Insights Why Is Quantum Mechanics So Difficult? - Comments

[bhobba]

Here a critique of Popper's interpretation of quantum mechanics and the claim that the propensity interpretation of probability resolves the foundational problems of the theory

Without even reading it, its fairly obvious calling probability propensity, plausibility or any other words you can think of, will not change anything.

What probability is, is defined by the Kolmogorov axioms.

The rest is simply philosophical waffle IMHO.

Those axioms all by themselves are enough, via the law of large numbers, to show Ballintines ensembles conceptually exist, which is all that required to justify his interpretation.

If you think of probability as some kind of plausibility then you get something like Copenhagen - although the law of large numbers still applies and you can also conceptually define ensembles if you wish.

I sometimes say guys with a background in applied math like me and philosophers sometimes talk past one another.

Here's an example from Rub's paper:
'The propensity interpretation may be understood as a generalization of the classical interpretation. Popper drops the restriction to "equally possible cases," assigning "weights" to the possibilities as "measures of the propensity, or tendency, of a possibility to realize itself upon repetition." He distinguishes probability statements from statistical statements. Probability statements refer to frequencies in virtual (infinite) sequences of well-defined experiments, and statistical statements refer to frequencies in actual (finite) sequences of experiments. Thus, the weights assigned to the possibilities are measures of conjectural virtual frequencies to be tested by actual statistical frequencies: "In proposing he propensity interpretation I propose to look upon probability statements as statements about some measure of a property (a physical property, comparable to symmetry or asymmetry) of the whole experimental arrangement; a measure, more precisely, of a virtual frequency'

My view is just like Fellers:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. instead we shall prove theorem's and show how they are applied'

Conceptual infinite ensembles are easily handled by simply assuming there is a very small probability below which it is indistinguishable in practical terms from zero. If you do that the law of large numbers leads to large, but finite ensembles.

For example we know there is a very small probability all the atoms in a room will go in the same direction at once and levitate a chair into the air - but in practice it never happens - we can safely assumes probabilities that small can be neglected - just like in calculus at an applied level we often think of dx as a small increment in x such that dx^2 can be ignored.

That's why guys with my background and those with a philosophical bent sometimes talk past each other.

Thanks
Bill

 

[microsansfil]

Chribella, D'Ariano and Perinotti http://arxiv.org/abs/1011.6451. So the question is does Ballentine's derivation work?

Alexei Grinbaum "THE SIGNIFICANCE OF INFORMATION IN QUANTUM THEORY"

Interest toward information-theoretic derivations of the formalism of quantum theory has been growing since early 1990s thanks to the emergence of the field of quantum computation.

In Part II we derive the formalism of quantum theory from information-theoretic axioms. After postulating such axioms, we analyze the twofold role of the observer as physical system and as informational agent. Quantum logical techniques are then introduced, and with their help we prove a series of results reconstructing the elements of the formalism. One of these results, a reconstruction theorem giving rise to the Hilbert space of the theory, marks a highlight of the dissertation. Completing the reconstruction, the Born rule and unitary time dynamics are obtained with the help of supplementary assumptions. We show how the twofold role of the observer leads to a description of measurement by POVM, an element essential in quantum computation.

Patrick

 

[bhobba]

Alexei Grinbaum "THE SIGNIFICANCE OF INFORMATION IN QUANTUM THEORY"

Can you detail the relevance to Atty's statement about Ballentine's interpretation?

Thanks
Bill

 

[microsansfil]

Can you detail the relevance to Atty's statement about Ballentine's interpretation?

Formal systems seem to be rigid because purely syntactic, but their semantics embedded in the axioms is unspoken. In MQ i agree with the point of view that axiomatization has to be based on postulates that can be precisely translated in mathematical terms but not vice versa. The Alexei Grinbaum's work is an example among others.

Patrick

 

[microsansfil]

about Ballentine's interpretation?

About : " So the question is does Ballentine's derivation work?" included I am my quote is simply a mistake of cut and paste.

What is the meaning of "work" in the context of interpretation ?

Patrick

 

[bhobba]

What is the meaning of "work" in the context of interpretation ?

Mate all I am asking is for you to detail the point you are trying to make because I am confused about it.

What Ballentine does is show the probability axioms are consistent with his two axioms. He calls probability propensity, but that's not really relevant; philosophers get caught up in that sort of thing but mathematically it the axioms whatever it is obeys that's important. He uses the Cox axioms, but they are equivalent to the Kolmogorov axioms.

That implies the existence of ensembles which is all that is required - its got nothing to do with the semantics of the situation.

Is that what you mean by information theoretic?

If so information theoretic is not what I would use - axiomatic based would be my description.

Added Later:
While I was penning the above you did another post that hopefully clarified what you had in mind. Will address that.

Thanks
Bill

 

[microsansfil]

What probability is, is defined by the Kolmogorov axioms.

The area of ​​relevance of a formal system is confined - by design - to the field of relevance of a hidden semantic, whose presence is unspoken.

indeed,there is a comparability between other formalism like Cox-Jaynes’s approach to probability and de Finetti. Yet as written http://www.siam.org/pdf/news/86.pdf

In summary, we see no substantive conflict between our system of probability and Kolmogorov’s as far as it goes;
rather, we have sought a deeper conceptual foundation which allows it to be extended to a wider class of applications, required by current problems of science.

Patrick

 

[microsansfil]

Mate all I am asking is for you to detail the point you are trying to make because I am confused about it.

What Ballentine does is show the probability axioms are consistent with his two axioms.

I don't know Ballentine "Point of view". Is it an oher interpretation of MQ or is it a new axiomatic of MQ ?

Patrick

 

[bhobba]

Formal systems seem to be rigid because purely syntactic, but their semantics embedded in the axioms is unspoken.

That's the whole point - they are semantic neutral.

Again - read what Feller said:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. Instead we shall prove theorem's and show how they are applied'

This is the modern view.

BTW when I say modern it developed during the 19th century where a more cavalier attitude caused problems (eg 1 - 1 + 1 - 1 ... converged in naive Fourier series) and permeated all of modern pure and applied math - including physics. Many say the pure guys went a bit too far, which led to a bit of good natured ribbing between applied and pure camps, but both have taken on the central lesson.

Thanks
Bill

 

[bhobba]

I don't know Ballentine "Point of view". Is it an oher interpretation of MQ or is it a new axiomatic of MQ ?

His view is similar to Popper - and would not be my choice of how to attack it.

The key point I am trying to get across is his arguments depend on the axioms - not how you interpret them.

I have already posted my derivation of the two axioms that starts with a single axiom:
'An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.'

That way you don't have to show its compatible with probability - its there right from the start - without any semantic baggage.

It's clearer IMHO what's going on that way.

Of course Ballentine isn't wrong - but as this thread shows it gets caught up in semantic baggage.

Thanks
Bill

 

[microsansfil]

That's the whole point - they are semantic neutral.

Why can we derive the formalism of quantum theory from information-theoretic axioms, design with other very different concept ? The foundation of mathematic can also be buid with Category theory rather then Set theory.

Built physics with the Wheeler's «it from bit» point of view is also an other modern view.


Patrick

 

[bhobba]

Why can we derive the formalism of quantum theory from information-theoretic axioms, design with other very different concept ?

Its just the way things are - many roads lead to Rome.

BTW that's not an endorsement of the validity of any approach I haven't studied in detail.

But many physical theories such as classical mechanics have different but equivalent starting points.

Take a look at the Cox and Kolmogorov axioms - they are equivalent. Its simply the nature of the beast.

Thanks
Bill

 

[microsansfil]

Its just the way things are - many roads lead to Rome.

Logician would say : "The sense fails in nonsense like rivers into the sea". This means that semantics are determined by the syntax.

Example :

A circle is a set of points with a fixed distance, called the radius, from a point called the center.

I understand your point of view.

Patrick

 

[bhobba]

This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.

As usual Frederk hit the nail on the head.

To apply it you need some rules to make sense of the math.

Its fairly obvious semantics won't resolve the type of issues Frederic pointed out.

That's where you need to add something like we ignore probabilities below a certain very small level as being irrelevant.

There are probably other ways, and discussing that may be interesting.

Thanks
Bill

 

[bhobba]

Logician would say : "The sense fails in nonsense like rivers into the sea"

That's probably a philosophical logician like Wittgenstein.

He had some well known debates about it with the mathematical logician, and very great mathematician (and Wittgenstien was equally as great - and - while not well known was actually well trained in the applied math of aeronautics - he started a Phd in it before being influenced by Russell and switched to philosophy) - Turing.

By 'it' I mean the foundations of applied math.

It was judged as a debate Wiggenstein may have won it - but later appraisal (by mathematicians of course ) gave it to Turing.

But this is getting into philosophy - which is off topic here.

If you want to pursue it the philosophy forums would be a better choice.

Thanks
Bill

 

[microsansfil]

If you want to pursue it the philosophy forums would be a better choice.

This is not a good argument.

No, behind there is the question about : can we reduce the physics to the mathematical axiomatic ( Proof theory ) ?

Patrick

 

[bhobba]

The mathematical theory of probability is now included in mathematical theory of measure.

Yea - Lebesgue integration and all that.

Fortunately in discussing the foundations of QM you don't need to worry about that because its enough to deal with finite discreet variables.

One then uses the Rigged Hilbert Space formalism to handle the continuous case.

Thanks
Bill

 

[bhobba]

This is not a good argument.

Its not an argument - its a statement of fact.

Philosophy is off-topic here.

If you go down that path, I will not respond, and the moderators will take action.

Discussing the modern axiomatic view of math would be on topic, the philosophy behind it, such as for example Wittgenstein's conventionalism, wouldn't.

Thanks
Bill

 

[microsansfil]

Philosophy is off-topic here.

This is why I'm not talking about philosphy. Why you see philosphy in my speech ? Is it a Straw man argument to impose your philosophy ?

The question is about axiomatize the physics

Discussing the modern axiomatic view of math would be on topicl

This is the point of the discussion.

Now this may be beyond the scope of this thread ?

Patrick

 

[bhobba]

Logician would say : "The sense fails in nonsense like rivers into the sea". This means that semantics are determined by the syntax.

I am not going to get into an argument about it - but stuff like the above IMHO is philosophy pure and simple.

I will not be drawn into it.

Thanks
Bill

 

[microsansfil]

I am not going to get into an argument about it - but stuff like the above IMHO is philosophy pure and simple.

It was a metaphor in response to your (you failed to write). I give a mathematical example in this context : http://en.wikipedia.org/wiki/Taxicab_geometry

Again Can we reduce physics to mathematical aximomatics ? Physical reduce itself to an applied science of mathematics?

Patrick

 

[bhobba]

Can we reduce physics to mathmematical aximomatics

You really need to start a new thread about that - its getting off topic.

But, as the only comment I will make here on it, attempts to do it, for example in QFT, leads to some extremely mind numbing math.

I used to ask questions like that in my degree.

The answer I got was I can give you some books that do just that - but you wouldn't read them.

He was right and it cured me.

BTW its nothing to do with semantics - its to do with rigour and reasonableness.

As an example it isn't hard to derive a Weiner process, but showing such actually exists is mathematically quite difficult. That's the difference between pure and applied math. Physically, because of the process it models, you believe it exists. But rigorously proving it is another matter.

Thanks
Bill

 

[stevendaryl]

This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.

I waffle back and forth about the importance of understanding what QM is all about. If you take the sensible point of view that QM is not the ultimate theory, but a "good enough" theory, then a lot of the debate about foundations seems beside the point. Whether you believe in collapse of the wave function or not, whether you believe in Many Worlds or not, whether you believe in Bohmian nonlocal interactions or not, it just doesn't matter. When it comes to applying QM, we pretty much all agree on how to do it. We have a recipe for applying QM, and that recipe tells us enough about the meaning of QM to get on with doing science. There are lots of puzzling aspects of the various interpretations: What's special about measurement? What's happening between observations? How do these nonlocal correlations come about? Etc. But if you take the point of view that QM is just an incomplete theory, with operational semantics, and not anything ultimate, then it's really not that important that it answer all those questions. If you don't expect it to answer those questions, then it hardly matters what interpretation of QM you use.

On the other hand, the thing that is puzzling about QM as an incomplete theory is that there are no hints as to the limits of its applicability. There are no hints as to what more complete theory might replace it.

 

[stevendaryl]

As an example it isn't hard to derive a Weiner process, but showing such actually exists is mathematically quite difficult. That's the difference between pure and applied math. Physically, because of the process it models, you believe it exists. But rigorously proving it is another matter.

I'm a little puzzled about the role of rigor in physics. It seems that there are times when there are rigorous proofs that a certain thing is impossible, and physicists go ahead and do it, anyway. The example that comes to mind is Haag's theorem. I don't complete understand it, but based on a very superficial understanding, it seems to be saying that the techniques that physicists use in QFT, namely, starting with the free particle Hilbert space and viewing particle interactions via perturbation theory, can't work. But physicists do it and seem to get reasonable results. So what exactly is Haag's theorem telling us?

 

[microsansfil]

You really need to start a new thread about that - its getting off topic.

OK

Your speech on the proselytism of Ballentine is on this topic ?Patrick

 

[bhobba]

It seems that there are times when there are rigorous proofs that a certain thing is impossible, and physicists go ahead and do it, anyway.

Mate that is a deep question I have no answer for.

Zee says, correctly, there are many good physicists with the technological ability to do things like long mind numbing computations. But that doesn't make a great physicist - it's the ability to see into the heart of a problem. They are magicians - you can't go where they go. There have only been a few - Feynman, Landau, Einstein, Von Neumann come to mind.

Many people marvel at the technical virtuosity of Von-Neumann, but what really set him apart and made great mathematicians like Poyla scared of him was this magical ability to see to the heart of things - "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.'

Feynman, no slouch in the Magician area himself, freely admitted Von-Neumann was his better.

Now we come to Einstein. Technically all those others I mentioned were way ahead of Einstein - they were all mathematical virtuosos. Not so Einstein - his math ability was quite ordinary - competent - but not spectacular. But his ability to see to the heart of an issue was above all those other greats - and that's what made him a greater physicist.

As they said about Feynman 'Feynman seemed to possesses a frightening ease with the substance behind the equations, like Albert Einstein at the same age, like the Soviet physicist Lev Landau—but few others.' That's the real key - the substance behind the math. Few have it - and its those that somehow, magically, know what to ignore, and what's important, that are great.

Thanks
Bill

 

[bhobba]

Your speech on the proselytism of Ballentine is on this topic ?

Have you actually been reading what I have been saying?

I have issues with Ballentine.

Its the best book on QM I have read - but perfect it aren't.

Look the exact divide between on and off topic is obviously a matter of opinion.

But I think most would say a discussion on the axiomatisation of physics is far wider than Why is QM So Difficult.

A discussion of exactly how Ballentine tackles the topic of QM would seem quite relevant

Its dead simple to start another thread - why get worried about it?

Thanks
Bill

 

[microsansfil]

Have you actually been reading what I have been saying?

I have issues with Ballentine.

The topic is about : Why Is Quantum Mechanics So Difficult ? isn't it ?

Perhaps that the possible divergence of view is an answer ?

Patrick

 

[bhobba]

Perhaps that the possible divergence of view is an answer ?

Indeed it is an answer - the semantic waffling of no actual mathematical content clouds the issue - as I have been discussing.

But the general axiomatisation of physics is beyond that.

Simply start a new thread.

It wouldn't be in the QM section - it would be in the general physics section.

Thanks
Bill

 

[bolbteppa]

Since I'm in the extremely small minority that dislikes Ballentine's book, let me say that I don't think the criticisms from Neumaier and Motl are that relevant to my point of view (although Neumaier and Motl may be correct, but I won't comment on that, since Ballentine's Ensemble interpretation itself appears to have changed between his famous erroneous review and the book, and Neumaier and Motl might be commeting on the review). Neither is the issue about the interpretation of probability important to me. Clearly, Copenhagen works despite its acknowledged problem of having to postulate an observer as fundamental. One cannot just declare that individual systems don't have states, or that collapse is wrong, since that would mean Copenhagen is wrong (Ballentine erroneously claims that Copenhagen is wrong, but my point if that even if we forgive him that, that does not fix his problems). The major approaches to interpretation never claim that Copenhagen is wrong. Rather, they seek to derive Copenhagen, but remove the observer as a fundamental component of the postulates. Ballentine doesn't even try to do that, and his theory has a Heisenberg cut, so it is not really an interpretation. Rather it is at best a derivation of Copenhagen or "Operational Quantum Theory" from axioms other than those found in Landau and Lifshitz, Shankar, Sakurai and Napolitano, Weinberg, or Nielsen and Chuang. Excellent examples in this spirit are those of Hardy http://arxiv.org/abs/quant-ph/0101012 or Chribella, D'Ariano and Perinotti http://arxiv.org/abs/1011.6451. So the question is does Ballentine's derivation work? I believe it doesn't, and that it is technically flawed.

The key question is whether Ballentine is able to derive his Eq 9.30. For comparison, one may see Laloe's treatment of the same equation in http://arxiv.org/abs/quant-ph/0209123, where it is Eq 37. If Ballentine did derive that equation, I think the other mistakes could be overlooked. If he did not, his interpretation has a hole and is not quantum mechanics.

Now should all approaches to interpretation be without flaw? No, but they should be clear where their flaws and issues are. For example, Wallace makes clear that the issue of how probability arises at all in Many-Worlds is still an issue, even if his derivation of the Born rule were to be correct. Similarly, there is the well known limitation that Bohmian Mechanics at present sits uncomfortably with exact Lorentz invariance. For the same reason, Landau and Lifshitz and Weinberg are excellent Copenhagen books because they explicitly point out the Heisenberg cut, rather than sweeping it under the rug.

Finally a bit of substance regarding this book. So Ballentine a) not only doesn't make the flaws explicit, b) he actually goes and claims Copenhagen is wrong? Mix that with c) You have to use a different system of probability (apparently equivalent after you do a ton of work and change your entire perspective of probability), d) you have to treat single particle systems in some weird way, & a potential e) your only benefit is fewer axioms at the expense of a less general form of QM, where as you say it's even questionable that he can achieve QM at all. I haven't read any of the guys bragging about Ballentine on here mention any of this stuff, these are such serious issues that I'm amazed tbh...

Why put yourself through such nonsense when you've got Landau, Dirac and Von Neumann sitting right there... I guess QM is so hard because people ignore the good books.

Thanks man