Discussion on the foundation of quantum mechanics

Question

This thread continues the discussion begun here about the measurement problem and its interpretation.

Answer 1

user1247 wrote: 

indexical uncertainty in the space of all worlds leads to subjective randomness. Their presence plays an important role.

What is the probability distribution on the space of all worlds underlying Everett's interpretation? Without an assumed distribution, nothing definite follows.  

Even given such a probability distribution, it only leads to subjective randomness in the space of all worlds, not in the single world we observe. To see this, simplify the space of all worlds to be just $[0,1]^n$, with uniform distribution. Then drawing worlds is a stochastic process, but each particular world is completely deterministic. 

For $n=10$, say, the world [1111100000] is just as likely or unlikely as [1110111110] or [0101010101] or [0110100110]. One therefore cannot deduce anything probabilistic about a single world from an assumed uncertainty about the space of all worlds. 

For more complex worlds we have the same paucity of deductions. Any inference from a probabilistic law to a particular realization is mathematically spurious, hence based on wishful thinking or silent assumptions.

Comments

Maybe someone else can jump in and explain what Arnold is saying, because I'm at a loss. That indexical uncertainty leads to subjective randomness in a multiverse is completely and totally uncontroversial. What is controversial are perhaps questions like the one I asked here that generated this thread and which I would be very interested in an answer to. Unfortunately Arnold's answer was essentially a non-sequitur as explained in the comment section there. 

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The question is: what's the measure? Suppose the transporter is malfunctioning, and produces 1 kirk a day forever, each on the same planet on the same rock. What is the probability distribution for Kirk to come out on such and so date? The question is only meaningful when you know what measure to assign to the different kirks, and it depends on how the transporter malfunctions, and on exactly how you can come and talk to the different kirks so produced.

This philosophical issue can be resolved with the kirk example and in many worlds, but it requires careful philosophy (which Everett did, and others have difficulty following). The main principle within quantum mechanics is that there is a unique measure which is picked out from the consideration that "small worlds have a small probability". Where "small" is by sum-square or compatible measure. The natural measure on worlds is the usual quantum probability.

For Kirks, the natural measure depends on exactly how YOU can come and talk to Kirk, so if you measure on day 30, and you talk to a Kirk at random, you find the probability for Kirk is 1/30. If you do so on another day, you find another probability. The internal probability for Kirk is only meaningful teleologically, if you define the full process and look backwards from an end-point where the process is completed, and you can account for all the kirks produced, and how they communicated with other people, and so on. This kind of headache is annoying philosophically, and the end result is just the same as "shut up and calculate".

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@Ron Maimon, yeah, the measure problem is an issue I'm aware of, but I think it is distinct from Arnold's more radical position that subjective probability cannot occur in a deterministic system. In the Kirk scenario there is subjective probability, regardless of how you want to argue about what that probability is. 

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 @user1247:

Arnold's more radical position that subjective probability cannot occur in a deterministic system. In the Kirk scenario there is subjective probability

There is (in the limit of infinitely many recursive clonings) a conditional probability P(red|person), but there is nothing subjective about it. It is determined in the usual objective way

P=lim(number of good cases/number of all cases), 

so calling it a subjective probability is a misnomer.

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I don't know what to say at this point other than that it is absolutely clear to me that you do not understand the concept, and that any other response from me would be redundant. Maybe it will perhaps spark some insight if I tell you that of course the experiment can be done in a way such that there is no possibility of objectively keeping track of indices. This isn't really relevant, but it may help you understand. For example the "original copy" is destroyed and two clones are produced at each step, such that there is no way to objectively keeping track of trajectories. It is entirely subjective. And any claim that it is really "objective" is silly in a logical positivist sense that I explained earlier, in that if you poll the participants they will describe subjective experiences that are analogous to the experiences of scientists measuring quantum observables: in each case what they see is purely random to within statistics.

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Any other word play of "objective" vs "subjective" is to divert attention from the point; namely, that, for all practical experimental purposes, if you find yourself to be in such a cloning experiment, you will observe certain observables to be purely random to the best of your ability to ever tell. 

Answer 2

@user1247 wrote: 

it is certainly a philosophical concept, just as the MWI is philosophy. [...] Maybe see Bostrom's "Anthropic Bias" for a review.

The foundations of quantum mechanics comes in two forms: those for physicists and those for the general public. The latter version is usually more fiction than fact, not to be taken serious as science, since arguments are oversimplified, and more emphasis is given to selling a story than to get it factually right. Its purpose is to raise curiosity, but to understand what is scientific about it, one has to look for less biased material.

Everett's work is physics, not philosophy. But Bostrom is a philosopher. His book Anthropic Bias can be found here. One can see from the preface that it is addressed to all intellectually active people, hence belongs to the second category. The level of the audience is specified as

a reader who doesn’t like formalism should not be deterred. There isn’t an excessive amount of mathematics; most of what there is, is elementary arithmetic and probability theory, and the results are conveyed verbally also.

His book is of course based on science, but shouldn't be taken as a serious source for claims about what is completely and totally uncontroversial.

Comments

You seem to be equivocating between "philosophy" and "pop-science/philosophy". The book by Bostrom is not "pop-science/philosophy", and I don't see how "a reader who doesn’t like formalism should not be deterred" is really relevant criticism of whether or not it should be taken seriously as a source (much of contemporary philosophy falls into the same category). The book happens to be a nice self-contained review of the basic concepts involving  "indexical uncertainty," and provides many authoritative citations you can follow up on if you don't like the pedigree of the book itself.  

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BTW "self-locating uncertainty" is another keyword, the article here by Carroll and Sebens is the first thing that comes up in google.

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@user1247: I didn't say Bostrom is "pop-science/philosophy" but

His book is of course based on science, but shouldn't be taken as a serious source for claims about what is completely and totally uncontroversial.

The first part acknowledges that it contains relevant stuff, in fact the book is his Ph.D. thesis in Philosophy. I would say the second part about any book on philosophy of science. I have read many - the whole point of writing such a book is to present the author's view of things. It is necessarily biased - namely by the author's view of what is natural to assume without saying so.

This author even knows about how different the results can be given different priors, and even a noninformative prior (such as the uniform one in your example in the other thread) is a very strong assumption, far too informative in the absence of knowledge.

The book happens to be a nice self-contained review of the basic concepts involving  "indexical uncertainty,"

Bostrom is not very careful in building up his conceptual basis. For example, I couldn't infer from looking at quite a number of occurrences of the term 'indexical' what it should mean. He explains nowhere, he assumes the reader to share his background knowledge. In the mean time I found a wikipedia article that explains something about this term, but still the precise meaning of 'indexical information' and 'indexical fact' remains unclear to me.

In any case, from what I seem to understand from Wikipedia, physics is not at all concerned with indexical information and indexical facts but only with objective information and objective facts. To use arguments involving it to make claims about emergent probabilities in physics is therefore very dubious.

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@Arnold, the "index" in "indexical" is very easy to understand from context, to the extent that I'm not sure I should take seriously your purported confusion. It refers to the "index" that identifies an element of an observer's reference class. The point being that one is uncertaint which "index" he or she is. For example if you clone a person into a red room and a blue room, before she opens her eyes she is uncertain which index she is, the one identifying the copy in the red room or the blue room. 

Your last paragraph indicates you may be more interested in playing rhetorical games that actually understanding anything about indexical uncertainty as it relates to the MWI. If you don't want to discuss philosophy of physics, then fine. But then don't say that the MWI cannot "be insightful" if you are not prepared to engage in a philosophical argument against the widely understood notion that the MWI is a deterministic theory that produces the subjective experience of pure randomness. This is an "insight" that has been assumed and integrated into the philosophy of physics discourse for decades.

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@user1247: be assured that I don't play games (I have no time for it), and that if I say I don't understand something then this is really the case. In your last explanation, I can sense something of what you might mean, but it is very nonstandard English language that you shouldn't expect to be understood without an explanation. Not to argue, but in order that you understand how foreign your concepts are for me, let me give the following example:

she is uncertain which index she is

Is the person an index? Am I an uncertain index if I wake up in the morning, before I have looked on my clock to find out which time it is? probably not. Thus the index issue is not clarified by your statement.

After all this, I guess that something (and not the person but the wall color or the clock time) is indexical, because it refers to a subject, and that the subject in question may be uncertain about something that depends on its current situation, which makes the corresponding information having indexical uncertainty. But I have to read this between the lines, as neither your philosopher nor wikipedia gives a clear definition of how the term is used. And what precisely the index is in all this is still unclear o me.

You don't need to reply to this, but please understand - there seems to be a huge barrier between what you and I are taking for granted. So please remove misunderstandings but don't get upset and accuse me based on lack of understanding.

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@ArnoldNeumaier: "Indexical uncertainty" is nonstandard usage only in that cloning observers is not a common situation people think about, Everett was the first to consider it. For a discussion of this within the field of philosophy, you can read Dennett's 1980 "Where am I?", which is reprinted in Hofstadter and Dennett's "The Mind's I". This also contains a passage on many-worlds somewhere inside, where the philosophical idea is expounded upon. The concept is as Bostrom explains it, but the concept of "indexical uncertainty" is a new one to Everett, and he identifies this as the source of quantum information. It's a very strange, very radical idea, but it is ultimately practically equivalent to Copenhagen with just a realist philosophy on top.

Answer 3

In mathematical physics we also have expansions of functions like $\psi=\sum a_i \phi_i$, but we do not call the coefficients $a_i$ "probability amplitudes". Therefore, the Born rule is not only due to the normalization condition and is not due to "our definition" ;-) . It has something to do with physics.

Comments

@Vladimir, I think you are missing the point. A connection between mathematical formalism and physics including state reduction to $\phi_i$ is taken for granted here, the question is what is M($a_i$) given the assumption of Hilbert space and that $a_i$ somehow encode the wave function density in a way that indexical uncertainty leads to probabilities that must sum to 1. 

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state reduction to $\phi_i$ is taken for granted here

It is very important that in QM one obtains one $\phi_i$ in each measurement, not $\psi$. This fact changes the meaning of coefficients $a_i$ and makes QM interpretation weird.

As to derivation of the Born rule, one can consider a scattering problem, when a flux $J$ falls on a scattering center. Then coefficients $|a_i|^2$ are "branching ratios" and this is literally derived rather than postulated. But for a low intensity flux $J$ (one particle in a beam), the branching ratios become inevitably probabilities. Thus, this interpretation is due to discreteness of the low intensity flux $J$: one incident particle - one measured outcome $\phi_i$. If $J$ were continuous till $J=0$, we would always have all outcomes together rather than one $\phi_i$.

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Vladimir, yes you are saying true things, but you aren't really addressing my question. 

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@user1247: OK, let me remind that so far we implied states $\psi$ and $\phi_i$ to be observable.

Now let us take a time-independent part of $\psi$, i.e., just a coordinate function $\tilde\psi(x)$. It can be expanded in sum of coordinate functions $\tilde\phi_i(x)$. For example, $\tilde\psi(x)$ is a ground state of the total Hamiltonian $H$: $H\tilde\psi=E_0\tilde\psi$, and those $\tilde\phi_n$ are the eigenfunctions of $H^{(0)}$: $H^{(0)}\tilde\phi_n=E^{(0)}_n\tilde\phi_n$: $\tilde\phi_i(x)=\sum_n a_n\tilde\phi_n$. When you measure the energy of the state $\psi$, you will never observe any $E^{(0)}_n$, nor $\tilde\phi_n$. The coefficients $a_n$ in this decomposition have another meaning despite Hilberst space and all that are in place here too.

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@Vladimir, it's not clear to me what you are getting at. Are you changing basis from H to H$^{(0)}$ (what is H$^{(0)}$)? Why don't you observe E$_n^{(0)}$ if you are making an energy measurement? If you are getting at the preferred basis problem, that's something of a tangent. The question is about the measure for the $a_i$ in front of the eigenfunctions of the experimental observable, not in some other basis. Also, I don't know what you are arguing against; Gleason's theorem is uncontroversial, no? 

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@user1247: One cannot make a measurement of the energy operator $H$ as only energy differences are meaningful. The observation of spectral lines indeed gives $E_j-E_k$ for certain Hamiltonians. But it is obvious that these are not measurements according to Born's rule. The latter applies only in very special (collapse) situations - most measurements are not of this kind.

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@Arnold, OK but I fail to see the relevance to the question.  Of course I'm only interested in experimental questions where the Born rule applies, since this is a discussion of the Born rule...

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I was just answering to your question in the comment. I can't say more about your original question than given in my answer, as I

1. don't have the thesis (maybe you can add a link to it),
2. wouldn't read it anyway after having seen the mistake he makes in the official publication - it is unlikely that his thesis is more carefully written than his publication, and
3. believe that the MWI explains nothing but is the worst kind of philosophy.

What would have to be explained is why, in the only world we can actually observe, the rules are as they are. But clearly, all the other worlds cannot contribute the slightest to this, as they never talk to each other. Thus all the math is spurious and can be dropped by Ockham's razor, without losing the slightest amount of insight. There are other strange things - it is never made clear when the current world splits; thus the only consistent interpretation would be that any arbitrary wave function describes one of the possible worlds. But this makes the theory vacuous....

See instead my view of the measurement problem for what I think is the appropriate way to look at the matter.

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@Arnold, "But clearly, all the other worlds cannot contribute the slightest to this" But of course this is wrong given that indexical uncertainty in the space of all worlds leads to subjective randomness. Their presence plays an important role. "Thus all the math is spurious and can be dropped by Ockham's razor, without losing the slightest amount of insight." You don't think solving the measurement problem within the framework of a real, local, deterministic theory could be insightful? In any case this is a complete tangent that has nothing to do with the narrow scope of my question.

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 @user1247:

I addressed your first statement in an answer below.

You don't think solving the measurement problem within the framework of a real, local, deterministic theory could be insightful? 

I believe that the dynamics of the universe (the only experimentally accessible one) is ultimately realistic and deterministic, though we currently do not yet understand how. It cannot be local though in the sense usually used in foundational work, as nonlocality is an experimentally verified fact. It can be local only in the sense used in quantum field theory.

But I don't understand what this question has to do with your immediately preceding sentence where you invoke subjective randomness. In a deterministic world, randomness enters objectively through chaos, not through subjects. The universe evolved according to the same laws as today even long before anything qualifying as a subject could exist.

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@ArnoldNeumaier, I am using 'local' the way I think it is ordinarily used is such discussions, ie Bohmian mechanics is non-local, Everettian is local, etc. Obviously of course there are Bell-like non-classical correlations that we are all aware of but I don't think those are called 'non-local' typically, but again this is another tangent.

> In a deterministic world, randomness enters objectively through chaos, not through subjects.

Deterministic chaos is not the pure randomness that we are talking about in QM. You seem to be equivocating, as I'm sure you quite realize this. Indexical uncertainty in principle leads to the subjective experience of pure randomness, this isn't something I think anyone disputes, so I am pretty confused about what we are arguing about.

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I was trying to make sense of your comment. Probably I don't know enough about current views on Everett's theory, I only read his published paper, a long time ago, and at that time it didn't make sense; so I tried to spell out here some of what I think is irrational about his theory, as i remember it. I try to express myself as clearly as I am able.

In one sentence you talk about randomness, in the next about determinism. How are the two related? How can deterministic dynamics contain any randomness apart from that induced by chaos? I never before heard of the term 'indexical uncertainty', where is the term explained? Wikipedia has no entry for it. Maybe I am the first one to dispute the subjective experience of pure randomness, but how can subjectiveness enter physics at a time when there was no subject? 

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scholar.google.com gives exactly 10 entries in the search for "indexical uncertainty" (with quotation marks present in the query), none of them in the physics literature. I conclude that it is not a physically well-defined concept but just a philosophical idea. I asked here for the precise meaning.

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@Arnold, there are many names for the same thing, it is certainly a philosophical concept, just as the MWI is philosophy. It's just the idea that if you clone an observer, that observer subjectively experiences "indexical uncertainty" until he or she identifies which clone he or she is, the same way Kirk experiencing a transporter malfunction that copies him to two places will have a 50% chance of finding himself in one or the other. The transporter copying him is a purely deterministic process, but his subjective experience is purely random. Maybe see Bostrom's "Anthropic Bias" for a review. I will respond to the question you opened when I get a chance.