Particle vs Wave Interpretations of QM

[Roberto Pavani]

see https://arxiv.org/pdf/2101.11052.

In that paper, Carroll and Lodman explicitly state that observers do not observe the global wavefunction; they observe only the branch they happen to be in.

That seems very close to the point that is confusing me. Statistical estimates and frequency counts are performed by observers within branches. The global wavefunction itself does not perform any estimation.

I am therefore trying to understand how the Born weights enter from the perspective of an observer who only has access to the history and data contained in a single branch.

Perhaps this is where my conceptual difficulty lies.

 

[Lord Jestocost]

From MathPages (https://www.mathpages.com/home/kmath701/kmath701.htm)

“Dialogue on Many Worlds

The following is a transcript of a dialogue on the so-called “Many Worlds Interpretation” (or MWI) of quantum mechanics.

Salviati: For an observer tossing a quantum coin, the MWI maintains unitarity by saying that following the toss we have [observer$1$, observer$2$] reporting [heads, tails]. The problem with this is that it does not account for why we don’t get [observer$1$, observer$2$] reporting [tails, heads]. All we can say is that we don't care about the difference – but try telling that to observer$1$ if "heads" means he loses his.

Sagredo: But Salviati, your observer labels $1$ and $2$ have no significance. An advocate of MWI would say that, if we have a sequence of 50 coin tosses, there are $2^{50}$ distinct observers, one for each possible sequence of outcomes, and we identify those observers with their respective sequences of outcomes. So it's self-contradictory to talk about permuting the observers to different sequences of outcomes. The sequences of outcomes are the observers. On the other hand, the multiplicity of these outcomes is strictly unobservable, in the empiricist sense, so one can argue that the unitarity is only a fantasy (to any given observer). In this sense MWI represents a highly rationalist – rather than empiricist – approach to science.”

At the end, MWI is conceptually very simple.

 

[Roberto Pavani]

there are $2^{50}$ distinct observers, one for each possible sequence of outcomes

Suppose the underlying quantum event is strongly asymmetric, for example

$P(A)=0.9,\qquad P(B)=0.1$.

There are still exactly $2^{50}$ observers, one associated with each outcome history.

Each observer only has access to the outcomes recorded in their own history and can perform a statistical estimate of $P(A)$ from those data:

$\hat P(A)=\frac{N_A}{N}$.

My difficulty is that the number of observers is unchanged, whereas the Born probabilities are radically different.

For $N=50$ trials, the Born expectation is about $45$ occurrences of $A$.

However, the fraction of all possible outcome histories containing at least 45 occurrences of $A$ is

$\frac{\sum_{k=45}^{50}\binom{50}{k}}{2^{50}}$,

which is extremely small.

So if observers are identified with outcome histories, I do not see how the observer can infer the Born rule.

Each observer estimates probabilities only from the observed frequencies within their own history.

 

[jbergman]

In MWI, the fundamental object is an evolving quantum state.

It always confuses me about how to think about MWI in the Heisenberg picture where the state is just an initial condition and operator's evolve. Is there any literature on the two?

I have a similar question about Bohmian Mechanics. It strikes me as a questionable interpretation if it depends on which picture you use to carry out calculations.

 

[PeterDonis]

aIn that paper, Carroll and Lodman explicitly state that observers do not observe the global wavefunction; they observe only the branch they happen to be in.

And this is done by trading on an ambiguity in the word "observers".

"Observation" and "measurement", as has already been said, are interactions that entangle the system being "observed" or "measured" with whatever is doing the "observing" or "measuring". In other words, they entangle different degrees of freedom in Hilbert space. Normally, when we use the term "observer" or "measuring device", we are referring to the degrees of freedom in Hilbert space that describe the thing doing the observing or measuring.

So, for example, if I measure the spin of a qubit and observe the result, there are degrees of freedom in my brain that are now entangled with the degrees of freedom in the qubit and the device that I used to measure its spin.

In such a situation, where we have a joint system composed of entangled subsystems, we don't normally use a word describing a subsystem to just refer to one branch of the wave function. For example, we don't say that the "qubit" is just one branch of the wave function. The "qubit" refers to degrees of freedom in Hilbert space, and in the wave function, those degrees of freedom are entangled with other degrees of freedom, and all of the branches of the entangled wave function are referred to, not just one. Similarly, the term "observer" would refer to the degrees of freedom in my brain, and all of the branches of the entangled wave function are referred to, not just one.

But in what you refer to in the paper, Caroll and Lodman are not using the term "observer" that way. They are using the term to refer to just one particular branch of the wave function. This sort of shifting of ground is (unfortunately) common among MWI proponents: they want to have it both ways. They want to assert that all of the branches of the wave function are "real", but then they want to talk as if only one is whenever it suits them better.

 

[PeterDonis]

I am therefore trying to understand how the Born weights enter from the perspective of an observer who only has access to the history and data contained in a single branch.

As has already been said, this is a major open issue (many skeptics would say the major open issue) with the MWI. There is no known answer that is generally accepted. Even MWI proponents don't all agree on one answer.

 

[jbergman]

That seems very close to the point that is confusing me. Statistical estimates and frequency counts are performed by observers within branches. The global wavefunction itself does not perform any estimation.

I am therefore trying to understand how the Born weights enter from the perspective of an observer who only has access to the history and data contained in a single branch.

Perhaps this is where my conceptual difficulty lies.

Experientially it's no different than what we observe. You estimate probabilities by flipping a quantum coin a lot of times. Even though you are only in one branch you will see let's say a billion outcomes in your world if you flip it enough.

Now with MWI, it opens the possibility for some very strange worlds where the coin is always heads. In that Universe they may believe the laws of physics are different but Carroll would say that branch has incredibly small weight so it doesn't matter.

It's a lot like Boltzmann Brains.

 

[Matterwave]

So if observers are identified with outcome histories, I do not see how the observer can infer the Born rule.

Each observer estimates probabilities only from the observed frequencies within their own history.

I agree with @PeterDonis that this is a major open issue in MWI.

Here is my understanding, which could be quite flawed, and I'd love for real MWI proponents (or Peter, who seems quite knowledgeable on the subject though not seemingly a MWI proponent) to come correct me~

One argument from the MWI perspective is to use "self-locating probability" to see the connection to the Born rule. The argument (mostly gathered this from Sean Carroll from various podcasts) goes that decoherence timescales is much shorter than human cognition time scales. Thus, there is a time period after decoherence but before the observer has recognized/realized the result of the experiment. The rational observer asks "what is the chance that "I" am part of branch A vs branch B?" (the self-locating probability). The answer given is that this probability corresponds to the Born rule. Each of the branches carry a weight according to the Born rule and this weight gives you the self-locating probability. (Of course, note that I put quotes around "I" because this "I" must now refer to an "I" in a branch and not the "I" across branches.)

So although it is true that every outcome exists as a branch in MWI, not every branch is created equal. The way to apply the Born rule is not via branch counting but via the branching weights. And although there are branches where the Born rule was grossly violated in the past, somehow "you" (again, the you in a branch, not the you across every branch) are unlikely to find yourself in one of those branches.

I am not sure how convinced I am by this argument. And I'm sure I have probably butchered parts of it and made various mistakes. But this seems to be the gist of what I hear. Would totally love corrections.

 

[Roberto Pavani]

In that Universe they may believe the laws of physics are different but Carroll would say that branch has incredibly small weight so it doesn't matter.

Let $K$ denote the number of real $(\text{observer},\text{counting history})$ pairs after $N$ binary measurements.

1. If $K = 2^N$, then for $p\neq\tfrac12$ most observers do not infer $p$. Since the multiplicity of histories is $\binom{N}{n}$, which is maximal near $n=N/2$, most observers estimate a frequency near $\tfrac12$.

2. If most observers do recover the Born value $p$ for $p\neq\tfrac12$, then $K$ must be larger than $2^N$.

3. If $(\text{observer},\text{counting history})$ pairs are not physically real entities, then $K$ can be arbitrary and need not have any physical significance.

 

[Roberto Pavani]

Each of the branches carry a weight according to the Born rule and this weight gives you the self-locating probability.

A real observer only has access to the outcomes observed in their own branch and can estimate probabilities only from those frequencies. The Born weights are not themselves observable data within the branch.

 

[jbergman]

Let $K$ denote the number of real $(\text{observer},\text{counting history})$ pairs after $N$ binary measurements.

1. If $K = 2^N$, then for $p\neq\tfrac12$ most observers do not infer $p$. Since the multiplicity of histories is $\binom{N}{n}$, which is maximal near $n=N/2$, most observers estimate a frequency near $\tfrac12$.

2. If most observers do recover the Born value $p$ for $p\neq\tfrac12$, then $K$ must be larger than $2^N$.

3. If $(\text{observer},\text{counting history})$ pairs are not physically real entities, then $K$ can be arbitrary and need not have any physical significance.

Well, this type of thinking doesn't quite work because you also have a physical model like for a classical coin toss. We don't always get an even number of heads as tails but with enough trials the number becomes small where the deviations from it.

But with MWI you end up with world counts that are so absurd that one would question the underlying physical model and that is just with one coin toss. It's really hard to imagine what happens in some worlds.

 

[PeterDonis]

The Born weights are not themselves observable data within the branch.

Yes, that's one of the reasons why trying to explain where the Born Rule comes from is a major open issue with the MWI.

And please bear in mind that, since it is an open issue in the relevant physics community, we are not going to solve it here. We can state what the issue is, and it looks like that's been done. But after that, there is not much point in further discussion.

 

[QuarkyMeson]

Let K denote the number of real (observer,counting history) pairs after N binary measurements.

1. If K=2N, then for p≠12 most observers do not infer p. Since the multiplicity of histories is (Nn), which is maximal near n=N/2, most observers estimate a frequency near 12.

2. If most observers do recover the Born value p for p≠12, then K must be larger than 2N.

3. If (observer,counting history) pairs are not physically real entities, then K can be arbitrary and need not have any physical significance.

Branch number is not uniquely defined by decoherence. Decoherence yields an approximate branch structure, not a count of observers or histories. so "most observers" arguments have no well-defined reference. The fact that an observer only sees outcomes within their own branch does not imply equal branch weighting, it only means the Born measure is part of the global theory rather than an observable inside any single branch. That is the same status probability has in any theory.

A real observer only has access to the outcomes observed in their own branch and can estimate probabilities only from those frequencies. The Born weights are not themselves observable data within the branch.

On the self-locating uncertainty, the observer does not read the branch weights from their environment. Rather, before learning the outcome, they assign credences to their possible branch-relative successors. Arguments based on epistemic separability claim that these credences should be proportional to the Born weights. Whether that succeeds is disputed, but the fact that the weights are not themselves branch local observables does not refute the argument.

Which is why:

As has already been said, this is a major open issue (many skeptics would say the major open issue) with the MWI. There is no known answer that is generally accepted. Even MWI proponents don't all agree on one answer.

 

[PeterDonis]

Branch number is not uniquely defined by decoherence. Decoherence yields an approximate branch structure, not a count of observers or histories.

While in general this is a caveat that needs to be kept in mind, for the particular case under discussion here, spin measurements on qubits, I don't think it's a concern: each measurement has only two possible outcomes, and they're easily distinguishable, so the "branch counting" that @Roberto Pavani is doing should work fine for that particular case.

 

[Roberto Pavani]

From the replies so far, I still have not managed to understand whether MWI worlds are regarded as physically real entities or merely as a useful way of describing the structure of the wavefunction.

Assuming they are physically real, my point was simply that for binary measurements with $p\neq\tfrac12$, the number of real observer-histories cannot be just $2^N$. If there were only $2^N$ such histories, their multiplicity would be governed by $\binom{N}{n}$, which is maximal near $n=N/2$, so most observers would infer a frequency near $\tfrac12$ rather than $p$.

 

[jbergman]

From the replies so far, I still have not managed to understand whether MWI worlds are regarded as physically real entities or merely as a useful way of describing the structure of the wavefunction.

Assuming they are physically real, my point was simply that for binary measurements with $p\neq\tfrac12$, the number of real observer-histories cannot be just $2^N$. If there were only $2^N$ such histories, their multiplicity would be governed by $\binom{N}{n}$, which is maximal near $n=N/2$, so most observers would infer a frequency near $\tfrac12$ rather than $p$.

To answer this specific question. I think where your assumptions are breaking down is that there is only one world for each outcome. This is discussed in Carroll's paper he actually assumes that you can thousands of worlds per coin flip so that you can approximate any rational number?

 

[PeterDonis]

From the replies so far, I still have not managed to understand whether MWI worlds are regarded as physically real entities or merely as a useful way of describing the structure of the wavefunction.

MWI proponents, as I understand it, say that the worlds are physically real--as a consequence of the more gerneral claim that the wave function, all of it, is physically real (and is in fact a complete description of physical reality).

my point was simply that for binary measurements with $p\neq\tfrac12$, the number of real observer-histories cannot be just $2^N$.

No, you have it backwards. The MWI is forced to say that the number of real observer-histories is $2^N$ in your scenario: that's what the straightforward math of the wave function says.

Your observation is simply that at least one MWI proponent argument for why the Born Rule works, that worlds (what you are calling "observer-histories") with relative frequencies that don't obey the Born Rule have negligible weights in the wave function, does not appear to work for the case where the probabilities of the different possible outcomes at each measurement (i.e., where each "branching" occurs) are not all equal.

I don't know if this argument has appeared in the literature; I would be interested to see if anyone can find a reference that discusses it.

 

[PeterDonis]

I think where your assumptions are breaking down is that there is only one world for each outcome.

I don't think he's making that mistake. What he is calling "multiplicity" is the number of worlds in which the relative frequency of the outcomes takes a particular value. In each of the worlds in which that relative frequency takes a particular value, the specific sequence of outcomes will be different--there is only one world for each of the $2^N$ possible sequences of outcomes for a binary measurement. But there can be multiple sequences of outcomes that give the same relative frequency (the same number of heads vs. tails in the quantum coin flips). That "multiplicity" is what @Roberto Pavani is describing.

 

[jbergman]

I don't think he's making that mistake. What he is calling "multiplicity" is the number of worlds in which the relative frequency of the outcomes takes a particular value. In each of the worlds in which that relative frequency takes a particular value, the specific sequence of outcomes will be different--there is only one world for each of the $2^N$ possible sequences of outcomes for a binary measurement. But there can be multiple sequences of outcomes that give the same relative frequency (the same number of heads vs. tails in the quantum coin flips). That "multiplicity" is what @Roberto Pavani is describing.

Let's forget about a sequence coin flips. If you have a quantum coin with a probability of 1/3 and perform a single experiment Carroll and Zurek both suggest that you don't just get two worlds after the experiment. Instead you would end up with some large number with the fraction of worlds with each outcome proportional to the probability. They argue this is how you go from 50/50 probabilities to general using envariance. It's a subtle point that always has slightly bothered me. Because it is just stated without much support. Some of the arguments for this are that one can generate decoherent worlds fairly easy with interactions unrelated to the coin flip.

https://arxiv.org/abs/1405.7907

I think this addresses the essence of Robert's question.

 

[PeterDonis]

Carroll and Zurek both suggest that you don't just get two worlds after the experiment. Instead you would end up with some large number with the fraction of worlds with each outcome proportional to the probability

Which is assuming the Born Rule--so as an argument for deriving the Born Rule, it's just arguing in a circle. (I think this general pattern has occurred more than once in the literature.)

 

[gentzen]

If you have a quantum coin with a probability of 1/3 and perform a single experiment Carroll and Zurek both suggest that you don't just get two worlds after the experiment. Instead you would end up with some large number with the fraction of worlds with each outcome proportional to the probability. They argue this is how you go from 50/50 probabilities to general using envariance.

Do you have a reference for Zurek? I know that he talks about envariance, but I would like to read where he talks about large numbers of worlds.

 

[PeterDonis]

this is how you go from 50/50 probabilities to general

But the "probabilities" here don't even make sense--at least, that's the problem that all this is supposed to solve, but Carroll et al in the paper you referenced don't solve it, they just assume it solved. They talk about "the probability of being in a given branch" or "the probability of observing a particular result"--but even talking that way assumes the Born Rule! The whole point is that in the MWI, there are no such probabilities--the observer ends up in every branch, and every possible result occurs. It doesn't even make sense to talk about the probability of one thing or the other happening--they all happen.

 

[QuarkyMeson]

While in general this is a caveat that needs to be kept in mind, for the particular case under discussion here, spin measurements on qubits, I don't think it's a concern: each measurement has only two possible outcomes, and they're easily distinguishable, so the "branch counting" that @Roberto Pavani is doing should work fine for that particular case.

Ah I remembered reading: https://arxiv.org/abs/2201.06087 and assumed even that was problematic, but I see your point.

I'm looking at it again and this:

Which is assuming the Born Rule--so as an argument for deriving the Born Rule, it's just arguing in a circle. (I think this general pattern has occurred more than once in the literature.)

Is probably also a fair criticism of Saunders paper.

From the replies so far, I still have not managed to understand whether MWI worlds are regarded as physically real entities or merely as a useful way of describing the structure of the wavefunction.

If you mean the mainstream MWI positions as I understand them, the worlds are regarded as real but emergent, decoherence defined structures in the universal wavefunction, not fundamental entities in and of themselves.

MWI is also not the only game in town for only unitary evolution though. There are several other interpretations that also worship at the church of the larger hilbert space but handle the measurement problem in different ways.

 

[PeterDonis]

here are several other interpretations that also worship at the church of the larger hilbert space

but handle the measurement problem in different ways.

Can you give any references?

 

[QuarkyMeson]

Can you give any references?

Ah I had things like the meta interpretation of the MWI Many Minds, along with the Modal Interp and Consistent histories in mind.

 

[jbergman]

Do you have a reference for Zurek? I know that he talks about envariance, but I would like to read where he talks about large numbers of worlds.

I will try to reply when I have more time later today.

 

[Matterwave]

https://arxiv.org/abs/1405.7907

I think this addresses the essence of Robert's question.

Not having read past eqn. 10, it really does seem to me like this paper is at least trying to resolve @Roberto Pavani concerns. Eqn 10 explicitly uses an example where the probabilites for a binary outcome are not 50:50.

So far, the paper has not been super convincing to me personally. But I need to find more time to read the rest of it.

 

[Roberto Pavani]

The "trick" of replacing a branch by multiple copies in order to recover the Born rule seems reasonable at first sight. One might imagine that nature somehow generates a multiplicity of branches so that the counting reproduces the Born weights.

However, probabilities need not be rational. For example, one can have $p=1/\sqrt{2}$. In that case, no finite counting can reproduce the Born value exactly. One must either accept rational approximations, introduce an infinite multiplicity of branches, or replace counting altogether by a measure on the branching structure.

Note that for $p \in \mathbb{R}\setminus\mathbb{Q}$ (for example $p=1/\sqrt{2}$), no finite branch multiplicity can reproduce the Born value exactly. An exact counting interpretation would therefore seem to require a continuum of branches.

This raises two related questions. First, what does it mean for two branches to be distinct worlds if they differ only infinitesimally? Second, how is such a continuum of branches represented within the single universal wavefunction $\Psi_U$ without introducing additional mathematical structure beyond $\Psi_U$ itself?

 

[Matterwave]

The "trick" of replacing a branch by multiple copies in order to recover the Born rule seems reasonable at first sight.

Would you let me know where you saw this trick? Is it from the paper jbergman posted?

 

[jbergman]

Would you let me know where you saw this trick? Is it from the paper jbergman posted?

This paper by Zurek may be a better reference. He discusses orthogonal microstates, https://doi.org/10.1103/PhysRevA.71.052105