In what sense does MWI fail to predict the Born Rule?

[DarMM]

Why do you think WMI cannot just assume the Born rule, interpreted as a measure of existence?

Let me be clear here. I am discussing the form of Many Worlds advocated by Wallace, Deutsch, Zurek (who no longer advocates it) and others, which is basically just the Hilbert Space of states and unitary evolution with the correct Hamiltonian. The claim being that this contains in some sense the Born rule as an effective subjective rule for observers. I find the current attempts at showing this mathematically circular.

I'm not really discussing versions of MWI where the Born rule is assumed, hence the focus on the proofs and I think from the thread title they wouldn't be the topic (they can't fail to predict the Born rule and the proofs don't concern them). I do have some thoughts on them, but perhaps another thread.

EDIT: I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

 

[akvadrako]

Let me be clear here. I am discussing the form of Many Worlds advocated by Wallace, Deutsch, Zurek (who no longer advocates it) and others, which is basically just the Hilbert Space of states and unitary evolution with the correct Hamiltonian. The claim being that this contains in some sense the Born rule as an effective subjective rule for observers. I find the current attempts at showing this mathematically circular.

I'm not really discussing versions of MWI where the Born rule is assumed, hence the focus on the proofs and I think from the thread title they wouldn't be the topic (they can't fail to predict the Born rule and the proofs don't concern them). I do have some thoughts on them, but perhaps another thread.

EDIT: I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

Thanks for the clarification, but I don't really think we are talking about different theories here. Work done in WMI uses the Born rule for calculations as always. Instead they are attempting to find the most acceptable axiomatic basis and a conceptual way to introduce probabilities into a deterministic theory. I would say those physicists thought they could derive the Born rule with a certain set of axioms; in at least some cases their derivations had flaws. But they didn't "have" to use those axioms and it turns out most of the derivations apply equally well to collapse theories.

Also, since all the derivations we have mentioned include more assumptions than Hilbert spaces and unitary evolution, it doesn't make much sense to say they are promoting it without extra assumptions.

 

[stevendaryl]

I will say that a common "promotion" of Many-Worlds is that it is just Unitary QM without extra assumptions, so I do think this is a fairly common version of MWI. As Kent's list above shows though there are many versions of MWI and I think it can be confusing as these are always presented as one interpretation.

I think that the Born rule doesn't by itself contradict the assumption that evolution is purely unitary. What you don't have, with purely unitary evolution, is the "collapse" after an observation is made.

 

[DarMM]

Thanks for the clarification, but I don't really think we are talking about different theories here.

I think different axioms are different theories, if you have different fundamental base assumptions, I would view that as a separate version of MWI. For example Zurek's version of MWI has no concepts like "Weights of Existence". Regardless of this point, I think a theory which derives the Born rule as an effective result is quite different from one where it is a base feature, regardless of if they share another feature, i.e. multiple classical worlds.

Also, since all the derivations we have mentioned include more assumptions than Hilbert spaces and unitary evolution, it doesn't make much sense to say they are promoting it without extra assumptions.

Wallace describes that as one of the virtues of Many-Worlds in his book. My point isn't how much sense it makes, it's that it is a claim made. He also doesn't view the assumptions in his proof as assumptions, more details of pure unitary MWI to be filled in, i.e. there will eventually be a pure unitary proof of decoherence.

I think it makes sense to say it as it is literally claimed in books and papers by MWI proponents.

I think it is a mistake to notice the extra assumptions and think this means they must not be claiming it is only the Hilbert Space + Unitary evolution part of QM. Unsubstantiated claims can easily be made, or people can view their assumptions as "obvious" or "trivial" details (or yet to be proved Lemmas), when in fact they change the physical picture.

This is also a widespread misunderstanding see this stack exchange question and its accepted answer:
https://physics.stackexchange.com/q...on-mwi-cannot-derive-the-born-rule-would-that

It is a commonly held and promoted view of MWI.

 

[DarMM]

I think that the Born rule doesn't by itself contradict the assumption that evolution is purely unitary. What you don't have, with purely unitary evolution, is the "collapse" after an observation is made.

I don't necessarily think it contradicts unitary evolution either, there's just no proof it's a consequence of unitary evolution.

 

[akvadrako]

I think different axioms are different theories, if you have different fundamental base assumptions, I would view that as a separate version of MWI.

So if I had a theory which assumed $a = b$ and $b = c$ and proved $a = c$, then I switched an axiom with the proof, would you call that a different theory?

I think it is a mistake to notice the extra assumptions and think this means they must not be claiming it is only the Hilbert Space + Unitary evolution part of QM. Unsubstantiated claims can easily be made, or people can view their assumptions as "obvious" or "trivial" details (or yet to be proved Lemmas), when in fact they change the physical picture.

If those assumptions change the physical picture, then they are mistaken; I think Wallace and Zurek would agree with that. The physical picture, at least the objective viewpoint of the whole system, should get all it's dynamics from unitary evolution. It doesn't even depend on decoherence. The additional thing which is needed, is some way to measure the system (not necessarily a particular way). Maybe it's so obvious it's overlooked, same as when numbers are assumed to be real instead of p-adic.

It's a common misunderstanding that WMI somehow depends upon a definition of world (or decoherence or branching structure). What defines WMI best is probably:

Without drawing on any external metaphysics or mathematics other than the standard rules of logic, EWG are able, from these postulates, to prove the following metatheorem: The mathematical formalism of the quantum theory is capable of yielding its own interpretation.

The further assumptions in the derivations are only used to show how a subjective agent within the WMI would experience it and if that would correspond with the Born rule. They are, as you said, (approximate) details for our convenience.

This is also a widespread misunderstanding.

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

 

[DarMM]

So if I had a theory which assumed $a = b$ and $b = c$ and proved $a = c$, then I switched an axiom with the proof, would you call that a different theory?

I don't really think this is analogous to having the Born rule as an effective approximate rule for agents versus having it as a primary aspect of your theory.

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

I didn't claim it needed to.

I'm discussing versions of MWI where no assumptions are made beyond unitary evolution and a derivation of the Born rule is attempted. If the Born rule isn't assumed, it needs to be derived.

Other versions of MWI do assume it and of course they don't need a derivation of it.

Again when discussing derivations of the Born rule I can only discuss those versions of MWI where it is derived and not assumed. This is not a "claim" that all versions of MWI need to derive the rule.

 

[akvadrako]

I don't really think this is analogous to having the Born rule as an effective approximate rule for agents versus having it as a primary aspect of your theory.

I think it's analogous - what matters is that the physical situation is the same and that is always the unitary evolution. It certainly can't depend on decoherence.

Again when discussing derivations of the Born rule I can only discuss those versions of MWI where it is derived and not assumed. This is not a "claim" that all versions of MWI need to derive the rule.

I'm saying these are not different versions of MWI, but different attempts to derive the Born rule for subjective observers, which doesn't define the theory - it works within it.

 

[DarMM]

I think it's analogous - what matters is that the physical situation is the same and that is always the unitary evolution. It certainly can't depend on decoherence.

Really? In Wallace's version the Born rule is only an effective result which can fail when certain features are absent, e.g. the control over the environment assumed by erasure for example.
In one form of Zurek's MWI, it would fail for example when a system includes no subsystems whose invariance under unitary transformations implied equal probabilities for their outcomes.

To me, these are both different from each other (different regimes where Born rule fails) and different from a theory where it isn't an effective rule, but assumed from the start.

 

[akvadrako]

Really? In Wallace's version the Born rule is only an effective result which can fail when certain features are absent, e.g. the control over the environment assumed by erasure for example.
In one form of Zurek's MWI, it would fail for example when a system includes no subsystems whose invariance under unitary transformations implied euqal probabilities for their outcomes.

To me, these are both different from each other (different regimes where Born rule fails) and different from a theory where it isn't an effective rule, but assumed from the start.

So these theories are about the experience of an subjective observer within the universal wavefunction. This seems to be necessarily an approximate concept and they make different assumptions to show it kinda makes sense. But WMI is not QBism - it isn't about the subjective experiences of observers.

It wouldn't make sense to say the evolution of the objective state depends upon control of the environment or choice of subsystems. And the physical situation is given fully by that state (assuming some fixed properties).

 

[DarMM]

I see now what you mean. I will need time to think about it, I'll get back to you after I cover Zurek's proof in more detail.

 

[A. Neumaier]

the Born rule doesn't by itself contradict the assumption that evolution is purely unitary.

Of course not.

The unitary evolution is postulated for a closed system, while the Born rule is claimed for an open, observed system. That makes all the difference, since they apply to disjoint regimes, and hence cannot contradict each other.

I don't necessarily think it contradicts unitary evolution either, there's just no proof it's a consequence of unitary evolution.

Under certain conditions, the Born rule for measuring a tiny subsystem by a large detector can be derived from statistical mechanics. (One doesn't need MWI, which only creates additional confusion.) We discussed this in the thread Collapse from unitarity.

 

[PeterDonis]

Is the journal peer reviewed ?

The Mentors have reviewed the paper and journal and it is an acceptable source for PF discussion.

 

[DarMM]

We start with a macroscopic system $\mathcal{S}$ in a state $|\psi\rangle$ and we have an apparatus that measures the basis $\{|\sigma_j\rangle\}$. There is also an environment $\mathcal{E}$.

The combined system-environment state is then assumed to evolve into Schmidt form after measurement:

$$\Psi_{\mathcal{S}\cdot\mathcal{E}} = \sum_i \alpha_i |\sigma_i\rangle |\eta_i\rangle$$

with $|\eta_i\rangle$ a orthonormal basis for the environment.

What we want to show is that the probability of measuring a particular outcome $j$ of the basis $|\sigma_j\rangle$ from the microstate $\psi$, denoted $p\left(j,\{\sigma_j\},\psi\right)$ is derivable from some property of the final state $\Psi_{\mathcal{S}\cdot\mathcal{E}}$. That is:

$$p\left(j,\{\sigma_j\},\psi\right) = F\left(\sigma_i,\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

and that this probability agrees with the Born rule.

Assumption 1, Environmental Noncontextuality, EN:
$$p\left(j,\{\sigma_j\},\psi\right) = F\left(\sigma_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

The probabilities do not depend on the environmental states after measurement.

Envariance is really just a consequence of this assumption. For if some unitary on the microscopic system can be undone by the environment and probabilities do not depend on the environment, then that unitary can have no effect on the probabilities.

Assumption 2, Perfect correlation, PC:
The chance of observing an environmental state $\eta_i$, denoted $G\left(\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$, obeys:
$$G\left(\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right) = F\left(\sigma_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

That is, in a Schmidt state the environment and the system are perfectly correlated.

Zurek's proof in the equal amplitude case then be seen more easily via looking at the state:

$$\psi = \sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)$$

for which the post-measurement Schmidt state is:

$$\Psi_{\mathcal{S}\cdot\mathcal{E}} = \sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)$$

So:

$$p\left(0,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right) = F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)\right)$$

Swapping the microscopic states can be undone by swapping the environment states, so by assumption 1 it has no effect on the system, hence:

$$F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)\right) = F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right)$$

In this new state the chance to measure the system in $|0\rangle$ is the same as the environment in $|1\rangle$, hence:
$$F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right) = G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right)$$

We can then use the observation that swaps have no effect once again, this time on the environment:
$$G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right) = G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right)$$

And once more use the correlation between system and environment:
$$G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right) = F\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right)$$

However this is just $p\left(1,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right)$, hence:

$$p\left(0,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right) = p\left(1,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right)$$

Note how this works. We use EN to swap environments, PC then let's us convert this to a probability for environment observations. The latter step is what allows us to alter it from a statement about $|0\rangle$ states to one about $|1\rangle$ states, we just have to use EN once more to reattach the system states in their original order.

So envariance alone only tells us we can swap environments in certain scenarios, but it will never allow us to convert a statement about one element of a basis to another, we need PC for that (which is B3 in my list above, you could also use B1 or B2).

Okay, onto problems next.

 

[DarMM]

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

Okay I have thought about this a bit more. Quite simply:

1. It has to derive the Born Rule as that is the empirical core of quantum mechanics. If unitary QM alone cannot obtain the Born rule, then it doesn't match experiment.
2. If you wish to alter Many-Worlds, so that it has a Born rule built-in, that is a different theory outside the scope of this thread, as it is no longer deriving the Born rule as a consequence of unitary evolution. That is not to say such theories will display an inconsistency between the Born rule and Unitary evolution.
   There is an interpretive difference between the Born rule as effective and the Born rule as primary.

I do think versions of MWI where the Born-rule is built in are quite different from just pure unitary QM where it is not. I think this can be demonstrated by trying to give a meaning to the assumed Born rule. You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

 

[stevendaryl]

I do think versions of MWI where the Born-rule is built in are quite different from just pure unitary QM where it is not. I think this can be demonstrated by trying to give a meaning to the assumed Born rule. You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

I would say that a pure wavefunction evolving as a deterministic function of time cannot possibly have a unique physical interpretation. Nothing physically follows from "There is a complex-valued function of 3N+1 dimensional space obeying such-and-such differential equation". You have to make psycho-physical assumptions (what is the relationship between the mathematical objects of the theory and our observations) before a theory can be said to have any empirical content whatsoever.

So if you're trying to derive the Born rule from pure unitary evolution, you have to make some interpretation assumptions to even make sense of the question: What is the probability of this or that happening? You have to at least identify what are the events that can either happen or not happen. Or it seems that way to me.

 

[akvadrako]

You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

It's not clear to me where our disagreement lies, but a theory without such a measure is incomplete, even disregarding the empirical connection. If have the state $\psi= 1|a\rangle + 0|b\rangle$, then to have a physical interpretation at all I need to be able to say something like $|a\rangle$ exists and $|b\rangle$ doesn't.

Anyone trying to derive the Born rule from just unitary QM must be assuming it has some measure. Otherwise it's hopeless from the start.

 

[DarMM]

It's not clear to me where our disagreement lies, but a theory without such a measure is incomplete, even disregarding the empirical connection. If have the state $\psi= 1|a\rangle + 0|b\rangle$, then to have a physical interpretation at all I need to be able to say something like $|a\rangle$ exists and $|b\rangle$ doesn't.

I think they'd all agree on this, as they would see it as a branch being absent/present, however this is an easy special case. What is the difference between:

$$\sqrt{\frac{1}{2}}|a\rangle + \sqrt{\frac{1}{2}}|b\rangle$$

and

$$\sqrt{\frac{1}{3}}|a\rangle + \sqrt{\frac{2}{3}}|b\rangle$$

within the "measure of existence" concept?

Different versions of Many Worlds would distinguish these two states in a different manner.

Anyone trying to derive the Born rule from just unitary QM must be assuming it has some measure. Otherwise it's hopeless from the start.

Wallace and Zurek are explicitly avoiding assuming any measure, that's the point of their derivations.

 

[akvadrako]

I think they'd all agree on this, as they would see it as a branch being absent/present, however this is an easy special case.

It's two different issues:

(1) Can you derive the Born rule with no additional assumptions? My claim is it's obviously not possible, otherwise you have no reason to disregard 0-weight terms.
(2) Do the different sets of assumptions agree?

What is the difference between:
$$\sqrt{\frac{1}{2}}|a\rangle + \sqrt{\frac{1}{2}}|b\rangle$$
and
$$\sqrt{\frac{1}{3}}|a\rangle + \sqrt{\frac{2}{3}}|b\rangle$$
within the "measure of existence" concept?

The straightforward version just says the measure of existence of a term is the absolute squared amplitude. It's not a priori the same as what an observer would experience.

 

[DarMM]

I would say that a pure wavefunction evolving as a deterministic function of time cannot possibly have a unique physical interpretation. Nothing physically follows from "There is a complex-valued function of 3N+1 dimensional space obeying such-and-such differential equation". You have to make psycho-physical assumptions (what is the relationship between the mathematical objects of the theory and our observations) before a theory can be said to have any empirical content whatsoever.

So if you're trying to derive the Born rule from pure unitary evolution, you have to make some interpretation assumptions to even make sense of the question: What is the probability of this or that happening? You have to at least identify what are the events that can either happen or not happen. Or it seems that way to me.

I would agree, I think in the back and forth of this something has become unclear, as I am not proposing just a complex wavefunction with no physical meaning is assumed in Many Worlds.

Neither Wallace and Zurek discuss a purely mathematical "unitary evolution only" QM, neither would any MW interpretation for the reasons you mention. They do give a physical meaning to the mathematics (I'm not going to try to explain the meaning they give, because certainly in Wallace's case, I find it confusing still).

What I should say is that the only part of QM they give a physical meaning to at the outset is the unitary evolution component.

An example might be better.

There are versions of MWI (such as Mandolesi's) where additional structures beyond just the unitary evolution are given a direct physical interpretation at the beginning. For example neither Wallace nor Zurek attribute a direct physical meaning to the magnitude of branch coefficients, Mandolesi gives them the direct meaning of "measure of resistance to interference". Other interpretations directly say the coefficient magnitudes are a "count" of how many copies of that state there are (how many worlds).

I am saying that the versions like Wallace and Zurek's, where the coefficient magnitudes have no direct physical meaning (aside from distinguishing states) are the ones that have to "derive" the connection with experiment known as the Born rule. Others do not.

 

[DarMM]

The straightforward version just says the measure of existence of a term is the absolute squared amplitude. It's not a priori the same as what an observer would experience.

What does that mean physically? That is, what is the interpretative difference between the two states I mentioned? Mandolesi would say in the second state the $|b\rangle$ branch is more resistant to interference, other versions would say there are more $|b\rangle$ branches/worlds in the second state than the first.

 

[DarMM]

I will say one thing however, I think there is an assumption that Many-Worlds is one interpretation and Wallace, Zurek, DeWitt etc are all talking about the same thing. Hence some of the confusion on this thread. They are not. They have completely different explanations for what the Born rule is (in some it is subjective, in others not), what a world is, etc

A read of Wallace's book and some MWI papers shows them directly disagreeing with each other and arguing over the physical meaning of terms. Even some versions that look very similar initially, e.g. Vaidmann and Carroll/Seben, have fundamental differences.

 

[akvadrako]

What does that mean physically? That is, what is the interpretative difference between the two states I mentioned? Mandolesi would say in the second state the $|b\rangle$ branch is more resistant to interference, other versions would say there are more $|b\rangle$ branches/worlds in the second state than the first.

That's a term from Vaidman and I don't see a clear physical meaning, though he does mention the significance of $|b\rangle$ being more resistant to interference. In terms of WMI, one can consider worlds as operators, so I would say it makes sense to treat the coefficients as how much it contributes to the total state, in the same sense as a cosmological measure.

I will say one thing however, I think there is an assumption that Many-Worlds is one interpretation and Wallace, Zurek, DeWitt etc are all talking about the same thing. Hence some of the confusion on this thread. They are not. They have completely different explanations for what the Born rule is (in some it is subjective, in others not), what a world is, etc

A read of Wallace's book and some MWI papers shows them directly disagreeing with each other and arguing over the physical meaning of terms. Even some versions that look very similar initially, e.g. Vaidmann and Carroll/Seben, have fundamental differences.

You can call them different interpretations, but they all agree that the physical (ontic) situation maps 1-to-1 onto unitary QM. That's what the whole Everett program is about. The extra structure given to it is just descriptive. There are different uses of the Born rule - one about subjective experience of agents and one about the objective coefficients of worlds, so that's a fair point to disagree about.

 

[akvadrako]

To expand on the zero-weight case for why a measure must be assumed, let's say we have this setup:

$|Y\rangle(-1|A\rangle) + |X\rangle(a|A\rangle + b|B\rangle)$

If you can manipulate this so the two parts in parentheses are together then you'll have:

$|?\rangle(-1|A\rangle + a|A\rangle + b|B\rangle)$

When $a = 1, b = 3$, the $|A\rangle$ branch weight becomes 0 and no longer exists, which is different then when $a = b = 2$. So even different non-zero branch weights must be given some different meaning.

 

[stevendaryl]

There is a notion of measure that is built-in to the definition of the Hilbert space of square-integrable functions. Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

I actually read an article once that tried to bootstrap from: "Two functions are equal if they only differ on a set of measure zero" to the Born rule. The argument went this way:

Imagine that there is some measurement that has two results: $U$ or $D$. The amplitude for one result is $\alpha$ and the amplitude for the other is $\beta$.

Now, imagine repeating the measurement over and over again, and recording it in some persistent storage.

Then the wave function of the whole universe can afterwards be written as a superposition of possible worlds, each of which has a different history of Us and Ds.

Some of those worlds will have records of Us and Ds that agree with the Born rule, some will not. The claim made was that in some kind of limit as the number of measurements goes to infinity, the measure of the set of "maverick" worlds (where the relative frequency disagrees with Born) goes to zero.

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

 

[Michael Price]

There is a notion of measure that is built-in to the definition of the Hilbert space of square-integrable functions. Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

I actually read an article once that tried to bootstrap from: "Two functions are equal if they only differ on a set of measure zero" to the Born rule. The argument went this way:

Imagine that there is some measurement that has two results: $U$ or $D$. The amplitude for one result is $\alpha$ and the amplitude for the other is $\beta$.

Now, imagine repeating the measurement over and over again, and recording it in some persistent storage.

Then the wave function of the whole universe can afterwards be written as a superposition of possible worlds, each of which has a different history of Us and Ds.

Some of those worlds will have records of Us and Ds that agree with the Born rule, some will not. The claim made was that in some kind of limit as the number of measurements goes to infinity, the measure of the set of "maverick" worlds (where the relative frequency disagrees with Born) goes to zero.

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

 

[stevendaryl]

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

Well, there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway. So I would not say that it's clear-cut that this argument says anything about the actual universe (with only finitely many measurements).

 

[Michael Price]

Well, there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway. So I would not say that it's clear-cut that this argument says anything about the actual universe (with only finitely many measurements).

And yet the worlds which obey the Born Rule are also infinitely old. Their norm tends to 1, whilst the norm of the worlds where the Born Rule is violated tends to 0. That has to be telling us something.

 

[A. Neumaier]

Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

But wave functions convey physics only up to an x-independent phase. Your distance doesn't and hence is physically meaningless.

 

[A. Neumaier]

there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway.

But finitely many measurements do not at all contribute to the mathematical ensemble mean which is always a mean over infinitely many cases - so according to the same argument as you employ here, measurements are completely unrelated to ensemble means! This shows that arguing with rigor about something that has only an approximate connection to reality can never prove anything about reality...