## Can anyone defend the Many Worlds Interpretation?

Quantum mechanics has many interpretations or classes of interpretations with internal splits: Copenhagen, Bohmian, spontaneous collapse, many worlds, transactional, etc. Because my take on the matter falls within the neo-Copenhagen family, I do not follow very closely interpretations which fall outside my interest. But although I disagree with non-Copenhagen interpretations, I do understand the approaches they take with only one exception: the many worlds interpretation (MWI). Not for the lack of trying but as far as I dug into it, MWI did not make any sense whatsoever to me (except Zurek's approach which technically is not MWI). So here is my challenge: can anyone defend MWI in a way that will answer the issue I will raise below?

The ground rule of any interpretation is first and foremost to recover the standard quantum mechanics predictions, otherwise it cannot call itself a "quantum mechanics interpretation". Quantum mechanics has this novel feature called the Born rule. Let me digress for a bit and expand on why this does not occur in classical physics. If you recall from prior posts, in configuration space in classical mechanics one encounters the Hamilton-Jacobi equation, while in quantum mechanics one has the Schrodinger equation. In classical physics in phase space we need both the position and momenta of a particle to specify the trajectory, and therefore it should come as no surprise that in configuration space where we only have positions there can be crossing trajectories in the Hamilton-Jacobi case. Therefore the information content attached to a configuration point is ambiguous in classical physics: no Born rule in classical physics. However in the quantum case in configuration space we can attach an information interpretation to the Schrodinger wavefunction known as the Born rule. Born rule shows that quantum mechanics is probabilistic and initial conditions are not required. (In the Bohmian case you can add initial conditions only in a contextual (parochial) way respecting an additional constrain called quantum equilibrium otherwise you violate Born rule).

Is MWI compatible with Born rule?

But what is MWI, and why is it considered? MWI supposes to solve the measurement problem without resorting to the collapse of the wavefunction.

Suppose we have two outcomes, say spin up and down. Once spin is measured up, a quick subsequent measurement confirms the result, and the same for down. Since the wavefunction respects the superposition principle we can derive a superposition of up and down with the measurement device pointing up for the spin up, and pointing down for spin down. In other words, we arrive at the famous half dead half alive Schrodinger cat which does not occur in nature. Everett noticed that there is a correlation between how the measurement device points and the spin value and he proposed that the world splits between different outcomes: in each outcome the observer is only aware of his own unique measurement result. One proponent of this narrative was Sydney Coleman!!! (I have a big respect for the late Sidney Coleman, but in this instance I think he was shooting from the hip.) I grant that MWI is an appealing idea, but does it stand up to close scrutiny?

People naturally objected to the idea of split personality or "the I problem" to which supporters can fire back with "you do not take quantum mechanics seriously enough to trust what it shows". Also there is a "preferred basis problem" because the split can happen on an infinite number of basis. But to me the most important problem is the treatment of probabilities and agreement with Born rule. I think it is safe to say that anyone agrees that the original Everett argument of why MWI obeys Born rule is not satisfactory. If Everett's derivation were correct, then there will not be that many new "derivations" of Born rule in the MWI framework. However I found no satisfactory derivation to date of Born rule in MWI. Moreover, the only thing that makes sense to me is branch counting and this is definitely violating Born rule - also no disagreement here.

But why I am not convinced by the proposals of deriving Born rule? A common criticism is that those derivations are circular. I assert something stronger: when not circular, Born rule derivations in MWI are mathematically incorrect. Let me show why.

I have discovered long ago that the simplest problems are the hardest, and I will use this here. Instead of muddling the water with convoluted arguments and examples, let's streamline the basic system to the max. So consider a source of electrons which fire only one particle say once a minute. Pass this through a Stern Gerlach device and select only the spin up branch. In other words, we prepare a source of single electrons with a known vertical spin. Then we pass our electron through a second Stern Gerlach device and we measure spin on say the x axis. Half the time we will get the positive spin x and half the negative spin x. In MWI both outcomes occur and I am split into two "me" each observing one definite outcome. So far so good, but now rotate one of the two devices by some angle theta. The statistics changes!!! but what does MWI predict? The world still splits in two and in one world I detect up and in the other one down. In other words, no changes and this is the root cause of why MWI makes no sense.

Now supporters of MWI are well aware of this fact and attempt to derive Born's rule regardless starting from more or less natural assumptions. Let's dig deeper into their claims.

First we need to collect more data to make up a meaningful statistics. For the first electron we have two branches: one up and one down: u d. For the second electron we have 4 branches: uu ud du dd, for the 3rd electron we have 8: uuu ... Now let's count in those branches how many spins are up and how many are down regardless of the actual order of the events:

1st run:                                           1u    1d
2nd run:                                     1uu  2ud    1dd
3rd run:                          1uuu    3uud     3udd       1ddd
4th run:                1uuuu    4uuud   6uudd    4uddd      1dddd

We get Pascal's triangle and the binomial coefficients. This is nothing like Born rule, and the frequentist approach in statistics is rejected by the MWI supporters. Instead they adopt the Bayesian approach. For simple problems like this, the frequentist and Bayesian approaches predict the same things so something else must be thrown in the mix: "the rational observer".  A "rational observer" would have expectations of probabilities before the actual experimental outcome is obtained, and MWI supporters contend that to a rational observer making rational decisions, while branching is incompatible with Born rule, the sane way for such a person to behave is as Born rule appears to be true. Something like: the Earth moves around the Sun, but to us it appears that the Sun moves around the Earth. This line of reasoning was introduced by Deutsch and continued by Wallace.

Several natural sounding principles were proposed to justify this apparent emergence of Born rule in the MWI world. Now Born rule deals with the complex coefficients in front of the ket basis, and those coefficients are simply ignored by  branching because this is what it means to to have a relative correlation between the wavefunction and the measurement device. To derive Born rule you must deal with those coefficients and moreover you must do it in an indirect way. The only indirect way possible is for your "natural sounding principle" to say something nontrivial about a superposition. And in the best case scenario what you actually say about the superposition is nothing but the Born rule in disguise and you have a circular argument

But it gets worse if you claim you broke the circularity: you become mathematically inconsistent. Here is why:

1. To prove Born rule in MWI you need to reject branch counting.
Why? Because Born rule's prediction changes with changing complex coefficients, but branch counting does not.

2. Branch counting arises as a particular case of Born rule. When? In the particular case when the complex coefficients are equal.

So the very act of proving even an apparent Born rule inherently contains a contradiction. All mathematically consistent proposals of deriving Born rule in MWI I am aware of are circular arguments and all their "natural sounding principle" respects branch counting as well.

In summary, coming back to my physical example with the electron source and the two S-G devices, because branching happens the same way regardless of the orientation of the two devices, there is a one to (uncountable infinite) many degeneracy problem which MWI cannot hope to solve by relative arguments alone. In the frequentist approach it is impossible to derive Born rule which acts as a removal of this degeneracy, and MWI supporters pin their hopes on derivations of an apparent Born rule by using some "natural principles". However all the derivations I studied so far are circular, and I know one by Tippler which is mathematically incorrect-maybe I should write a rebuttal to that one, it was published last year. Moreover they cannot reject branch counting because this follows from Born rule when all the scalar coefficients are equal. If you claim you reject  branch counting you are killing your "apparent" Born rule too.

I am challenging MWI supporters to present a valid non-circular derivation of Born rule (either real or apparent). I don't have the time to closely follow MWI developments and maybe there is a recent proposal I missed which can stand up to scrutiny. However I contend it can't be done for the reasons outlined above.

1. Right. But I am just infinitesimally curious whether anyone will ever reasonably address these issues because the answer is pretty obviously No, it is not possible to address them.

When one looks through all this positive talk about MWI, it becomes clear that it is just a "shut up and never calculate anything" interpretation that simply can't be connected with any observations.

The basic reason is simple: physics must calculate *something* S - some numbers or data - and there are only two ways how this S may be connected with observations (one may reduce all the observations to blinking red and green light or whatever like that). And the question 1 "how to connect the calculable numbers with the observations in the most complete or predictive form of the theory" only has two possible answers:

1a) One may in principle calculate the individual observations (like positions of planets) exactly, and say that all other a priori possible observations are wrong

1b) One may only calculate numbers whose functions are probabilities of one outcome or another.

There just can't be any "third way". This is a totally well-defined yes/no question, unlike lots of the fog that MWI champions like to spread, and the answers 1a) and 1b) produce classical mechanics and quantum mechanics, respectively.

MWI champions want the wave function to be "something in between" a classical field and a function of classical distributions but there simply cannot be "anything in between". Either the precious unique outcomes are in principle calculable, it's what arises from the calculations (if the laws of physics had some classical random generators, the calculations would be uncertain, but it would still be true that they're the objects that the equations calculate); or not - in which case, the only other thing we may calculate about the observations are the statistical distributions. In the latter case, the probabilities must be *fundamental* because of the definition of the latter case. If they were not fundamental, we would be back to deterministic classical physics.

1. I've had this exchange with Martin Schnabl, my older ex-classmate in Prague who is a boss of a physics institute. He was asked similar questions. He fell in love with the MWI memes when he saw a wave in undergraduate QM partially transmitted, partially reflected from a barrier. It got split to two worlds. Except that this "sharp separation" rarely occurs in compact enough systems (solids etc.) where all the possible outcomes form a continuum. In those general cases, he doesn't know - and there cannot exist - any way to "split the wave function" into pieces. He only relies on some geometric intuition that the big Hilbert space is "geometric" and one may associate parts of the wave function with "regions" which are "far from each other" but that's simply nonsense. One may write the wave function as a sum in a deeply infinite number of ways and call them "pieces" or "different worlds" but the theory doesn't give any preferred separation.

The questions how we wants to calculate the probabilities - and *all* predictions of quantum mechanics are basically probabilities - led to absolutely no even remotely promising answer and we had several hours for that. In QM, there has to be an observer who can feel something and he must know it, so he asks the "right questions". When he doesn't ask full questions (pick the observables etc.), quantum mechanics cannot produce any specific answer. It's as simple as that.

If the system is complicated, decoherence effects etc. may be used to show that only questions in which the information "classically copies itself" which lead to the nearly diagonal form of the density matrix are defensible. Something where no density matrix is ever diagonalized cannot be perceived as clear classical outcomes. But this is a negative selection. This usage of decoherence shows that some questions aren't fit for observations.But it is not "positive" in the sense that it would actively make any collapse. Decoherence doesn't make any collapse. The equations are unchanged and the unitary equations don't collapse. The decoherence still produces nonzero probabilities for many possible outcomes and an observer is needed to perceive one of them.

When decoherence is present, it's OK to use the basic classical logic for the decohered properties because the potential for reinterference between them goes away - that's what the decoherence means. But that doesn't mean that they lose their probabilistic character. They don't. The precise outcome isn't predictable, only the probabilities are (in QM).

And no decoherence is ever "perfect" so this introduction of the assumption that the outcome is sharp is always just an approximation, something that introduces tiny but nonzero errors. Quantum mechanics' equations are nice and they don't introduce any errors. If there were a collapse, it would be a Ghirardi-Rimini-Weber-like theory, not MWI, anyway, which is also wrong.

But if there are no calculable numbers that are interpreted as subjective probabilities; and if there's no collapse at all, then there are no calculable numbers by the theory connected to any observations someone can make at all. It's really trivial to see. The wave function's intrinsically (subjectively) probabilistic meaning in QM is a necessary condition for the "constantly spreading/splitting" character of the wave function - producing superpositions of macroscopically different cats etc. - not to contradict our sharp observations.

2. It's silly to deny this robust logic, it's much more robust, controllable, and well-defined than any pro-MWI paper that's been ever written. The papers mostly do really elementary errors, like denying that the phases matter. All the relative phases in QM matter up to an observation. Right before the observation, the relative phases between the would-be future outcomes become irrelevant. But that may only happen if the coming measurement is fully specified. There's always a potential for doing a completely different measurements for which the phases (in a basis) are always as important as the absolute values.

Now add all the nonsense about quantum computers by Deutsch. They're fast because they work in many worlds. This is complete rubbish. Quantum computation is a process in which decoherence must be totally or maximally avoided. Because the growth of the number of worlds must only occur during "measurements" which require some decoherence, it's clear that during a quantum computation, the number of worlds must stay the same for the calculation to work at all.So it is obviously *not* exponentially large. And the number of worlds at the end is one because we may destroy the quantum computer. If we interpreted random parts of the quantum computer state vector as "different worlds", it would be silly because those worlds get reunified - the whole ability of the quantum computer to produce right results arises from the interference between these parts of the wave function. And there can't be an interference between "different worlds" which is why it is totally silly to describe any parts of the state vector of a quantum computer as different worlds.

Now, one must also realize that - as Feynman's path integral makes obvious - *all* histories are really possible. In the path integral, they integrand always has the absolute value so all histories, including histories in which the Earth jumps to Jupiter for a second and gets back, are equally likely at the very microscopic level. The only reason why we don't observe these very unlikely things is the destructive interference - the phase oscillates too quickly for the intuitively bad histories so the actual probabilities of well-defined questions result from averaging out and they're basically zero. The MWI advocates implicitly assume that only some "reasonably likely" outcomes exist as worlds but this is bullshit. All worlds exist. There can't be any sharp boundary between reasonably likely histories and very unlikely ones - because the probability is continuous.

There are also many physicists who just say "MWI" but what they mean is that there isn't any added new mechanism like Bohm or the GRW objective collapse. But they don't really think hard whether what they believe has anything to do with Everett's thesis (as opposed with Heisenberg's and other papers that began QM, or Dirac's textbook) or whether it has many worlds. They only want to say that the nice QM equations always work nicely, for objects small and big. Well they do work but how they work - the answer is that they work according to the Copenhagen-like rules.

3. > am just infinitesimally curious whether anyone will ever reasonably address these issues because the answer is pretty obviously No, it is not possible to address them.

What I said above is basically the pattern which emerged after closely looking at various proposals. I must say the proposals are quite clever and at first sight they do not appear circular. Then you see that branch counting is satisfied as well and by looking of how the statistics changes you will be able to uncover an innocuous looking statement injected in the chain of reasoning which is nothing but Born rule. This is usually quite hard to spot and moreover it is easy to make the honest mistake to introduce Born rule in this fashion. I think Deutsch and Wallace were honest people making innocent mistakes (unlike other people I know who publish garbage knowing full well what they submit is incorrect). Still that does not make them right.

But hey, I can be wrong, and if so prove me wrong! Show me the money. Show me a derivation of Born rule in MWI in which I cannot found either a mathematical mistake or a subtle insertion of Born rule in your reasoning chain.

2. >Decoherence doesn't make any collapse. The equations are unchanged and the unitary equations don't collapse. The decoherence still produces nonzero probabilities for many possible outcomes and an observer is needed to perceive one of them.

Decoherence is used in MWI to solve the basis ambiguity problem.

I think one can reject the three leading non-Copenhagen interpretations: MWI, Bohmian, GRW by different arguments:

GRW-by experimental observations; GRW predicts things different than QM and experiments are becoming precise enough to distinguish GRW from QM

Bohmian: adds a hidden variable (particle position) which while can be made compatible for one experiment it has problems for subsequent experiments

MWI: one cannot derive Born rule in MWI

3. Thanks Florin for bringing up this subject and Lubos for participating in it.Apart from all the theoretical arguments both of you brought about, I dislike MWI because splitting universes sounds too much metaphysical and arbitrary to me.I do not mind metaphysics as such.My religion (Hindu) is full of that!! But it should not be mixed up with physics!!I am also puzzled by the fact that some of the prominent physicists, like Gell Mann, Wilczek and others believe in MWI. But they do not participate in blogs. So I have no way of arguing with them! Weinberg frankly admits that he does not like any interpretation. Hopefully someone will join in arguments with you so I can find out his/her reasons!

1. Thanks Kashyap,

I don't oppose any interpretation for personal reasons, I respect nature for what it actually is. If MWI is the true story, so be it, but am yet to see this interpretation get off the ground. The MWI community writes more papers than I can properly read and analyze and there is a very remote possibility a valid non-circular Born rule derivation is available. If so, please defend it on my blog. I will give it my full attention and I will either accept it or point out where the mistake or circularity is introduced.

I think some well known physicists like MWI as a reaction to classical bias because MWI promises this pure QM story. I think that was the case of Sidney Coleman. However those physicists never thought seriously about the foundation of QM.

QM interpretation area is filled with opposing camps and everyone emphasizes their approach and intuition. But now that QM is mathematically derived step by step in constructive fashion, its proper interpretation should also follow as a mathematical theorem. This is what I am working on. The first step is to find the mathematical issues with existing interpretations. This should put all of them to rest in a way that will be accepted by their supporters.

4. Florin,
I wonder if you're aware (you probably are) of Ruth Kastner's argument that Quantum Darwinism contains a similar circularity when trying to solve preferred basis--assuming the existence of a well-defined measurement apparatus where none exists? I know you think Zurek's approach could maybe someday solve the measurement problem, but you also seem to think that, regardless of this, preferred basis is still a problem for MWI. Am I understanding you correctly?
Eric Hamilton.

5. Hi Eric, indeed I am aware of it, Ruth explained it to me last spring (she also had a guest post on this blog about her book). I think it was Bell who first noticed the basis problem in MWI, but decoherence has taken much of the pressure off the subject by now. In my view there is a disconnect between the decoherence view and the MWI view: on one hand decoherence is this mundane ubiquitous process, and on the other hand the world split is this extraordinary event. Why is the split occurring *only* after decoherence? If you want to be even handed, the split (or the "collapse") can happen at any time and you get GRW. GRW is more natural than MWI.

I like Zurek's approach regardless of its success or not in solving the measurement problem because he has some cogent observations. In fact I am combining some of his ideas with mine to arrive of a categorical solution of the measurement problem and I will present it in due time at my blog. I think I found the natural and unique mathematical way to eliminate treating the macroscopic measurement device as a ket (meaning no superposition of macroscopic objects), and to express mathematically the update of information (the collapse) which happens after measurement.

Best,
Florin

6. Thanks for the clarification--your position is kind of what I thought it was. I also thought it was interesting that you mentioned the "'I' problem" of MWI. I think this is actually a valid criticism (not just aesthetic) because unless consciousness can be truly reduced to the Schrodinger (or any) equation, MWI will either be wrong or incomplete. But then that might be a little Qbist of me!
Eric.

1. The "I problem" is a real problem but arguing it with a MWI supporter will get you nowhere. Invariably the discussion leads to: you do not trust QM to believe what it tells you.

By focusing on Born rule I am selecting a topic of concern which can be decided one way or the other.

7. Hi Florin,
What is your opinion about Sean Carroll and collaborators' derivation of Born rule? Does that also assume circular arguments?

1. Dear Kashyap, you haven't asked me but there's some literature on that stuff. The Carroll-Sebens "derivation" has one followup that discusses the content in detail, one by Adrian Kent, which explains that the paper by Carroll-Sebens is a collection of internally inconsistent ill-defined and vague assertions taken out of thin air. There is absolutely no valuable content in those papers.

2. Thanks Lubos. We are old buddies on your blog!

3. Hi Kashyap,

First I don't quite know what to call Carroll's proposal, perhaps MWI^2 (MWI squared) because he seems to have something new: a MWI split within a MWI branch. Adrian Kent's criticism from the link in Lubos' reply is rather flimsy and if I were a MWI believer I could very easily dismiss it: the jump from eq 2 to 3 is simply distributivity regardless of how I interpret it.

There is ambiguity in Caroll's proposal: does a world branch split internally or not? There is language to suggest that the branch split happens, but there is also language that suggests that the usual (Everett) MWI picture is considered.

Criticizing Carroll's proposal requires the ambiguity to be settled first. Until that time I do not consider it a fully formed proposal. It is possible that Carroll already removed this ambiguity in recent papers I am not aware of yet. After the ambiguity is resolved, my arguments from the post above applies.

8. Hello Florin,

As a proponent of the many-world approach I have a couple of responses to your arguments here.

Firstly, a specific point: your claim about mathematical inconsistency doesn't hold up. It's true that in cases of equal coefficients, the Born rule probabilities coincide with the probabilities that would be derived from branch counting given an appropriate decomposition. But that doesn't mean an Everettian who embraces the Born rule but rejects branch counting is being inconsistent. Even a stopped clock is right twice a day - but we shouldn't use it to tell the time. In the same way, Everettians should simply say that branch counting sometimes gives the right answer by coincidence even though it is not the correct rule to use.

Now to the more general point. You challenge Everettians to produce a derivation of the Born rule. While some (Deutch, Wallace, Carroll, Sebens) do try to do this, I agree with Saunders, Papineau and others that in fact no such derivation is necessary for the many-worlds approach to be adequate. Everettians can simply take it as a primitive claim of the theory that the Born rule specifies probabilities (or their functional equivalents). Saying that probabilities are given by the squared-amplitudes is a theoretical identification, like saying that water=H20. Such identifications don't have to be derived but are judged on their theoretical consequences. To compare: no other interpretation can derive the Born rule either! They all just postulate it as part of the theory. There's no reason why Everettians should not do the same.

all best
Al

1. Hello Al.

You have a very interesting take on the issues, and I’ll try to reply. Let me start by quoting Wallace from “Elegance and Enigma”, in the “Big Issues” chapter on page 55:

“As it happens, I *do* think the measurement problem is solvable within ordinary quantum mechanics: I think the Everett (“many worlds”) interpretation solves it in a fully satisfactory way, and while I think there are some philosophical puzzles thrown up by that solution—mostly concerned with probability and with emergence—that would benefit more thought, I wouldn’t call them *pressing*.”

Now I contrast this sentence with Kelvin’s two clouds speech from 1900:

“The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds.”

Clouds or not, pressing or not, the issues of probability and Born rule emergence sinks the whole MWI approach and I am putting the spotlight on the core issues of MWI.

You state: “To compare: no other interpretation can derive the Born rule either! They all just postulate it as part of the theory.” This is simply false. Gleason’s theorem derives Born rule given its assumption: the existence of a probability function respecting some natural requirements. No such thing exists in MWI and that is why Born rule is not derivable in MWI. And this is why all known valid derivations in MWI are circular.

You also state: “There's no reason why Everettians should not do the same.” This is easier said than done. If you postulate Born rule you have to explain probabilities in MWI and this is where you run into trouble. Show me how you do it. Take the S-G example from the post and show what it means for the probabilities to change when I rotate the device.

First “your claim about mathematical inconsistency doesn't hold up.” Perhaps, but I see no real rebuttal of my claim.

“But that doesn't mean an Everettian who embraces the Born rule but rejects branch counting is being inconsistent”. This is a wishful ex cathedra statement without proof. As far as I can tell, the only proof offered was:

“Everettians should simply say that branch counting sometimes gives the right answer by coincidence even though it is not the correct rule to use.” I agree with this statement, but this does not reject my claim.

Here is my claim again: “when not circular, Born rule derivations in MWI are mathematically incorrect”. All the evidence so far is consistent with my claim: I know only invalid or valid circular derivations of Born rule in MWI. I think on this aspect there should be no debate. The real test of my claim is based on future attempts to derive the Born rule. My claim is falsifiable: show me a valid non-circular derivation of Born rule in MWI and I stand corrected. I already rebutted above: “There's no reason why Everettians should not do the same.” and therefore the only way to disprove/reject my claim is to produce a valid non-circular derivation of Born rule in MWI.

“Everettians can simply take it as a primitive claim of the theory that the Born rule specifies probabilities (or their functional equivalents).”

Again, easier said than done. Take the S-G example from the post and show what it means for the probabilities to change when I rotate the device.

“Saying that probabilities are given by the squared-amplitudes is a theoretical identification, like saying that water=H20. Such identifications don't have to be derived but are judged on their theoretical consequences.”

This is fine with me. What are the consequences in my example from the main text? I run the experiment N times and I get N+1 worlds with distinct total numbers of up and down. I rotate the device and repeat the same experiment N times: I get the same N+1 worlds with distinct total numbers of up and down. *No change*. The value of the coefficients simply do not matter.

Best,
Florin

2. Hi Florin,

Thanks for the reply. Some responses below:

"You state: “To compare: no other interpretation can derive the Born rule either! They all just postulate it as part of the theory.” This is simply false. Gleason’s theorem derives Born rule given its assumption: the existence of a probability function respecting some natural requirements. No such thing exists in MWI..."

I don't understand why you say this. Gleason's theorem is just as applicable to MWI as it is to other interpretations. The assumption of the existence of a probability function is just as strong an assumption in e.g. Copenhagen as it is in Everett.

"You also state: “There's no reason why Everettians should not do the same.” This is easier said than done. If you postulate Born rule you have to explain probabilities in MWI and this is where you run into trouble. Show me how you do it. Take the S-G example from the post and show what it means for the probabilities to change when I rotate the device."

For Everettians just like for others, for the probabilities to change is just for the squared-amplitude co-efficients to change - no more, no less. I don't see the difference, or the special problem for Everettians.

"“Everettians should simply say that branch counting sometimes gives the right answer by coincidence even though it is not the correct rule to use.” I agree with this statement, but this does not reject my claim."

I had misunderstood your argument, then. It seemed to me that you were arguing that since branch counting sometimes agrees with the Born rule, Everettians could not consistently maintain that branch counting was incorrect and the Born rule was correct. I pointed out that this would be a bad argument, using the clock analogy. But if your point is just that all derivations of the Born rule are either incorrect or circular, then my objection does not apply.

As it happens I'm agnostic about whether Everettians can derive the Born rule. I'm not sure whether the existing derivations work, and I'm not sure whether other derivations might arise in the future. I thought you were arguing that no derivation could possibly work, and that was the argument I was objecting to.

"I run the experiment N times and I get N+1 worlds with distinct total numbers of up and down. I rotate the device and repeat the same experiment N times: I get the same N+1 worlds with distinct total numbers of up and down. *No change*."

No change in the number of worlds, no, but Everettians don't have to agree that this means there are no changes at all. They can say that there are changes in the coefficients so there are changes in the probabilities.

"The value of the coefficients simply do not matter."

What is the argument for this? All the Everettians I know would deny it. It seems to me that you're presupposing that the only thing that could possibly be relevant to probabilities for Everettians is the branch numbers. And I don't think there's any reason to make that presupposition.

all best
Al

3. Hi Al,

“The assumption of the existence of a probability function is just as strong an assumption in e.g. Copenhagen as it is in Everett.”

I think this is the main contention point. I do not know what probabilities mean in MWI.

Let’s consider this: let’s have an unfair (quantum) coin (corresponding to a vertical spin 1/2 measurement) which lands 80% heads and 20% tails. In a non-MWI description (Copenhagen, etc), when I flip the coin I get either heads or tails, and if I repeat it I get on average 80 heads and 20 tails out on 100 tries (up to the expected statistical fluctuations; if you want to reduce them repeat say 1 million times). Now in an Everettian description, at each flip the worlds splits and in each split I get both the heads and tails outcomes. How can I define probabilities in an Everettian description for my unfair coin?

To me the probabilities in this case should let me know if the coin is fair or not.

If I do the same thing with a fair coin, I get the very same outcomes and word splits as in the unfair case. *In other words, in an Everettian universe outcomes by themselves cannot tell a fair from an unfair coin.* And so the concept of probability in an Everettian universe is meaningless.

All other points are secondary to this point and I will not reply to them to preserve clarity. How can you tell apart a fair from an unfair (quantum) coin in an Everettian universe?

Best,
Florin

PS: please note my operational meaning attached to probabilities. I can say probabilities in MWI just as I can say flux capacitor, but without a concrete way to test its experimental consequences, the concept is vacuous, just meaningless marks on paper.

4. PPS: My argument above is not an argument against MWI, and I believe Wallace would reply in the same way. It is only an argument against the contention that Born rule need not be derived in MWI. Once the need for deriving Born rule in MWI is established, we need to look at the validity of such a proof. I contend that if you give me such a proof I will either
(a) find a mathematical mistake in it
or
(b) find a place where you assume Born rule and hence your derivation is circular

If I cannot do either (a) or (b) I will admit defeat.

Since I managed to derive QM from physical principles, I am confident in my QM knowledge to put this challenge to the MWI community. I can also bet you cannot defeat Bell correlations using LHV, or square the circle, but I don't think there are sane takes on those bets.

5. Hi Florin,

Thanks for the reply. Given your claims about operational meaning, I think I now see where you're coming from with this critique. It may be that operationalism/frequentism about probability is what we disagree about, rather than anything specifically quantum.

"Let’s consider this: let’s have an unfair (quantum) coin (corresponding to a vertical spin 1/2 measurement) which lands 80% heads and 20% tails. In a non-MWI description (Copenhagen, etc), when I flip the coin I get either heads or tails, and if I repeat it I get on average 80 heads and 20 tails out on 100 tries (up to the expected statistical fluctuations; if you want to reduce them repeat say 1 million times). Now in an Everettian description, at each flip the worlds splits and in each split I get both the heads and tails outcomes. How can I define probabilities in an Everettian description for my unfair coin?"

You can't define probabilities in terms of frequencies given Everettian QM. But I say that you can't define them in terms of frequencies at all, no matter what your interpretation. No matter how times you toss your biased coin, there's always a chance that they'll all come up heads. Sure, this chance gets very small as you toss it many times, but the fact remains: probabilities don't align with relative frequencies automatically, but *only with very high probability*. Since the connection between probabilities and frequencies itself presupposes probability, it can't be used to define probability. That's true in all interpretations.

"*In other words, in an Everettian universe outcomes by themselves cannot tell a fair from an unfair coin.* And so the concept of probability in an Everettian universe is meaningless."

Again, I agree, but I think that probabilities cannot and should not be reduced to 'outcomes by themselves'. We need to bring in the branch weights also.

"How can you tell apart a fair from an unfair (quantum) coin in an Everettian universe?"

You can do the same as in the non-Everettian case. Toss it many times, observe the frequencies. In the non-Everettian case, worlds in which frequencies approach probabilities get high probability. In the Everettian case, branches in which frequencies approach branch weights get high branch weight. Both approaches are just as circular as one another.

By the way, I think Wallace would agree with everything I say above. Look at Saunders, 'What is Probability?', in the 'Many Worlds?' edited collection, for a more detailed presentation of this sort of position.

all best
Al

6. Hi Al,

Sorry for the delay in response, I was very busy working on a paper.

You state:
“You can't define probabilities in terms of frequencies given Everettian QM. But I say that you can't define them in terms of frequencies at all, no matter what your interpretation. No matter how times you toss your biased coin, there's always a chance that they'll all come up heads. Sure, this chance gets very small as you toss it many times, but the fact remains: probabilities don't align with relative frequencies automatically, but *only with very high probability*. Since the connection between probabilities and frequencies itself presupposes probability, it can't be used to define probability. That's true in all interpretations.”

This is why I emphasized the operational point of view. Vaidman asks about self-location, I ask a similar question: can you detect the biased coin in a MWI universe? Suppose I work in a factory producing coins and my job is that of quality assurance: eliminate the biased coins. How do I do it in a MWI universe?

“You can do the same as in the non-Everettian case. Toss it many times, observe the frequencies. […] In the Everettian case, branches in which frequencies approach branch weights get high branch weight.”

I believe we are at an impasse. I do not understand your statement: “branches in which frequencies approach branch weights get high branch weight”. How does a branch get a branch weight? What does this mean? What is a branch weight?

Best,
Florin

7. Hi Florin,

Branch weight is just an Everettian term for the coefficient, or squared-amplitude, associated with the branch. So a branch gets a high branch weight by a high coefficient being associated with that branch, and a set of branches gets high total weight by the sum of the coefficients of each branch in the set being high.

So, my claim is that as the biased coin is tossed many times, the sum of the coefficients associated with the set of branches in which the relative frequency of Heads approaches 0.8 itself approaches 1. This is what Everett's original argument shows. And it's no more or less convincing than the equivalent argument in a non-Everettian interpretation: as the biased coin is tossed many times, the probability of the relative frequency of heads approaching 0.8 itself approaches 1.

As I say, all this is a basic part of the modern Everettian approach due to Saunders, Wallace, etc. They also try to go beyond this using the decision-theoretic arguments, but I maintain that those additional arguments are optional and not required for probability to make sense. We don't require any such additional arguments in the non-Everettian context.

all best
Al

8. Hi Al,

Thank you for your reply. You state: “a branch gets a high branch weight by a high coefficient being associated with that branch”

Now I understand your point, but this statement is circular in my opinion. You stated something above about circularity.

Do you agree this particular statement is circular?

Best,
Florin

9. Hi - that was just a part of a definition of the term 'branch weight', which you'd asked for! So, I don't think it's fair to call it circular - unless all definitions are circular.
all best
Al

10. Now I got the definition, but the circularity criticism is not on the definition but on the action: "a branch gets a..." How does a branch get its weight? This is what I contend it is circular. Do you agree with this statement?

I flip a biased coin which lands with probability p heads, and (1-p) tails. In MWI there is a universe where the coin lands heads, and a universe where the coin lands tails. The probability p information is lost in MWI as I always have two branches. And it is lost every single time I repeat the coin flipping. It does not matter if I flip the coin once or a million times. To preserve the p information you have to add it by hand in MWI in a circular way. Your statement is basically this: our universe (meaning our branch) gets the p weight because p is associated with our universe.

Best,
Florin

11. Thanks, Florin. I didn't mean anything special by 'gets a branch weight' - I just meant that the branch has a particular property. So I don't think that it's circular, but perhaps there are other problems with the claim.

In fact I think I understand your worry a bit better now. Your concern is that it makes no sense to say that a particular universe is characterized by a parameter like the branch weight / squared-amplitude / co-efficient (or whatever you want to call it). I personally see no problem with this. The branch weight is just a property that the universe has. Quantum mechanics itself describes that property and tells us how to calculate it. Perhaps your concern is that it can't be directly observed - I agree with that. But quantum mechanics still tells us how to estimate it through measurement of relative frequencies as I described above.

You say 'information is lost in MWI' - I don't see why this should be. The information about the coefficients isn't encoded in the number of branches, no. But the information doesn't disappear either. Why should it? It's encoded in the coefficient of each branch. There is no mechanism in QM where these coefficients get deleted, or set to zero, or whatever, on a measurement. A process like that is added by some interpretations, but not by Everettians.

all best,
Al

12. Hi Al,

The branch weight is meta-information about the branch: suppose I have two branches and I attach a number to the branch 1 say 0.8 and to branch 2: 0.2. But I can also attach some other meta-information to the branches like color, or a bank account number, or toothferryness. To make them meaningful there should be a way to associate those meta-information to experimental outcomes because this is what an experimentalist actually measures. Toothferiness is an absurd notion but I am making a point. In the case of MWI if the associated weight number is .8 the way to associate the weight with the experimental outcomes should be that if I flip a coin 1 million times, I will get 800 thousands head and 200 thousands tail plus or minus one thousand.

This association should be 1-to-1. The problem in MWI is that the prescription is not 1-to-1. No matter which weight I choose, the branches are always the same given by the binomial coefficients.

To take this to the extreme, suppose the branch weight for heads is 0.999999999… and the one for tails is 10 to the power of -100, both numbers adding up to 1. If I flip this coin one million times I will get for sure all heads (the odds against it are 1 to 10 to the power 94. For comparison the mass of the visible universe measured in electron masses is less than 10 to the power of 90, four orders of magnitude smaller). But in MWI only 1 out of 2 to the power 1 million (2 to the power of one thousand is 10 to the power of 300) cases I will get all heads. This is absurd.

Best,
Florin

13. Hi Florin,

I don't see any reason to accept your claim that 'This association should be 1-to-1.' What is the argument for this?

Similarly, I don't see why the example you give at the end is absurd. There's just no reason to think that branch numbers have anything to do with probabilities, since they're not physically fundamental.

As Saunders and Wallace have argued, and as Gleason's theorem shows in the Everettian context, there's something physically special about the branch weights which distinguishes them from 'toothferiness' and makes them appropriate candidates to be identified with probabilities.

all best
Al

14. Hi Al,

“I don't see any reason to accept your claim that 'This association should be 1-to-1.' What is the argument for this?”

I explained why you need this: to have a well-defined meaning of probabilities, otherwise the concept is vacuous just like unicorns or tooth ferries. Physics is an experimental science and you need to attach experimental consequences to concepts. No 1-to-1 association means no experimental distinction between weights and no way to tell apart a fair from a biased coin. Simply stating that you watch the outcomes to determine a fair from an unfair coin is not good enough. This has to follow from theory and it does not because there is no 1-to-1 association. If it does not follow from theory, then fundamentally MWI is not different than: “In the beginning God created the heavens and the earth.” Is MWI faith based, or is it a falsifiable scientific theory?

“Similarly, I don't see why the example you give at the end is absurd. There's just no reason to think that branch numbers have anything to do with probabilities, since they're not physically fundamental. ”

It is absurd because you get odds impossible to be realized in our universe.

“As Saunders and Wallace have argued, and as Gleason's theorem shows in the Everettian context, there's something physically special about the branch weights which distinguishes them from 'toothferiness' and makes them appropriate candidates to be identified with probabilities. ”

I never had the pleasure of meeting Saunders, but I listen to many discussions Wallace took part in it and I never comprehended his arguments. If they will humor me and repeat their arguments here in this blog, I will give them my full attention and I will reply to them. But since you are a MWI expert, I presume you can repeat their arguments here.

Best,
Florin

15. Hi Florin,

Sorry for the extensive delay. I think we've reached an impasse now, since I simply don't understand why you continue to insist that proportions of numbers worlds and probabilities have to go together. The only reason I can imagine for thinking this is the assumption that we should treat our world as randomly picked from all the worlds there are. But quantum mechanics is not like a lottery machine, so there's no reason to treat all the individual worlds that exist as equally probable.

You talk about falsifiability, but I don't see how the assumption that probabilities is given by number of worlds is any more or less falsifiable than the assumption that probability is given by branch weights. It's not like we can directly measure numbers of worlds either.

Thanks for the interesting discussion, hope to get to talk more in person someday.

all best,
Al

16. Hi Al,

Indeed I don't think we can make any more progress and we can agree to disagree. I bet we will meet in person one day and I am looking forward to it.

All Best,
Florin

9. Hi Florin,
Can you summarize Gleason's assumptions for lazy people like me (!)
who do not want to read his papers? Are they reasonable and acceptable right away?

1. Hi Kashyap,

Here it is (from Ingemat Bengtsson and Karol Zyczkowski book: Geometry of Quantum States, page 143):

Assumption 1: normalization: The elements $$|e_i \rangle$$ of every orthonormal basis are assigned probabilities such that:
$$\sum_{i=1}^{N} p_i |e_i\rangle \langle e_i| = 1$$

Assumption 2: non-contextuality: Every vector is an element of many orthonormal bases. The probability of its ray is independent of how the remaining vectors of the basis are chosen.

Gleason's theorem: under the conditions stated, and provided the dimension of N of the Hilbert space obeys N>2, there exists a density matrix $$\rho$$ such that $$p_i = Tr (\rho |e_i \rangle \langle e_i|$$)

10. This is interesting. Would you have any objection to me sharing this with my friends on Google plus?

1. Not at all, please be my guest.