Why MWI cannot explain the Born rule

[Demystifier]

If you allow that every branch has an amplitude and the "likelihood" is in some way monotonically related to the absolute value of the amplitude, you can naturally deduce that the relationship has to take form of the Born rule, $P_i = |c_i|^2 / \sum_k |c_k|^2$.

The question is: WHY would likelihood be monotonically related to the absolute value of the amplitude?

Let me use a classical analogy. Assume that we have two houses: a big one with 100 apartments, and a small one with only 2 apartments. These two houses are analogs of two worlds.

Now we ask the question: Which house is more probable, the big one or the small one? It is not even clear what this question means. Nevertheless, one meaningfull answer is that both houses are equally probable. On the other hand, the Born rule corresponds to the claim that the big house is 50 times more probable than the small one. But if the two houses are ALL we have, then it is difficult to justify such a claim.

Now consider a modified question: What is the probability that Jack lives in the big house? Now it is reasonable to assume that each APARTMENT has equal a priori probability to be the Jack's home, which implies that probability that Jack lives in the big house is 50 times larger than probability that Jack lives in the small house. In other words, by introducing an additional "hidden" variable (called Jack) that lives in one of the worlds, the Born rule attains a natural explanation. Needless to say, Jack is an analog of the pointlike particle in the Bohmian interpretation.

 

[pellman]

You are right. However, most literature on that issue is not very easy to read. My intention is to present and discuss this issue in a simpler way, so that all physicists can easily understand it.

Looking forward to reading it. If you do write that paper, shoot me a PM or email when its done.

 

[Demystifier]

Looking forward to reading it. If you do write that paper, shoot me a PM or email when its done.

Well, I didn't say that I will write a paper. But I didn't say that I will not write it either.

By the way, what do you think about my analogy in #71? In my view, this IS the simplest way (though certainly not the most rigorous one) to explain my opinion on that stuff.

 

[Hurkyl]

Let me use another classical analogy.

Suppose we have an ordinary six-sided die. I ask the question "are the sides equally likely?" What does that question really mean?

Of course, I'm sure you have a ready answer for that, based on the classical frequentist interpretation of probabilities -- mutter something about propensities, repeated experiments, argue that, for each positive epsilon, the propensity of "The proportion of times '1' comes up is in $(1/6 - \epsilon, 1/6 + \epsilon)$" converges to 1 as the number of experiments goes to infinity, and then we go and do controlled empiricial trials until we're satisfied we're close close enough to infinity to believe the answer.


I offer the following theorem:

Let $\psi$ be a state a Hilbert space $H$. Let $\psi^{(n)} := \psi \otimes \psi \otimes \cdots \otimes \psi$ (n times).

Let $M_{n, p, \epsilon}$ be a "toy frequency counting operator" on $H^{(n)} \otimes \mathbf{C}^2$. (C² is the space of pure qubit states) -- for some basis {e_m} of H, M is given by

• If the proportion of the $a_k$ equal to 1 is in $(p - \epsilon, p + \epsilon)$, then
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m \oplus 1\rangle$$
• Otherwise,
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle$$

We can construct a function $f_{p, \epsilon, \psi}$ that maps each positive integer n to the state space of a qubit given by taking the partial trace of $M \left(\psi^{(n)} \otimes |0\rangle)$ -- heuristically, this function tells us the (probably mixed!) quantum state of the result of toy measurement if we do n trials.

Finally, define $P(e_n ; \psi) = p$ if and only if, for every positive epsilon, we have $\lim_{n \to +\infty} f_{p, \epsilon, \psi}(n)$ is the pure state $|1\rangle$.

Theorem: P as defined above is given by the Born rule.


Is the above mildly complex? Sure -- but if you aren't going to handwave over everything, isn't the corresponding argument in classical mechanics similarly complex?

 

[pellman]

By the way, what do you think about my analogy in #71?

Not entirely sure I follow it. Is the analogy such that it corresponds to an observation/experiment in which result BIG HOUSE is 50/2 times more likely than result LITTLE HOUSE?

In other words, by introducing an additional "hidden" variable (called Jack) that lives in one of the worlds, the Born rule attains a natural explanation. Needless to say, Jack is an analog of the pointlike particle in the Bohmian interpretation.

Are we going for the Many-worlds-Bohmian interpretation now? Ladies and gentlemen, the captain has now turned on the "fasten seat belt" sign. Passengers should return to their seats.

 

[Fredrik]

Let me use another classical analogy.

Suppose we have an ordinary six-sided die. I ask the question "are the sides equally likely?" What does that question really mean?

Of course, I'm sure you have a ready answer for that, based on the classical frequentist interpretation of probabilities -- mutter something about propensities, repeated experiments, argue that, for each positive epsilon, the propensity of "The proportion of times '1' comes up is in $(1/6 - \epsilon, 1/6 + \epsilon)$" converges to 1 as the number of experiments goes to infinity, and then we go and do controlled empiricial trials until we're satisfied we're close close enough to infinity to believe the answer.

I do have an answer, but it's not that one. I think both the question and that answer are unscientific at best, and nonsense at worst. The question is asking for the correct definition of something undefined. That can't ever make sense. And that answer asserts the existence of a "limit" which is "defined" by a description of an act that can't be performed. That's just more nonsense.

This is the short version of my answer: Probability in mathematics is a number assigned by a probability measure. If we try to associate such a number with something in the real world, we're no longer doing pure mathematics. We're either doing science or pseudoscience.

A scientific theory associates numbers assigned by a probability measure with possible results of experiments. The definition of science requires that we test theories by comparing those numbers to relative frequencies in the results of a large but finite number of almost identical experiments.

If someone claims that probability "is" something more than this, they're making a statement about the real world, not about mathematics, and that means that the statement must satisfy the definition of a theory to be worthy of further consideration.

See also #11 in this thread, and my posts in the thread I linked to there.

I offer the following theorem:

Let $\psi$ be a state a Hilbert space $H$. Let $\psi^{(n)} := \psi \otimes \psi \otimes \cdots \otimes \psi$ (n times).

Let $M_{n, p, \epsilon}$ be a "toy frequency counting operator" on $H^{(n)} \otimes \mathbf{C}^2$. (C² is the space of pure qubit states) -- for some basis {e_m} of H, M is given by

• If the proportion of the $a_k$ equal to 1 is in $(p - \epsilon, p + \epsilon)$, then
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m \oplus 1\rangle$$
• Otherwise,
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle$$

We can construct a function $f_{p, \epsilon, \psi}$ that maps each positive integer n to the state space of a qubit given by taking the partial trace of $M \left(\psi^{(n)} \otimes |0\rangle)$ -- heuristically, this function tells us the (probably mixed!) quantum state of the result of toy measurement if we do n trials.

Finally, define $P(e_n ; \psi) = p$ if and only if, for every positive epsilon, we have $\lim_{n \to +\infty} f_{p, \epsilon, \psi}(n)$ is the pure state $|1\rangle$.

Theorem: P as defined above is given by the Born rule.

I don't understand what you're doing here, but see #13 in the thread I linked to above for a comment about the 1968 article by Jim Hartle that (I think) introduced frequency operators.

 

[Count Iblis]

The argument by Hartle allows you to replace the Born rule by the weaker rule that says that measuring an observable of a system if the system is in an eigenstate of that observable, will yield the corresponding eigenvalue with certainty.

 

[Fredrik]

The argument by Hartle allows you to replace the Born rule by the weaker rule that says that measuring an observable of a system if the system is in an eigenstate of that observable, will yield the corresponding eigenvalue with certainty.

I don't think that's correct. This is what some Wikipedia article said, right? I suspect that whoever came to that conclusion believed that every system is always in an eigenstate. (An idea which as you know has been thoroughly disproved by Bell inequality violations). It seems to me that all Hartle did was to prove that for each eigenstate |k>, there's an operator that has every $|s\rangle\otimes|s\rangle\otimes\cdots$ (where |s> is an arbitrary state) as an eigenvector with eigenvalue $|\langle k|s\rangle|^2$.

It's natural to interpret this operator as a frequency operator when it acts on a tensor product of eigenstates, but it doesn't seem to have a natural interpretation when it's acting on $|s\rangle\otimes|s\rangle\otimes\cdots$ with |s> arbitrary. The only way I can justify the interpretation as a frequency operator is to use the Born rule, i.e. the rule he's trying to prove.

I would also say that Hartle used the Born rule before that, when he assumed that the Hilbert space of a system is the tensor product of the Hilbert spaces of its subsystems. I don't think that assumption can be justified without appealing to the Born rule.

So my opinion is that Hartle's article is completely useless. If I'm wrong, I'd like to know it, so feel free to try to convince me. Post #18 in the other thread (linked to above) has a direct link to the article. My comments are in #13.

 

[Demystifier]

Not entirely sure I follow it. Is the analogy such that it corresponds to an observation/experiment in which result BIG HOUSE is 50/2 times more likely than result LITTLE HOUSE?

No, but 100/2=50.

Are we going for the Many-worlds-Bohmian interpretation now?

Not necessarily, but why not to mention it if is related to MWI and to the Born rule.

 

[pellman]

No, but 100/2=50.

Right. That's what I meant.

Not necessarily, but why not to mention it if is related to MWI and to the Born rule.

No reason. I was kidding. I lean towards it myself.

 

[pellman]

Let $\psi$ be a state a Hilbert space $H$. Let $\psi^{(n)} := \psi \otimes \psi \otimes \cdots \otimes \psi$ (n times).

Let $M_{n, p, \epsilon}$ be a "toy frequency counting operator" on $H^{(n)} \otimes \mathbf{C}^2$. (C² is the space of pure qubit states) -- for some basis {e_m} of H, M is given by

• If the proportion of the $a_k$ equal to 1 is in $(p - \epsilon, p + \epsilon)$, then
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m \oplus 1\rangle$$
• Otherwise,
  $$M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle = M|e_{a_1}, e_{a_2}, \ldots, e_{a_n}, m\rangle$$

We can construct a function $f_{p, \epsilon, \psi}$ that maps each positive integer n to the state space of a qubit given by taking the partial trace of $M \left(\psi^{(n)} \otimes |0\rangle)$ -- heuristically, this function tells us the (probably mixed!) quantum state of the result of toy measurement if we do n trials.

Finally, define $P(e_n ; \psi) = p$ if and only if, for every positive epsilon, we have $\lim_{n \to +\infty} f_{p, \epsilon, \psi}(n)$ is the pure state $|1\rangle$.

Theorem: P as defined above is given by the Born rule.

I might be able to understand this with a little help. What are the $$a_k$$?

 

[Count Iblis]

I don't think that's correct. This is what some Wikipedia article said, right? I suspect that whoever came to that conclusion believed that every system is always in an eigenstate. (An idea which as you know has been thoroughly disproved by Bell inequality violations). It seems to me that all Hartle did was to prove that for each eigenstate |k>, there's an operator that has every $|s\rangle\otimes|s\rangle\otimes\cdots$ (where |s> is an arbitrary state) as an eigenvector with eigenvalue $|\langle k|s\rangle|^2$.

It's natural to interpret this operator as a frequency operator when it acts on a tensor product of eigenstates, but it doesn't seem to have a natural interpretation when it's acting on $|s\rangle\otimes|s\rangle\otimes\cdots$ with |s> arbitrary. The only way I can justify the interpretation as a frequency operator is to use the Born rule, i.e. the rule he's trying to prove.

I would also say that Hartle used the Born rule before that, when he assumed that the Hilbert space of a system is the tensor product of the Hilbert spaces of its subsystems. I don't think that assumption can be justified without appealing to the Born rule.

So my opinion is that Hartle's article is completely useless. If I'm wrong, I'd like to know it, so feel free to try to convince me. Post #18 in the other thread (linked to above) has a direct link to the article. My comments are in #13.

There is another discussion of Hartle's arguments http://arxiv.org/abs/hep-th/0606062" where it is suggested that you need to introduce a discrete Hilbert space to fix the derivation. The problem they address is the fact that the number of factors in the tensor product is always finite in practice, so the a-typical branches don't exactly get a zero norm.

About the assumptions, I don't think one assumes the Born rule, only the L^2 norm. The notion of a probability is eliminated from the postulates. The MWI is supposed to be a deterministic theory, so talking about probabilities in the postulates is unnatural. Instread, you can attempt to replace them by statements about certainties, like that measuring a system in an eigentate will yield a certain outcome with certainty.
This combined with cutting away zero norm sectors of Hilbert space gives you the Born rule.

 

[Fredrik]

There is another discussion of Hartle's arguments http://arxiv.org/abs/hep-th/0606062" where it is suggested that you need to introduce a discrete Hilbert space to fix the derivation.

They don't address the issues I mentioned, so they're probably not aware of them. If they're right about the need for a discrete state space, they've just found another problem with an article that was already useless. It also seems to me that this article is much worse than Hartle's. Their summary of Hartle's argument looks like complete crackpot nonsense to me.

This isn't meant as criticism against you. I'm getting a bit upset about the fact that this article got published, but I don't think you had anything to do with that. I actually hope I'm wrong about this article, because it's quite sad if articles as bad as I think this one is can get published.

About the assumptions, I don't think one assumes the Born rule, only the L^2 norm.

How does that justify the use of the tensor product, or the interpretation of Hartle's operator as a frequency operator?

The MWI is supposed to be a deterministic theory, so talking about probabilities in the postulates is unnatural.

You get a deterministic model if you just remove the Born rule, but it's not a theory since we don't have a way to interpret the mathematics as predictions about results of experiments. It also doesn't include anything about "worlds", since the Born rule is what tells us that there's something in the mathematics that we can think of as worlds. It doesn't even include a description in terms of subsystems, since the Born rule is what justifies the use of the tensor product.

Instread, you can attempt to replace them by statements about certainties, like that measuring a system in an eigentate will yield a certain outcome with certainty.

I haven't seen any reason to believe that. I haven't even seen an attempt to prove it. I just have a vague memory of seeing that claim in a Wikipedia article that I wasn't able to find when I tried yesterday, and that used Hartle's article as a reference. But I don't think Hartle's results could be described that way even if his argument had been valid.

This combined with cutting away zero norm sectors of Hilbert space gives you the Born rule.

There are already no such sectors in the individual system Hilbert space, so they're clearly talking about the tensor product of infinitely many copies of that space. The removal of the zero norm "vectors" that get included by accident when we take the tensor product of infinitely many copies of a Hilbert space is necessary to ensure that the result is a Hilbert space. It doesn't have anything to do with the Born rule.

 

[PTM19]

The minimal set of assumptions defining MWI is:
1. Psi is a solution of a linear deterministic equation.
2. Psi represents an objectively real entity.

This is of course not enough to define MWI, for that one has to redefine what "objectively real entity" means. Normal definition clearly doesn't include parallel universes (If I sold you an "objectively real" car and later claimed it's in a parallel universe you would likely be upset). Without parallel universes the interpretation based on above assumptions is simply ruled out by experiment. To avoid being ruled out MWI requires a lot of additional postulates which add parallel unobservable worlds and so on.

The point being that MWI is in no sense a minimal interpretation - on the contrary it is the most extremely baroque of interpretations, as it necessitates belief in an immense number of unobservable universes, what's more it cannot even be falsified making it unscientific. MWI is the most extreme violation of Ockham's razor principle one can imagine.

The minimal interpretation is an ensemble interpretation (without PIV, described here for example (thx for link Fredrik) https://www.physicsforums.com/showthread.php?t=360268&page=1) since it doesn't require one to believe in anything beside what can be experimentally verified and agrees with all experimental evidence.

 

[RUTA]

The problem I have with MWI is that you can’t know whether your physics is legit. Suppose the “real” experimental distribution is 50-50, someone occupies a universe that is 100-0 and someone else 0-100 (with others at all possible distributions in between). So, theoretical physicists in those universes (and most others in between) who predict the “real” 50-50 distribution will not get tenure in physics, but will have to try for tenure in philosophy or religious studies. Physicists in those universes who predict the empirical, but “wrong,” results are heralded and rewarded. I don’t understand why anyone in physics would subscribe to the MWI ontology.

 

[Hurkyl]

To avoid being ruled out MWI requires a lot of additional postulates

 

[Fredrik]

The point being that MWI is in no sense a minimal interpretation - on the contrary it is the most extremely baroque of interpretations, as it necessitates belief in an immense number of unobservable universes,

I addressed those claims in my reply to you in the thread in the philosophy forum (and also earlier in this thread).

what's more it cannot even be falsified making it unscientific.

True, but the same thing holds for every claim that some theory describes what actually happens. This is because experiments can't tell us anything except how accurate a theory's predictions are.

MWI is the most extreme violation of Ockham's razor principle one can imagine.

If there's anything I'm sure about here, it's that this claim is completely wrong.

The minimal interpretation is an ensemble interpretation...

See my comment in the philosophy forum.

 

[PTM19]

For example it has to postulate the existence of parallel universes and that those universes are unobservable. The concept and existence of parallel universes certainly does not follow from those two axioms, it has to be postulated in their interpretation.

 

[Fredrik]

For example it has to postulate the existence of parallel universes and that those universes are unobservable. The concept and existence of parallel universes certainly does not follow from those two axioms, it has to be postulated in their interpretation.

Those two axioms don't even say anything about one world, because they don't make any predictions about (probabilities of) results of experiments.

 

[dmtr]

The problem I have with MWI is that you can’t know whether your physics is legit. Suppose the “real” experimental distribution is 50-50, someone occupies a universe that is 100-0 and someone else 0-100 (with others at all possible distributions in between). So, theoretical physicists in those universes (and most others in between) who predict the “real” 50-50 distribution will not get tenure in physics, but will have to try for tenure in philosophy or religious studies. Physicists in those universes who predict the empirical, but “wrong,” results are heralded and rewarded. I don’t understand why anyone in physics would subscribe to the MWI ontology.

I don't see any problem, as long as the symmetric apparatus gives you symmetric probabilities (probability amplitudes to be precise). Let me illustrate this with this simple example: I'll go to the website generating random numbers from the physical origin and generate two groups of numbers from 0 to 1. After that I'll sum them. Now 0 would correspond to 0:100; 1 would correspond to 50:50 and 2 would correspond to 100:0.

Now, you wouldn't be surprised to see some zeros, and twos? Right?
Here are the numbers:
GR1: 1 0 0 1 0 0 0 1 1 1
GR2: 0 1 1 0 0 0 0 0 1 0
SUM: 1 1 1 1 0 0 0 1 2 1

 

[Count Iblis]

Fredrik:

There are already no such sectors in the individual system Hilbert space, so they're clearly talking about the tensor product of infinitely many copies of that space. The removal of the zero norm "vectors" that get included by accident when we take the tensor product of infinitely many copies of a Hilbert space is necessary to ensure that the result is a Hilbert space. It doesn't have anything to do with the Born rule.

You get the Born rule from this by considering the "frequency operator". Only the states that have the correct statistics have a non-zero norm and they are eigenvectors of the operator. So, the "certainty rule" then implies that you will observe the statistics as given by the Born rule (at least when you consider an infinite numbers of copies of the system).

 

[RUTA]

I don't see any problem, as long as the symmetric apparatus gives you symmetric probabilities (probability amplitudes to be precise).

You don't know if it's symmetric or not. You can't know. All you know is the history of your particular branch. You don't know if the statistics you observe are the overall "correct" statistics, but you can guess they're probably not.

 

[dmtr]

You don't know if it's symmetric or not. You can't know. All you know is the history of your particular branch. You don't know if the statistics you observe are the overall "correct" statistics, but you can guess they're probably not.

But I do observe the "correct" statistics. And it corresponds nicely to the symmetric apparatus. In the example above there is roughly the same number of '0's and '1's in the original sequences. I can also see that the SUM: 1 1 1 1 0 0 0 1 2 1 contains more '1's than '0's (or '2's). This corresponds nicely to the: '50/50' world is more probable than 100/0 (or 0/100).

And it doesn't look like this "correct" statistics excludes MWI in any way. Why I-who-saw-"1" should consider myself any better from I-who-saw-"0"? From the symmetry considerations I shouldn't.

 

[RUTA]

But I do observe the "correct" statistics. And it corresponds nicely to the symmetric apparatus. In the example above there is roughly the same number of '0's and '1's in the original sequences. I can also see that the SUM: 1 1 1 1 0 0 0 1 2 1 contains more '1's than '0's (or '2's). This corresponds nicely to the: '50/50' world is more probable than 100/0 (or 0/100).

And it doesn't look like this "correct" statistics excludes MWI in any way. Why I-who-saw-"1" should consider myself any better from I-who-saw-"0"? From the symmetry considerations I shouldn't.

You're missing the point. Your results are only one path in the bifurcated tree. You have no idea where your particular history resides in the tree because you only have access to your single history.

 

[dmtr]

You're missing the point. Your results are only one path in the bifurcated tree. You have no idea where your particular history resides in the tree because you only have access to your single history.

I don't see how is that different from say, your position in the spatial dimension. You only have access to the single position, yet that doesn't stop you from using symmetries, homogeneity or isotropy.

 

[Demystifier]

For example it has to postulate the existence of parallel universes and that those universes are unobservable. The concept and existence of parallel universes certainly does not follow from those two axioms, it has to be postulated in their interpretation.

This is simply wrong. The existence of parallel universes DOES follow from those two axioms. But you cannot understand it without understanding the theory of decoherence (which, by the way, is not only a theory, but also an experimental fact.)

 

[yossell]

The existence of parallel universes DOES follow from those two axioms.

Just a reminder of the axioms:
1. Psi is a solution of a linear deterministic equation.
2. Psi represents an objectively real entity.

Existence of many worlds simply follow from these - can't even imagine how the derivation would go unless you're packing a lot of background ASSUMPTIONS into what counts as an 'objectively real entity', or unless you mean something much less by 'many worlds' than the literal existence of many worlds. Already given example of mathematical platonist who believes psi is a solution of linear deterministic equation, and who believes psi reps an objectively real entity - but no many worlds theorist. Still waiting for the proper argument to be elaborated on.

 

[Hurkyl]

Hrm. A question to make sure everyone is on the same page.



Let's say I'm using coordinates and arithmetic as a means of studying Euclidean geometry.

Would you say that this does, or does not, require more assumptions than just the postulates of Euclidean geometry?

Would you say that the arithmetic properties of coordinates do, or do not, follow from the axioms of Euclidean geometry

 

[yossell]

Let's say I'm using coordinates and arithmetic as a means of studying Euclidean geometry.

Would you say that this does, or does not, require more assumptions than just the postulates of Euclidean geometry?

Would you say that the arithmetic properties of coordinates do, or do not, follow from the axioms of Euclidean geometry

Interesting question - I'm not sure where you're going with this but, spidey sense tingling and feeling there's a trap... I'm going to hedge an answer:

Some presentations use coordinate systems in their actual presentation of Euclidean geometry - they're built in at the start. But Euclid himself (I understand) used geometric rather than arithmetic predicates - in his system, he talked directly of lines, points and planes rather than triples of reals. Hilbert, Foundations of Geometry, gives a more modern version, involving predicates of betweenness and congruence, and modified to allow for things like the completeness of the real line to be formulated - something I'm not sure the original Euclidean system was capable of formulating.

If, by 'axioms of Euclidean geometry' you had in mind a formulation which involved only such geometric predicates, then my first reaction is to say that we did need more than just Euclidean geometry - we would also need axioms setting up the properties of real numbers, and triples of real numbers, and something linking the axioms about reals with the axioms about lines and points.

The reason, however, that I hedge, is because I believe that in Hilbert's formulation of Euclidean geometry, one can "model" the numbers and do number theory: it's possible to "define" + and x (in the sense that they have the right logical properties). If you're some kind of structuralist about numbers, this might all one needs to think one can have coordinates in Eucidean geometry. But off the top of my head, I'm not sure how much coordinate geometry can be recovered in this way.

 

[Hurkyl]

But off the top of my head, I'm not sure how much coordinate geometry can be recovered in this way.

The theory of real numbers turns out to be equivalent to the theory of Euclidean geometry, with one direction working pretty much just as you described.



(disclaimer: I am a mathematician not a physicist, and I don't have a particularly deep knowledge of what I'm going to say, so take it with a grain of salt)

Everything I've understood about MWI works in the same way -- "parallel worlds" and such are simply ideas built out of the quantum mechanical state space, which are used to describe the behavior of states.

(and they could also be used to study arbitrary vector spaces, or state spaces of algebras, and so forth -- but I have absolutely no idea if it would ever be a useful thing to do)


The only new supposition of MWI is that the process of describing reality by quantum states evolving unitarily might continue to be applicable when we include measuring devices and observers into the quantum system -- something interpretations like Copenhagen reject a priori. Since this is a novel and powerful feature, a lot of effort is put into studying it. But if it really cannot be done, MWI would still be applicable to the domains where quantum states evolving unitarily works.



The reason I ask my question is I agree that MWI is more complex -- without simplifying assumptions like wavefunction collapse, it is technically more complicated, thus prompting new theoretical concepts to study it, and a lot of people equate these new theoretical concepts with extra assumptions and multiplying entities. (But I think that point of mine is not directed at you, but at others in the thread. I think the part of this most applicable to you is the gray parenthetical in the middle)

 

[RUTA]

I don't see how is that different from say, your position in the spatial dimension. You only have access to the single position, yet that doesn't stop you from using symmetries, homogeneity or isotropy.

It's very different because you have access to information from the space that surrounds you. You have no such access to "other" universes, by definition (if you have access, they're not "other," they're part of this one).

 

[Fredrik]

The existence of parallel universes DOES follow from those two axioms. But you cannot understand it without understanding the theory of decoherence (which, by the way, is not only a theory, but also an experimental fact.)

The thing is, decoherence uses more than those two axioms. It uses the Born rule implicitly, by taking the Hilbert space to be a tensor product, and by computing the "reduced" density matrix as a partial trace of the state operator of the universe.

Without the possibility to do decoherence calculations, the only way to define the worlds is to say that given a basis (any basis) for the Hilbert space of the universe, each basis vector represents a world. To go beyond that, we need the Born rule, and a way to express the Hilbert space as a tensor product. Those things make decoherence a meaningful concept.

I have previously said that decoherence defines the worlds. I no longer think that that's the most appropriate way to define the worlds. What decoherence does is to single out a basis that defines interesting worlds. If my understanding of decoherence ideas is accurate (it might not be), any other basis defines worlds where the subsystems can't contain stable records of the states of other subsystems (such as a memory in the brain of a physicist). If well-defined memory states is an essential part of what consciousness is, the worlds identified by decoherence are the only ones that can contain conscious observers.

 

[yossell]

Thanks Hurkyl

I'm neither a mathematician nor a physicist - so take what I say with a siberian salt mine. And I think I take the main point in your post.

Everything I've understood about MWI works in the same way -- "parallel worlds" and such are simply ideas built out of the quantum mechanical state space, which are used to describe the behavior of states.

Maybe this is just my own conceptual stumbling block, but I have problems seeing the parallel (no pun intended) clearly.

The ontology and ideology of Hilbert's axiomatisation of geometry is clear - the quantifiers of the theory range over points and regions and there are primitive predicates 'between' and 'congruent'. I take these predicates to be (reasonably) physically clear and meaningful, that gives the theory its *physical* content and makes it more than pure maths or logic.

From a purely logical point of view, of course, we don't really care what these predicates mean - the predicates may be replaced with formal letters xByz and xyCongzw for all we care. But if we want the theory to have more than formal properties then (I think) it is because 'point' and 'line' and 'region' correspond to genuine physical geometric objects, and 'Between', and cong to genuine physically geometric relations that we can see the theory as having some genuine physical content.

Given this, when we embed arithmetic into our Hilbertian theory is finding geometric structures which are isomorphic (in the clear model-theoretic sense) to mathematical ones. A statement in the language of Peano arithmetic becomes equivalent to a statement about geometric lines and regions. A mathematical statement, a statement in the language of Peano arithmetic, gets reintepreted as a statement about geometric entities.

Now, when it comes to mathematical and logical issues, I think this kind of thing is probably fine, because it's not clear that there's anything to our conception of mathematical objects other than something formal or structural.

But it's not clear to me how things go when we're dealing with terms that are supposed to have physical significance. The worry is this: the interpretation of the terms can play a role in solving the relevant problem and so matters of interpretation need to be tracked in a way that they don't in the more formal cases.

For instance: a no-collapse theorist may want to explain how it is that we experience a cat which is determinately dead or alive, even though, by his lights, there is no collapse. He solves his problem by postulating many worlds - there are literally two cats and two observers, each one having experience a cat in a determinate state. (Not defending this move, just noting that it offers a solution to a problem). The many worlds, the many observers, the many cats - they may not be fundamental, but they have to be there for this version of the solution to go through. But if all we're doing is dropping the collapse postulate, then I'm not sure where many anythings come in. And then, this particular solution of this problem would not be available to him.

 

[Fredrik]

Hurkyl: The axioms we're discussing aren't mathematical axioms. They are statements that describe how things in a mathematical model correspond to things and the real world. So I don't really see the point of your analogy with Euclidean geometry.

 

[Fredrik]

You get the Born rule from this by considering the "frequency operator". Only the states that have the correct statistics have a non-zero norm and they are eigenvectors of the operator. So, the "certainty rule" then implies that you will observe the statistics as given by the Born rule (at least when you consider an infinite numbers of copies of the system).

I decided to take another look at the article you referenced. First they're saying that the Born rule tells us that the probability of a sequence of measurement results [itex]i_1,\dots,i_N[/tex] is

$$|\langle i_1|s\rangle|^2\cdots|\langle i_N|s\rangle|^2=|\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle|^2$$

Then they're saying that "Everett noted" that "it follows" that the probability of a particular sequence [itex]i_1,\dots,i_N[/tex] is low if $|\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle|$ is small, so they're giving Everett credit for knowing how to take the square root of the square of a positive real number.

Then they claim that in the "formal limit" N→∞, $|\langle i_1|\otimes\cdots\otimes\langle i_N|\ |s\rangle\otimes\cdots\otimes|s\rangle|\rightarrow 0$ if the sequence is not statistically typical. What does that even mean? If |s> isn't orthogonal to any of the eigenstates, we're just talking about a product of N numbers in the open interval (0,1) in the N→∞ limit (or whatever they have in mind when they say "formal limit"). Is it even possible for the result not to go to 0? Maybe if the Nth factor goes to 1 as N→∞ limit, but in that case the probabilities of the possible results in a single experiment don't add up to 1, and the |s> we started with has infinite norm!

So after just using the Born rule, and then saying that "this implies"...something that appears to be complete nonsense, they claim that this means that "the Born rule is a consequence of excluding zero norm states from the Hilbert space"! (Note that the definition of a Hilbert space already excludes them).

Are you saying this is not nonsense? As always, if I have misunderstood something, I'd like to know.

I notice that you didn't adress any of my arguments (1, 2) against the whole frequency operator approach. As far as I can tell, it's completely circular even for finite tensor products, and the N→∞ limit isn't going to make that problem go away.