## The Quantum-Classical Debate:

Today I will reply to the arguments Andrei made last week in his guest post. But let me first start by giving out the answer to the questions from two posts ago. First, on why there is no uncertainty relationship for spin, from any two non-commuting observables A and B one obtains:

$$\sigma_A \sigma_B \geq \frac{1}{2} |\langle [A,B] \rangle|$$

The key point of Heisenberg uncertainty relationship for position and momenta is to be pedantic and observe that the commutator is proportional with the identity operator $$I$$:

$$[x,p] = i \hbar I$$

which is not true in the case of spin. Because of this, in the spin case there exists states for which the right hand side value $$|\langle [A,B] \rangle|$$ is zero. The position-momenta uncertainty remains bounded from below for any state.

For the other question on the apparent violation of the uncertainty principle. here is what Heisenberg stated:

"If the velocity of the electron is at first known, and the position then exactly measured, the position of the electron for times previous to the position measurement may be calculated. For these past times, δpδq is smaller than the usual bound." and "the uncertainty relation does not hold for the past." I think this is not a well known or appreciated fact by the majority of physics community.

Then Heisenberg pointed out that these values can never be used as initial conditions in a prediction about the future behavior of the electron.

Now back to answering Andrei's challenge to quantum mechanics, Andrei discussed 3 points:

Objection 1: Classical, local theories have been ruled out by Bell’s theorem.
Objection 2: Classical theories cannot explain single-particle interference (double slit experiment), quantum tunneling, the stability of atoms or energy quantification in atoms or molecules.
Objection 3: Even if one could elude the previous points, there is no reason to pursue classical theories because quantum mechanics perfectly predicts all observed phenomena.

Let's analyze them in turn.

On Objection 1, I agree that classical, local theories have been ruled out by Bell’s theorem with only one loophole left: super-deterministic theories pursued by 't Hooft. Any statistical theory obeying Kolmogorov's axioms respects Bell's inequalities. It is interesting to see how different realistic quantum mechanics interpretations escape Kolmogorov's axioms: Bohmian interpretation is contextual while quantum mechanics in phase space uses negative probabilities.

On superdeterminism, one needs to deny free will and this is a very tall order. While I (and anyone else) cannot give a rigorous definition of free will, I know that I have it. Andrei contends that classical theory is deterministic. While true, this is both an insufficient and an irrelevant argument. Superdeterminism is only a pre-requisite step: you need to obtain from it quantum correlations, and so far I am not aware of any successful model. Second, determinism does not imply superdeterminism because the existence of chaotic evolution equations. Predicting weather is a classical example. I do not think superdeterminism has any chance of success.

On Objection 2, I again agree with its statement. Quantum mechanics arose out of the inability of classical mechanics to explain atomic phenomena. But instead of expanding on this let's reply to the concrete arguments Andrei raised. Let's start with:

" This is all nice, but classical physics is not the same thing as Newtonian physics of the rigid body. Let’s consider a better classical approximation of the electron, a charged bullet. The slits are made of some material that will necessarily contain a large number of charged “bullets”. As the test bullet travels, its trajectory will be determined by the field generated by the slitted barrier. The field will be a function of position/momenta of the “bullets” in the barrier. But the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit, so the effect with both holes open is NOT the sum of the effects with each hole open alone."

This argument is wrong on two counts. First, one can make an interference experiment with neutrons where the neutrons not passing through the slits will be simply absorbed. Using electrically neutral particles renders irrelevant Andrei's objection. Second, "the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit" is incorrect as shown by a simple order of magnitude analysis. The electric fields near the slit are relevant on an atomic distance scale, while the distance between slits is macroscopic. You are looking at about seven order of magnitude difference in the ratio of relevant distances which translates in terms of force into a ten to minus fourteen order of magnitude effect. But the interference pattern is macroscopic and the difference between two Gaussian distributions vs. interference pattern cannot be explained away by a force fourteen of orders of magnitude smaller than what it is needed.

The next objection is appealing to Yves Couder's results. Those are interesting experiments, but they are not a confirmation of quantum mechanics emergence from classical physics and I know no one who claims it so. As such the argument is irrelevant to the current debate.

On the free fall atomic model, I did not read those papers so do not know if the claims are correct or not. The author may simply have proposed a model and analyzed the consequences and never claimed that his model describes nature.  Based on the general information available it is immediately clear that that model has nothing to do with reality. Also peer review is no magic bullet for avoiding publishing incorrect results. In my area of expertise in the last 12 months I read 2 published papers which were pure unadulterated garbage: the errors were not subtle, but blatant and packaged in a dishonest way to bamboozle the reader. Moreover I have concrete proof that the authors were aware of their mistakes when they published it. Physics is not immune to charlatans, crooks, and incompetent reviewers.

On the ionic crystals argument, without quantum mechanics the collection of electrons and nuclei will behave like a plasma and not like a crystal. This is moderately easy to test: create a computer simulation of say 1000 atoms and use Maxwell's and Newtonian equations of motion only to model the interaction. Then try to find an initial configuration which will be stable. I think there is none. Prove me wrong with such a model and I'll concede this point.

On quantum tunneling, the argument is pure handwaiving. Let me make an analogy. I know how my microwave works. But an alternative explanation might be that little Oompa-Loompas inside it are heating the food and I cannot see them because they move really fast.  The point is that the argument needs to have more predictive power than a fuzzy non-committal: "A new theory could predict a much stronger force." Show me the money. Propose such a theory and then we can discuss its merits. I am not asking something impossible. Regarding tunneling, quantum mechanics provides testable predictions which were confirmed experimentally. I only hold any alternative theory to the same level of experimental confirmation.

On Objection 3, I somewhat disagree with it. It is worthwhile to pursue non-quantum toy theories to better understand quantum mechanics, but not to search for an alternative to quantum mechanics.

Let me answer the four sub-points raised by Andrei:

a.       If nature is not probabilistic after all, there is much to be discovered. Detailed mechanism behind quantum phenomena should be revealed, bringing out a deeper understanding of our universe, and maybe new physical effects.

There is no "sub-quantum" or "hidden variable" explanation to quantum effects. Quantum mechanics is at the core of Nature, and my work is about proving this rigorously and not as a result of personal beliefs.

b.      Quantum theories are not well equipped to describe the universe as a whole. There is no observer outside the universe, no measurement can be performed on it, not even in principle.

This is a sterile objection to quantum mechanics and this cannot be used to justified a realistic alternative theory. First, even in quantum mechanics there are realistic interpretations. Second, epistemic interpretation like Qbism avoids this because they only talk about Bayesian probabilities. This objection only applies to naively using quantum mechanics in cosmology. Loop quantum gravity is unaffected by this objection.

c.       Due to its inability to provide an objective description of reality, quantum mechanics may not be able to solve the cosmological constant problem. A theory that states clearly “what’s there” could provide a much better estimate of the vacuum energy. After all we are not interested in what energy someone could find by performing a measurement on the vacuum, but what the vacuum consists of, when no one is there to pump energy into it.

The statement underscores a deep misunderstanding of the vacuum. Vacuum is actually a very violent place filled with virtual particles due to interplay of relativity and quantum mechanics. See this poor quality but brilliant video of a Sidney Coleman lecture to understand why merging them is not a trivial thing as one may naively expect that adding symmetries to quantum mechanics always results in simplifications. See also the QCD "Lava Lamp" which was shown at the 2004 Nobel Physics lecture. The cosmological constant problem is not a problem of quantum mechanics. I am not an expert in string theory but I know it has at least a solution to this problem (I don't know if it got rid of Susskind's "Rube Goldberg" label).

d.      Quantum mechanics requires an infinitely large instrument to measure a variable with infinite precision. When gravity is taken into account, it follows that local, perfectly defined properties cannot exist, because, beyond a certain mass, the instrument would collapse into a black hole.

The same argument can be used in the case of classical deterministic physics.

As you can see I disagree with the points Andrei was making, but nevertheless I want to thank him for participating in this debate and I look forward to discuss his replies in the commenting section of this post. I think such debates are useful, and I feel that the professional community of physicists is not doing a good job in engaging the public or explaining what it is doing. Physicists are very busy people trying to get ahead in a very competitive field. However the outside world usually experiences an arrogant wall of silence.

1. Rebuttal - Part 1

Dear Florin,

I would like first to clear two important issues:

1. My opening post is about classical, deterministic, field theories in general. It is about a class of possible theories, not about a specific theory. So, when I am speaking about a field I do not refer to the electric field, or magnetic field, or a pressure field, or gravitational field. I am just speaking about theories which define a function, let’s call it Phi (x, t), ascribing to each point in space and time a set of vectors/scalars. Let’s not go into more detail and just see how this class of theories can deal with the arguments I presented.

2. My claim is not that classical field theories are true. I may believe that, but I am not prepared to defend such a belief. As I have clearly specified in the title I only argue for compatibility between them and QM’s predictions and the way I did this was to show that the non-existence arguments fail. So, by asking me to explain how some specific quantitative results are recovered from classical theories, you are trying to shift the burden of proof.

In light of the two clarifications above, I will answer to your rebuttal.
“On Objection 1, I agree that classical, local theories have been ruled out by Bell’s theorem with only one loophole left: super-deterministic theories pursued by 't Hooft.”

Classical field theories are also superdeterministic. 't Hooft’s model (the cellular automaton) is just a special case of a discrete field theory. As you let the lattice unit go to zero, you regain the continuous case. Therefore Bell’s theorem is irrelevant for them as well.

“Any statistical theory obeying Kolmogorov's axioms respects Bell's inequalities.”

Are classical field theories obeying those axioms?

“Bohmian interpretation is contextual”

So are classical field theories.

“On superdeterminism, one needs to deny free will and this is a very tall order. While I (and anyone else) cannot give a rigorous definition of free will, I know that I have it.”

Free-will is an incoherent concept, and it implies non-determinism, rendering your argument logically fallacious (circular reasoning).

“Superdeterminism is only a pre-requisite step: you need to obtain from it quantum correlations, and so far I am not aware of any successful model.”

Here is one point where you try shifting the burden of proof. You have admitted earlier that superdeterministic theories are not excluded by Bell’s theorem. This is all I want to prove here. My claim is that classical field theories are possible, not that they are true (even though they could be).

“Second, determinism does not imply superdeterminism because the existence of chaotic evolution equations.”

I have made clear in my opening statement that classical field theories imply correlations between distant systems (Alice and Bob and the source of particles). This is what superdeterminism means (as far as I know). If you have some other idea of superdeterminism in mind, please make it clear so I can adjust my arguments accordingly.

Chaos does not preclude correlations, just makes them more difficult to notice. In a deterministic field theory there is always a perfect correlation between all particles, which is another way of saying that the trajectory of each particle is a function of the state of all other particles.

Objection 2

“This argument is wrong on two counts. First, one can make an interference experiment with neutrons where the neutrons not passing through the slits will be simply absorbed.”

There is no such thing as “simple absorption”; all interactions are mediated by fields. Neutrons, which are composed of 3 electrically charged quarks couple with all known fields (gravitational, electromagnetic, week and strong). Hopefully you wouldn’t want to claim that they are like drops of glue, sticking as they encounter a nucleus.

1. Dear Andrei,

Indeed, I think we have a disagreement on the burden of proof. If you propose a new theory the burden of proof is on you. But if you want to justify researching an alternative, you are free to do whatever you want, physics is not a dictatorship. However, if you request money for grants, the burden of proof is again on you.
“Classical field theories are also superdeterministic.” Determinism is not superdeterminism. Superdeterminism is like in the Matrix: “You've already made the choice, now you need to understand it.”
“Are classical field theories obeying those axioms?” Yes.
“So are classical field theories.” No. See http://www.mth.kcl.ac.uk/~streater/lostcauses.html#I for definition of contextual.
“You have admitted earlier that superdeterministic theories are not excluded by Bell’s theorem. This is all I want to prove here.” It is well known that Bell’s theorem has this superdeterministc loophole.

2. Rebuttal - Part 3

Andrei: “Quantum theories are not well equipped to describe the universe as a whole. There is no observer outside the universe, no measurement can be performed on it, not even in principle.”

Florin: “This is a sterile objection to quantum mechanics and this cannot be used to justify a realistic alternative theory. First, even in quantum mechanics there are realistic interpretations.”

Those are either non-local (Bohm) or inconsistent (many worlds).

“Second, epistemic interpretation like Qbism avoids this because they only talk about Bayesian probabilities.”

QBism is worse than Copenhagen.

1. It speaks about the subjective experiences of observers. There has to be plenty of them outside the universe.

2. It is based on decision theory, which presupposes a brain that works in the way our brain works. Explaining the properties of an electron based on some hardly understood properties of a brain is as fallacious as creationist’s appeal to an all-powerful god. A proper scientific approach explains complex systems (like a brain) in terms of simpler constituents (particles), not the other way around. This objection would apply as well to the “free-will” assumption in Bell’s theorem.

3. Its “local” character is a joke. The theory is supposed to be local because it all boils down to the experience of one observer. So, if you put a QBist on a Klingon war-bird and take him to Alpha-Centauri and back in 5 minutes he will still claim that he lives in a local universe because it is after all just another experience in his consciousness! There is also no evidence that this “experience” is a local process.

“Loop quantum gravity is unaffected by this objection.”

As far as I can tell LQG has been falsified for some time. So now, that it is dead, it is really unaffected by anything. Anyway, I am not arguing that classical field theories are compulsory, just worthy of more research.

“Vacuum is actually a very violent place filled with virtual particles due to interplay of relativity and quantum mechanics.”

If this is really the case, then you need to explain the “small discrepancy” between the prediction of QM and observed value, of about 120 orders of magnitude. Or you can go anthropic. The problem has not been solved by string theory.

Andrei: “Quantum mechanics requires an infinitely large instrument to measure a variable with infinite precision. When gravity is taken into account, it follows that local, perfectly defined properties cannot exist, because, beyond a certain mass, the instrument would collapse into a black hole.”

Florin: “The same argument can be used in the case of classical deterministic physics.”

No, it cannot. According to a classical theory the particle is there, with well-defined properties, it does not matter if you measure them or not. There are no unpredictable quantum fluctuations either, so you need not use an infinitely large instrument.

In the end, I would also like to point out that, because "the uncertainty relation does not hold for the past." one cannot argue for the fundamental probabilistic character of QM based on uncertainty. Just another argument against a classical universe that bites the dust!

Andrei

1. “QBism is worse than Copenhagen.” No, it is a variant of Copenhagen.

“It speaks about the subjective experiences of observers.” In probability theory you have the frequentist approach and the Bayesian approach. Their predictions are identical. Qbism takes the Bayesian point of view and attempts to make sense of the world from within the world.

“It is based on decision theory” No, it is based on Bayes update of information theorem https://en.wikipedia.org/wiki/Bayes%27_theorem

“Its “local” character is a joke.” – I agree.

“As far as I can tell LQG has been falsified for some time.” – This is irrelevant for the discussion purpose. Compare LQG with Wheeler DeWitt.

“If this is really the case, then you need to explain the “small discrepancy” between the prediction of QM and observed value, of about 120 orders of magnitude.” The most natural value is actually zero. String theory does have a contrived solution to it. Vacuum polarization is another effect of the vacuum.
“No, it cannot. According to a classical theory the particle is there, with well-defined properties, it does not matter if you measure them or not. There are no unpredictable quantum fluctuations either, so you need not use an infinitely large instrument.” GR is a classical theory. The resolution of a microscope is given by the wavelength of light. Ultra short distances require ultra short wavelengths which demand very high energy, to the point that you will create a black hole.

“I would also like to point out that, because "the uncertainty relation does not hold for the past." one cannot argue for the fundamental probabilistic character of QM based on uncertainty.”
The probabilistic character of QM arises out of non-commutativity. It is this non-commutativity which demands the matrices in Heisenberg’s formalism, and those matrices (after diagonalization) contain the predictions for all experiments, not a single one. Hence the probabilistic character.

3. Andrei, thank you for your rebuttal. There is quite a lot of material and it will take me some time to reply. I'll try to do it this weekend but I cannot make hard promises due to other commitments, but I will reply to all your points.

4. Dear Florin,

For some reason the part 2 of my rebuttal did not get published, so it may appear that I have not responded to some points. I will upload it again on Monday. I do not have access to the text until then.

Andrei

1. Your part 2 is not on the server and must have gotten lost during posting. Looking forward to it.

5. Rebuttal - Part 2-1

“Using electrically neutral particles renders irrelevant Andrei's objection. Second, "the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit" is incorrect as shown by a simple order of magnitude analysis. The electric fields near the slit are relevant on an atomic distance scale, while the distance between slits is macroscopic. You are looking at about seven order of magnitude difference in the ratio of relevant distances which translates in terms of force into a ten to minus fourteen order of magnitude effect.”

As I have specified in the beginning of my rebuttal, I am not speaking of the electric field, but about some field Phi (x, t). Let’s discuss the general issues leaving for now the details regarding the strength of the forces involved, etc.

“But the interference pattern is macroscopic and the difference between two Gaussian distributions vs. interference pattern cannot be explained away by a force fourteen of orders of magnitude smaller than what it is needed.”

The force has nothing to do with this, but let’s just agree that for a generic field, Phi (with whatever strength you want), Feynman’s analysis is incorrect.

“The next objection is appealing to Yves Couder's results. Those are interesting experiments, but they are not a confirmation of quantum mechanics emergence from classical physics and I know no one who claims it so.”

Again, I did not make the claim that you imply. Those experiments only prove that Feynman’s analysis about how probabilities should add in “ANY” classical theory is incorrect.

“On the free fall atomic model, I did not read those papers so do not know if the claims are correct or not. The author may simply have proposed a model and analyzed the consequences and never claimed that his model describes nature. Based on the general information available it is immediately clear that that model has nothing to do with reality.”

What do you mean by “general information”? Can you point me to some paper discussing the classical atom with spin? If you cannot, you should drop this argument.

1. Dear Andrei, Sorry for the long delay. On the free fall atomic model all I had access to was the Wikipedia information on it. From there it is clear it does not describe reality.

On Yves Couder's example, the position is completely classical obeying Bell's inequalities. As such Feynman's argument is still valid.

6. Rebuttal - Part 2-2

“On the ionic crystals argument, without quantum mechanics the collection of electrons and nuclei will behave like a plasma and not like a crystal. This is moderately easy to test: create a computer simulation of say 1000 atoms and use Maxwell's and Newtonian equations of motion only to model the interaction. Then try to find an initial configuration which will be stable. I think there is none. Prove me wrong with such a model and I'll concede this point.”

A classical simulation of NaCl crystal growth:

http://tresen.vscht.cz/fch/tul/jz302065w.pdf

does not seem to produce such a plasma.

Anyway, you can take a system of charged metal spheres covered with some dielectric. Would such a system produce a “plasma” too?

“On quantum tunneling, the argument is pure handwaiving. Let me make an analogy. I know how my microwave works. But an alternative explanation might be that little Oompa-Loompas inside it are heating the food and I cannot see them because they move really fast. The point is that the argument needs to have more predictive power than a fuzzy non-committal: "A new theory could predict a much stronger force." Show me the money.”

Again, shifting of the burden of proof. If you think that tunneling rules out only some specific classical theories, then be careful and make that clear. Do not say “no classical theory can explain tunneling” and then argue from some details that are not general to any classical theory, like the strength of the electrostatic or weak force.

“There is no "sub-quantum" or "hidden variable" explanation to quantum effects. Quantum mechanics is at the core of Nature, and my work is about proving this rigorously and not as a result of personal beliefs.”

This is a declaration of personal believes, nothing to argue against.

Andrei

1. Thank you for the simulation paper. I am not an expert in those kinds of modeling so I cannot asses what they use inside them. It is very likely they use some sort of effective effects which arise out of QM. What I asked was different. use only Maxwell and Newton. If they had used only those, then I concede the point.

Again we disagree on the burden of proof. It the job of the challenger of an existing paradigm to prove he has something worthwhile.

On the no sub-quantum, see my next post which is in preparation right now.

7. I was proposing last June or July that the PBR result was correct in that D > 0, but that the strict result that D = 1 was not. As such while the ψ-epistemic interpretations are wrong, it is uncertain in that 0 < D <= 1. My argument centered around the uncertainty principle for spin.

For noncommuting operators X and Y we have that

ΔXΔY = ħ/2|< |[X, Y]| >|.

We may set X = L_y and Y = L_y and we have that [L_x, L_y] = ħL_z and = l(l + 1) – m^2 and so

ΔXΔY = ½ħ^2(l(l + 1) – m^2).

This means the uncertainty principle is within a discrete set of possible values, rather than a continuous distribution. My argument centered around this.

This idea was a bit of a sideline to what I have been working on, which is to find quotient group realizations of fields on a horizon that correlated with locality in spacetime with one dimension larger. The approach I took to this was a quotient group realization, and the G/H = K quotient would give the exact PBR result for H = id.

I came here and see that you are presenting something that is similar to what I was proposing last summer.

8. Hi Lawrence, see http://fmoldove.blogspot.com/2015/09/heisenbergs-matrix-mechanics-and.html

Originally I was reading Beltrametti and Cassineli to prepare for the post. Shortly after that we had a discussion where you challenged me on the discrete spectra case. I have to admit I had to think a bit to find the correct answer. Then I thought the problem would be of interest to a larger audience.

I still maintain there is no universal uncertainty principle for spin, only uncertainties for given states.

1. That would imply a different sort of relationship between the state and the operator.

I may return to that little project, which is something I thought of with respect to working on quotient groups for unitary qubits with black holes.

LC

2. Lawrence, I don't quite follow what you are saying. Do you say there is an uncertainty relationship for spin for any states? If so I can find a counterexample.

9. What I am saying is that this seems to imply some subtle difference in the relationship operators and states. I am not sure what that is.

LC

1. Same comment as in the next post: another novel idea, pursue it and see where it leads.

2. The idea is in relationship to quotient groups and SLOCC for qubit-black hole correspondence. I also think this has something to do with BMS supertranslations. The elementary case is the conformal theory with SO(2,1)/U(1), which is a special case with SL(2n,C)/SU(n). Another case is the case of the quotient group between the 9 rays in the F4 Kochen-Specker vs the 8 rays in the set of possible rays, 8 = 2^3, that can construct a classical ray trace path. The additional dimension is ~ R, that reduces the 9 KS rays to the 8 classical rays.

This is a bit of a side calculation, and it has some bearing upon the PBR result. The PBR result has been accused of having some ontology built into it. If 0 < D[ψ, φ] < 1 then all quantum interpretations have holes in them.

LC

3. The idea is in relationship to quotient groups and SLOCC for qubit-black hole correspondence. I also think this has something to do with BMS supertranslations. The elementary case is the conformal theory with SO(2,1)/U(1), which is a special case with SL(2n,C)/SU(n). Another case is the case of the quotient group between the 9 rays in the F4 Kochen-Specker vs the 8 rays in the set of possible rays, 8 = 2^3, that can construct a classical ray trace path. The additional dimension is ~ R, that reduces the 9 KS rays to the 8 classical rays.

This is a bit of a side calculation, and it has some bearing upon the PBR result. The PBR result has been accused of having some ontology built into it. If 0 < D[ψ, φ] < 1 then all quantum interpretations have holes in them.

LC