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Classical and Quantum Mechanics via Lie algebras

by A. Neumaier
Tags: algebras, classical, mechanics, quantum
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A. Neumaier
#1
Apr15-11, 04:54 AM
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I'd like to open a discussion thread for version 2 of the draft of my book ''Classical and Quantum Mechanics via Lie algebras'', available online at http://lanl.arxiv.org/abs/0810.1019 , and for the associated thermal interpretation of quantum mechanics, espoused in the book.

The goal of the thread is to obtain reader's feedback that helps me to improve the presentation while I work towards a version for publication.


Abstract: The book fulfils the didactical purpose of showing that
-- quantum mechanics and classical mechanics are much more similar than can be seen from the usual presentations of the subject;
-- in a very significant sense, theoretical classical and quantum mechanics is nothing but applied Lie algebra;
-- quantum mechanics has a common sense interpretation once one takes the thermodynamical findings of statistical mechanics serious in the foundations.

The content of the book is fully mainstream, covering hundreds of publications by others (301 references, too numerous to include them into this opening post), including many references to basic experiments. However, the selection and presentation of the material is very different from what one can find elsewhere.

The importance of the topic is obvious. With exception of the thermal interpretation, nothing is new about the scientific content. The presentation of the book is in intelligible English, complemented by LaTeX (some of it only intelligible by intelligent readers). With exception of historical evidence (and perhaps oversights), everything is defined or derived with mathematical rigor. The empirical equivalence of the presented material to standard mechanics is manifest, and almost the whole body of experimental physics supports the theory presented. [If this paragraph sounds a bit crackpottish - I am required to state all these things in order to conform to the submission rules.]


The thermal interpretation of quantum mechanics was presented first last year in a lecture whose slides (Slide 23-34 define the interpretative core) are available at http://www.mat.univie.ac.at/~neum/ms/optslides.pdf , which in turn is based on insights from Sections 8.4 and 10.3-10.5 of version 2 (or Sections 5.4 and 7.3-7.5 of version 1) of the above book.A short exposition is given in the entry ''Foundations independent of measurements'' of Chapter of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/ph...aq.html#found0 . See also the following PF posts:

http://www.physicsforums.com/showthr...36#post3246436
http://www.physicsforums.com/showthr...39#post3187039
http://www.physicsforums.com/showthr...47#post3193747
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qsa
#2
Apr18-11, 09:02 PM
P: 362
how does the thermal arise for a single electron.
A. Neumaier
#3
Apr20-11, 10:38 AM
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Quote Quote by qsa View Post
how does the thermal arise for a single electron.
The thermal interpretation does not only apply to thermal states; it is completely general.

I suggest that you begin by reading the slides http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
where the interpretation is explained for a single photon. The electron is essentially the same, except for the different state space.

A. Neumaier
#4
Apr20-11, 11:51 AM
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P: 1,943
Classical and Quantum Mechanics via Lie algebras

This post contains replies to posts 1-7 from http://www.physicsforums.com/showthread.php?t=490677
I'll comment on the later posts at another time.

Quote Quote by rodsika View Post
can you pls create a more layman friendly introduction to it like describing in detail how it explains buckyball made up of 430 atoms that can still interfere with itself?
Quote Quote by rodsika View Post
One major idea being put forth by Neumaier is that fields are now primary and particles are just momentum of the fields. So particle concept is now outdated and there is no sense of thinking of the double slit experiment as particle that moves in between the emitter and detector but more like field interacting in between (perhaps like Feynmann interaction vortexes). If that is true, then the buckyball composing of 430 atoms can be considered as field too.. but does it make sense to think in terms of 430 atom buckyball field??
If a beam of buckyballs is coherently prepared so that it shows interference effects (which is not easy at all - typical buckyball beams are not coherent)
then its quantum field oscillates in a wave manner.




Quote Quote by unusualname View Post
A Neumaier is clearly a very erudite and impressive scholar, but his interpretation is unlikely to be correct as modern experiments urge a fundamental probabilistic character to Nature which his interpretation is not in agreement with.
Why not?
Quote Quote by unusualname View Post
He is a genius clinging to old school deterministic ideas about nature
The thermal interpretation explains the conditions under which a stochastic interpretation of QM is valid, the conditions under which a determinstic interpretation is valid, and the conditions under which neither is valid.
See Chapter 10.3-5 of version 2 of the draft of my book.

No other interpretation can do that.
Quote Quote by unusualname View Post
And also I'm not sure he should be allowed to promote such a non peer reviewed philosophy so strongly on these forums.
about 10 of my over 1300 contributions to PF mention the thermal interpretation. Everything else is orthodox physics.
Quote Quote by unusualname View Post
the interpretation stuff is really not scientific
Before making such unsupported negative comments you should first read and understand what my interpretation consists of.




Quote Quote by rodsika View Post
Arnold, can you please write a Wikipedia article about it and give the basics
I write here, and leave Wikipedia to those active there.
Quote Quote by rodsika View Post
as your articles are so vague and disorganized.
I suggest that you begin by reading the slides http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
where the interpretation is explained. They contain the material of a well-organized lecture in which nothing is left vague.




Quote Quote by rogerl View Post
Neumaier gives example of beam of photon. But a beam of photon has real wave, whereas electron wave is just probability wave. So one can't argue whether before observation, a beam of photon is there or not. It has real wave. Whereas matter waves are not actual waves.
Coherent electron waves are as real as coherent photon waves, as the many interference experiments show. In QFT they are actual waves in the most real sense of the word.
Quote Quote by rogerl View Post
Feynman mentions in his book "The Strange Story of Light and Matter" about reflections of light. He said that in reflections in a glass, 4% of the photons are always deflected. How do the photons know how to be 4% Feynmann asks. What I want to know is, can reflections be done with matter wave too like electron wave such that you can also see 4% of electron being deflected?
Electron fields are not so different from photon fields as you might think. There is a whole discipline of physics called electron optics http://en.wikipedia.org/wiki/Electron_optics where the optical properties of electrons are studied in close analogy to the optical properties of light.



Quote Quote by JesseM View Post
would the field operators at a given point in spacetime give the probabilities or expectation values for the outcome of some measurement at that point?
The field operator evaluated at a point x in space-time defines the probability distribution of the magnitude of the quantum field at that point. But field operators are only distributions, not ordinary functions, which means that only smeared values of the field make sense. What is measurable is usually the expectation value of the field smeared over the region in space time determined by the active part of the detector and the duration of the measurement.
Quote Quote by JesseM View Post
Neumaier goes on to say that expectation values are the basic elements of his interpretation:
Yes. In the thermal interpretation, _all_ measurements are interpreted as measurements of expectation values.
Quote Quote by JesseM View Post
But if he wants to avoid collapse, how does he go from expectation values to actual measured values of microscopic systems? (or macroscopic 'pointer states' associated with those measurements)
Here is the main difference to the traditional interpretations.
The thermal interpretation doesn't change anything in the theory. What is changed is _only_ the interpretation of measurements. All measurements are primarily measurements of the macroscopic object that is actually inspected when measuring something.

To be valid, any inference about the value of some microscopic object must be
(ideally) backed up by an argument that the microscopic object influences the macroscopic object in a way that the observed macroscopic behavior results.
strangerep
#5
Apr21-11, 04:38 AM
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Quote Quote by A. Neumaier View Post
In the thermal interpretation, _all_ measurements are interpreted as measurements of expectation values.
Leaving aside the interpretation, and looking instead at how you formulate QM,
starting with an algebra of observables, then states and relying on a list of
axiomatic (Whittle-style) properties of expectations, I'm wondering whether one
can indeed account for all features of QM that way...

Consider the Cauchy (or Breit-Wigner) distribution that gives the probability
distribution of the lifetime of unstable particles. The usual expectation, variance,
etc, are undefined for that distribution but it's clearly an important part of quantum
physics. How then do you get a Breit-Wigner distribution if you've started the
theory from expectations?
A. Neumaier
#6
Apr21-11, 11:04 AM
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Quote Quote by strangerep View Post
Leaving aside the interpretation, and looking instead at how you formulate QM,
starting with an algebra of observables, then states and relying on a list of
axiomatic (Whittle-style) properties of expectations, I'm wondering whether one
can indeed account for all features of QM that way...

Consider the Cauchy (or Breit-Wigner) distribution that gives the probability
distribution of the lifetime of unstable particles. The usual expectation, variance,
etc, are undefined for that distribution but it's clearly an important part of quantum
physics. How then do you get a Breit-Wigner distribution if you've started the
theory from expectations?
Only _bounded_ quantities _must_ have an expectation. For unbounded quantities the expectation need not exist. Thus (as in C^*-algebras), one can always go to the exponentials e^{is A} of a self-adjoint but unbounded quantity. in probability theory, the function defined by the expectations f(s):=<e^{isa}> is called the characteristic function of A. it completely characterizes the distribution of functions of A, and is the right thing to study in case of cauchy-distributed A.
A. Neumaier
#7
Apr21-11, 11:40 AM
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Quote Quote by A. Neumaier View Post
This post contains replies to posts 1-7 from http://www.physicsforums.com/showthread.php?t=490677
I'll comment on the later posts at another time.
This post contains replies to the remaining posts from http://www.physicsforums.com/showthread.php?t=490677



Quote Quote by JesseM View Post
But Neumaier says that his interpretation "acknowledges that there is only one world", and that it "is consistent with assigning a well-defined (though largely unknown) state to the whole universe", shouldn't that mean the interpretation has to give more than just a collection of probabilities for different states at different points in spacetime, since in our "one world" we see "states" consisting of definite outcomes rather than just probabilities?
We see definite outcomes whenever we look at a system large enough that the assumptions of statistical mechanics apply. In particular, this holds for all the things that are _actually_ measured, such as pointers of instruments, colors of pixels on a screen, developped photographic plates, sounds in a Geiger counter,
currents in a photodetector.

We _infer_ from these raw measurements properties of systems that we cannot ''read'' directly, and the inference is as good or as bad as the causal link provided by quantum mechanical theory in the respective case.
Quote Quote by JesseM View Post
But there are plenty of cases where var(A) would be large even for macroscopic systems, like the state of macroscopic "pointers" which show the results of experiments on quantum particles
Yes. This implies that the macroscopic pointer gives only unreliable information about the quantum particle, unless many repeated measurements are made under sufficiently idenitcal conditions.





Quote Quote by rogerl View Post
Do you guys agree that the Ensemble Interpretation (a requirement for Neumaier Interpretation) is already falsified?
The thermal interpretation does not _require_ the ensemble interpretation.
Instead, it gives the conditions under which an ensemble interpretation is valid.
See Section 10.3 of my book.
Quote Quote by rogerl View Post
I presume that the Ensemble Interpretation is the same as the Statistical Interpretation?
Yes.
Quote Quote by rogerl View Post
Both these can't handle single system. But Neumaier Interpretation (actually not an Interpretation but just a QFT way of looking at it or from a QFT point of view) can handle single system. Why is that Neumaier's can handle single system while the Ensemble and Statistical can't since they are identical? What are the differences?
The difference is that I discard the so-called eigenvalue-eigenstate link,
and give the quantum expectation a different interpretation. See Sections 10.3-10.5 of my book.






Quote Quote by JesseM View Post
if Neumaier's interpretation only gives probability distributions for such macro-states rather than definite values,
It gives definite values to macroobservables of macrostate, within some tiny uncertainly level.
Quote Quote by JesseM View Post
the simulation yielding a series of macroscopic pointer states whose overall statistics should match the results of analogous experiments performed in the real world. If we require that the simulation be a "local" one
A valid simulation must be as nonlocal as QM itself.
Quote Quote by JesseM View Post
So, I think it's misleading to call Neumaier's interpretion a "local" one
It isn't local; I nowhere claimed that. The thermal interpretation shares all nonlocal features with orthodox quantum mechanics.
qsa
#8
Apr21-11, 12:36 PM
P: 362
Quote Quote by A. Neumaier View Post
The thermal interpretation does not only apply to thermal states; it is completely general.

I suggest that you begin by reading the slides http://www.mat.univie.ac.at/~neum/ms/optslides.pdf
where the interpretation is explained for a single photon. The electron is essentially the same, except for the different state space.
Thanks. I still have more questions but let me ask you this. I understand you that you claim that the wavefunction and its derived probability is not real in standard QM interpretation, it is just a mashinary to interpret experiments. But, in QFT and QM any time we want to calculate something (like energy) we take the whole wavefunction into account which to me it says that all aspects of wavefunction (at least the probabilities) is real and it exists at the same time. Am I saying things correctly?
Rap
#9
Apr21-11, 11:44 PM
P: 789
Quote from chapter A5 section 1 -

"Note that a measurement does not need a conscious observer.
A measurement is any permanent record of an event, whether or not
anyone has seen it. Thus the terabytes of collision data collected
by CERN are measurements, although most of them have never been
looked at by anybody."

Consider a "Schroedinger's cat" scenario - the particles generating the tracks and the "permanent record" are inside an isolated box - How is the "permanent record" described by a scientist outside the box? I expect it would be a mixed state, but does this mixed state constitute a measurement?
A. Neumaier
#10
Apr26-11, 07:07 AM
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Quote Quote by qsa View Post
I understand you that you claim that the wavefunction and its derived probability is not real in standard QM interpretation, it is just a mashinery to interpret experiments. But, in QFT and QM any time we want to calculate something (like energy) we take the whole wavefunction into account which to me it says that all aspects of wavefunction (at least the probabilities) is real and it exists at the same time.
In the standard interpretation probabilities are not observables, but propensities for observing something.
This makes the standard ontology quite weird.

In the thermal interpretation (as in real life), many expectations are measurable (to some limited degree of accuracy), and so are many probabilities (as expectations of projectors). Thus these are real in the thermal interpretation, making things much more intelligible.
A. Neumaier
#11
Apr26-11, 07:11 AM
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Quote Quote by Rap View Post
Quote from chapter A5 section 1 -

"Note that a measurement does not need a conscious observer.
A measurement is any permanent record of an event, whether or not
anyone has seen it. Thus the terabytes of collision data collected
by CERN are measurements, although most of them have never been
looked at by anybody."

Consider a "Schroedinger's cat" scenario - the particles generating the tracks and the "permanent record" are inside an isolated box - How is the "permanent record" described by a scientist outside the box? I expect it would be a mixed state, but does this mixed state constitute a measurement?
A permanent record is typically a macroscopic state in local equilibrium, changing so slowly that the techniques of statistical mechanics are applicable.

Whether it constitutes a measurement depends on one's definition of a measurement. The thermal interpretation has the huge advantage that one doesn't need to know what a measurement is, and still has a perfectly valid interpretation. Measurement is a difficult subject, so it should not figure in the foundations.
Phrak
#12
Apr26-11, 07:30 AM
P: 4,512
Quote Quote by A. Neumaier;3247879[B
Abstract:[/B] The book fulfils the didactical purpose of showing that
-- quantum mechanics and classical mechanics are much more similar than can be seen from the usual presentations of the subject;
-- in a very significant sense, theoretical classical and quantum mechanics is nothing but applied Lie algebra;
-- quantum mechanics has a common sense interpretation once one takes the thermodynamical findings of statistical mechanics serious in the foundations.
I have positve criticism of your abstract in presentation. Consider me just a reader, trying to understand and see if I am interested in reading further.

didactical
I'm a physicist not a philosopher, so I need some added reminder or hint about what didactic means.

-- quantum mechanics and classical mechanics are much more similar than can be seen from the usual presentations of the subject;
Give an example with motivational argument to show similarity. Get rid of the (--) and use standard English.

-- in a very significant sense, theoretical classical and quantum mechanics is [sic.: are] nothing but applied Lie algebra;
Again, show by comparison of the usage of Lie algebra's in classical and quantum physics to motivate comparison.

-- quantum mechanics has a common sense interpretation once one takes the thermodynamical findings of statistical mechanics serious in the foundations.
This one is a giant step. You need more motivation for making this claim within your abstract. Give a few more details and defer it to the main body of the text using a phrase such as "as will be demonstarted". This should be a good half of your abstract.

I hope this helps on the frontier of written communication.
A. Neumaier
#13
Apr26-11, 07:51 AM
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Quote Quote by Phrak View Post
I'm a physicist not a philosopher, so I need some added reminder or hint about what didactic means.
http://en.wikipedia.org/wiki/Didactic_method
Quote Quote by Phrak View Post
Give an example with motivational argument to show similarity.
This is done leisurely in Chapter 1 of the book, and comes across at many other places in the book. The abstract is not intended to make looking at the actual text. superfluous. It just summarizes what someone can expect to get out from reading the book.
Quote Quote by Phrak View Post
Again, show by comparison of the usage of Lie algebra's in classical and quantum physics to motivate comparison.
In a nutshell:
Poisson brackets define Lie algebras. Commutators define Lie algebras.

But to appreciate how far this goes, you need to study the book, not the abstract. It will be worth your time.
Quote Quote by Phrak View Post
This one is a giant step. You need more motivation for making this claim. Give a few more details and defer it to the main body of the text using a phrase such as "as will be demonstrated". This should be a good half of your abstract.
The main body of the text is the whole book. Innovations on old subjects can usually not be described in a few paragraphs. Please criticise the book (or the FAQ, or the lecture quoted), not the abstract.
A. Neumaier
#14
May2-11, 09:09 AM
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Quote Quote by A. Neumaier View Post
An electron behaves as a particle only in situations where an approximate semiclassical description is applicable. This is _not_ the case for a tightly bound electron such as in an ordinary atom or molecule, but it is often the case for an electron in a beam, when one doesn't aske too detailed questions. (Electron beams in full generality are treated in electron optics, where the Dirac equation is treated as a classical field.)

Depending on the preparation, you may regard it as a particle before it passes the screen, but afterwards no longer - passing the screen turns the electron into a delocalized object - the more delocalized the further away from the slits.

Indeed. There is no quantum mystery.

Not as a fundamental quantum law. But the Born rule has limited validity, which can be discussed in the framework of the thermal interpretation.

Talking about it hear is the first part of spreading it. I am gathering experience in how people respond and what must be said to make the case. For publishing it, the Scientific American is not the right place - this is a journal for expaining things to the interested public, not for describing new results to the scientific community.

But sooner or later, my thermal interpretation will be published - a full, convincing account of it is not written in a few pages, but takes time.

And in 10 years time, it will be the consensus of the scientific community, since unlike with all other interpretations, there is nothing weird about the thermal interpretation.
Quote Quote by Varon View Post
Let us focus on the double slit experiment as Feymann said it's the main mystery. If it's solved, the entire quantum mystery solved.

I can't understand what you meant by "passing the screen turns the electron into a delocalized object". You said the electron is a particle before it passes the screen. Since it is already a particle, how can it turned into a delocalized particle at the screen?
The electron is always a quantum field. The quantum field can be regarded to describe a particle if and only if the field has a nonzero expectation only in a region small compared to the whole system considered. Thus we may say that the field is a particle as long as this condition is satisfied. Because of the dispersion of the field caused by the slits, this condition stops to be satisfied almost immediately after the field (with support large enough to cover both slits) passed the double slit. Thus it is no longer justified to talk about a particle.

The situation is similar as with a sphere of glass. If you throw it, you may regard it as a particle. But if it hits an obstacle and fragmentizes, it is no longer localized enogh to deserve the name of a particle.
Quote Quote by Varon View Post

Let's go from the beginner in the emission. So the electron is emitted. You believed it travels as particle? But where does it pass, the left or right slit? And what caused the interferences in the screen. Standard explanation says it interferes with itself because it is a wave after it is emitted.. and only become a particle at the detection screen. Pls. elaborate what happened to your electron after emission.. before it reaches the slits.. after it exits the slits and after detection in the screen.
The field passes the doulbe slit in a fashion similar as a water wave would do, except with quantum corrections.
Varon
#15
May2-11, 09:29 AM
P: 525
Quote Quote by A. Neumaier View Post
The electron is always a quantum field. The quantum field can be regarded to describe a particle if and only if the field has a nonzero expectation only in a region small compared to the whole system considered. Thus we may say that the field is a particle as long as this condition is satisfied. Because of the dispersion of the field caused by the slits, this condition stops to be satisfied almost immediately after the field (with support large enough to cover both slits) passed the double slit. Thus it is no longer justified to talk about a particle.

The situation is similar as with a sphere of glass. If you throw it, you may regard it as a particle. But if it hits an obstacle and fragmentizes, it is no longer localized enogh to deserve the name of a particle.

The field passes the doulbe slit in a fashion similar as a water wave would do, except with quantum corrections.
Interesting. But how come the detector detects one electron and not the fragmentized parts (after passing thru the slits)?
A. Neumaier
#16
May2-11, 11:19 AM
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Quote Quote by Varon View Post
Interesting. But how come the detector detects one electron and not the fragmentized parts (after passing thru the slits)?
The quantum field does not fragmentize like a broken glass sphere. It just expands into a superposition of two spherical waves. The outer electrons of the detector respond to the incident quantum field by an approximate Poisson process with rate proportional to the incident density. This accounts correctly for the simple statistics obtained for an ordinary electron beam. See post #4 of this thread, and the longer discussion of the case of photons in
http://www.physicsforums.com/showthr...39#post3187039
and in the thread
http://www.physicsforums.com/showthread.php?t=480072
Varon
#17
May2-11, 05:32 PM
P: 525
Quote Quote by A. Neumaier View Post
The quantum field does not fragmentize like a broken glass sphere. It just expands into a superposition of two spherical waves. The outer electrons of the detector respond to the incident quantum field by an approximate Poisson process with rate proportional to the incident density. This accounts correctly for the simple statistics obtained for an ordinary electron beam. See post #4 of this thread, and the longer discussion of the case of photons in
http://www.physicsforums.com/showthr...39#post3187039
and in the thread
http://www.physicsforums.com/showthread.php?t=480072
What "outer electrons"? I'm talking about single electron. So when your single electron is emitted. It is a particle before reaching the slits. After it reach the slits. The single electron become delocalized or spread into a field. Now the mystery is how the detector is able to detect single electron again. So don't talk about "outer electrons" in the detector because we are only dealing with a single-electron at a time double slit experiment. Pls. explain what goes on between the slits and the detector and how the detector only detects one electron. My example is not the same as your electron beam and statistics for the ensemble. I'm referring to a single electron emission and detection.
Rap
#18
May2-11, 07:30 PM
P: 789
Quote Quote by Varon View Post
Pls. explain what goes on between the slits and the detector
This is an improper question. Only in classical physics can you describe what goes on, because you can make a measurement of what goes on which does not affect the outcome of the experiment. In QM, you cannot, so you cannot know what "goes on". To ask a question for which there is no answer is improper.


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