Insights Why Is Quantum Mechanics So Difficult? - Comments

[bhobba]

it seems to me that you have also an other possibility develop by E.T.Jaynes "probability theory as an extension of logic". In this context probability is not reduce to random variables.

That's the Bayesian view where its how plausible something is.

What you do is come up with reasonable axioms on what plausibility should be like - these are the so called Cox axioms. They are logically equivalent to the Kolmogorov axioms where exactly what probability is is left undefined.

Ballentine bases it on those axioms but called it propensity - which isn't really how Coxes axioms are usually viewed. It's logically sound since its equivalent to Kolomogorovs axioms - just a bit different.

In applied math what's usually done is simply to associate this abstract thing called probability defined by the Kolmogerov axioms with independent events. Then you have this thing called the law of large numbers (and its a theorem derivable from those axioms) which basically says if you do a large number of trials the proportion of outcomes tends toward the probability. That's how you make concrete this abstract thing and its certainly how I suspect most people tend to view it.

Basically what Ballentine does it look at probability as a kind of propensity obeying the Cox axioms. Then he uses the law of large numbers to justify his ensemble idea.

There is no logical issues with this, but personally I wouldn't have used propensity - simply an undefined thing as per Kolomogorov's axioms.

But really its no big deal.

Thanks
Bill

 

[microsansfil]

just a bit different.

In his book E.T Jaynes write.

Foundations: From many years of experience with its applications in hundreds of real problems, our views on the foundations of probability theory have evolved into something quite complex, which cannot be described in any such simplistic terms as \pro-this" or \anti-that." For example, our system of probability could hardly be more different from that of Kolmogorov, in style,philosophy, and purpose. What we consider to be fully half of probability theory as it is needed in current applications the principles for assigning probabilities by logical analysis of incomplete information|is not present at all in the Kolmogorov system.

As noted in Appendix A, each of his axioms turns out to be, for all practical purposes, derivable from the Polya-Cox desiderata of rationality and consistency. In short, we regard our system of probability as not contradicting Kolmogorov's; but rather seeking a deeper logical foundation that permits its extension in the directions that are needed for modern applications. In this endeavor, many problems have been solved, and those still unsolved appear where we should naturally expect them: in breaking into new ground.

However, our system of probability differs conceptually from that of Kolmogorov in that we do not interpret propositions in terms of sets, but we do interpret probability distributions as carriers of incomplete information. Partly as a result, our system has analytical resources not present at all in the Kolmogorov system. This enables us to formulate and solve many problems- particularly the so-called "ill posed" problems and "generalized inverse" problems - that would be considered outside the scope of probability theory according to the Kolmogorov system. These problems are just the ones of greatest interest in current applications.

E.T Jaynes purposefully do not use the term “random variable”, as it is a much too restrictive a notion, and carries with it all the baggage of the Kolmogorov approach to probability theory, but a random variable seem to be an example of an unknown/incomplete information.

Possible point of view : Quantum mechanics is basically a mathematical recipe on how to construct physical models. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics.

For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion.

Patrick

 

[bolbteppa]

Sure they need not remain homogeneous - but the conceptualisation is they do - its a straw man argument.

Many, many books explain the validity of the frequentest interpretation when backed by the Kolmogorov axioms eg
https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

Having looked through Feller, he actually doesn't claim that the frequency interpretation of probability is justified by Kolmogorov's axioms, and just to be clear - if such a passage actually existed then it would imply both me and Ballentine are wrong when we say frequentist probability is flawed. Ballentine mentions this issue uniqueness of the limit on page 32:

One of the oldest interpretations is the limit frequency interpretation. If the conditioning event C can lead to either A or ∼A, and if in n repetitions of such a situation the event A occurs m times, then it is asserted that P(A|C) = limn→∞(m/n). This provides not only an interpretation of probability, but also a definition of probability in terms of a numerical frequency ratio. Hence the axioms of abstract probability theory can be derived as theorems of the frequency theory. In spite of its superficial appeal, the limit frequency interpretation has been widely discarded, primarily because there is no assurance that the above limit really exists for the actual sequences of events to which one wishes to apply probability theory.

The defects of the limit frequency interpretation are avoided without losing its attractive features in the propensity interpretation. The probability P(A|C) is interpreted as a measure of the tendency, or propensity, of the physical conditions describe by C to produce the result A. It differs logically from the older limit-frequency theory in that probability is interpreted, but not redefined or derived from anything more fundamental. It remains, mathematically, a fundamental undefined term, with its relationship to frequency emerging, suitably qualified, in a theorem. It also differs from the frequency theory in viewing probability (propensity) as a characteristic of the physical situation C that may potentially give rise to a sequence of events, rather than as a property (frequency) of an actual sequence of events.

Calling my argument a strawman argument is calling Ballentine's argument a strawman argument. I notice you only focused on homogeneity, but what about the issue of uniqueness of the limit that me and Ballentine brought up?

He does use propensity - but I think he uses it simply as synonymous with probability - most certainly in the equations he writes that's its meaning.

As the quote from Ballentine given above shows, it seems he uses this word as a way to give the closest thing to a frequentist interpretation possible, but qualifies this by saying it's merely a word given to a theorem proven from Cox's axioms. That's an extremely important distinction in the sense that, logically, it's very different from taking the crass frequentist interpretation that you implied, and doubly important since you are claiming both that frequentist probability can be justified by Kolmogorov's axioms and that Ballentine is taking a frequentist interpretation when he clearly says he isn't...

So he's not using frequentist probability, he's using Cox's probability axioms and just interpreting some theorems in a way that lies closest to a frequentist interpretation possible. That's fine, but had I not checked that out I'd be left with a completely wrong impression of Ballentine based on this thread.

To be frank I don't even understand Lubos's criticism - mind carefully explaining it to me?

All I'm going off is the conclusion which is that all we really get from the ensemble interpretation is a restricted and modest view of the power of QM. Hopefully someone who understands it fully will be able to challenge it.

You should. But what leaves me scratching my head is you seem to have all these issues with it - but haven't gone to the trouble to actually study it. I could understand that if it was generally considered crank rubbish - but it isn't. Its a very well respected standard textbook. It is possible for sources of that nature to have issues - and it does have a couple - but they are very minor.

Well I want to find out about the book, which is why I'm posting. Thus far I have been given the impression that it's based on frequentist probability, and been told such a position can be justified by Kolmogorov's axioms, when in fact the book explicitly says it's not based on frequentist probability and actually uses Cox's axioms. Then we have the two main issues, one about the theory applying to a single particle, which may be more complicated than Neumaier implied http://physics.stackexchange.com/a/15553/25851 and also Lubos' claim that all we really get from the ensemble interpretation anyway is just a restricted and modest view of the power of QM. Sounds awfully unappealing at this stage.

From all the comments on Ballentine I've read on here that stick in my head, the only benefit compared to Landau is that a) it's easier than Landau, b) you can prove one or two things Landau assumes (though apparently at the price of a less general form of QM) as long as you take a different interpretation of QM to that of Landau, an interpretation that, at best, is ultimately no more justifiable than Landau's perspective, and at worse is less general. In that light, it seems like the book is a waste of time, but I'm happy to be wrong.

 

[stevendaryl]

I mentioned--either in this thread, or another--the "propensity" interpretation of probabilities, but in my opinion, it's not an interpretation, at all. It's just another word for "probability". Maybe it's supposed to be that part of probability that is left over after all probabilities due to ignorance are stripped away. So in the context of QM in the density-matrix approach, pure states represent propensities, while mixed states combine propensities and subjective probabilities?

 

[microsansfil]

So in the context of QM in the density-matrix approach, pure states represent propensities, while mixed states combine propensities and subjective probabilities?

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, by http://www.philosophy.umd.edu/people/bub.


Patrick

 

[Fredrik]

what about the issue of uniqueness of the limit that me and Ballentine brought up?

It's not an issue. The assignment of probabilities in the purely mathematical part of the theory, is just an assignment of relative sizes to subsets. These assignments tell us nothing about the real world on their own. That's why the theory consists of the mathematics and a set of correspondence rules that tell us how to interpret the mathematics as predictions about results of experiments. Those rules tell us that the relative frequency of a particular result in a long sequence of identical measurements, will be equal to the probability that has been assigned to (a subset that represents) that particular result.

The correspondence rules can't just say that probabilities are propensities, because we need to know how to test the accuracy of the theory's predictions. If we can't, it's not a theory.

The non-existence of a limit wouldn't be relevant even if we had a theory that has a chance of being exactly right, because

1. You can't perform an infinite sequence of measurements.
2. The measurements won't be perfectly accurate.
3. The measurements won't be identical.
4. If a very long sequence of identical measurements would (for example) sometimes go into the interval 1.000000001-1000000002 and then hop around inside it, and in another experiment go into the interval 1.0000000005-10000000006 and then hop around inside it, the conclusion would be that somewhere around the tenth decimal, we're hitting the limits of the theory's domain of validity. This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.

 

[bhobba]

That's an extremely important distinction in the sense that, logically, it's very different from taking the crass frequentist interpretation that you implied

That's the precise problem. Ballentine and I are not advocating a 'crass' frequency interpretation. We are advocating the modern version where it is based on the Kolmogorov axioms (or equivalent) and applying the law of large numbers.

It matters not if you call it propensity, plausibility, or leave it it semantically open, it implies exactly the same thing.

Thanks
Bill

 

[atyy]

From all the comments on Ballentine I've read on here that stick in my head, the only benefit compared to Landau is that a) it's easier than Landau, b) you can prove one or two things Landau assumes (though apparently at the price of a less general form of QM) as long as you take a different interpretation of QM to that of Landau, an interpretation that, at best, is ultimately no more justifiable than Landau's perspective, and at worse is less general. In that light, it seems like the book is a waste of time, but I'm happy to be wrong.

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.

 

[microsansfil]

We are advocating the modern version where it is based on the Kolmogorov axioms (or equivalent)

This leave with the impression that Kolmogorov’s axiomatization was born full grown. Kolmogorov only translates probability concept, well known many years later, into an axiomatic/formal mathematical language. The mathematical theory of probability is now included in mathematical theory of measure.

The measurement theory is the branch of mathematics that deals with measured spaces and is the axiomatic foundation of probability theory.

The basic intuition in probability theory remain the notion of randomness based on the notion of random variable.

There are certain ‘non commutative’ versions that have their origins in quantum mechanics, for instance K. R.
Parthasarathy (an introduction to quantum stochastic calculus), that are generalizations of the Kolmogorov Model.

Patrick

 

[bhobba]

since Ballentine's Ensemble interpretation itself appears to have changed between his famous erroneous review

It did.

He had to take on board Kochen-Specker.

He assumed, initially (in his original review article), it had the property when measured. Kochen-Specker says you can't do that. Fredrick put his finger on it - originally it was basically BM in disguise.

However with decoherence you can do that - but of course it still doesn't fully resolve the measurement problem - which looked like was his hope.

Thanks
Bill

 

[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

 

[microsansfil]

But the general axiomatisation of physics is beyond that.

But In the spécific of QM axiomatic is only your speech ? From the same axiomatic we can build different semantics. In mathematics is Model theory. The link between semantic and syntax is build by Gödel's completeness theorem.

Patrick

 

[atyy]

But In the spécific of QM axiomatic is only your speech ? From the same axiomatic we can build different semantics. In mathematics is Model theory. The link between semantic and syntax is build by Gödel's completeness theorem.

Yes, the derivations must put in some "semantics", or rather "physics". Semantics is the assignment of sets (and to use sets we have to have natural language) to meaningless symbols and grammar. Physics is the assignment of things we see and things we do to meaningless symbols and grammar. Even Euclidean geometry has different physical interpretations because of the duality between lines and points in the theory, so a physical line can correspond to a point in the theory. The derivations of Hardy or Chiribella et al start from the same physics background as standard Copenhagen - we assume a commonsense macroscopic world, and we know what a measurement (a little black box that takes an input and gives an output). They are alternative axioms for Copenhagen, in the same sense that the Hilbert action, the Palatini action and the Einstein field equations are different axioms for the same classical theory of gravity.

 

[bhobba]

he actually goes and claims Copenhagen is wrong?

Yes that's an error - one of its, fortunately, minor ones.

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.

You mean Von-Neumann's thrashing of the Dirac Delta function that Ballentine rectifies? Things have moved on a lot since that classic was penned.

I am not going into the issues with the others, but will point out Ballentine is the only one of those that explains the true foundation of Schroedinger's equation etc - the symmetries of the POR.

Otherwise it looks basically like it's pulled out of a hat.

Dirac comes closest with his algebraic approach to Poisson Brackets but it doesn't explain why it holds. The POR is a general law applicable to all physics.

Thanks
Bill

 

[atyy]

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

OK, maybe I was a bit hard on Ballentine claiming that Copenhagen is wrong. Strictly, speaking he only claims that his caricature of Copenhagen is wrong. But as you can see, even bhobba who likes the book makes far stronger criticisms of Ballentine's earlier interpretation - the claim that the earlier Ensemble Interpretation is secretly Bohmian is very strong criticism. Nothing wrong with being Bohmian of course, but the assumption should be stated clearly. Ballentine is vague enough, and doesn't even mention the Heisenberg cut, unlike Landau and Lifshitz or Weinberg, that I don't know if I agree with bhobba. But yes, if Ballentine is secretly Bohmian that would make a lot of sense, since one would then not need to add an assumption that proper and improper mixtures are equivalent, an assumption Ballentine makes in his book but fails to state. It also seems that Ballentine is secretly Many-Worlds, since he seems to want to have unitary evolution of the wave function and nothing else. Maybe he is secretly Bohmian Many-Worlds, which is possible, since Bohmian mechanics has unitary evolution of the wave function.

 

[bhobba]

Strictly, speaking he only claims that his caricature of Copenhagen is wrong. But as you can see, even bhobba who likes the book makes far stronger criticisms of Ballentine's earlier interpretation - the claim that the earlier Ensemble Interpretation is secretly Bohmian is very strong criticism.

Very true. BTW the BM thing is fixed in the book - but at a cost.

Don't get me wrong.

It has issues eg I think that propensity stuff is a crock of the proverbial - I wouldn't touch it with a barge pole.

But you have to look at it overall.

His explanation of the math, for example, is simply a cut above, even giving an overview of the important Rigged Hilbert Space formalism.

Thanks
Bill

 

[TrickyDicky]

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.

And if there were you couldn't mention them here anyway, so there are reasons to waffle on about beside the point interpretational debates.

 

[atyy]

Very true. BTW the BM thing is fixed in the book - but at a cost.

Don't get me wrong.

It has issues eg I think that propensity stuff is a crock of the proverbial - I wouldn't touch it with a barge pole.

But you have to look at it overall.

His explanation of the math, for example, is simply a cut above, even giving an overview of the important Rigged Hilbert Space formalism.

Thanks
Bill

Yes, I agree that Ballentine's presentation of the symmetries in the first few chapters is valuable, and hard to find elsewhere. So I would say use Ballentine for the "maths" (I put it in quotes because he presents it in a nice physicky way, which I don't know if strict mathematicians will like), but not so much for the interpretation, which is (at best) Copenhagen renamed.

 

[stevendaryl]

And if there were you couldn't mention them here anyway, so there are reasons to waffle on about beside the point interpretational debates.

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

 

[atyy]

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

Yes, there are two sorts of theories: those which can be a theory of some universe, and so experiment, and experiment alone tell us it must break down (eg. Newtonian gravity), while there are others where the theory itself tells us it must breakdown (eg. QED, if there is no asymptotic safety). Copenhagen itself suggests QM must breakdown, since Copenhagen typically does not acknowledge a wave function of the universe. Interpretations such as Bohmian Mechanics would place QM together with QED, and so far these are the only interpretations that are known to be without technical flaw (except maybe for chiral interactions). Bohmian Mechanics says that QM must break down, because it requires the quantum equilibrium condition, which is analogous to equilibrium in statistical mechanics. For the ensembles to emerge from a single reality, there has to be non-equilibrium in reality, but not detectable over the resolutions that we are able to access at the moment. If pure Many-Worlds works, then QM could conceivably be a theory of some universe, just like Newtonian gravity.

 

[TrickyDicky]

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

I'm not sure what you mean by saying that QM is very different in this regard, what are you calling QM exactly? Because the endless interpretational debates are mostly about "Schroedinger's QM", that you cite as an example of theory for which we we know what it means to break down.