The Time Symmetry Debate in Quantum Theory

[lugita15]

Undecidable? You are confused about Gödel's incompleteness theorems, which are about the inherent limitations of all axiomatic systems capable of doing arithmetic

Yes, but it's about the inherent *arithmetical* limitations of axiomatic systems capable of doing Peano arithmetic. First of all, it's not clear why a physical theory would need to be capable of doing Peano arithmetic. And even if there were such a theory, Godel's theorem would not prevent you from using the theory to predict the position and momentum of all the particles in the universe, for all time. So Godel's theorem doesn't really place physical limitations on a physical theory.

the consistency of arithmetic is provably impossible.

Sorry, did you mean that the consistency of arithmetic is provably impossible to prove?

 

[audioloop]

In my opinion,
it's experiments which ultimately decides whether a theory needs to be corrected or replaced. Our understanding has nothing to do with this.
Nature decides.
We observe. And we try to understand what nature is telling us. If/when there will come an experiment that disagrees with QM, it will become famous, no doubt about it.

welll said.

Observation of a kilogram-scale oscillator near
its quantum ground state.
New Journal of Physics 11 (7): 073032
Abbott, B. et al.
http://eprints.gla.ac.uk/32707/1/ID32707.pdf

Quantum Upsizing
Aspelmeyer, Schwab, Zeilinger.
http://fqxi.org/data/articles/Schwab_Asp_Zeil.pdf


.

 

[DevilsAvocado]

Godel's theorem would not prevent you from using the theory to predict the position and momentum of all the particles in the universe, for all time. So Godel's theorem doesn't really place physical limitations on a physical theory.

Of course not, Gödel doesn’t prevent you from calculating anything; I could count the number of sheep in the universe and that would be perfectly doable. But the reason I mentioned Gödel was in relation to the completeness of QM (but now I regret it), and AFAIK the foundation of physics rest on mathematics.

Any formal system that is strong enough to formulate its own absence of axiomatic contradiction can prove its own consistency if – and only if – it is inconsistent. Since theorems are derived from a set of axioms, to be embodied in some general principle that makes it part of a larger theory – it looks like Gödel has something to say about this enchilada... when it comes to completeness.

But what do I know...

Maybe Professor Mark Colyvan can explain it better:

KURT GÖDEL AND THE LIMITS OF MATHEMATICS - Professor Mark Colyvan
https://www.youtube.com/watch?v=92Gdhr7dd_I

Sorry, did you mean that the consistency of arithmetic is provably impossible to prove?

I think so; it is not possible to find a totally adequate set of axioms for arithmetic:
http://math.mind-crafts.com/godels_incompleteness_theorems.php


... I feel guilty, this is not what OP asked about and this will be my last comment on this ...

 

[TrickyDicky]

... I feel guilty, this is not what OP asked about and this will be my last comment on this ...

Hey, no problem, I actually agree with your take on Godel and find your comments pertinent.

 

[audioloop]

Steve Giddings
http://www.edge.org/response-detail/23857

"These principles clash when pushed to the extreme—the sharpest version of the problem arises when we collide two particles at sufficient energy to form a black hole. Here, we encounter the famed black hole information problem: if the incoming particles start in a pure quantum state, Hawking's calculation predicts that the black hole evaporates into a mixed, thermal-like final state, with a massive loss of quantum information. This would violate—and thus doom—quantum mechanics."

 

[DevilsAvocado]

Hey, no problem, I actually agree with your take on Godel and find your comments pertinent.

Thanks! Well, fasten your seatbelt, here we go!

Sir Roger Penrose held a talk at GoogleTechTalks about conscious understanding, where he discussed Gödel’s theorem, quantum mechanics and the human brain. Very interesting!

It’s quite long so I fixed direct links to different parts. Notice that Penrose claim QM “is wrong in some sense”, but I think he really mean “not the whole story”... his view is that there has to be a radical new way of looking at quantum mechanics which will make almost no difference (hence QM is correct but not complete) in the same way general relativity makes almost no difference to Newtonian physics but it’s a completely different framework, and this is what Penrose suspects will happen also to QM.

Who knows...

Conscious Understanding: What is its Physical Basis?
https://www.youtube.com/watch?v=f477FnTe1M0

• @19:09 – Gödel’s theorem
• @39:40 – Something non-computational in mathematical understanding & physical laws
• @43:10 – A non-computable toy model universe
• @48:49 – Computable classical physics & (non-)computable quantum mechanics
• @55:31 – The measurement problem; a sign quantum mechanics is not right at all levels
• @57:34 – Non-computable quantum processes in the human brain (microtubules)
• @1:10:00 – Q&A
• @1:15:25 – Quantum mechanics is incomplete

Regarding Gödel, @1:53:53 Penrose get a question on discrete computation and continues computation and the human spectrum between true/false, and then mention his colleague Professor Tim Palmer who works with stochastic physics and climate modeling and furthermore has put forward a quite interesting hypothesis, the Invariant Set Postulate: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity (which of course is not a rigorous physical theory at this stage but still very interesting), which seems to satisfy both Bohr and Einstein, and the key feature of this idea is that it is not a new interpretation – like a ‘QM overcoat’ – but a new ‘backbone’ that QM could rest on; the hypothesis suggests the existence of a state space, within which a smaller (fractal) subset of state space is embedded. There’s an introduction on Phys.org and the paper is published on Proceedings of the Royal Society A and arXiv.org.

Click to watch the zoom sequence

Tim Palmer: "The invariant set hypothesis"
https://www.youtube.com/watch?v=Ciduvyv7ToE

 

[bhobba]

Notice that Penrose claim QM “is wrong in some sense”, but I think he really mean “not the whole story”... his view is that there has to be a radical new way of looking at quantum mechanics which will make almost no difference (hence QM is correct but not complete) in the same way general relativity makes almost no difference to Newtonian physics but it’s a completely different framework, and this is what Penrose suspects will happen also to QM. Who knows...

First thanks for posting that - very enjoyable.

Its the same view Einstein had (it's wrong to think Einstein disagreed with QM - he thought it merely incomplete - not incorrect - many people seem to forget that - possibly because his views changed a bit from, his early struggles with Bohr, to the publishing of the EPR paradox) and I think Weinberg holds to it as well.

There is no doubt Rogers views are very interesting and thought provoking - I have read many of his books such as the Emperors New Mind. I even held to his view about the literal existence of the Platonic realm for a while to explain issues in Wigner's famous essay:
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

But was ultimately swayed by Murray Gell-Manns View:
http://www.ted.com/talks/murray_gell_mann_on_beauty_and_truth_in_physics.html

My personal view for what its worth is QM is complete and its simply one of two possible probabilistic theories that follow from some very reasonable assumptions:
http://arxiv.org/pdf/0911.0695v1.pdf

It would seem that there are only two reasonable alternatives - standard probability theory and QM. The difference is entanglement or having continuous transformations between the outcomes of observations (the so called pure states). You can't do either with standard probability theory.

Still - who knows what the future will bring.

Thanks
Bill

 

[DevilsAvocado]

First thanks for posting that - very enjoyable.

You are welcome, glad you liked it!

Its the same view Einstein had (it's wrong to think Einstein disagreed with QM - he thought it merely incomplete - not incorrect - many people seem to forget that - possibly because his views changed a bit from, his early struggles with Bohr, to the publishing of the EPR paradox) and I think Weinberg holds to it as well.

Agreed, there seems to be some confusion regarding Einstein’s later ideas.

There is no doubt Rogers views are very interesting and thought provoking - I have read many of his books such as the Emperors New Mind. I even held to his view about the literal existence of the Platonic realm for a while to explain issues in Wigner's famous essay:
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

But was ultimately swayed by Murray Gell-Manns View:
http://www.ted.com/talks/murray_gell_mann_on_beauty_and_truth_in_physics.html

Thank you for the links, Murray Gell-Mann is just splendid! And of course he is right – you have to be blind not to see the “mysterious” link between nature and mathematics. The questions is: Is mathematics a fundamental part of nature (at the deepest level), or are we just incredible lucky to have invented this marvelous “nature-compatible-tool”?

I have absolutely no idea... but if mathematics is a fundamental part of nature and Gödel is right – then nature must be inconsistent!

And maybe she is...

My very personal thoughts on this, goes something like this: The human brain obeys the laws of nature. Humans are undoubtedly inconsistent. Something in the laws of nature must allow human thinking to be inconsistent, even if the laws themselves are perfectly consistent. When humans think about nature they utilize the laws of nature, and that fact will strongly influence what ideas humans could have about the laws of nature. Humans are not prefect but very creative. When humans invented the tool of mathematics (which at a later stage helped us understand physics) it was a mix of inconsistency, creativity and the laws of nature – but it was not perfect/complete!

And this explains some of the ‘situation’ today... maybe... perhaps... what do I know...

Agreed, Penrose is ‘provocative’, but brilliant as he is, linking his ideas to people like Anirban Bandyopadhyay (microtubules) could turn out to be a mistake. I’m only a layman, but flashy videos and no papers don’t really convince me Bandyopadhyay has found something exceptionally extraordinary...

It would seem that there are only two reasonable alternatives - standard probability theory and QM. The difference is entanglement or having continuous transformations between the outcomes of observations (the so called pure states). You can't do either with standard probability theory.

Sounds reasonable, the thing that has interested me is the words of Bell:

For me then this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation and fundamental relativity. That is to say, we have an apparent incompatibility, at the deepest level, between the two fundamental pillars of contemporary theory...

I.e. we have two great contemporary theories, that are empirically tested – and they don’t match!?

So, what’s going on here...?

If you (and maybe TrickyDicky) are interested in the background to Gödel’s theorems, here’s the full lecture by Professor Mark Colyvan:

Key thinkers: Kurt Gödel and the Limits of Mathematics. Mark Colyvan (p1)
https://www.youtube.com/watch?v=bYpSVSGBxis


Key Thinkers: Kurt Gödel and the Limits of Mathematics. Mark Colyvan (p2)
https://www.youtube.com/watch?v=CCac2oP4XB8



P.S: Isn’t this just amazing... David Hilbert who wanted mathematics to be formulated on a solid and complete logical foundation – the same man who introduced the concept of a Hilbert space, an indispensable tool in quantum mechanics – was crushed by Kurt Gödel, whom later become a very close friend of Einstein... what a thriller! ;)

 

[bhobba]

Regarding the incompatibility between relativity and GR:

we have two great contemporary theories, that are empirically tested – and they don’t match!? So, what’s going on here...?

Yea - I suspect Bell wasn't aware of the latest developments in the area, in particular that EFT shows QM and Relativity are not incompatible:
http://arxiv.org/abs/1209.3511

Thanks
Bill

 

[DevilsAvocado]

Yea - I suspect Bell wasn't aware of the latest developments in the area, in particular that EFT shows QM and Relativity are not incompatible:

Thanks, very interesting, but it’s not the final answer is it?

http://arxiv.org/abs/1209.3511]The[/PLAIN] effective field theory has limits to its validity, most notably it is limited to scales below the Planck energy, and does not resolve all of the issues of quantum gravity. However, effective field theory has shown that general relativity and quantum mechanics do in fact go together fine at ordinary scales where both are valid. GR behaves like an ordinary field theory over those scales. This is important progress. We still have work to do in order to understand gravity and the other interactions at extreme scales.

I think that Bell’s primary concern was not gravity but that his theorem established an essential conflict between the well-tested empirical predictions of quantum theory and Relativistic Local Causality (i.e. SR).

Okay, you can ‘escape’ this problem by accepting either the Many Worlds Interpretation or Superdeterminism (=absence of free will), but neither feels like a tasty final answer...

 

[T0mr]

This last property is not a problem, because everybody knows that classical statistical physics is not some basic theory from which we would like to derive deterministic models. Everybody knows it is the other way around: the classical statistical physics is built upon already available deterministic classical mechanics and probability theory.

Thank you Jano for your views, I feel you are right on with your observations. Good observation about the use of classical equations of motion for alpha particles in Rutherford experiment too.

My main problem with the QM is that it claims these probabilities are fundamental. Take the hydrogen atom in ground state. The probability distribution gives a spherically symmetric pattern where probabilities are concentrated around the Bohr radius. But at the Bohr radius there is a spherical shell were the probability of finding an electron at any position on the shell is completely random. You have the same chance of finding the electron anywhere on that spherical shell. The same goes for any spherical shell at radius greater or less than the Bohr radius.

In my mind two interpretations of the electron can be given at this point:

1.) If you interpret the electron position to be described only by probabilities at any instant, then it must be the case that to maintain equal probabilities over individual shells over some period of time, the electron position must be discontinuous in space.

2.) If instead you interpret the electron path to be continuous then the particle must not be more fundamentally described by a probability distribution which could only acts as a statistical analysis. Since we have assumed the particle has a set path whether we can observe that path or not.

Both 1 and 2 assume that one particle is always one particle and that the particle must exist somewhere in space at all times. With these assumptions we can observe that the particle must either follow a continuous path or be discontinuous in space.

Option 1 has obvious problems such as the infinite velocity required for the electron to make discontinuous jumps through space. So QM could not logically say that what these probability distributions really represent are particle positions in time. Then what do the probability distributions represent? We are only left with option 2 in which the paths of particles are always continuous through space and so cannot have been actually represented by spatial probability distributions. In the case of an orbiting electron at the Bohr radius, the electron must move from one point in space to adjacent points and can never move to points not adjacent to its current position. So, a probability distribution tells us nothing about the particles actual path and yet we must assume that continuous path through space exists. Therefore the probability distribution for a particle can only represent the state of our knowledge about where the particle can be not being aware of its path which is exactly how a statistical approach works.

Then the question becomes what is deciding the path the electron is taking. It cannot be the probability distribution because by 1 that will necessarily cause the path to be discontinuous. So we are forced to conclude, at least I am, that we are missing something that would explain the continuous path a particle must have.

So what is the only thing that we have that can represent a continuous path through space? This leaves me to think there must be deterministic functions governing the path of particles. Otherwise we must violate one or more of the above assumptions: that one particle is always one particle, the particle must always exist in space, and the particle cannot move instantaneously through space.

By assuming QM is the final theory we are forced to propose things like by the measurement of a particle the particle becomes real and/or the wave function collapses or multiple universes must exist etc. and so on and on. The existence of such propositions as legitimate theories is a big hint that something is fundamentally wrong with QM.

 

[DrChinese]

...

By assuming QM is the final theory we are forced to propose things like by the measurement of a particle the particle becomes real and/or the wave function collapses or multiple universes must exist etc. and so on and on. The existence of such propositions as legitimate theories is a big hint that something is fundamentally wrong with QM.

But that is precisely the point as far as I am concerned. There is nothing wrong with the theory other than you don't care for the philosophical implications. The kind of quantum (non-continuous) leaps you refer to do appear to occur. Such quantum non-locality is evidenced in many ways, and there are hundreds of papers documenting such.

http://arxiv.org/find/all/1/ti:+EXACT+non_locality/0/1/0/all/0/1?per_page=100
http://arxiv.org/find/all/1/AND+abs:+EXACT+non_locality+abs:+experiment/0/1/0/all/0/1?per_page=100

So I see it as hinting at the correctness of QM, not that QM is wrong! In fact, I would say that anyone wanting a strictly local (and presumably realistic) version of QM is the one missing hints . Except that experimental evidence is much stronger than a hint. There are no experimental misses at this point on the predictions of QM.

 

[Jano L.]

1.) If you interpret the electron position to be described only by probabilities at any instant, then it must be the case that to maintain equal probabilities over individual shells over some period of time, the electron position must be discontinuous in space.

I am not sure what you mean by " the electron position must be discontinuous in space." The electron jumping instantaneously from one point to another? If so, I do not see how it follows from the probabilistic description. In the theory of the Brownian motion, the particle can have similar probability distribution, but its motion is continuous.

 

[T0mr]

There is nothing wrong with the theory other than you don't care for the philosophical implications. The kind of quantum (non-continuous) leaps you refer to do appear to occur. Such quantum non-locality is evidenced in many ways, and there are hundreds of papers documenting such.

I never said that entanglement does not violate locality. It certainly appears to be the case. The question is now why can't a deterministic theory be used to explain the instantaneous determination in entangled particle properties. It is not really necessary that we limit our set of laws to describe the interactions between billiard balls. That is a very simplistic view. Why can't the instantaneous determination of spin of entangled particles be based in deterministic law?

I am not sure what you mean by " the electron position must be discontinuous in space." The electron jumping instantaneously from one point to another? If so, I do not see how it follows from the probabilistic description. In the theory of the Brownian motion, the particle can have similar probability distribution, but its motion is continuous.

I was thinking about the probability distribution as a whole but if you only consider adjacent points in space as possibilities around the electron then you could come up with something similar to Brownian motion. I guess that should have been option 3. Is that how these paths are interpreted to behave or should we determine points based on the entire distribution at any point in time regardless of where the particle might be located?

Now that you say that. I think I have to fall back from the original argument to another position. How can probability determine positions at all? What is it to be governed by probability and how can concrete positions or continuous paths be determined completely by probability? For the case similar to Brownian motion, what is picking the next position in the chain of positions if all we are given is probability. You need some kind of selection process to determine points and if everything that interacts with a particle is governed by probability how is it possible to construct a path? Or what is picking the next point in the path of an electron whose position is described by a probability distribution. Essentially what I am saying is that probabilities cannot pick points only deterministic functions can.

 

[bhobba]

I never said that entanglement does not violate locality. It certainly appears to be the case. The question is now why can't a deterministic theory be used to explain the instantaneous determination in entangled particle properties.

Well it can eg Bohmian Mechanics. But as far as the formalism of QM goes it is unnatural being contextual ie the outcome depends on what else you are measuring at the same time. Why exactly do you want to make assumptions that are a bit kludgey? That's got nothing to do if BM is correct or not (that is a matter for experiment and until experiment can decide there is no way to tell what nature chose), the issue though is why is determinism more appealing than contextuality?

Determinism is simply a special case of probabilistic theories - the only allowed probabilities are zero and one. Some fairly basic considerations indicate there are really only two basic ways to model physical systems - standard probability theory and QM:
http://arxiv.org/pdf/0911.0695v1.pdf

And QM is the only one that allows entanglement.

Thanks
Bill

 

[bhobba]

I am not sure what you mean by " the electron position must be discontinuous in space." The electron jumping instantaneously from one point to another?

Indeed. Exactly what electrons are doing when not being observed is a matter of interpretation - many interpretations say it has no property unless observed so thinking it jumps from one point to another, instantaneously or otherwise, is rather meaningless.

Thanks
Bill

 

[TrickyDicky]

My main problem with the QM is that it claims these probabilities are fundamental. Take the hydrogen atom in ground state. The probability distribution gives a spherically symmetric pattern where probabilities are concentrated around the Bohr radius. But at the Bohr radius there is a spherical shell were the probability of finding an electron at any position on the shell is completely random. You have the same chance of finding the electron anywhere on that spherical shell. The same goes for any spherical shell at radius greater or less than the Bohr radius.

In my mind two interpretations of the electron can be given at this point:

1.) If you interpret the electron position to be described only by probabilities at any instant, then it must be the case that to maintain equal probabilities over individual shells over some period of time, the electron position must be discontinuous in space.

2.) If instead you interpret the electron path to be continuous then the particle must not be more fundamentally described by a probability distribution which could only acts as a statistical analysis. Since we have assumed the particle has a set path whether we can observe that path or not.

Both 1 and 2 assume that one particle is always one particle and that the particle must exist somewhere in space at all times. With these assumptions we can observe that the particle must either follow a continuous path or be discontinuous in space.

Option 1 has obvious problems such as the infinite velocity required for the electron to make discontinuous jumps through space. So QM could not logically say that what these probability distributions really represent are particle positions in time. Then what do the probability distributions represent? We are only left with option 2 in which the paths of particles are always continuous through space and so cannot have been actually represented by spatial probability distributions. In the case of an orbiting electron at the Bohr radius, the electron must move from one point in space to adjacent points and can never move to points not adjacent to its current position. So, a probability distribution tells us nothing about the particles actual path and yet we must assume that continuous path through space exists. Therefore the probability distribution for a particle can only represent the state of our knowledge about where the particle can be not being aware of its path which is exactly how a statistical approach works.

Then the question becomes what is deciding the path the electron is taking. It cannot be the probability distribution because by 1 that will necessarily cause the path to be discontinuous. So we are forced to conclude, at least I am, that we are missing something that would explain the continuous path a particle must have.

So what is the only thing that we have that can represent a continuous path through space? This leaves me to think there must be deterministic functions governing the path of particles. Otherwise we must violate one or more of the above assumptions: that one particle is always one particle, the particle must always exist in space, and the particle cannot move instantaneously through space.

By assuming QM is the final theory we are forced to propose things like by the measurement of a particle the particle becomes real and/or the wave function collapses or multiple universes must exist etc. and so on and on. The existence of such propositions as legitimate theories is a big hint that something is fundamentally wrong with QM.

I never said that entanglement does not violate locality. It certainly appears to be the case. The question is now why can't a deterministic theory be used to explain the instantaneous determination in entangled particle properties. It is not really necessary that we limit our set of laws to describe the interactions between billiard balls. That is a very simplistic view. Why can't the instantaneous determination of spin of entangled particles be based in deterministic law?



I was thinking about the probability distribution as a whole but if you only consider adjacent points in space as possibilities around the electron then you could come up with something similar to Brownian motion. I guess that should have been option 3. Is that how these paths are interpreted to behave or should we determine points based on the entire distribution at any point in time regardless of where the particle might be located?

Now that you say that. I think I have to fall back from the original argument to another position. How can probability determine positions at all? What is it to be governed by probability and how can concrete positions or continuous paths be determined completely by probability? For the case similar to Brownian motion, what is picking the next position in the chain of positions if all we are given is probability. You need some kind of selection process to determine points and if everything that interacts with a particle is governed by probability how is it possible to construct a path? Or what is picking the next point in the path of an electron whose position is described by a probability distribution. Essentially what I am saying is that probabilities cannot pick points only deterministic functions can.

As I have insisted in previous posts but apparently nobody noticed this argument (maybe it deserves a thread of its own but I think it is quite related to QM incompleteness issues), it seems to me like the measurement problem and the probabilistic uncertainties associated to QM (and I'm concentrating here on NRQM since everybody seems to be referring to it anyway) all rest on holding on to the particle picture as fundamental. Why should the wave function be describing both the position and momentum of a quantum as if it was a classical particle? why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field? Do we think in terms of heat particles when dealing with the heat equation? Doesn't it make more sesnse to think in terms of relativistic quantum fields whose localized behaviour is subject to probabilistic analysis?

 

[bhobba]

As I have insisted in previous posts but apparently nobody noticed this argument (maybe it deserves a thread of its own but I think it is quite related to QM incompleteness issues), it seems to me like the measurement problem and the probabilistic uncertainties associated to QM (and I'm concentrating here on NRQM since everybody seems to be referring to it anyway) all rest on holding on to the particle picture as fundamental. Why should the wave function be describing both the position and momentum of a quantum as if it was a classical particle? why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field? Do we think in terms of heat particles when dealing with the heat equation? Doesn't it make more sesnse to think in terms of relativistic quantum fields whose localized behaviour is subject to probabilistic analysis?

I think you need to read chapter 3 of Ballentine - Quantum Mechanics - A Modern Development. It's got nothing to do with a particle picture other than position as an observable exists (which is an indisputable experimental fact) but symmetries.

Thanks
Bill

 

[TrickyDicky]

I think you need to read chapter 3 of Ballentine - Quantum Mechanics - A Modern Development. It's got nothing to do with a particle picture other than position as an observable exists (which is an indisputable experimental fact) but symmetries.

Thanks
Bill

I don't have it handy. Can you give the basic points where Ballentine argues that puzzlement about QM not giving deterministic trajectories but amplitude probabilities has nothing to do with insisting about classical particle behaviour?

Besides, observavility of fields interactions positions is not what I mean by classical particle behaviour.

 

[bhobba]

I don't have it handy. Can you give the basic points where Ballentine argues that puzzlement about QM not giving deterministic trajectories but amplitude probabilities has nothing to do with insisting about classical particle behaviour?

Purely from the Principle Of Relativity that the laws of physics are the same in all inertial reference frames you can derive - not assume - but derive Schrodinger's equation, the definition and existence of the momentum operator and all the stuff that other textbooks assume. This is because of the symmetries implied by the POR. Its the exact analogue of the least action formulation of classical mechanics. As you will find in Landau - Classical Mechanics that is the case there as well - the existence of momentum, Newtons laws etc all follow from symmetry.

If you haven't read Landau read it:
http://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifgarbagez-Mechanics.pdf
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.

The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.

The difficulty with this approach, and the reason why this book is not a beginner's book, is that to the follow symmetric arguments, one really has to have already mastered vector calculus. Ideally, you should be able to transform coordinate in your sleep, perform integrals without missing a beat, whether they be line, area, or path, and differentiate functions in many dimensions. The arguments are not sloppy, as some have claimed - it only seems so if you have not mastered vector calculus.

Tradition says that in Plato's academy was engraved the phrase, "Let no one ignorant of geometry enter here", so should the modern theoretical physicist, with Landau's bible in hand, march under the arches engraved with the words "Let no one ignorant of symmetry enter here".'

After reading that read Ballentine. You will see its got nothing to do with a particle picture but symmetries. These force the equations on us purely from the fact position is an observable.

When you understand this you will wonder what the connection is - answer - Feynman's path integral approach. This is why classical mechanics is as it is - and both are deeply, no very deeply, determined by symmetry. This is unbelievably beautiful mathematically and once I understood it it revolutionized my view of nature - as I think it will for anyone exposed to it.

Also get Susskinds new book:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20

At a more basic mathematical level in the sense he develops the math as you go along he explains the same stuff as Landau. Like Euclid Landau and Susskind looked on physical beauty bare - once you understand it so will you.

Enough said - end of rant.

Thanks
Bill

 

[Jano L.]

Determinism is simply a special case of probabilistic theories - the only allowed probabilities are zero and one.

That works only for discrete space of events. If the space of events is continuous (like in classical mechanics), we cannot restrict the probability values in probabilistic model only to 0 and 1.

 

[bhobba]

That works only for discrete space of events. If the space of events is continuous (like in classical mechanics), we cannot restrict the probability values in probabilistic model only to 0 and 1.

Yes you can - the predicted values have a value of 1 - all the rest zero. You will have to use Dirac Delta functions - but hey welcome to applied math.

What you don't have is a continuous transition which is an issue in QM but not classically.

Thanks
Bill

 

[Jano L.]

Why should the wave function be describing both the position and momentum of a quantum as if it was a classical particle? why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field?

The basic reason is that it is the most natural and simplest interpretation in quantum chemistry.

To paraphrase Slater, after the success of de Broglie wave hypothesis for individual electrons, people wondered, what to do for multi-particle system, like the helium atom or hydrogen molecule. Should we introduce different $\psi$ functions for each different particle (field-like idea of the electron) or should the many-particle system be described by one big $\psi$ function?

It turned out that the close analogy to Hamilton-Jacobi theory works well even for many particle systems, i.e. one $\psi$ function describes the whole system.

Why are often electrons thought of as particles and not fields? Well, the particle is much simpler concept and it is quite natural if you look at the equation which people use to calculate properties of the atoms and molecules:

$$
\partial_t \psi(\mathbf r_1, \mathbf r_2, ...) = \frac{1}{i\hbar} \left[\sum_a \frac{\hat{\mathbf p}_a^2}{2m_a} + U(\mathbf r_1, \mathbf r_2, ...)\right] \psi
$$

The function $\psi$ is defined on configuration space of the system of particles. The variables $\mathbf r_1, ...$ are possible positions of those particles.

 

[Jano L.]

Yes you can - the predicted values have a value of 1 - all the rest zero. You will have to use Dirac Delta functions - but hey welcome to applied math.

Probabilistic models on continuous spaces usually work with regular probability distributions. What you suggested would require a probabilistic model which would assign singular probability to one event in continuous space as a result of the calculation. Can you give an example of such model ?

 

[bhobba]

Probabilistic models on continuous spaces usually work with regular probability distributions. What you suggested would require a probabilistic model which would assign singular probability to one event in continuous space as a result of the calculation. Can you give an example of such model ?

The probability distribution of finding something at position x with certainty is delta(x-x') where delta is the Dirac Delta function. x in that equation is the positions predicted by classical mechanics.

Thanks
Bill

 

[Jano L.]

I think the problem is in the following. Probabilities are measures and measures do not capture the essential character of certainty that determinism gives us.

In your example with delta distribution, if the probability distribution on space of x' is $\delta(x'-x)$, the only thing we can infer from it is that the probability that the particle is out of x is 0.

We cannot infer that the particle is at x with certainty. It can still get out of x, as long as it spends infinitely less time there. However, if it can get out of x, it can have some significant influence there.

In a deterministic model, if the particle is at x, it really is there and never gets anywhere else; it cannot have influence at other points than at x.

 

[bhobba]

We cannot infer that the particle is at x with certainty. It can still get out of x, as long as it spends infinitely less time there.

I have zero idea what you are trying to say. It has zero probability of being anywhere other than x because, for any point other than x, you can always find points near x to take the integral around that will give zero. But if it includes x you get one. That has one and only one interpretation - the particle is with a dead cert at x. And the only probabilities that enter into it are zero and one - just like I asserted.

Thanks
Bill

 

[Jano L.]

That has one and only one interpretation - the particle is with a dead cert at x.

There is difference between probability 1 and absolute certainty.

Probability 1 means only that the particle will be found out of x insignificant number of times, in other words, measure of such cases is zero. Still, in a long enough series of measurements, one may find particle out of x million times.

Absolute certainty of x would require that the particle will never be found out of x. Such kind of certainty cannot be described by measure on the event space, but it can be described by statements like "particle will be found at x", used in deterministic models.

 

[DevilsAvocado]

1.) If you interpret the electron position to be described only by probabilities at any instant, then it must be the case that to maintain equal probabilities over individual shells over some period of time, the electron position must be discontinuous in space.

2.) If instead you interpret the electron path to be continuous then the particle must not be more fundamentally described by a probability distribution which could only acts as a statistical analysis. Since we have assumed the particle has a set path whether we can observe that path or not.

Both 1 and 2 assume that one particle is always one particle and that the particle must exist somewhere in space at all times. With these assumptions we can observe that the particle must either follow a continuous path or be discontinuous in space.

I might be missing something here; but isn’t the problem that you treat the electron as a particle instead of a spherical cloud of probability, which we know must be correct since a rotating charge classically orbiting around the nucleus, would constantly lose energy in form of electromagnetic radiation, and finally collapse into the nucleus...

 

[DevilsAvocado]

Why can't the instantaneous determination of spin of entangled particles be based in deterministic law?

A non-local deterministic theory can, but a local deterministic theory cannot.

By the way, isn’t this talk about determinism a bit ‘superfluous’...? I mean, the Schrödinger equation is perfectly deterministic...

Isn’t the real problem that we don’t know exactly (in mathematical terms) what happens at measurement, when QM suddenly ‘transforms’ into a probabilistic theory?

Who cares if it’s deterministic or probabilistic? As long as we fully understand what’s going on.

(no problem to start ‘stochastic’ unpredictable chaos in a deterministic system)

 

[DevilsAvocado]

why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field? Do we think in terms of heat particles when dealing with the heat equation?

I think the reason is – you can’t put sunlight in your pocket, but a marble you can.

 

[DevilsAvocado]

I don't have it handy. Can you give the basic points where Ballentine argues that puzzlement about QM not giving deterministic trajectories but amplitude probabilities has nothing to do with insisting about classical particle behaviour?

I don’t think Ballentine do away with the measurement problem in chapter 3:

Kinematics and Dynamics
The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators. This will be done in the present chapter. The fundamental physical variables, such as linear and angular momentum, are closely related to space–time symmetry transformations. The study of these transformations serves a dual purpose: a fundamental one by identifying the operators for important dynamical variables, and a practical one by introducing the concepts and techniques of symmetry transformations.

Where’s the “symmetry” in this?

https://www.youtube.com/watch?v=ZJ-0PBRuthc

 

[bhobba]

There is difference between probability 1 and absolute certainty.

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

In my example let's for simplicity consider the 1 dimensional case. If you pick a point other than x you can find an interval that doesn't have x in it but includes that point. The probability of the particle being in that interval is integral delta (x-x') over that interval - which is zero. It can't be in that interval so can't be at the point. Do the same thing but at x and you get 1 - it must be in that interval. This means it must be at x.

The ability to define only 0 and 1 and such being deterministic is one of the key ingredients in the Kochen-Specker theorem:
http://en.wikipedia.org/wiki/Kochen–Specker_theorem

Thanks
Bill

 

[bhobba]

Isn’t the real problem that we don’t know exactly (in mathematical terms) what happens at measurement, when QM suddenly ‘transforms’ into a probabilistic theory?

Its probabilistic from the outset. What is deterministic is the state that allows you calculate probabilities.

Who cares if it’s deterministic or probabilistic? As long as we fully understand what’s going on.

That's the whole problem with QM - what one person thinks is a full understanding to another is an anathema.

Thanks
Bill

 

[T0mr]

I might be missing something here; but isn’t the problem that you treat the electron as a particle instead of a spherical cloud of probability, which we know must be correct since a rotating charge classically orbiting around the nucleus, would constantly lose energy in form of electromagnetic radiation, and finally collapse into the nucleus...

The electron in orbit was given as an example of a possible path. I have read about this argument before. That an classically orbiting electron should emit radiation presumably because it is a charged object and an accelerated charged object (changing direction) will emit electromagnetic radiation. Yet if you were to put opposite charge on two spheres, one light and one heavy, and then set the lighter in orbit (in space) around the heavier would the two spheres not act just as the two body problem for gravitational force. Or take for instance an electromagnet, as electrons move at relatively similar speeds around the coils, no electromagnetic fields are emitted just magnetic fields. Magnets do not die out rapidly. So that argument as being the motivation for adopting a completely probabilistic approach to the atom seems to me to be inadequate.

By the way, isn’t this talk about determinism a bit ‘superfluous’...? I mean, the Schrödinger equation is perfectly deterministic...

A wave function can be deterministic in the sense that it can determine the probabilities of a property of a quantum object. But it really isn't deterministic in the sense that it can determine what those properties actually are. For example, a wave function will not give the position of the object (or position of object's center) even when given the classical initial conditions required. A wave function gives many objects and the concern may be with only one object. If you ask it where the center of an electron is at time t, it will reply "I do not know but here are an infinite number of possible options." That kind of process really is not in the spirit of determinism. A deterministic function in my mind is something that is one to one, not one to infinity(with odds).

 

[bhobba]

I don’t think Ballentine do away with the measurement problem in chapter 3:

That was not my assertion - my assertion was its not based on a particle model.

Where’s the “symmetry” in this?

The symmetry is in the laws of physics. It doesn't matter where you do the experiment, in what direction its done or when you do it the same laws of physics apply and the same outcome will occur. To be specific since this is QM the probabilities will be the same. This implies physical laws and is one of the very deep things modern physics has taught us - to be specific it was one of the great insights of Wigner and part of the reason he got a Nobel prize. For classical physics it was the great mathematician Emily Noether that discovered it. It was Feynman that showed how the two were related.

This is in fact the defining property of an inertial frame - the Earth isn't exactly inertial but for many practical purposes such as this experiment it is.

Thanks
Bill

 

[TrickyDicky]

That was not my assertion - my assertion was its not based on a particle model.

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.
Now oddly enough you try to refute this by highlighting the similarities of the classical mechanics model(which is a particle model isn't it?) with quantum mechanics.

The symmetry is in the laws of physics. It doesn't matter where you do the experiment, in what direction its done or when you do it the same laws of physics apply and the same outcome will occur. To be specific since this is QM the probabilities will be the same. This implies physical laws and is one of the very deep things modern physics has taught us - to be specific it was one of the great insights of Wigner and part of the reason he got a Nobel prize. For classical physics it was the great mathematician Emily Noether that discovered it. It was Feynman that showed how the two were related.

This is in fact the defining property of an inertial frame - the Earth isn't exactly inertial but for many practical purposes such as this experiment it is.

I'm not one to be convinced of the mathematical beauty of symmetries, but in this thread we are actually centering on departures of quantum theory from those classical symmetries.
If you are so fascinated by symmetries you surely must feel how awkwardly those symmetries are spoiled in QFT/QM by a lot of things that can be reduced to the measurment problem for brevity and the fact that the nice path integral formulation in order to obtain sane results (to many decimal places) needs to recurr to arbitrary procedures like regularization that are neither physically nor mathematically justified, yeah I know, welcome to applied mathematics but I thought you had some fondness of symmetries and mathematical beauty.
So you make a rather strange mix of demanding the aesthetic value of symmetries one side and practical purposes that are only fulfilled thru rather ugly manouvers (in terms of mathematical rigour that is). Not to mention the incompatibility between GR and QM. Certainly some symmetry is not right here.

To insist in the example I used earlier, the Schrodinger equation is basically a heat equation with a Wick rotation, this introduces time reversibility thru i but again there is nothing fundamental in the equation that makes us use the particle picture other than being used to it from classical mechanics and the practical matters that Jano L. mentioned, that have nothing to do with fundamental issues. That practical use is compatible with considering those local observables simply as excitations of quantum relativistic fields. What I was pointing out was that IMO as soon as one stops thinking about particles with trajectories (that is with simultaneous position and momentum) the probabilistic issues and the measurement problem/collpse of wf or even entanglement and other "quantum weird stuff" gets downgraded just by concentrating on the fields picture.

 

[bhobba]

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.

I think you are missing my point. QM does not insist on analogies with a classical particle model. All it assumes is position is an observable - which is a fact. The rest follows from symmetries - no other assumptions at all. It's forced on us - no escaping it. For example momentum exists because the laws of physics are space translation invariant which means it's generator has certain properties and those properties imply the momentum operator. No particle assumption - yet momentum exists. The same with energy. No assumption about it at all yet time translation symmetry, similar to momentum, means it exists and implies the Schrodinger equation. It's a very deep insight.

This is a very important point and I think we need to get it sorted before discussing other stuff.

BTW its true QFT does shed considerable light on QM - see for example:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

But that's not because a particle analogy was chosen - symmetries force it onto us - no escaping.

Thanks
Bill

 

[TrickyDicky]

I think you are missing my point. QM does not insist on analogies with a classical particle model. All it assumes is position is an observable - which is a fact. The rest follows from symmetries - no other assumptions at all. It's forced on us - no escaping it. For example momentum exists because the laws of physics are space translation invariant which means it's generator has certain properties and those properties imply the momentum operator. No particle assumption - yet momentum exists. The same with energy. No assumption about it at all yet time translation symmetry, similar to momentum, means it exists and implies the Schrodinger equation. It's a very deep insight.

This is a very important point and I think we need to get it sorted before discussing other stuff.

BTW its true QFT does shed considerable light on QM - see for example:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

But that's not because a particle analogy was chosen - symmetries force it onto us - no escaping.

I can't argue with someone that while insisting on classical symmetries like time and space translation keeps saying there are no analogies with classical mechanics(even after recommending me the classic by Landau to have a better understanding of QM) or that classical Newtonian mechanics are unrelated to a particle picture.

 

[bhobba]

I can't argue with someone that while insisting on classical symmetries like time and space translation keeps saying there are no analogies with classical mechanics(even after recommending me the classic by Landau to have a better understanding of QM) or that classical Newtonian mechanics are unrelated to a particle picture.

Sigh. It looks like there is a schism here that's difficult to overcome. For example 'classical symmetries' - there is nothing classical about the POR that these symmetries derive from - its true relativistically, QM, classically, all sorts of ways. It is these symmetries that determine the dynamics of QM - not an analogy to anything. Even if you know nothing of classical physics exactly the same equations result.

The reason I recommended Landau is it shows classical mechanics is really about symmetry so its no surprise QM is also about it. In fact the reason classical mechanics is about symmetry is because QM is about symmetry. Its not the other way around - QM is not based on classical analogies - it based on the implications of symmetry and that implies that classical mechanics is also. In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian. But as Feynman showed the Lagrangian follows from the rules governing states and observables.

Thanks
Bill

 

[stevendaryl]

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator O always produces an eigenvalue of O.
4. The probability that a measurement of an observable corresponding to operator O produces outcome o_i is given by:
   | \langle \psi | P_i |\psi \rangle |^2, where P_i is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue o_i.
5. Immediately after the measurement, if the result was o_i, then the system will be in state P_i | \psi \rangle.
6. Between measurements, the system evolves according to equation:
   \dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

 

[bhobba]

[*]Between measurements, the system evolves according to equation:
\dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

 

[TrickyDicky]

Sigh. It looks like there is a schism here that's difficult to overcome. For example 'classical symmetries' - there is nothing classical about the POR that these symmetries derive from - its true relativistically, QM, classically, all sorts of ways.
It is these symmetries that determine the dynamics of QM - not an analogy to anything. Even if you know nothing of classical physics exactly the same equations result.

The reason I recommended Landau is it shows classical mechanics is really about symmetry so its no surprise QM is also about it. In fact the reason classical mechanics is about symmetry is because QM is about symmetry. Its not the other way around - QM is not based on classical analogies - it based on the implications of symmetry and that implies that classical mechanics is also. In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian. But as Feynman showed the Lagrangian follows from the rules governing states and observables.

Thanks
Bill

It's no use to keep circling around this. I'm not disagreeing with this rather with the way you present it anyway.
Instead and since I think you favor the statistical interpretation, QFT is also related to statistical field theory by a Wick rotation. I sometimes entertain myself relating ensemble properties to field properties.

 

[TrickyDicky]

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator O always produces an eigenvalue of O.
4. The probability that a measurement of an observable corresponding to operator O produces outcome o_i is given by:
   | \langle \psi | P_i |\psi \rangle |^2, where P_i is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue o_i.
5. Immediately after the measurement, if the result was o_i, then the system will be in state P_i | \psi \rangle.
6. Between measurements, the system evolves according to equation:
   \dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle (in units where h-bar = 1), where H is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

 

[TrickyDicky]

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

 

[bhobba]

Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't get that at all.

For example check out the link I posted before:
http://arxiv.org/pdf/0911.0695v1.pdf

Precisely where in that axiomatic development is point particles assumed?

Thanks
Bill

 

[bhobba]

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

 

[TrickyDicky]

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

Thus my "whether one considers it a historical accident or not".

 

[stevendaryl]

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

 

[TrickyDicky]

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.
Quote by Quantum Mechanics - A Modern Development, Leslie E. Ballentine (1998)
Kinematics and Dynamics:
"The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators."
It is in those correspondence rules that the problem arises, and depending on how one interprets the Born rule, for instance, you might have a smaller or bigger problem.
I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?
When you were talking about observables were you thinking about them in terms of properties of particles, fields...?