I Modern View of Quantum Phenomena

[bhobba]

This is not, strictly speaking, a discussion of interpretations per se.

We often see discussions based on QM as it was understood during the early days and the famous Einstein-Bohr debates. The problem with this is that things in QM have advanced tremendously since then, and the 'weirdness' that puzzles those attempting to understand QM has changed.

I recently came across a synopsis of these advances, allowing those interested in interpretational issues to understand the modern view.

https://rreece.github.io/talks/pdf/2017-09-24-RReece-Fields-before-particles.pdf

It is advanced, but I tagged it as an I-level. Beginners may not understand the details, but they should get a sense of the modern perspective.

It also provides background to my current view, based on Wienberg's Folk Theorem, that QM is the EFT that any theory will look like at large enough distances. It is very mathematically sophisticated, but after years of thinking about interpretations, I have come to believe there is no 'simple' way to understand QM. It is, by its very nature, very advanced mathematically.

The real 'mystery' is why QM is based on operators, and complex space ones at that. I have posted a heuristic view of why, but it is just that a heuristic. That said - is it a mystery? What was it Newton said - hypotheses non fingo (Latin for "I frame no hypotheses", or "I contrive no hypotheses"). Is QM any different?

Thanks
Bill

 

[Peter Morgan]

The real 'mystery' is why QM is based on operators, and complex space ones at that.

One answer to the complex structure question comes from an article by Leon Cohen in Foundations of Physics 1988, "Rules of Probability in Quantum Mechanics" (paywalled), which is available on the author's Academia page. The argument in brief is that we use both probability measures and their Fourier transforms, characteristic functions. We could work with the sine and cosine components of the Fourier transform explicitly, but —as in signal analysis, electrical engineering, et cetera— it's mathematically much more convenient to introduce a complex structure.
Operators give us a very natural way to encode a random variable's sample space and σ-algebra as the eigenvalues of an operator and as a Projection-Valued Measure. Indeed, we can introduce a commutative operator algebra model of the Kolmogorov axioms. A noncommutative operator algebra can be thought of as a classically natural extension that encodes multiple experimental contexts in a single formalism. I refined the description I offer for that in a colloquium I gave by Zoom for NSU Dhaka on May 18th, "A Dataset & Signal Analysis Interpretation of Quantum Mechanics" (EDIT: I attach a PDF of the slides for that, because why not.)

 

[Peter Morgan]

after years of thinking about interpretations, I have come to believe there is no 'simple' way to understand QM. It is, by its very nature, very advanced mathematically.

Although that's definitely true when we consider details, I think signal analysis offers a starting point for a more accessible understanding. In particular, signal analysis includes Fourier transforms of signals over time, for which Wigner functions (called time-frequency analysis) are so classically natural that there are many YouTube videos explaining the HUP in terms of that idea. Signal analysis is also closer to field theory because it uses functions of time, in a way that is very different from classical states on phase space. I discuss that aspect of signal analysis in the introduction of Annals of Physics 2020, "An algebraic approach to Koopman classical mechanics" (that's an arXiv link, DOI there).
I'm told by a first year Yale undergraduate who came to my talk at Yale on May 1st and who then dived into other videos on my YouTube channel that the Dhaka NSU colloquium I link to in my previous comment is the most accessible of what she has seen. One aspect that I emphasize more in that talk is that I break away from an ensemble interpretation, in which an operator corresponds to a particle property; instead, I take an operator's set of eigenvalues to correspond to the set of data values contained in a Dataset. Since axioms of QM typically begin with something like "a Hilbert space corresponds to a system" and I do not mention systems or their properties whatsoever, this is a huge change. Some philosophers dislike such an instrumental way to ground QM more firmly in experiment, but that grounding allows a more realist interpretation of the structures we can develop.
I'll stop there. If you're interested, you'll say so, otherwise my apologies for landing so much on PF this morning.

ps: I enjoyed the PDF you posted enough that I wrote to the author through Academia. Thank you for it.

 

[A. Neumaier]

The real 'mystery' is why QM is based on operators, and complex space ones at that.

This a mystery only for those who think the discrete quantum information approach carries the essence of quantum mechanics, and forgot the origins of modern quantum mechanics 100 years ago:

Heisenberg discovered in 1925 that the observed Rydberg-Ritz combination principle is encoded in the canonical commutation rule [p,q]=i\hbar, which necessitates both complex numbers and operators.

 

[robphy]

Possibly interesting:

• On the relation between classical and quantum observables
  Abhay Ashtekar
  Comm. Math. Phys. 71(1): 59-64 (1980).
  https://projecteuclid.org/journals/...l-and-quantum-observables/cmp/1103907394.full
  
  
  
• Quantum Mechanics as Quantum Measure Theory
  Rafael D. Sorkin
  Mod.Phys.Lett. A9 (1994) 3119-3128
  https://arxiv.org/abs/gr-qc/9401003
  https://doi.org/10.1142/S021773239400294X

 

[Lord Jestocost]

The real 'mystery' is why QM is based on operators, and complex space ones at that. I have posted a heuristic view of why, but it is just that a heuristic. That said - is it a mystery? What was it Newton said - hypotheses non fingo (Latin for "I frame no hypotheses", or "I contrive no hypotheses"). Is QM any different?

One reads from the “THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY” by Sir Isaac Newton (translated into English by Andrew Motte):

“But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses; for whatever is not deduced from the phænomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phænomena, and afterwards rendered general by induction.”

As Aage Bohr, Ben R. Mottelson and Ole Ulfbeck put it in "The Principle Underlying Quantum Mechanics":
"In fact, the quantum mechanical formalism was discovered by ingenious guesswork which was given an interpretation in terms of probabilities for the results of measurements." The orthodox version of quantum mechanics frames no hypotheses as it is a calculational recipe, designed in the last resort to predict the probabilities of various directly observed macroscopic outcomes - phenomena.

 

[Peter Morgan]

Possibly interesting:

• On the relation between classical and quantum observables
  Abhay Ashtekar, Comm. Math. Phys. 71(1): 59-64 (1980).
• Quantum Mechanics as Quantum Measure Theory
  Rafael D. Sorkin, Mod.Phys.Lett. A9 (1994) 3119-3128

Thank you for pointing out those two articles, neither of which I had seen before.
Ashtekar's article inches towards something like the ideas I develop in the article I mentioned above, "An algebraic approach to Koopman classical mechanics", but I think it is not ultimately much different from geometric quantization (he hints at that when he says in his introduction, "these questions are rather elementary, and the answers probably well known among experts in geometrical quantum mechanics"). In contrast, we can instead find isomorphisms between Classical Mechanics and Quantum Mechanics, if we are willing to abandon the Correspondence Principle, which almost nobody was ready for in 1980. Koopman's Hilbert space formalism for CM has been developed enough, however, since Sudarshan pointed out that it is a useful tool for discussing chaotic dynamical systems in Pramana 1976, that the ground is now rather differently prepared (some of the practical aspect of that development can be found in an article in SIAM Review 2022, "Modern Koopman Theory for Dynamical Systems").
A second consideration is that Ashtekar's introduction of Planck's constant is —as is generally found in the literature— not classically motivated. As I point out already in my Annals of Physics 2020, and more clearly on slides 7-8 in the NSU Dhaka talk, if we adopt a more classical starting point, I think we have to ask how quantum noise and thermal noise are different. Quantum thermodynamics gives a clear answer that can be adopted naturally into classical physics: the quantum vacuum (where quantum fluctuations live, so to speak) is Lorentz invariant, whereas thermal fluctuations are not invariant under boost transformations. Planck's constant becomes the amplitude of Lorentz invariant quantum fluctuations, very comparably to kT being the amplitude of thermal fluctuations.
I won't rehearse here more of the discussion in my Annals of Physics 2020 and in my NSU Dhaka talk.

 

[RUTA]

This is not, strictly speaking, a discussion of interpretations per se.

We often see discussions based on QM as it was understood during the early days and the famous Einstein-Bohr debates. The problem with this is that things in QM have advanced tremendously since then, and the 'weirdness' that puzzles those attempting to understand QM has changed.

I recently came across a synopsis of these advances, allowing those interested in interpretational issues to understand the modern view.

https://rreece.github.io/talks/pdf/2017-09-24-RReece-Fields-before-particles.pdf

It is advanced, but I tagged it as an I-level. Beginners may not understand the details, but they should get a sense of the modern perspective.

It also provides background to my current view, based on Wienberg's Folk Theorem, that QM is the EFT that any theory will look like at large enough distances. It is very mathematically sophisticated, but after years of thinking about interpretations, I have come to believe there is no 'simple' way to understand QM. It is, by its very nature, very advanced mathematically.

The real 'mystery' is why QM is based on operators, and complex space ones at that. I have posted a heuristic view of why, but it is just that a heuristic. That said - is it a mystery? What was it Newton said - hypotheses non fingo (Latin for "I frame no hypotheses", or "I contrive no hypotheses"). Is QM any different?

Thanks
Bill

I read all 51 slides in that presentation, very comprehensive. However, it is missing the axiomatic reconstruction of QM via information-theoretic principles completed in 2011. I think that's the newest view of QM.

 

[Peter Morgan]

I read all 51 slides in that presentation, very comprehensive. However, it is missing the axiomatic reconstruction of QM via information-theoretic principles completed in 2011. I think that's the newest view of QM.

VERY good to see you here, @RUTA! Though it is a notable new approach, I think the information-theoretic approaches introduce axioms that I don't find obvious enough for them to be axioms.
It's not yet ready for prime-time, but please consider thinking of "A Dataset & Signal Analysis Interpretation of Quantum Field Theory" as perhaps the notable 2026 newcomer on the block. We can pull back from the abstraction of information as a basis for the axioms by taking actually recorded datasets (in computer memory or in a lab notebook) as a grounding for axioms that do not mention particles or systems or their properties, leaving those to be derived when they can be (or not to be derived when they are, as so often, problematic).
[I like to include "Signal Analysis" as a familiar hook in my titles, but it's partly a bait and switch insofar as we notice after a while that we have to work with stochastic processes as a signal analysis made more elaborate by introducing probability theory and, one step further, by introducing generalized probability theory. If I said "stochastic processes", there would still be that further step to generalized probability theory.]

 

[A. Neumaier]

the axiomatic reconstruction of QM via information-theoretic principles completed in 2011. I think that's the newest view of QM.

It does not even reconstruct the canonical commutation rules for a harmonic oscillator, hence is far from a useful reconstruction.

 

[RUTA]

It took me a lot of study and application before I realized what the quantum reconstruction had accomplished. On the surface it doesn't look like much, but it's all in there. Let me give you an executive summary. It took an entire book (Einstein's Entanglement) to explain why this is true, so don't feel bad if you don't see it from this simple summary. Here is an open access paper that serves as a 17-page summary. Or you can watch this 11-min YouTube video.

Special relativity is introduced as follows in:
Physics for Scientists and Engineers with Modern Physics 5e
R. Knight
Pearson, San Francisco
p. 1057 (2022)

Quantum mechanics could be introduced in analogous fashion:

 

[A. Neumaier]

It took me a lot of study and application before I realized what the quantum reconstruction had accomplished. On the surface it doesn't look like much, but it's all in there. Let me give you an executive summary. It took an entire book (Einstein's Entanglement) to explain why this is true, so don't feel bad if you don't see it from this simple summary. Here is an open access paper that serves as a 17-page summary. Or you can watch this 11-min YouTube video.

Special relativity is introduced as follows in:
Physics for Scientists and Engineers with Modern Physics 5e
R. Knight
Pearson, San Francisco
p. 1057 (2022)
View attachment 364606

Quantum mechanics could be introduced in analogous fashion:

View attachment 364608

The Schrödinger equation is nonrelativistic, hence violates your principle!

 

[RUTA]

The Schrödinger equation is nonrelativistic, hence violates your principle!

You're confusing the relativity principle with the theory of relativity. Newtonian mechanics obeys the relativity principle and it's certainly not relativistic. Newton made a big deal out of that, even repeating Galileo's argument.

 

[PeterDonis]

Newtonian mechanics obeys the relativity principle

It obeys a relativity principle, but it's the wrong one, as we now know by experiment. It obeys the Galilean relativity principle, but we now know by experiment that the Lorentzian relativity principle is the correct one.

 

[RUTA]

It obeys a relativity principle, but it's the wrong one, as we now know by experiment. It obeys the Galilean relativity principle, but we now know by experiment that the Lorentzian relativity principle is the correct one.

As appears in Serway & Jewett: Einstein's relativity principle, "The laws of physics must be the same in all inertial reference frames" is generalized from Galileo's relativity principle, "The laws of mechanics must be the same in all inertial frames of reference." Since "the laws of mechanics" is a subset of the "laws of physics," anything that obeys Galileo's version also obeys Einstein's version.

 

[pines-demon]

Since "the laws of mechanics" is a subset of the "laws of physics," anything that obeys Galileo's version also obeys Einstein's version.

I guess I can try to understand what you mean here. But reading it for the first time, it looks like a complete fallacy. What you can say is that in their domain of validity, the laws of non-relativistic mechanics follow Galilean relativity, but this is just an approximation.

 

[gentzen]

It took me a lot of study and application before I realized what the quantum reconstruction had accomplished. On the surface it doesn't look like much, but it's all in there.

Do you believe that you can communicate your "realization" to me (or PeterDonis, or PeroK, or ...)?

Let me give you an executive summary. It took an entire book (Einstein's Entanglement) to explain why this is true, so don't feel bad if you don't see it from this simple summary.

Do you believe that I would get your "realization", if I would read your entire book?

Here is an open access paper that serves as a 17-page summary. Or you can watch this 11-min YouTube video.

I guess there is a decent chance that after reading your 17-page paper, I would once again have trouble to see what has changed compared to your previous papers? Or do you disagree?

And I already learned that your 11-min video ends with referring me to part 2 and 3, and part 2 ends with referring me to part 3, but part 3 doesn't exist yet. And independent, I guess there is still a decent chance that I won't get what should be different this time, compared to your older videos or papers.

OK, I did notice one thing which I really like: You mention Bob Coecke and his latest book, in part 1. For him, I really agree that reading his older longer book is currently the only way (for me) to really learn and understand what he found. The book has a summary at the end of each chapter, but already those summaries no longer did the trick for me, even so they really do summarize the important parts of the content.

 

[PeterDonis]

anything that obeys Galileo's version also obeys Einstein's version

Only if you ignore all the actual experimental results.

 

[RUTA]

Only if you ignore all the actual experimental results.

I think you're confusing the relativity principle with the light postulate. The relativity principle holds for G4 and M4, but the light postulate only holds in M4.

 

[robphy]

Conservation of Galilean-momentum holds in Galilean relativity.
Conservation of relativistic-momentum holds in Special relativity.

Conservation of Galilean-momentum does not hold in Special relativity.

Conservation of Galilean-momentum is only a small relative-velocity approximation of Conservation of relativistic-momentum.

 

[RUTA]

I guess I can try to understand what you mean here. But reading it for the first time, it looks like a complete fallacy. What you can say is that in their domain of validity, the laws of non-relativistic mechanics follow Galilean relativity, but this is just an approximation.

Again, don't confuse the relativity principle with the theory of relativity.

 

[RUTA]

Conservation of Galilean-momentum holds in Galilean relativity.
Conservation of relativistic-momentum holds in Special relativity.

Conservation of Galilean-momentum does not hold in Special relativity.

Conservation of Galilean-momentum is only a small relative-velocity approximation of Conservation of relativistic-momentum.

Again, don't conflate the relativity principle with the theory of relativity.

 

[RUTA]

Do you believe that you can communicate your "realization" to me (or PeterDonis, or PeroK, or ...)?

Do you believe that I would get your "realization", if I would read your entire book?

I guess there is a decent chance that after reading your 17-page paper, I would once again have trouble to see what has changed compared to your previous papers? Or do you disagree?

And I already learned that your 11-min video ends with referring me to part 2 and 3, and part 2 ends with referring me to part 3, but part 3 doesn't exist yet. And independent, I guess there is still a decent chance that I won't get what should be different this time, compared to your older videos or papers.

OK, I did notice one thing which I really like: You mention Bob Coecke and his latest book, in part 1. For him, I really agree that reading his older longer book is currently the only way (for me) to really learn and understand what he found. The book has a summary at the end of each chapter, but already those summaries no longer did the trick for me, even so they really do summarize the important parts of the content.

Everyone requires different experiences to "understand" something; the term means different things to different people. Physicists typically play with the math to see how it maps to physical phenomena and once they see that they feel like they "understand" the physics. Given that superposition, complementarity and entanglement are so widely applied today in physics, most physicists feel like they understand quantum mechanics, so they don't care about the axiomatic reconstruction of QM via information-theoretic principles. Those of us in quantum foundations want a bit more (see Becker's book "What is Real?" for example), so we struggle with the interpretations program. The quantum reconstruction program offers us a principle alternative to the quagmire that is the interpretations program.

So, I'm not sure what of the three options I posted will suffice for you. I'm sorry about not having episodes 3-5 posted on YouTube yet. They're ready to be recorded, but I broke my back three weeks ago and I can't stand or sit upright for more than 30 min, plus I can't reach or lift anything, so it might be another week or two before I can get them recorded and posted. Try the paper and if that doesn't work, go to the book. If you read the book, check the Preface to see which chapters you should skip based on your interest.

 

[PeterDonis]

I think you're confusing the relativity principle with the light postulate.

No, I'm not. I'm pointing out that "the relativity principle", as far as physics is concerned, is not one thing. Galilean invariance is not the same thing as Lorentz invariance.

 

[RUTA]

No, I'm not. I'm pointing out that "the relativity principle", as far as physics is concerned, is not one thing. Galilean invariance is not the same thing as Lorentz invariance.

Galilean transformations between inertial reference frames in G4 are time translations, spatial translations, spatial rotations, and Galilean boosts. Poincare transformations between inertial reference frames in M4 are time translations, spatial translations, spatial rotations, and Lorentz boosts. The only difference is in their boosts and that has nothing to do with the relativity principle, that difference is due to the light postulate.

Note that it is the observer-independence of h under spatial rotations (e.g., spin-1/2 and photon polarization qubits) and spatial translations (e.g., double-slit and Mach–Zehnder qubits) that leads to the Hilbert space kinematics for QM.

 

[PeterDonis]

The only difference is in their boosts

But the Lorentz boosts cannot be invariantly separated from the spatial rotations. So it's the entire Lorentz group that's different.

that has nothing to do with the relativity principle, that difference is due to the light postulate.

Sorry, but I disagree. You can't separate out the two things that way. They're inseparably linked.

Note that it is the observer-independence of h under spatial rotations (e.g., spin-1/2 and photon polarization qubits) and spatial translations (e.g., double-slit and Mach–Zehnder qubits) that leads to the Hilbert space kinematics for QM.

And since, as above, spatial rotations can't be invariantly separated from Lorentz boosts, this works differently for the two relativity principles. The only way to limit things as you state them here is to go to the non-relativistic approximation, which is useless if you're talking about foundations.

 

[pines-demon]

Galilean transformations between inertial reference frames in G4 are time translations, spatial translations, spatial rotations, and Galilean boosts. Poincare transformations between inertial reference frames in M4 are time translations, spatial translations, spatial rotations, and Lorentz boosts. The only difference is in their boosts and that has nothing to do with the relativity principle, that difference is due to the light postulate.

Note that it is the observer-independence of h under spatial rotations (e.g., spin-1/2 and photon polarization qubits) and spatial translations (e.g., double-slit and Mach–Zehnder qubits) that leads to the Hilbert space kinematics for QM.

You seems to want to say that something can follow the first postulate of relativity independent if it is a relativistic or a non-relativistic equation. Fine, but no need to claim that what follows Galilean transformation follows Lorentzian ones.

 

[RUTA]

But the Lorentz boosts cannot be invariantly separated from the spatial rotations. So it's the entire Lorentz group that's different.

Lorentz boosts stand alone as a transformation between inertial reference frames in M4. You're conflating group structure with functionality. Spacetime translations are a group. SO(3) is a group. Lorentz boosts with SO(3) form the restricted Lorentz group. These are the subgroups of the Poincare group.

Sorry, but I disagree. You can't separate out the two things that way. They're inseparably linked.

The relativity principle says: The laws of physics are the same in all inertial reference frames. It stands as stated regardless of what transformations you use between inertial reference frames. If that wasn't true, then the relativity principle and light postulate wouldn't both be needed for special relativity. As John Norton wrote, "Maxwell’s theory entails the constancy of the speed of light and that constancy, along with the principle of relativity, entails the relativity of simultaneity." It is the relativity of simultaneity that differentiates G4 and M4.

And since, as above, spatial rotations can't be invariantly separated from Lorentz boosts, this works differently for the two relativity principles. The only way to limit things as you state them here is to go to the non-relativistic approximation, which is useless if you're talking about foundations.

Lorentz boosts need SO(3) to close as a group, but the converse is not true. SO(3) is a subgroup of both the Galilean and Poincare transformations between inertial reference frames. And, again, the relativity principle doesn't say anything about transformations between inertial reference frames. To get those transformations, you need to add an additional postulate.

 

[RUTA]

You seems to want to say that something can follow the first postulate of relativity independent if it is a relativistic or a non-relativistic equation.

That is exactly correct. Here are two more quotes from Norton:

the principle of relativity tells us that we recover a full description of a moving asteroid with its satellite by merely taking the easy case of the asteroid at rest and setting it into uniform motion by means of a Galilean transformation.

While not present by name, the principle of relativity has always been an essential part of Newtonian physics. According to Copernican cosmology, the earth spins on its axis and orbits the sun. Somehow Newtonian physics must answer the ancient objection that such motions should be revealed in ordinary experience if theyare real. Yet, absent astronomical observations, there is no evidence of this motion. All processes on earth proceed just as if the earth were at rest. That lack of evidence, theNewtonian answers, is just what is expected. The earth’s motions are inertial to very good approximation; the curvature of the trajectory of a spot on the earth’s surface issmall, requiring 12 hours to reverse its direction. So, by the conformity of Newtonian mechanics to the principle of relativity, we know that all mechanical processes on the moving earth will proceed just as if the earth were at rest. The principle of relativity is a commonplace of modern life as well. All processes within an airplane cabin, cruising rapidly but inertially, proceed exactly as they would at the hangar. We do not need to adjust our technique in pouring coffee for the speed of the airplane. The coffee is not left behind by the plane’s motion when it is poured from the pot.

 

[PeterDonis]

Lorentz boosts stand alone as a transformation between inertial reference frames in M4

No, they don't. They're not a group in 3+1 spacetime; they're not closed under composition. Only in 1+1 spacetime do Lorentz boosts by themselves form a group.

The relativity principle says

I understand that your version of it says that. I just don't agree with your version. I don't think you can just ignore the fundamental difference between Galilean invariance and Lorentz invariance, or say that it's not part of the relativity principle.

Lorentz boosts need SO(3) to close as a group

Which means your claim that I quoted at the start of this post is false, as I said.

the converse is not true.

Spatial rotations about a specific point in a specific inertial frame are a group, yes. But you have to pick a frame--or, equivalently, you have to pick a particular spacelike hypersurface of constant time for the rotations to operate in. If you change frames in Minkowski spacetime, you change which set of transformations are "spatial rotations", because you change which spacelike hypersurfaces are surfaces of constant time. So I stand by my statement that in Minkowski spacetime you cannot invariantly separate spatial rotations and boosts.