the explanation.

But, isn't the mathematical structure, underlying quantum theory, supposed to be taken axiomatically? Why do you say that it's based on Born's rule? For example, i could argue that, based on how the interactions of nature act on subsystems (which is experimentally confirmed) i am led to axiomatically consider the tensor product structure of subsystems. I do not pre-assume Born's rule, it's just the way nature works. Isn't that analogous to asking, "Why Hilbert space?" ?

 

[Demystifier]

JK423, can you reply to my post #26? Thanks!

 

[JK423]

I have! #28. I was waiting for your reply!

 

[Fredrik]

The central idea of a MWI is that the state vector of the universe describes everything that's happening. If it does, then we should be able to explain why experimenters get results that are consistent with the Born rule. To even begin do something like that, we need to add assumptions about how to deal with subsystems. So we add an assumption (the tensor product stuff) that can be derived from the Born rule. Then we can try to recover the Born rule from that.

Zurek has claimed that he has done that. I have been really bothered by the apparent circularity that comes from the fact that the tensor product stuff can be derived using the Born rule. But right now I'm not sure that's even a problem. I need to think about this some more.

There is however another issue with Zurek's derivation and similar attempts, the issue of why states are to be associated with probabilities in the first place. That may be a more serious problem. I need to think about that too.

 

[JK423]

To my current understanding, the tensor product structure (TPS) has nothing to do with Born's rule. Since both Born's rule and TPS are correct, it's natural that you may derive one from the other (since they seem not to be independent), but it's wrong to say that TPS is a consequence of Born's rule since TPS would be true even if Born wasn't born! The opposite would be correct.The mathematical structure (i.e. TPS) i think is more fundamental than empirical rules (i.e. Born's rule).

The real problem, i think, is the other thing that you say, that there has to be some association with probabilities. My understanding ends here, i need to study Zurek's work more to understand what he's done.

 

[Fredrik]

Does the CI assume that the observer is classical (or better, non-quantum mechanical)? I think yes, because if not then we are lead to Everett's view (which simply says that everything is quantum mechanical).

Niels Bohr liked to emphasize that a measurement by definition has a result. The result is indicated by some component of the measuring device. The possible final states of the indicator component must be easily distinguishable by a human. If not, we wouldn't consider what just happened a "measurement". This means that the experiment must make the quantum state of the indicator component for all practical purposes indistinguishable from a classical superposition.

It doesn't mean that the indicator component doesn't have a quantum state. It just means that if there's an experiment in which it behaves in a noticeably non-classical way, we would consider it a specimen, not a measuring device.

These are statements about what sort of thing we would consider a "measurement". They say very little (if anything at all) about the properties of measuring devices or the domain of validity of quantum mechanics.

 

[Fredrik]

...it's wrong to say that TPS is a consequence of Born's rule since TPS would be true even if Born wasn't born! The opposite would be correct.The mathematical structure (i.e. TPS) i think is more fundamental than empirical rules (i.e. Born's rule).

What makes you say that? I don't see any reason to think so.

The tensor product stuff isn't even part of any specific quantum theory. It's just a prescription for how to define new quantum theories from existing ones.

 

[JK423]

Niels Bohr liked to emphasize that a measurement by definition has a result. The result is indicated by some component of the measuring device. The possible final states of the indicator component must be easily distinguishable by a human. If not, we wouldn't consider what just happened a "measurement". This means that the experiment must make the quantum state of the indicator component for all practical purposes indistinguishable from a classical superposition.

It doesn't mean that the indicator component doesn't have a quantum state. It just means that if there's an experiment in which such an indicator component behaves in a noticeably non-classical way, we would consider it a specimen, not a measuring device.

These are statements about what sort of thing we would consider a "measurement". It says very little (if anything at all) about the properties of measuring devices or the domain of validity of quantum mechanics.

It seems like you're right, i confirmed this from a work of Zurek's where he explicitly mentions that

Indeed, since the ultimate components of classical objects are quantum, Bohr emphasized that the boundary must be moveable, so that even the human nervous system could be regarded as quantum, provided that suitable classical devices to detect its quantum features were available.

This is a really strange point of view! Bohr was Everettian without even knowing it .

What makes you say that? I don't see any reason to think so.
The tensor product stuff isn't even part of any specific quantum theory. It's just a prescription for how to define new quantum theories from existing ones.

In nature, particle states interact via TPS. Why is this fact dependent on what you will find IF you make measurements? The quantum state exists and evolves even if you don't measure it, it's independent of the Born's rule. Now, the fact that IF you measure it you will indeed find Born's rule is due to the fact that in nature particles interact the way they do, not the other way around.

 

[Demystifier]

Before trying to answer your question, i need to know if the following hypothesis that i make is correct:
Does the CI assume that the observer is classical (or better, non-quantum mechanical)? I think yes, because if not then we are lead to Everett's view (which simply says that everything is quantum mechanical). If the observer is assumed to be non-quantum mechanical, then doesn't this mean that quantum mechanics fail at some point? Isn't the failure of quantum mechanics, in the description of the observer, in principle testable?

Yes is my answer to all these questions. But I would still like to challenge you to propose a CONCRETE thought (gedanken) experiment where the difference would be seen explicitly.

 

[Demystifier]

I have! #28. I was waiting for your reply!

Sorry, I haven't noticed it. See my reply in the post #55 above!

 

[JK423]

Yes is my answer to all these questions. But I would still like to challenge you to propose a CONCRETE thought (gedanken) experiment where the difference would be seen explicitly.

I would give it a lot of thought if that was the case, but as you can see from the posts above, it's not. Copenhagen interpretation does not say that the observer is non-quantum mechanical. For CI, an observer is quantum if he is being observed but something else if he is observing. All this is so vague that i am not sure what CI is about, and i don't know what to prove! In order to prove something, i need to know the rules.. and the rules seem so vague in CI.

 

[Demystifier]

I would give it a lot of thought if that was the case, but as you can see from the posts above, it's not. Copenhagen interpretation does not say that the observer is non-quantum mechanical. For CI, an observer is quantum if he is being observed but something else if he is observing. All this is so vague that i am not sure what CI is about, and i don't know what to prove! In order to prove something, i need to know the rules.. and the rules seem so vague in CI.

So compared with your first post on this thread, now you have changed your opinion. First you thought that CI is a theory different from MWI, now you think that CI is not even a well defined theory. Am I right?

If so, then I rephrase my challenge. Propose a thought experiment for which the measurable predictions of MWI are unambiguous, while those of CI are not!

 

[JK423]

So compared with your first post on this thread, now you have changed your opinion. First you thought that CI is a theory different from MWI, now you think that CI is not even a well defined theory. Am I right?

If so, then I rephrase my challenge. Propose a thought experiment for which the measurable predictions of MWI are unambiguous, while those of CI are not!

Yes you're right. To my current understanding CI is just MWI without explicitely saying it.. Otherwise it's nonsense.
How is it possible that an observer collapses wavepackets, but when observed he is quantum and he doesn't actually collapse anything, just unitary evolution?
If you accept that QM holds universally, even to observers, then all this is nonsense.

I need you to give me a clear definition of what CI is.. Can you? Then i'll try to take on your challenge
This is necessary, because otherwise i may assume something that leads to the response "hey, CI doesn't say that".

 

[Fredrik]

I need you to give me a clear definition of what CI is.. Can you? Then i'll try to take on your challenge
This is necessary, because otherwise i may assume something that leads to the response "hey, CI doesn't say that".

There is no definition of the CI that wouldn't make a lot of people go "hey, CI doesn't say that".

Yes you're right. To my current understanding CI is just MWI without explicitely saying it.. Otherwise it's nonsense.

It's clear that you're making the assumption that QM describes what's happening to the system even at times between state preparation and measurement. There's nothing in QM that forces us to make that assumption. I would say that this assumption is the starting point of a definition of a MWI, so what you're saying sounds to me like "if we assume the MWI, then it's nonsense to also assume something that contradicts it". This is obviously true, but I need to point out that you're making an assumption that isn't necessary.

It's also possible that QM isn't a description of what's actually happening. I would take that as the definition of the CI, and also as the definition of an ensemble interpretation, because that just seems to be a different way to say the same thing. This interpretation says that QM is just a set of rules that assigns probabilities to possible results of experiments. Edit: Since it doesn't make any claims about what is "actually happening", it's questionable if it should be called an interpretation at all.

How is it possible that an observer collapses wavepackets, but when observed he is quantum and he doesn't actually collapse anything, just unitary evolution?

Unitary evolution only applies to systems that are isolated from their environments. Observers, by definition, are not. If a system has an environment A that's isolated from its environment B, then "system+A" evolves unitarily. This sort of thing is taken into account in decoherence calculations.