Insights Some sins in physics didactics - comments

[A. Neumaier]

We are unfortunate because we cannot form a mental picture for the electron

Mental pictures have nothing to do with the senses. I have a mental picture of the electron but also of a 4-dimensional cube. On the other hand, our senses do not give a classical picture of the world; this classical picture can be perceived not by our senses but only by the mind, only for less than 400 years, and by people without school education not at all.

I didn't claim you post was wrong (it is just an opinion, not a collection of facts), but posted an opposing opinion that makes much more sense to me.

In complete darkness we can see a single photon hitting our eye, since the eye has an excellent resolution.

The shape of a photon is very flexible, in typical quantum optics experiments it has the form of one or (after passing a beam splitter) several rays. Its most general shape can be the energy density of any solution of the homogeneous Maxwell equation. The electron in an isolated hydrogen atom is shaped like a fuzzy ball - one can compute its charge density to verify this. Its most general shape is (ignoring radiative corrections) that of the charge density of any solution of the homogeneous Dirac equation.

Every classical system in Nature is just a simplified (slightly approximate) version of the corresponding quantum system, and the motion of the planetary system is well described by Ehrenfest's theorem together with the quantum Hamiltonian for planets attracted by an inverse square law form.

 

[atyy]

Every classical system in Nature is just a simplified (slightly approximate) version of the corresponding quantum system, and the motion of the planetary system is well described by Ehrenfest's theorem together with the quantum Hamiltonian for planets attracted by an inverse square law form.

Ehrenfest's theorem is the way it is most often explained. But one thing I don't understand is that Ehrenfest's theorem seem to me to doesn't include sequential measurements, which are necessary for observing a classical trajectory. Would a more proper way to get a continuously observed trajectory be to repeatedly observe and then collapse the wave function, say something like this approach http://arxiv.org/abs/quant-ph/0512192 to getting cloud chamber trajectrories?

 

[A. Neumaier]

Ehrenfest's theorem doesn't involve the notion of measurement, hence can be interpreted independent of it. It includes the notion of an ensemble mean.

According to quantum field theory, the reason is that there is only one quantum field (of each kind), given by $\phi(t,x)$, say. We cannot obtain averages of it by repeated measurements as in experimentally performable repeated measurements either time passes, or the experiment is performed in different places. Thus averages correspond to weighted sums over fields at different arguments, rather than to different realizations of the field. Thus the ensemble means are at best (as Gibbs indeed introduced them before quantum mechanics was born) averages over fictitious repetitions that justify the application of the statistical calculus for their computation. But they are properties of the individual field - since there is only one of each kind.

For example, quantum field correlations (2-point functions) are effectively classical observables; indeed, in kinetic theory they appear as the classical variables of the Kadanoff-Baym equations, approximate dynamical equations for the 2-point functions. After a Wigner transform and some further approximation (averaging over small cells in phase space), these turn into the classical variables of the Boltzmann equation. After integration over momenta and some further approximation (averaging over small cells in phase space), these turn into the classical variables of the Navier-Stokes equation, hydromechanic equations that - as every engineer knows - describe the behavior of macroscopic fluids. For macroscopic solids, one can use similar approximations to arrive at the equations of elasticity theory. The most detailed classical level, the Kadanoff-Baym equations, still contain the unsmeared ensemble means of field products.

Now all macroscopic objects are objects describable by hydromechanics and elasticity theory; so their classical variables have the same interpretation. Thus the quantum-mechanical ensemble averages are classical variables. Moreover, because of the law of large numbers, $$\langle f(x)\rangle \approx f(\langle x\rangle)$$ for any sufficiently smooth function $f$ of not too many variables. (These caveats are needed because high dimensions and highly nonlinear functions don't behave so well under the law of large numbers.) Thus we get from Ehrenfest's theorem the standard classical equations of motion for macroscopic objects.

 

[A. Neumaier]

@atyy: Note that neither quantum jumps nor any other form of state reduction is needed in my explanation.

 

[atyy]

Ehrenfest's theorem doesn't involve the notion of measurement, hence can be interpreted independent of it. It includes the notion of an ensemble mean.

According to quantum field theory, http://arnold-neumaier.at/physfaq/cei/ . The reason is that there is only one quantum field (of each kind), given by $\phi(t,x)$, say. We cannot obtain averages of it by repeated measurements as in experimentally performable repeated measurements either time passes, or the experiment is performed in different places. Thus averages correspond to weighted sums over fields at different arguments, rather than to different realizations of the field. Thus the ensemble means are at best (as Gibbs indeed introduced them before quantum mechanics was born) averages over fictitious repetitions that justify the application of the statistical calculus for their computation. But they are properties of the individual field - since there is only one of each kind.

For example, quantum field correlations (2-point functions) are effectively classical observables; indeed, in kinetic theory they appear as the classical variables of the Kadanoff-Baym equations, approximate dynamical equations for the 2-point functions. After a Wigner transform and some further approximation (averaging over small cells in phase space), these turn into the classical variables of the Boltzmann equation. After integration over momenta and some further approximation (averaging over small cells in phase space), these turn into the classical variables of the Navier-Stokes equation, hydromechanic equations that - as every engineer knows - describe the behavior of macroscopic fluids. For macroscopic solids, one can use similar approximations to arrive at the equations of elasticity theory. The most detailed classical level, the Kadanoff-Baym equations, still contain the unsmeared ensemble means of field products.

Now all macroscopic objects are objects describable by hydromechanics and elasticity theory; so their classical variables have the same interpretation. Thus the quantum-mechanical ensemble averages are classical variables. Moreover, because of the law of large numbers, $$\langle f(x)\rangle \approx f(\langle x\rangle)$$ for any sufficiently smooth function $f$ of not too many variables. (These caveats are needed because high dimensions and highly nonlinear functions don't behave so well under the law of large numbers.) Thus we get from Ehrenfest's theorem the standard classical equations of motion for macroscopic objects.

@atyy: Note that neither quantum jumps nor any other form of state reduction is needed in my explanation.

That seems to be quite an original interpretation. In the usual view, the initial state of the system is the same on each run of the experiment, and one gets different outcomes because quantum mechanics only predicts probabilities, so we use a large number of runs. In your interpretation, it seems that everything is deterministic,so the random outcome on each run of the experiment is due to the initial state of the system being different on each trial?

 

[Ken G]

I always find discussions about interpretations to be quite interesting and insightful, but I do end up concluding that physics is not really a formal endeavor. Mathematics is formal, and physics borrows from mathematics in important and interesting ways, but physics is itself not formalizable. I think this is because we do not understand, nor ever include, the roles our minds our playing when we do physics. We know we don't include this, and we hope it doesn't matter that we don't include this, but the fact that we don't include it is an impediment to formalization in ways that do not appear in mathematics because mathematics is purely syntactic. Physics doesn't work as a purely syntactic exercise, it is something we actually use, so we have to know how to use it.

 

[atyy]

I always find discussions about interpretations to be quite interesting and insightful, but I do end up concluding that physics is not really a formal endeavor. Mathematics is formal, and physics borrows from mathematics in important and interesting ways, but physics is itself not formalizable. I think this is because we do not understand, nor ever include, the roles our minds our playing when we do physics. We know we don't include this, and we hope it doesn't matter that we don't include this, but the fact that we don't include it is an impediment to formalization in ways that do not appear in mathematics because mathematics is purely syntactic. Physics doesn't work as a purely syntactic exercise, it is something we actually use, so we have to know how to use it.

Mathematics conceived as syntax is essentially physics, since what does one mean by syntax? It requires one to know what one means by the "same symbol", which is of course a question of psychology and physics. Another way to see this is that syntax is essentially about what computers can do, which is physics.

 

[Ken G]

That's an interesting turn on the situation, but I think what you are saying is that mathematics is not formalizable either, because it requires having a mathematician to say "yes, that's correct." That part of math is never formalizable, because if the mathematician is following a program, you need another program to say "yes, that is the correct program for saying what is correct." And so on. The syntax is inside of that, that part outside the syntax doesn't count as it is simply assumed.

What I meant as the non-syntactic element was the recourse to nature. One never knows how nature will respond to a given experiment, and what theory will accommodate the new discovery is not something you can formalize in the program. It is essentially the input of creativity, or genius, and if we could formalize that, we wouldn't need to wait for the next one to come along!

 

[Ken G]

Thanks for the "like", though I must say you raise a disturbingly valid point-- could we ever program "Watson" to do physics, call it the "Einstein" program? Would it be able to suggest experiments and new theories, perhaps suggesting possible new unification schemes? Will we do science like that in a hundred years, where scientists become drones of the Einstein program, carrying out experiments that we are instructed to attempt, with no need for us to try and be creative or intuitive because the Einstein program has already prioritized all the possible directions for inquiry? Then doing physics will indeed feel like an exercise in pure syntax, a distressing possibility but I cannot say it won't come true! (Perhaps then the "genius" will be in finding the proper syntax for the Einstein code!)

The relevance to the issue of whether or not it is a didactic sin to teach "old" quantum notions like Bohr atoms and wave/particle duality is that if we turn physics over to the Einstein program, we won't need to worry about any didactic sins at all, because we won't need insight or intuition, we will only need to know how to run an experiment and check a theory handed to us by the Einstein program! So what this means is, there is close connection between pedagogical issues like what is a didactic sin, and the whole endeavor of science as a process of human insight and intuition, rather than simply a process of finding more predictive and more unifying theories that predict more observations. Somehow there is a connection between the process of advancing science, and the aesthetics of doing science in the first place. So what we regard as a didactic sin should be connected to what we regard as proper scientific aesthetics-- not that the latter is a simple topic!

 

[A. Neumaier]

That seems to be quite an original interpretation. In the usual view, the initial state of the system is the same on each run of the experiment, and one gets different outcomes because quantum mechanics only predicts probabilities, so we use a large number of runs. In your interpretation, it seems that everything is deterministic,so the random outcome on each run of the experiment is due to the initial state of the system being different on each trial?

One can repeat the experiment many times only for microscopic systems, since the assumptions underlying the statistical interpretation is that one can prepare a system independently and identically many times. It is impossible to do this for a macroscopic system, let alone for a quantum field that extends from the Earth to the sun.

Yes, in QFT everything is deterministic; God doesn't play dice since he created world according to a QFT. The randomness is in the inability to reproduce identical quantum conditions for a macroscopic system, together with the inherent chaoticity of the kinetic, hydrodynamic and elasticity equations for macroscopic matter.

For the system under discussion in the main part of this thread, it is the randomness in the photodetector that is responsible for the indeterminism.

 

[A. Neumaier]

we do not understand, nor ever include, the roles our minds our playing when we do physics

Our minds are part of the initial state of the collection of quantum fields.

 

[Ken G]

We might choose to model our minds that way, but it does not imply that our minds are that. For one thing, it has never been demonstrated that modeling our minds that way offers any advantages, but it is quite clear that the reverse arrangement, whereby we say that our minds come up with the quantum-field model, offers valuable modeling advantages (for example, advantages that usher in the issue of "didactic sins").

 

[A. Neumaier]

I meant ''our minds are ...'' in the same sense that we say "water is $H_2O$''. It is the way a physicist must consider it in order to say something physical about it.

Clearly, whatever we can observe about the mind is an observation of macroscopic matter and hence observed by means of an observation of the corresponding quantum fields. Which mental picture we form is a different matter - samalkhaiat probably cannot form a mental picture of the mind, as mind is as unobservable as the electron (we cannot see, hear, feel, smell or taste it), but we other mortals have our own mental pictures of it, which may or may not differ a lot from the scientific picture based on the physics we know.

In particular, that some part of the quantum fields that make up the universe, localized in a human head, can think about quantum fields is not more peculiar than that other parts of the same quantum fields that make up the universe, localized in a computer box, can play chess. The latter was unthinkable 100 years ago; within the 100 years to come computers will be able to do mathematics at the research level. As one can easily observe, mankind is making itself dispensable for every activity that it understands well enough, and this trend is easy to extrapolate into the future. I wouldn't be surprised if 20 years from now a computer could get a Ph.D. in mathematics at Princeton University, say. My research group is working towards making this happen; though it is difficult to predict a precise time frame.

 

[Ken G]

I wouldn't be surprised if 20 years from now a computer could get a Ph.D. in mathematics at Princeton University, say. My research group is working towards making this happen; though it is difficult to predict a precise timeframe.

Interesting-- say the "Euler" code, rather than "Watson" or "Einstein." It's an important question-- if we could create a code that can take a set of axioms in some syntactic form, and generate in some kind of order of increasing complexity all the theorems, again in syntactic form, that can be proven from those axioms, would we be satisfied by this? It speaks to the question of why we do math-- do we just want to know what theorems are logically equivalent to what axioms, or do we wish to understand something? That gets us back to the OP and what is a "didactic sin," in terms of what is a crime against understanding. I'm not sure that mathematical proofs are just our best means at arriving at the destination of theorems, or physical laws-- it seems to me how we get there is important too. (Indeed, that's what my signature statement below is about.)

In the case of physics, we might imagine some "Einstein" code that generates unifying theories and tells us how to test them by experiment. Then we carry out the experiments, which can be viewed as running a kind of "Nature" program that determines the outcome of the experiment. In such a situation, we might feel like nothing but messengers, carrying the outputs from the Einstein and Nature programs back and forth like the operator in Searle's "Chinese Box." It seems to me we would be watching the progress of science, without actually participating in it, and more importantly, without really gaining any understanding-- even if we do watch the creation of tremendous predictive power, and technological advancement. There's something about science, and perhaps mathematics too, that is different from that.

 

[atyy]

One can repeat the experiment many times only for microscopic systems, since the assumptions underlying the statistical interpretation is that one can prepare a system independently and identically many times. It is impossible to do this for a macroscopic system, let alone for a quantum field that extends from the Earth to the sun.

Yes, in QFT everything is deterministic; God doesn't play dice since he world was created according to a QFT. The randomness is in the inability to reproduce identical quantum conditions for a macroscopic system, together with the inherent chaoticity of the kinetic, hydrodynamic and elasticity equations for macroscopic matter.

For the system under discussion in the main pat of this thread, it is the randomness in the photodetector that is responsible for the indeterminism.

Hmmm, I'm skeptical just because it seems so non-standard. Are there any references where I could read the details?

The other reason I'm skeptical is that it seems that QFT can in principle solve the measurement problem (remove the observer that the usual Copenhagen-type interpretation needs). However, non-relativistic QM can also be formulated in the second quantized language, so presumable non-relativistic QFT would also have a deterministic interpretation consistent with observable non-relativistic physics?

 

[A. Neumaier]

The opposite of didactical sin is didactical virtue - the ability to impart understanding, ultimately to the point that those taught can convince themselves of the truth of a claim by someone else. This means building upon the understanding that is already there and adding structure that helps to properly think about the topic to be taught.

The controversy in this thread is about what ''proper thinking" about quantum mechanics entails. I found that I had to unlearn quite a lot to reach my present understanding; a better start than what the textbooks tell could have saved me a lot of work. On the other hand, one has to be careful what to throw away. As samalkhaiat mentioned, Bohr-Sommerfeld quantization is still useful today. Indeed, in its modern generalization it gives the correct result whenever a system is completely integrable (and a good first approximation when it is nearly so); this is the reason why it worked so well for the hydrogen atom (which is completely integrable in several of its incarnations). But one should throw out the idea that Bohr-Sommerfeld quantization works because of a planetary model in miniature. Thus when telling the history one should immediately add that Bohr obtained a correct result (fortunately for the early QM) although his model is in most aspects unacceptable by modern standards.

 

[A. Neumaier]

Are there any references where I could read the details?

I had already given a link in my answer; following it you'll enter a new world view. Nothing is published, though - it saves me a lot of time not to prepare every insight for publication. I am collecting the material for a book. A preliminary version of my book is here - Chapters 8-10 make the case for my interpretation (though to be more elementary I avoid there to talk about quantum fields). According to my publishing contract, the final version of the book should be published in about two years from now.

non-relativistic QFT would also have a deterministic interpretation consistent with observable non-relativistic physics

Yes, it has; nothing in my arguments depends on relativity - it doesn't even depend on fields; just on being macroscopic. Neglecting most of the particles to get a tiny quantum system is the source of the randomness when observing a tiny system; as the system gets bigger, the noise mostly cancels out if you look only at the macroscopic variables. These macroscopic variables happen to be fields - but my book only treats the equilibrium case where the fields have constant values.

One has this intrinsic source of randomness in every chaotic deterministic dynamics (even in small ones such as the Lorenz system): The tiniest approximation (and neglecting something always forces an approximation) is immensely magnified and changes the results after a short time to an extent that only statistical information remains reliably predictable. This is the reason both for randomness in quantum mechanics and for the success of statistical mechanics.

 

[Ken G]

I had already given a link in my answer; following it you'll enter a new world view. Nothing is published, though - it saves me a lot of time not to prepare every insight for publication. I am collecting the material for a book. A preliminary version of my book is here - Chapters 8-10 make the case for my interpretation (though to be more elementary I avoid there to talk about quantum fields).

It is certainly a splendid accomplishment!

 

[Spinnor]

I had already given a link in my answer; following it you'll enter a new world view. Nothing is published, though - it saves me a lot of time not to prepare every insight for publication. I am collecting the material for a book. A preliminary version of my book is here - Chapters 8-10 make the case for my interpretation (though to be more elementary I avoid there to talk about quantum fields). According to my publishing contract, the final version of the book should be published in about two years from now.

I look forward to going through your book, thanks! From the preface of your book,

"The book originated as course notes from a course given by the first author in fall 2007, ..."

Do you by chance have video or audio recordings of your lectures that you would share?

 

[A. Neumaier]

Do you by chance have video or audio recordings of your lectures that you would share?

I have no recordings; sorry. But (since discussion of unpublished research is discouragaged here on PF) you are welcome to ask questions regarding the content here, if they are significant, while comments on typos, suggestions for improvement, etc. are best sent to me by email (collecting them for a while before sending them).

 

[TrickyDicky]

The controversy in this thread is about what ''proper thinking" about quantum mechanics entails. I found that I had to unlearn quite a lot to reach my present understanding; a better start than what the textbooks tell could have saved me a lot of work. On the other hand, one has to be careful what to throw away.

Yes, stumbling upon the right starting point is essential..

 

[Spinnor]

I have no recordings; sorry. But (since discussion of unpublished research is discouragaged here on PF) you are welcome to ask questions regarding the content here, if they are significant, while comments on typos, suggestions for improvement, etc. are best sent to me by email (collecting them for a while before sending them).

They would have been good lectures I'm sure. Have you given any talks that were recorded on this material, maybe time for one?

Thanks!

 

[strangerep]

I look forward to going through [Arnold's] book,

Good luck. I have been through many drafts of it, (mostly learning less well known ways of applying math to physics along the way). Although I have learned many things in the process I must admit that I still fail to grok Arnold's interpretation of QM. One stumbling block is that Arnold's book does not discuss Bell's theorem nor its cousins, so all the standard objections about hidden variables flood into my mind when I hear an interpretation that sounds deterministic. Thus I retreat away from philosophy to the comparative safety of minimal SUAC.

 

[Spinnor]

I still fail to grok Arnold's interpretation of QM.

As you are no dummy I'm sure I will have even more trouble. Maybe Mr. Neumaier has something that is already written up that outlines his research, on the level of Scientific American?

 

[vanhees71]

I'd rather call it SUACM=Shut up, calculate, and measure! That, closed to a circle, is physics ;-).

 

[samalkhaiat]

... Which mental picture we form is a different matter - samalkhaiat probably cannot form a mental picture of the mind, as mind is as unobservable as the electron (we cannot see, hear, feel, smell or taste it), but we other mortals have our own mental pictures of it, which may or may not differ a lot from the scientific picture based on the physics we know.

You are entitled to your opinions. Samalkhaiat, like almost everybody else, distinguishes between mental picturing from mental construction. Mathematics and mathematical models are abstract mental constructions which we (i.e. our brains) can not provide spatial or/and temporal pictures of them. Good for you, if you can SEE mathematics or SEE the mathematical representation of the electron.

 

[Ken G]

I'd rather call it SUACM=Shut up, calculate, and measure! That, closed to a circle, is physics ;-).

Ah, but I never knew a physicist who really did that. It sounds too much like the "messenger" I alluded to above-- imagine there really was an "Einstein" program that took all the available data and used it to test a search protocol of various theories, ordered by complexity. The program throws out theories that fail, and adjusts the parameters of theories that succeed, and then outputs new experimental tests that are needed to push the theories into new domains. Then you the physicist set up the experiments that the Einstein program suggests, and report the outcomes to the program, which further culls its theories and suggests new tests. Progress in physics rapidly accelerates, as the program is capable of searching a vast space of possibilities very quickly.

Then you decide to further increase efficiency by creating a "Faraday" program that takes the Einstein outputs directly and assembles robotic experiments per the Einstein requirements, and feeds the outcomes right back into the Einstein code. You the physicist just sit back and watch the outcome, which is a set of theoretical equations and models ordered in regard to complexity and accuracy. After awhile you find the equations and models have become too difficult for you to understand what they are saying, so you create a "Scientific American" program to create pedagogical explanations of the Einstein outputs, in some sense "dumbed down" to translate it from the syntactic machine language to a semantic human language, but without the deeper understanding necessary to come up with the theory in the first place because it is actually derived in a different language. You sit back with great pride in your accoplishment-- a fully automated SUACAM system!

But then you realize you have exactly the same relationship to that system as non-scientists have with our current system. You have turned yourself into a non-physicist, in the name of doing SUACAM as efficiently as possible. So there has to be something more than SUACAM in physics!

 

[Jimster41]

I think I know the answer but why is the bound electron quantized as a matter of course?

Is the point of the article that the discrete scattering probability of the quantized bound electron when bombarded with light (EM radiation) can be explained without also a-priori quantization of that radiation into "photons"?

 

[vanhees71]

Yes, that's the point of the article. Einstein's famous formula is entirely derived from a model, where light is described by a classical em. wave, not by the quantum field. At Einstein's time the only observable fact that makes the quantized field necessary is the Planck radiation law, contradicting the classical equipartition theorem, leading to the UV catastrophe of the older theories of black-body radition.

The electron is "quantized", because a bound state belongs, by definition, to the descrete spectrum of the Hamilton operator. E.g., you can take a hydrogen atom as a simple but very important example, which can be solved exactly (neglecting radiation corrections, for which you also need the quantization of the em. field leading to the Lamb shift, which can be calculated very accurately using perturbation theory).

 

[vanhees71]

But then you realize you have exactly the same relationship to that system as non-scientists have with our current system. You have turned yourself into a non-physicist, in the name of doing SUACAM as efficiently as possible. So there has to be something more than SUACAM in physics!

I don't understand, what you mean. It's the very foundation of the scientific method to have a model (or even a theory) of (or a certain part of) nature, leading to quantitative predictions for the outcome of experiments. Then you plan your experiment to check whether the prediction is right. Either it is, and you haven't learned anything new or there is a discrepancy, and you have to refine your model, leading to new predictions and new experiments to check them. Science is a process, and I'm not sure, whether this will ever stop culminating in a final "theory of everything".