Classical and Quantum Mechanics via Lie algebras

A. Neumaier

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

qsa

how does the thermal arise for a single electron.

A. Neumaier

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

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.

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.




Why not?
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.
about 10 of my over 1300 contributions to PF mention the thermal interpretation. Everything else is orthodox physics.
Before making such unsupported negative comments you should first read and understand what my interpretation consists of.




I write here, and leave Wikipedia to those active there.
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.




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.
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.



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.
Yes. In the thermal interpretation, _all_ measurements are interpreted as measurements of expectation values.
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

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

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

This post contains replies to the remaining posts from http://www.physicsforums.com/showthread.php?t=490677



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.
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.





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.
Yes.
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.






It gives definite values to macroobservables of macrostate, within some tiny uncertainly level.
A valid simulation must be as nonlocal as QM itself.
It isn't local; I nowhere claimed that. The thermal interpretation shares all nonlocal features with orthodox quantum mechanics.

qsa

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

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

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

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

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.

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

Give an example with motivational argument to show similarity. Get rid of the (--) and use standard English.

Again, show by comparison of the usage of Lie algebra's in classical and quantum physics to motivate comparison.

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

http://en.wikipedia.org/wiki/Didactic_method
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.
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.
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

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.

Varon

Interesting. But how come the detector detects one electron and not the fragmentized parts (after passing thru the slits)?

A. Neumaier

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

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.