Some sins in physics didactics - comments

[A. Neumaier]

What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.

See the new thread https://www.physicsforums.com/threads/818386/

 

[samalkhaiat]

I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space $\mathcal{H}$. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit $\lim_{\hbar \to 0} \mathcal{H}$, even though the classical limit of the operators algebra in $\mathcal{H}$ is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation $i \partial_{t} | \psi \rangle = H | \psi \rangle$. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function $\langle x | \psi \rangle$) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce $\mathcal{H}$, subsets of $\mathcal{H}$ representing pure states, the algebra of bounded operators $\mathcal{B}(\mathcal{H})$, etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator $| \psi \rangle \langle \psi |$), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

 

[A. Neumaier]

... the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

We (the blessed creatures) see huge quantum systems such as the sun, whose light and heat is produced by quantum processes, as well as small quantum systems such as single photons. We smell and taste molecules described by small but complex quantum systems, and we touch solids and liquids, large quantum systems described by elasticity equations and fluid dynamics, whose characteristics are computed from quantum statistical mechanics.

All these are covered by the axioms of quantum mechanics. My favorite set of axioms is described here.

 

[atyy]

I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space $\mathcal{H}$. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit $\lim_{\hbar \to 0} \mathcal{H}$, even though the classical limit of the operators algebra in $\mathcal{H}$ is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation $i \partial_{t} | \psi \rangle = H | \psi \rangle$. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function $\langle x | \psi \rangle$) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce $\mathcal{H}$, subsets of $\mathcal{H}$ representing pure states, the algebra of bounded operators $\mathcal{B}(\mathcal{H})$, etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator $| \psi \rangle \langle \psi |$), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.

 

[samalkhaiat]

We (the blessed creatures) see huge quantum systems such as the sun

Yes, thanks for its hugeness. So, why can't we explain the sun orbital motion using QM?

as well as small quantum systems such as single photons.

Did we? What does a single photon look-like? Is it rounded like football?
A glass full of liquid Helium is very much a quantum system, but to our senses it is no more that a glass full of very cold liquid .
Dear sir, my post contains no inaccurate or confusing statement, and by the piece you quoted I meant the following: We evolved to sense the macroscopic (classical) world and invented language to describe what we see, hear, feel, smell and taste. We are unfortunate because we cannot form a mental picture for the electron but, thanks to mathematics, we can live with that misfortune.

 

[samalkhaiat]

My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.

I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.

 

[atyy]

I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.

The first two are correct. What was wrong with my remark out the commutation relations?

 

[samalkhaiat]

What is it "correct" about "position is particle and momentum is wave"? What "particle nature" does an abstract complex vector space can possibly have? The statement "wave-particle duality is formalised by commutation relations" can not be proven. The statement that I made about the commutation relations can be proven

 

[atyy]

What is it "correct" about "position is particle and momentum is wave"? What "particle nature" does an abstract complex vector space can possibly have? The statement "wave-particle duality is formalised by commutation relations" can not be proven. The statement that I made about the commutation relations can be proven

Yes, the language is not standard, but I hope to convince you it can be correct. The idea is that "wave-particle duality" which is a vague heuristic in old quantum theory is still worth teaching, because there are several things in the proper theory which can be seen as formalizations of the heuristic.

So by "position is particle and momentum is wave", I just mean that in the position basis, the position eigenfunction is localized like a particle, while the momentum eigenfunction is a sinusoidal wave. Since this is captured by the commutation relation between the position and momentum operators, this is one way in which wave-particle duality is formalized.

Another formalization is that in non-relativistic quantum mechanics, the Hilbert space is constructed by thinking about discrete entities called particles. For example, the Hilbert space for two particles is constructed as the tensor product of the one particle spaces. Or in quantum field theory in the second quantized language, the Fock space is again constructed by thinking about discrete entities that are called particles. Then the notion of wave enters in that the Schroedinger equation in non-relativistic quantum mechanics, or the equation of motion for the operators in the Heisenberg picture of quantum field theory in the second quantized language is a wave equation. So we have both particle and wave aspects in the construction of the theory. The important point is that these are not classical particles, but quantum particles which do not have trajectories except in appropriate limits.

 

[atyy]

So, why can't we explain the sun orbital motion using QM?

Although I have never seen a calculation done, I believe in principle we can explain the sun's orbital motion using QM. The main difficulty is that QM is a statistical theory, so ideally we would like to have multiple independent preparations so that frequentist reasoning becomes easy. However, we have only one sun on which we can make sequential observations. In this case, what one would like is that the probability for the observed trajectory is sharply peaked around the classical trajectory. So what one does is measure some observable that corresponds to a rough estimate of position, which collapses the wave function, and then one makes another measurement that corresponds to a rough estimate of position. We do this repeatedly, and we should get a probability distribution over observed trajectories. That distribution should be sharply peaked around to observed orbit of the sun.

 

[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 ;-).