Revisiting the Born Rule: Implications for Cosmology

[Count Iblis]

http://arxiv.org/abs/0903.4888"

The Born rule may be stated mathematically as the rule that probabilities in quantum theory are expectation values of a complete orthogonal set of projection operators. This rule works for single laboratory settings in which the observer can distinguish all the different possible outcomes corresponding to the projection operators. However, theories of inflation suggest that the universe may be so large that any laboratory, no matter how precisely it is defined by its internal state, may exist in a large number of very distantly separated copies throughout the vast universe. In this case, no observer within the universe can distinguish all possible outcomes for all copies of the laboratory. Then normalized probabilities for the local outcomes that can be locally distinguished cannot be given by the expectation values of any projection operators. Thus the Born rule dies and must be replaced by another rule for observational probabilities in cosmology. The freedom of what this new rule is to be is the measure problem in cosmology. A particular volume-averaged form is proposed.

 

[Fra]

> The Born Rule has died

If it wasn't already, it's about time.

As is usually pointed out in papers on the measurement problem of cosmology quantum theory as an intrinsic measurement theory makes no sense.

Measurement requires a context, and when you study isolated subsystems embedded in large environment this context is usually fixed at least within the timescales of the duration of any interactions with the subsystem. In this case quantum theory, born rule and state vectors and the usual statistical foundations make sense.

But several of the premises for such a measurement theory aren't near valid for open models, and cosmological models. What fails is IMO not just the born rule itself, but the physical foundation for objective basis for observed statistics.

But perhaps this isn't a Quantum physics question, I think it's more a beyond standard model question.

/Fredrik

 

[humanino]

This title is ridiculous. The Born rule is quite alive and useful. People know the hypothesis used when deriving it.

 

[ZapperZ]

Isn't this like saying Newton Laws are dead just because they don't work in certain regimes? Did anyone informed any of those engineers that built your houses and your buildings and your bridges?

Zz.

 

[humanino]

Isn't this like saying Newton Laws are dead just because they don't work in certain regimes? Did anyone informed any of those engineers that built your houses and your buildings and your bridges?

My thoughts exactly. I wanted to suggest how people would have felt if Einstein would have entitled his paper "Newton died (again)". Is the fact that he did not an indication that he understood the "subtlety" ?

 

[D H]

Isn't this like saying Newton Laws are dead just because they don't work in certain regimes?

It's worse than that. Those certain regimes where Newton's Laws don't apply are observable. We can assess how well (or how poorly) Newtonian mechanics performs. Observation is what separates science from non-science. This

"However, theories of inflation suggest that the universe may be so large that any laboratory, no matter how precisely it is defined by its internal state, may exist in a large number of very distantly separated copies throughout the vast universe. In this case, no observer within the universe can distinguish all possible outcomes for all copies of the laboratory.​

is not science.

 

[Count Iblis]

Isn't this like saying Newton Laws are dead just because they don't work in certain regimes? Did anyone informed any of those engineers that built your houses and your buildings and your bridges?

Zz.

I don't think so, because Page is still assuming that Nature is fundamentaly described by quantum mechanics. So, it is a failure of the Born Rule within quantum mechanics.

 

[ZapperZ]

I don't think so, because Page is still assuming that Nature is fundamentaly described by quantum mechanics. So, it is a failure of the Born Rule within quantum mechanics.

It's my turn to say "I don't think so". You only have to read his abstract:

This rule works for single laboratory settings in which the observer can distinguish all the different possible outcomes corresponding to the projection operators.

This is no different than the applicability of Newton's Laws.

Zz.

 

[xepma]

Motl has quite a post on this one (or, actually, on a follow up on the paper you mentioned). Havent read it completely through though.

http://motls.blogspot.com/2009/07/don-page-born-again.html

 

[Fra]

Another title might be better I agree, but I didn't choose to focus on a poor choice of title.

QM formalism (including the born rule) is indeed very successful for atomic and particle physics. No sane human would deny that. But that isn't the problem either. It's the unification problems in physics, and to make sense out of statistics and rules of inference of statistical measures, when the ordinary statistical data simply can not be realized physically.

I think from a conceptual and foundational physics point of view in cases where the context of measurement is not as solid as in a particle lab, the assumptions behind quantum theory as a framwork, in particular the notion of state of the universe, multiverses and the meaning of statistics in cases where it's obvious that the statistical data simply can't be produced physically not even in a thought experiment, some of the old abstractions in quantum theory such as universal contexts for state spaces and born rules are not good enough.

But then that's not a question within quantum physics, it's rather a question in a wider context which seeks to extend the standard framework which is the home of the standard model becuse the question questions the framework itself.

Edit: About Newtons laws, the Newtonian realist/determinist way of thinking, is also a "dead conceptual framework" to me, in the sense I referred to at least. That doesn't mean that framework hasn't produced still going strong theories.

Since the paper mentiones cosmological measuremnt problem, I assumed that the OT referred to the foundational problems implicit in quantum mechanics when you add GR. It's still true that wether the problem is in QM or elsewhere, or both, is still a matter of opinon. My personal opinon is that thevery basic structure of QM, including the hilbert space structure and born rule with is not satisfactory as a unquestionable starting point.

/Fredrik

 

[Count Iblis]

This is no different than the applicability of Newton's Laws.

Within a hypothetical universe described exactly by classical mechanics, Newton's Laws would not fail. But, Page points out how Born's rule within a universe exactly described by quantum mechanics can fail. See also here:

http://arxiv.org/abs/0907.4152

 

[Fra]

"However, theories of inflation suggest that the universe may be so large that any laboratory, no matter how precisely it is defined by its internal state, may exist in a large number of very distantly separated copies throughout the vast universe. In this case, no observer within the universe can distinguish all possible outcomes for all copies of the laboratory.​

I didn't read the paper in detail, I just commented what I think was one of the general points. Lee Smolin has some interesting ponderings (http://pirsa.org/08100049/) in one of this talks on the reality of law about the notion of "exhausting all possibilities of the future". There are analogies to biology where in effect the state space evolves, and thus the notion of a state vector just evolving in a fixed state space makes no sense, except in cases where you are ina stable environment, defining a context and studies a subsystem. This is exactly when QM makes sense.

In cosmological models, or in general, open systems, this idealisations isn't a valid constraint since I don't see how any possible inference from physical experiment can establish it with certainty. Thus we might need a new way of undertaning the meaning of statistics and state spaces, when there simply is no fixed background context.

However I didn't understnad why the paper highlights the born rule?? The born rule in isolation can hardly be the prolbem, it's the entire notion of timeless state space structure that is the problem.

How can we infer a timeless structure, when the inference process itself takes time? Moreover there are no infinite information sinks we can instantly access.

The entire objection of timeless structures, are insignificant when you study relatively shortlived subsystems. So then QM makes sense. But it might be limiting case, the question is then what is the general case?

/Fredrik

 

[yossell]

While I agree that reports of the rule's death may be exaggerated, I don't fully follow the analogies that are being made here with Newtonian Physics.

Strictly speaking, Newtonian Physics is false. Quantum Mechanics and Special Relativity tell us where it breaks down. But in most situations that we're interested in, it's so close to the truth in its predictions (and maybe close to the truth in some stronger sense too) that we're all happy to keep on using. So, in the sense that there's a good reason to keep using it in most situations, Newtonian Physics never dies. So we use it, but we're aware that it's an approximation to better theories that we have.

What's the analogy that people have in mind? Here are two possibilities:

(A) Is there a better quantum theory that doesn't include Born's rule? That is, Quantum Mechanics is bettered by Quantum Field theory, which doesn't have the Born rule, and the Born rule can be understood as an approximation to something in QFT?

(B) Is it that the Born rule itself is not part of Quantum Mechanics proper, but is actually something added to core Quantum Mechanics that enables us to approximate to something in certain situations (perhaps, typical measurement situations?)

I don't know enough about (A), but if something like QFT was the better theory, that would really helpful for me to know.

I'm worried about (B) because the Born rule seemed to be a way of connecting the pure mathematics (complex amplitudes) with physical interpretation (probabilities/expectation values), and therefore seems quite different from what we normally do when we make a physical approximation

On reflection, there may be ways to make (B) work out, but before worrying about it I would find it helpful if anyone could say more about the analogy.

Thanks

 

[Fra]

(B) Is it that the Born rule itself is not part of Quantum Mechanics proper, but is actually something added to core Quantum Mechanics that enables us to approximate to something in certain situations (perhaps, typical measurement situations?)

As I learned QM, the born postulate is definitely part of the core QM. It's one of the postulates/axioms of QM. There are alternative ways to introduce QM, but then instead you postulate other things, such as relations between operators, which is not more plausible IMO. So as far as I'm concerned it's one coherent package but you might invent different ways to introduce it.

If the author of the paper suggest keeping all other structure of QM, but JUST replace the born postulate by a more complex, nonlinear inference of statistical measures then I do not believe in that idea. I think there is a problem, but it's much worse and just replacing the born postulate is unlikely to solve anything, or to make sense.

To me the notion of a hilbert space, encoding all information, and the time evolution is simpyl a unitary evolution in this timless space, is what makes no sense to me. It only makes sense in a birds view, which has no physical justification IMHO.

/Fredrik

 

[Haelfix]

Don Page is a great physicist, but sometimes he really goes off the deep end. I don't understand his arguments at all and he certainly hasn't proposed an alternative.

If he wants to play with nonlinear modifications to QM, by all means, join the line. Thats been tried literally thousands of times, without success.

 

[Count Iblis]

Well, perhaps Page's awkward notation is obsuring his message, but what he saying is very easy to understand.

All that Page does is point out that if you have more than one copy of an observer described by some pure state describing the entire universe, then the expectation value of the projection operator corresponding to the observer making a certain observation does not yield the probability of the observer making that observation. This violates the conventional formulation of the Born rule.

Using my own favorite notation, this boils down to the following.
Suppose we have one observer in a universe described by a pure state. The most general pure state is of the form:


|psi> = |O_1>|psi_1> + |O_2>|psi_2> + ...

where the |O_i> are orthonormal state vectors describing the observer being aware of some subjective information i (if the observer has done some measurement, the index i will then also specify the result of the measurement). The |psi_i> are un-normalized states that describes the rest of the universe. Then , the Born rule is equivalent to saying that the probability for an observer to find himself in the state |O_i> is <psi_i|psi_i> and that can in turn be formulated as the expectation value of the projecton operator |O_i><O_i| in the state |psi>.

So, just like in the usual Copenhagen formulation of QM, where you treat the observer as an external agent, we can formulate probabilities as averages of projection operators. But if you have more than one observer you don't have a projecton operator anymore whose average will always yield the probability of an observation.

You can e.g. consider the state:

|psi> = |O_1>|O_2>|psi_1>

Which means that we have two observers who have different information and then psi_1> specifies the state of the rest of the universe (it then also specifies where the observers are located, in the example used by Page in his latest article, the location is itself in some superposition of |O1> being in region 1 and |O_2> in region 2 and the other way around.). Since |psi> is normalized, it follows that|psi_1> is normalized. Then, clearly the expectation value of |O_1><O_1| is equal to 1, as is |O_2><O_2| so they can't be the probabilities of the two possible observations being made.

Page shows in detail that fixing this problem requires one to abandon the Born Rule.

 

[Fra]

I think I sense something specific that I feel is wrong about that analysis. I will try to read the paper later and respond. The problem is that you make use of an implicit external observer when you talk about the wavefunction of the universe. That is IMHO the first problem. There is probably a different way out, where the apparent inconsistency implies interactions between the different observers. It looks like page isn't seeing it like that.

I will try to read the paper later.

/Fredrik

 

[Haelfix]

I agree C Iblis, that's my interpretation of what he's saying also, I just have no idea why he thinks that its true.

This gets into some thorny details about how much we really want to trust the idea of an universe wide 'state'. Otoh I have no idea why he thinks this isn't already true in a small universe. Plenty of identical particles floating around that can act as measurement devices (say an environment of mostly identical electrons).

Anyway he more or less states my objection here:

"Note that in this paper I am not assuming that observations
are eigenstates or eigenvalues of Hermitian operators,
or that they correspond to subspaces of a Hilbert
space. Locally they may have that form, but the fact that
the location of the observation is indeterminate (cannot
be known by the frogs making the observation) means
that observations are not globally eigenstates or eigenvalues
or subspaces of a Hilbert space. Therefore, theorems
such as Gleason’s [62] that may be taken to imply
the uniqueness of Born’s rule need not apply to the frog
observations being considered in this paper."

This makes no sense to me whatsoever. I also don't really know what 'global' means here anyway, its completely illdefined in quantum mechanics in curved spacetimes as a general rule of thumb. Even if you did posit another form, he completely violates the principle of clustering (distant experiments are uncorrelated) which is an axiom and a necessary condition for most of modern physics.

 

[Fra]

This gets into some thorny details about how much we really want to trust the idea of an universe wide 'state'. Otoh I have no idea why he thinks this isn't already true in a small universe. Plenty of identical particles floating around that can act as measurement devices (say an environment of mostly identical electrons).

I agree these things are keys.

IMHO at least, the notion of objective state of the universe is not a scientific notion. It makes no sense. Neither does the notion of ensembles of possible universes. This temptation comes from logic that is fine for closed/confined systems, such as whatever is going on in a particle accelerator as observed from an embedding context (laboratory).

But there is a logic as to why it effectively makes sense when you study small subsystems. IMO, it has to do with relative complexy of the context, vs environment. In effect a relative inertia between context and measurement. The laboratory context is incredibly massive as compared to any imaginable high energy experiment, and there is plenty of support for the statistics.

This is completely assymmetric from the inside-out observation, where the basis for any statistics or information is bounded.

/Fredrik

 

[Fra]

While I share the opinion that current QM formalism has structural problems and that several problems in future physics might suggest we need something else, I do not connect to the specific reasoning this paper presents.

Page still makes use of a birds view in several ways, that more or less puts me off from his argumentation. Since my personal objections to this don't mix with bird-view reasoning. I am looking for a stronger inside view, and then still keeping bird view reasoning isn't radical enough for my taste.

- He makes some kind of distinction between laboratory and observer, mentioning also concsioussness. I do not connect to what he mean. IMHO, the laboratory context and the observers are connected. To me an observer need not be conscious in the biological sense. Also in each specific formulation, there is only one observer, the one implicit. other observer, seen from the point of view of the implicit observer (which is a bird in page's reasoning; but I think there is no such thing, there are only frogs) are simply parts of the environment and not special from any matter system interacting/observing it's environment.

- the notion of identical laboratories, is mixing birds and frogs views. It's common in some information theoretic treatments to use arguments like "two different observers with the same information will act consistently". The point is if you think that the information of the observers is making up it's identiy, two observers that have the same information are necessarily indistinguishable. You might even insist that it's the one and same observer. He also appears to distinguish spatial information from other information. I think there must be a uniform treatise here and that different information also lables spatial extension, and the difference also could feed interactions between the views.

- He also seems to talk about "mutually exclusive set of all possible observations". Now I am not sure what he means but without further elaboration that makes no sense to me. I think this set must be depend on time, and that there can hardly be an observable timeless set of all possible observations (ie including all future observations). So this brings in the problem of time as well into this. Timeless set of possibilibites can be found effectively in some special cases. But in a general case that simply doesn't make sense to me at least. It is strongly realist minded, and is consistent with a realist view of physical law that I don't think complies with a good scientific method. I think the set of possibilities are necessarily evolving without a reference to any realist bird views.

I think Page's not radical enough reasoning leads to the more specific suggestion that born's rule is the problem. But I think it's not just borns rule that is the problem, it's more than that.

My main general objection is that standard quantum mechanics is not an intrinsic (inside-view) measurement theory. It is a measurement theory that lives in an fixed external context. And with this I mean in a general way, not only a fixed background spacetime which I consider to be a special case of the background context. It seems Pages also is on this line, but his analysis and suggestions are I think too conservative with respect to current framework.

/Fredrik

 

[Dmitry67]

Quite interesting, thank you

It reminds me about the question I failed to reply - about the meaning of the Born rule in MWI. What exactly is meant by saying that 'the probability of that event is 1/1000000000', because there are always the observers who experience these branches of reality, and for these observers the reality is as real, as for the 'mainstream' ones.

Another thought is that say there is system A. In infinite universe, there is always a copy of A with the exact landscape around for the distance X. X is finite. So even A and B are indistinguishable in the beginning, they become different after X/c.

And another thought: any single-history theory in infinite universe becomes very MWI-like. Do you agree?

 

[Fra]

I don't know if this was addressed to someone particular but here are some random comments.

It reminds me about the question I failed to reply - about the meaning of the Born rule in MWI. What exactly is meant by saying that 'the probability of that event is 1/1000000000', because there are always the observers who experience these branches of reality, and for these observers the reality is as real, as for the 'mainstream' ones.

Not defining what a probability is - physically, in terms of information, is a common trait of many types of reasoning with bird views. In rovelli's RQM for example, he adds a footnose avoiding discussing the physical meaning of "probability".

I'm not MWI, but I guess what I adhere too could be thought of as a relational many view interpretation, and in that picture, probability and action are related measures. There are only subjective probabilities, and the observable consequence of two observers disagreeing upon probability numbers, imples a physical interacting. But then note that I picture here that an observer is not an outside observer like in QM, it's an inside player - ie it's part of the system. So it could be one atom observing another one. Their mutual interactions are (this is my personal conjecture) can be inferred from their diverging informations.

/Fredrik

 

[Dmitry67]

Fra, why Rovelli did not want to discuss what the probability is?
I want to understand his motivation.
I think this is very deep question and it can't be avoided

 

[Fra]

Fra, why Rovelli did not want to discuss what the probability is?
I want to understand his motivation.
I think this is very deep question and it can't be avoided

I clearly can't speak for Rovelli, but my impression of his reasoning is based both on my starting to read his QG book, and several of this papers on RQM, partial observables etc all the stuff that has come to be typical Rovelli.

Actually Rovelli's reasoning, up to the point where he avoids going deeper into probability and quantum mechanics, is great.

I have too look up the references. I don't know which paper or if it was in his book, but he never really elaborates on this. There is a footnote just noting that he doesn't want to discuss this. My impression is that I think he doesn't see how such a discussion can be constructive.

In this attempt to find a relationa interpretation of QM, he also says that he doesn't aim to change to reformulate QM, just to find a relational description that will make it's application to GR more natural.

I am not sure that Rovelli avoided the discussion because he really think it was not relevant, I suspect it could be because he felt that when writing that, his ambition was not to solve the fundamental question of QM, it was just to find a relational interpretation that served his purpose to apply it to GR.

Anyway, I found that reading his reasoning these omission are critical and can not be considered useless interpretational issues only.

I'll try to locate the original phrasings.

In short, the point where I disagree with Rovelli, does lead to a different model type. I've personally come to the personal conclusion that the lack of structural realism, bird view etc, can only maintain a reasonable consistency when you instead adopt the idea of evolving laws and symmetries.

This is a completely different reasoning fom rovelli uses.

All things, his avoidance of probability basis, his idea of timeless laws, the idea of diffeomorphism invarance are all related symptoms IMO of a particular way of reasoning.

I commented on rovelli's reasoning from my personal perspective in past threads in ways that at least relate to the current discussion. Here is one https://www.physicsforums.com/showthread.php?t=220841

The direction I'm personally onto is more relational than what rovelli describes as relational. Rovelli says there are no objective relations, only relational relations that are communicated between obserers, but indespite of this he flatly claims that this communication obeys QM.

I agree that the physical meaning probability is a deep key question. I think anyone who thinks this is a silly question, and just refers to statistics in repeating identical setups didn't understand the objection. this is why my personal quests starts with an attempt of instrinsic construction of information measures. In this loops, there is an unavoidable circularity, but this turns out not to be a problem, but I interpret it to be identified with evolution and time. This holds a deeper connection beteen probabiltiy measures and action measures. Sometime the difficulty of defining with certainty a measure of state, imples that the measure evolves, and this differential progress defines an action measure - or that is my conjecture, still to be worked out of course.

/Fredrik

 

[Fra]

On page 13 in his RQM paper (http://xxx.tau.ac.il/abs/quant-ph/9609002v2) Rovelli writes

"The next question is the extent to which the information (6) about the set of questions c determines the outcome of an additional question Q. There are two extreme possibilities: that Q is fully determined by (6), or that it is fully independent, namely that the probability of getting a yes answer is 1/2. In addition, there is a range of intermediate possibilities: The outcome of Q may be determined probabilistically by sc. The second postulate states explicitly that there are questions that are non-determined. Define, in general, as p(Q,Q(i)c) the probability that a yes answer to Q will follow the string s(i) c . Given two complete families of information sc and sb, we can then consider the probabilities‡"

In the Footnote he writes

"‡ I do not wish to enter here the debate on the meaning of
probability in quantum mechanics. I think that the shift of
perspective I am suggesting is meaningful in the framework
of an objective definition of probability, tied to the notion of
repeated measurements, as well as in the context of subjective
probability, or any variant of this, if one does not accept
Jayne’s criticisms of the last."

IMO, he is avoiding a large portion of the core problem by this passage.

/Fredrik