## CHSH inequality and the rejection of realism

I will start now a series on Bell's theorem and its importance to quantum foundations. Today I will talk about the Clauser, Horne, Shimony, and Holt inequality and its implication.

Quantum mechanics is a probabilistic theory which does not make predictions of the outcome of individual experiments, but makes statistical predictions instead. This opens the door to consider "subquantic" or "hidden variable" theories which would be able to restore full determinism. However, even with the statistical nature of quantum mechanics predictions there is something more which can be investigated: correlations. What distinguishes quantum from classical mechanics is how the observables of a composite system are related to the observables of the individual systems. In the quantum case there is an additional term related to the generators of the Lie algebras of each individual systems and this in turn prevents the neat factorization of the Hermitean observables of the composite system. It is this lack of factorization which prevents in general the factorization of the quantum states. In literature this goes under the (bad) name nonlocality.

Now suppose we have two spatially separated laboratories which receive from a common source pairs of photons. The "left lab" L chooses to measure the polarization of the photons on two directions $$\alpha$$ an $$\gamma$$, while the "right lab" R chooses to measure the polarization of the photons on two directions $$\beta$$ an $$\delta$$, Let's call the outcome of the experiments: $$a, b, c, d$$ for the directions of measurement $$\alpha, \beta\, \gamma, \delta$$, respectively. The values $$a, b, c, d$$ can take are +1 or -1.

Now let us compute the following expression:

$$C=(a+c)b-(a-c)d$$

Suppose $$(a+c) = 0$$, then $$(a-c)=\pm 2$$ and so $$C=\pm 2$$
Similarly if $$(a-c) = 0$$, then $$(a+c)=\pm 2$$ and again $$C=\pm 2$$

Either way $$C=\pm 2$$

Now suppose we have many runs of the experiment and for each run $$i$$we get:

$$a_i b_i + b_i c_i + c_i d_i - d_i a_i = \pm 2$$

from which we deduce on average that:

$$|\langle ab\rangle + \langle bc\rangle + \langle cd\rangle - \langle da\rangle|\leq 2$$

This is the CHSH famous inequality. Now under appropriate circumstances nature violates this inequality:

in an experiment with photons the average correlation between measurements on two distinct directions $$\alpha, \beta$$ is: $$\cos 2(\alpha - \beta)$$ and the inequality to be obeyed is:

$$| \cos 2(\alpha - \beta) + \cos 2(\beta - \gamma) + \cos 2(\gamma - \delta) - \cos 2(\delta - \alpha)| \leq 2$$

but if the angle differences are at 22.5 degrees we get that $$2\sqrt{2} \leq 2$$ so what is going on here?

A natural first objection is that not all 4 measurements can be simultaneously be performed and so we are reasoning counterfactually. But in N runs of the experiment we get 2N experimental results and there is a finite number of ways we can fill in the missing 2N data and in each counterfactual way of filling in the unmeasured data the CHSH inequality is still obeyed.

A second potential objection is that there is no free will and there is a conspiracy going on which prevents an unbiased choice of the 4 directions. There is no counteragument for this objection except that I know I have free will. If free will does not exists then mankind has much deeper troubles than explaining quantum mechanics: try to explain morality and justify the existence of the judicial system.

The introduction of bias can affect correlations and if the detection rate depends on the angle, then for appropriate dependencies one can obtain the quantum correlations. This is the so-called detection loophole, However, if such a dependency exists, it can be tested in additional experiments and the introduction of angle dependency only for Bell test experiments is indefensible. Loophole free Bell experiments while important to push the boundary of experimental technology have no scientific importance and they count only towards experimentalist's bragging rights.

Another way to obtain correlations above 2 is by appealing to contextuality: for example the value of $$a$$ when measured by lab L when lab R measures $$b$$ may not be the same when lab R measures $$d$$. While quantum mechanics is contextual, in this case such an argument means that the the choice lab R makes influences the result of measurement at lab L which is spatially separated!!!

Last, if the values of $$a, b, c, d$$ do not exist prior to measurement, this decouples again the value of $$a$$ when lab R measures $$b$$ from the value of $$a$$ when lab R measures $$d$$.

Assuming free will is true, we have only two choices at out disposal to be able to obtain correlations above 2:
• measurement in a spatially separated lab affects the outcome on the remote lab
• the outcome of measurement does not exist before measurement.
The first choice is taken by dBB theory because the quantum potential changes instantaneously and the second option is advocated by the Copenhagen camp. (I am excluding the MWI proposal because in it there is no valid derivation of Born rule. I am also excluding collapse models because they are a departure from quantum mechanics and experiments will soon be able to reject them).

Now here is the catch: the two labs need not be spatially separated and one experiment in lab L can unambiguously happen before the experiment in lab R. When the R lab measurement takes place it cannot affect the outcome in the L lab because that is in the past and already happened!

But can the first measurement affect the second one? In dBB this is possible as long as the first particle and its quantum potential is still around to "guide" the second particle. However, if after the first measurement the first particle is annihilated by its antiparticle then its quantum potential vanishes. The behavior of quantum potential after annihilation  is a reason why a relativistic second quantization dBB theory is not possible: either the quantum potential sticks around and messes up subsequent measurements, or vanishes and then the correlations cannot occur in the case above. (dBB supporters pin their hopes on a "future to be discovered" relativistic dBB quantum field theory which never materialized and cannot exists for several reasons.)

So from the two choices above only one remains valid:

the outcome of measurement does not exist before measurement

Realism is rejected by Bell's theorem. However in literature Bell's result is presented instead as a rejection of locality. But this is an abuse of language: locality=state factorization. Nature and quantum mechanics are incompatible with a state factorization. State factorization is just factorization, not locality. Rejection of realism is the only viable option left.

1. Dear Florin,

As you might expect, I strongly disagree with your conclusion, “Rejection of realism is the only viable option left.”

This is my rebuttal:

Modern classical theories (general relativity, our only theory of space and time and classical electromagnetism) are incompatible with free-will (as required by Bell’s theorem).

The assumption of free will is therefore identical with an a-priori denial of the most interesting local realistic theories (classical field theories). As I am not buying into this assumption I feel no need to lose any more time on Bell and I will go and demolish your only remaining argument:

“I know I have free will”

This type of argument, known under the name of “the argument from personal experience” is often put forward by second-class religious debaters. “God exists because he speaks to me!” – they say. Unfortunately for you, in the case of free will I can actually prove you wrong. I will show that it is impossible for you to actually know from subjective experience that you have free-will. So, let’s take a classical free-willed decision, tee over coffee. Let’s assume, for the sake of the argument that the decision is determined by the blood pressure in the relevant portion of the brain. It’s easy to see that you wouldn’t be able to notice this correlation because the pressure of the blood in your brain is not an information available to your consciousness. You simply cannot know the reason for your decision so claiming that you know it cannot be based on some deterministic process is absurd. There are experiments showing that even when you are determined to act in a certain way by using magnetic fields on your brain, you still feel that you acted “freely”.

Morality and justice can be easily explained without free-will. They are practical adaptations for living in society.

I also need to remind you that I have already proven that locality is incompatible with realism (in the comment section of “Correlations and Entanglement swapping take 2”. Till now you did not point to any flaw in that argument. So, local non-realism is double-busted.

Andrei

1. Andrei,

If you want to deny it you must do so on a solid basis. What is this basis?

Florin

2. Florin,

I reject free will on the basis that there is no evidence for it. The only so-called evidence (your personal experience) have been demolished in my previous post. If you disagree, put forward that strong evidence so that we can debate it. Otherwise, drop it.

It is your burden of proof to support your assumptions, not mine to find evidence against. You didn't even properly defined the concept.

Andrei

3. Andei,

Let me beg the difference. The basis is as follows: QM is a theory about nature which withstood all experimental tests. As such we must consider QM correct. QM has several interpretations and none of them deny free will. If you want to deny free will you have to come up with your brand new QM interpretation. The burden of proof is on you.

Florin

4. Florin,

"QM is a theory about nature which withstood all experimental tests."

True.

"As such we must consider QM correct."

Not exactly, but I agree.

"QM has several interpretations and none of them deny free will."

False. MWI and Bohm are strictly deterministic.

"If you want to deny free will you have to come up with your brand new QM interpretation."

Nonsense. By this line of reasoning if QM does not deny the existence of Santa Claus it means that we should believe in him.

"The burden of proof is on you."

No, it is not. Just take a look at the Wikipedia page on the subject "Philosophic burden of proof". It reads:

"When two parties are in a discussion and one asserts a claim that the other disputes, the one who asserts has a burden of proof to justify or substantiate that claim. An argument from ignorance occurs when either a proposition is assumed to be true because it has not yet been proved false or a proposition is assumed to be false because it has not yet been proved true. This has the effect of shifting the burden of proof to the person criticizing the proposition."

Just let this incoherent concept (free-will) rest in peace, Florin.

Andrei

5. The loss of reality in QM is the loss of a certain type of reality. If you have an entangled state from two spins, there is no ontological reality to the state of those spins. This means that if the experimenter makes a measurement of those spins there was no a prior reality to those spins. What is real though is that the total quantum information before and after the measurement is the same.

LC

6. "MWI and Bohm are strictly deterministic."

Deterministic but not conspiratorial to explain the QM correlations.

""If you want to deny free will you have to come up with your brand new QM interpretation."

Nonsense. By this line of reasoning if QM does not deny the existence of Santa Claus it means that we should believe in him."

Straw man argument. What we are talking here is the explanation of QM correlations, not Santa Claus. You contend lack of free will is at the root of explaining QM correlations in a realistic way. Show me how. That is why the burden of proof is on you (or anybody else making similar claims). 't Hooft does not object to this burden of proof and he is attempting to build a cellular automata model of QM. This is the correct approach and you have to do the same if you want your arguments to hold water, otherwise they are just handwaving.

It is not up to me to prove to you your approach is unworkable-I have no time or energy to do this. It is up to you to convince the quantum community you are right.

7. Florin,

You did not attempt to rebutt my argument where I have proven that locality is incompatible with realism (in the comment section of “Correlations and Entanglement swapping take 2”.

Based only on the assumption of locality I have shown that:

C1: Physics must be realistic
C2: Physics must be deterministic
C3: Free-will is false

So, as you can see, I have already proven (not assumed) everything you ask from me. I have proven that my position must be right and yours is wrong.

I need also remind you that you didn't atempt to save your only argument in favour of free-will (personal experience) so I can only assume that you dropped it.

Now, in your original post you say:

"A second potential objection is that there is no free will and there is a conspiracy going on which prevents an unbiased choice of the 4 directions. There is no counteragument for this objection except that I know I have free will."

When speaking about "conspiracy" you assume that one should expect the measurement to be "unbiased". I see no good reason to take that assumption seriously. Just like in the case of free-will this is just an unfounded assumption.

Regarding the so-called straw-man argument, It is nothing like straw-man. I have simply applied your faulty reasoning to a perfectly valid example so that you can see its ridiculous consequences. You claimed that If P is true and P does not imply Q it folows that Q is true. You claim that it works only for some P and Q but not for others. Sorry, this is not how logic works.

Andrei

8. Andrei,

Sorry for the delay. You state: "I have already proven (not assumed) everything you ask from me."

If this is the case, should I not be the judge of this? Don't you have to prove it to my satisfaction?

Florin

9. Florin,

I will repost my argument here:

Description of the experiment:

a. Two entangeled particles are created at time T0.

b. One of the particles (P2) is sent in a distant place. It arrives there at T1. The other one (P1) remains in the lab. Bob writes down his opinion about the spins’ existence.

c. Bob measures the spin of P1 on Z. at time T3

d. At some time after measurement (T4) Bob writes down again his opinion about the spins’ existence.

Lubos maintains that Bob’s first statement (S1) written at time T1 and Bob’s second statement (S2) written at time T4 are both true. Let’s write them here:

S1: The spins do not exist, they will be created at the time of the measurement. (non-local non-realism)

S2: The spins existed before the measurement. Just like in the case of Bertlmann's socks the measurement revealed their pre-existing values. (local realism).

Ok, let’s develop the argument further by choosing locality (S2) over non-realism (S1). After all, there is plenty evidence for locality but no evidence for non-realism. Here Lubos thinks there is a problem because, if Bob changes his mind at T1 and measures the spin on X or on Y he should ascribe reality to all spin components leading to a contradiction with the experimental results as Bell’s theorem shows. This implication is true so we have to choose between:

S2: The spins existed before the measurement. Just like in the case of Bertlmann's socks the measurement revealed their pre-existing values. (local realism).

S3: The experimenter has free-will (he can change his mind) (non-determinism)

This choice is easy again. There is no good evidence for free will so it has to be rejected. And, as a bonus, Bell’s theorem goes down the drain too (no free-will, no theorem). So we can actually ascribe reality to all spin components and we are free of all contradictions.

So, until now, I have proven that realism + determinism must be true if locality is true. No other assumptions have been made. Einstein was right. Bohr was wrong. There is no doubt about it. QM is incomplete and non-fundamental. This is probably the explanation why quantizing gravity is so hard.

I am looking forward for your rebuttal.

Andrei

10. Andrei,

The argument needs refinement. First, can you take Lubos out of your argument? The argument is about nature not Lubos being right or wrong. Second I don't get S2. How can after measurement say that "The spins existed before the measurement." Do you want to say The spins exist now? I don't get it. Third, I don't get why S2 is equivalent with locality.

Florin

11. Florin,

OK, the refined argument folows:

1. Description of the experiment:

a. Two entangeled particles are created at time T0.

b. One of the particles (P2) is sent in a distant place. It arrives there at T1. The other one (P1) remains in the lab.

c. Bob measures the spin of P1 on Z. at time T2

2. Discussion

It is an experimental fact that the spin of P2 on Z, if measured, will be perfectly anticorrelated with the spin of P1 on Z.

There are two possible explanations of the experimental outcome:

S1: The spins of P1 and P2 do not exist, they are created at the time of the measurement (non-realism).

S2: The spins existed before the measurement. Just like in the case of Bertlmann's socks the measurement revealed their pre-existing values. (realism).

We can see that S1 implies non-locality because the creation of the spin of the distant particle, P2, is caused by the measurement on P1. S2 is local description.

Ok, let’s develop the argument further by choosing locality (S2) over non-realism (S1). After all, there is plenty evidence for locality but no evidence for non-realism. There is a problem however because, if Bob changes his mind at T2 and measures the spin on X or on Y he should ascribe reality to all spin components leading to a contradiction with the experimental results as Bell’s theorem shows. This means we have to choose between:

S2: The spins existed before the measurement. Just like in the case of Bertlmann's socks the measurement revealed their pre-existing values. (local realism).

S3: The experimenter has free-will (he can change his mind) (non-determinism)

This choice is easy again. There is no good evidence for free will so it has to be rejected. And, as a bonus, Bell’s theorem goes down the drain too (no free-will, no theorem). So we can actually ascribe reality to all spin components and we are free of all contradictions.

So, until now, I have proven that realism + determinism must be true if locality is true. No other assumptions have been made.

Andrei

PS

" I don't get why S2 is equivalent with locality"

S2 is a clearly local description. The spins of P1 and P2 exist since the time the particles themselves are created and this is the reason they are found anticorrelated.

12. Andrei,

There are several problems with your argument. The first one occurs at this sentence:

"We can see that S1 implies non-locality because the creation of the spin of the distant particle, P2, is caused by the measurement on P1. S2 is local description."

S1 does not imply nonlocality. Measuring P1 does not determine the value of measurement on P2. The proper description is not nonlocality but contextuality which is weaker than nonlocality.

Then you have this: "but no evidence for non-realism." Quite the opposite. Take a look at the K-S theorem (https://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem). The values of the spin cannot be assigned independent of the measuring device setting. This is contextuality again.

There are other subsequent problems with your argument, but we should clarify the earlier ones from above before proceeding with the rest of the argument.

Florin

13. Florin,

"S1 does not imply nonlocality. Measuring P1 does not determine the value of measurement on P2."

Yes, it does. According to non-realism, the spin of P1 is created at time T2, when the measurement occurs. On the other hand, QM predicts and the experiments confirm, that the spin of P2 is also determined. If the spin of P1 is "created" as 1/2, the spin of P2 is also "created" as -1/2.

"The proper description is not nonlocality but contextuality which is weaker than nonlocality."

I agree with that, however, the non-realistic description (the creation of spins during measurement) is a non-local process. If you deny this please explain, in a local way, how the spin of P2 is prevented to be "created" as +1/2 once the spin of P1 has been created as +1/2. How do you account for the fact that the measurement of P1 gives a genuinely random result while the subsequent measurement of P2 does not (is completely determined).

"The values of the spin cannot be assigned independent of the measuring device setting."

This is true, indeed, but this is not in any way a proof against realism, on the contrary.

Andrei

14. "Measuring P1 does not determine the value of measurement on P2."
"Yes, it does."

No, it does not. Only if you happen to measure on the same (or opposite) direction on the distant particle you can predict with certainly the experimental value. But this follows from a conservation law since the total spin was zero. If you choose to measure on any other direction the outcome is not fixed: sometimes you get +1 sometimes -1. Only the overall statistical distribution obeys QM prediction.

15. Florin,

"Only if you happen to measure on the same (or opposite) direction on the distant particle you can predict with certainly the experimental value."

I have clearly specified that we only measure on Z (same direction) so let's only discuss this case (for now).

"But this follows from a conservation law since the total spin was zero."

OK, but how is this conservation law implemented locally using your "creation" mechanism? This is the crucial question. You cannot say that the spins were opposite since the particles were emitted, because you maintain that they did not exist at that time. You also reject some other hidden variable that could "program" the measurements to give opposite values. You measure P1 on Z and the result is randomly "created" at time T2. How does this information become available at P2?

Andrei

16. Andrei,

Sorry for the delay. Here are my answers:

"This is the crucial question."
indeed it is.

"You cannot say that the spins were opposite since the particles were emitted, because you maintain that they did not exist at that time."
Correct. And I have good reason for this position: take an isotropic distribution of opposite spins and compute the EPRB correlations. (this is the most natural thing to assume if the spins exist before measurement and the total spin is zero) You get something which is in disagreement with both QM and nature.

"You also reject some other hidden variable that could "program" the measurements to give opposite values."
Correct again. Any local hidden variables would spoil the correlation observed in nature. This is the essence of Bell's theorem.

"You measure P1 on Z and the result is randomly "created" at time T2. How does this information become available at P2?"

This is the million dollar question. Let me start by quoting Bell quoting Bohr:
"the impossibility of any sharp distinction between the behavior of atomic objects and the interaction with the measuring instrument which serve to define the conditions under which the phenomena appear".

Let me translate: the measuring device plays an essential role in defining the measurement context which in turn affects the statistics. In other words: contextuality. Here is what I stated at the end of the last post:
"the observer (but not consciousness) does play an active role in generating the experimental outcome. This active role happens even when parts of the composed system are out of causal reach because quantum mechanics is blind to space-time separations."

At this time you may feel unsatisfied with the contextuality answer. However I want to point three things:
-I have provided an actual mechanism inside QM by which contextuality does affect the outcomes. Read my Grothendieck group posts. The basic idea is that various outcomes are physically indistinguishable and there is a mechanism which breaks the degeneracy.
-in the case of special relativity, what make time stretch and distance shrink? Is there an invisible hand of God doing it? Are there dynamical explanations for this? No, the answer is kinematic. Seeking dynamical explanations in this case is a lost cause: there are none. Similarly seeking dynamical explanations of "How does this information become available at P2?" is a lost cause as well. There are no such explanations possible (although Bohmian supporters contend that there is such an explanation.).
-in your particular case when only one outcome is possible this follows from spin conservation. This is the only possible explanation. I contend that Einstein reality criterion is false: "If, without in any way perturbing the system, one can predict with certainty the value of a physical quantity referred to it, then there truly exists an element of physical reality corresponding to such quantity" I have an unpolished draft paper showing that this reality criterion is false. At some point I'll revisit and publish it.

17. Florin,

"take an isotropic distribution of opposite spins and compute the EPRB correlations. (this is the most natural thing to assume if the spins exist before measurement and the total spin is zero) You get something which is in disagreement with both QM and nature."

I disagree with this "naturalness" assumption. It is only true if there is no interaction between the source and detectors and if you assume that the initial states of source and detectors are not correlated. In classical electromagnetism those two assumptions are false. There exist electromagnetic and gravitational interaction between the source and detectors (actually between their electrons and quarks) and at the beginning of the experiment the states are correlated as a result of this interaction.

"Any local hidden variables would spoil the correlation observed in nature. This is the essence of Bell's theorem."

Not if you drop the unsupported "free-will" assumption. Didn't we settle this? If not, I need to ask you again to provide evidence for it.

"the measuring device plays an essential role in defining the measurement context which in turn affects the statistics. In other words: contextuality."

I totally agree with this.

"This active role happens even when parts of the composed system are out of causal reach because quantum mechanics is blind to space-time separations."

Exactly.

"At this time you may feel unsatisfied with the contextuality answer."

Of course, because contextuality is the experimentally observed reality that needs to be explained, not the explanation.

I fully agree that physics is contextual, I also point out that classical field theories are contextual and they provide a natural, local and parsimonious explanation for the observed experimental results. On the contrary, the non-realist scenario (where spins are created at the time of measurement) cannot provide any local explanation.

"I have provided an actual mechanism inside QM by which contextuality does affect the outcomes. Read my Grothendieck group posts. The basic idea is that various outcomes are physically indistinguishable and there is a mechanism which breaks the degeneracy"

Sorry, but "contextuality" cannot affect anything. Contextuality is the effect of how the laws of physics act. If you can show how randomly created spins at the time of the measurements manage to produce the observed experimental results, then you have provided the required explanation. Just renaming "correlations" as "contextuality" explains nothing. It's like saying that a particle cannot be accelerated at more than c because of locality. The true explanation comes from the equations of relativity that show that you would need an infinite amount of energy. Locality is another name for the fact that particles (or other physical entities) cannot exceed the speed of light.

To be continued

Andrei

18. "-in the case of special relativity, what make time stretch and distance shrink? Is there an invisible hand of God doing it? Are there dynamical explanations for this? No, the answer is kinematic. Seeking dynamical explanations in this case is a lost cause: there are none."

Wrong here. All those effects are a result of the constancy of the speed of light in all reference frames. Indeed, you need to accept that c is a constant without a deeper explanation, but at least it is a brute fact of nature.

"Similarly seeking dynamical explanations of "How does this information become available at P2?" is a lost cause as well. There are no such explanations possible"

Of course there is a natural, local and parsimonious explanation. The information goes together with the particles in the form of their spin. In other words, realism. Exactly as I have pointed out, non-realism fails to do the same so it has been falsified.

"in your particular case when only one outcome is possible this follows from spin conservation."

In this case you have to agree that your proposed "creation mechanism" fails to agree with the conservation laws as well. If the spins at P1 and P2 are randomly created locally during the measurements you would expect a different statistics than the observed one.

Also, I don't fully agree with how you apply momentum conservation here. Total angular momentum must be conserved, but this includes the instruments, observers, etc. It is not a-priori necessary that the entangled pair has a conserved angular momentum.

"I contend that Einstein reality criterion is false"

I have not used this criterion in my argument, so I do not feel necessary to defend it.

Andrei

2. "What distinguishes quantum from classical mechanics is how the observables of a composite system are related to the observables of the individual systems."

I can't imagine what the interpretation could be so that the statement is true. Composite systems allowing observables for subsystem A; and for subsystem B - have the property that any observable on A commutes with any observable on B.

When they commute, their mutual relationship is exactly the *same* as in classical physics. Classical physics *is* the special case of quantum mechanics when commutators are zero.

So all the "novelties" of quantum mechanics are only the novelties that appear inside A or inside B - where the commutators between observables are nonzero.

The non-factorizability of the quantum state is not a new aspect of QM in any sense. It is just the complexification - quantum description - of the correlations in probability distributions P(x1,x2) that are obviously possible in classical physics, too. Nothing conceptual is changing about the existence of these correlations or non-factorization in QM.

It's similar with other sentences of yours. About 1/2 of them are upside down. What you write is never quite right.

1. "I can't imagine what the interpretation could be so that the statement is true."
This requires no interpretation. What I state is a well established mathematical fact in the algebraic formulation of QM.

"Classical physics *is* the special case of quantum mechanics when commutators are zero."
I have trouble classifying this statement as either poetic or ignorant. There is a kernel of truth to it but the generalization is simply preposterous in so many ways. Think of CM in Hamiltonian formulation: it involves position and momenta which commute. In QM position and momenta do not commute. But now force the position and momenta to commute in QM and you get QM in phase space. What is changed in this case is the Poisson bracket. Again you are painting a way too simplistic picture.

"Nothing conceptual is changing about [..] non-factorization in QM." Huh, really? Show me a non-factorizable state in CM for spatially separated physical systems!!! Do you understand local realism?

"What you write is never quite right."
Gee, I get the same feeling from you.

2. Florin,

"Show me a non-factorizable state in CM for spatially separated physical systems!!!"

Does Earth and Sun qualify as "spatially separated"?

Andrei

3. Can you say exactly what you believe is wrong with Lubos's statement? It's also my understanding that probability theory can be generalised and formulated algebraically and that [(probabilistic) CM] QM is what you get when you apply it to [non] commuting random variables (and that this is an idea which has rather more than a kernel of truth to it).

"But now force the position and momenta to commute in QM and you get QM in phase space."

You get the Koopman-von Neumann formulation of classical mechanics, surely?

4. "You get the Koopman-von Neumann formulation of classical mechanics, surely?"

Yes, but this is quarter the story. both QM and CM have phase space and Hilbert space representations. In phase space you get ordinary CM and the Moyal bracket for QM. In Hilbert space you get ordinary QM and K-vN formulation of QM. Originally QM started from a different handling of the Ps and Qs and you have the Dirac problem: replace the Poisson bracket with the commutator, etc. At close inspection Dirac's problem turned out not to have a solution. This led to the development of rigorous qantization techniques like geometric quantization (and several others). All Poisson spaces are quantizable and this is a highly nontrivial mathematical result proven with hints from string theory (and by the way it lead to a Fields medal).

The area of the relationship between QM and CM is vast and to state that it all boils down to noncommutation is a mathematical caricature. It's like saying that all Newton did was to stand under that tree where the apple fell on his head and the rest easily followed.

5. "I have trouble classifying this statement as either poetic or ignorant. There is a kernel of truth to it but the generalization is simply preposterous in so many ways. Think of CM in Hamiltonian formulation..."

Sorry, Florin, what I wrote is a deep, well-known, and essential truth. The difference between classical and quantum mechanics is *exactly* the nonzero value of commutators in quantum mechanics - in other words, the difference and the only difference between classical and quantum mechanics is the uncertainty principle. It has been known since 1925 and you shouldn't have passed the #1 undergraduate course - if you have ever attended one - if you really misunderstand this basic point.

"Show me a non-factorizable state in CM for spatially separated physical systems."

Bertlemann's socks? The state is described by the probability distribution that has 0%,50%,50%,0% for left-and-right-green, left-green right-red, left-red right-green, left-and-right-read. The state, P(colorLEFT, colorRIGHT) isn't factorized.

Almost *every* composite classical system composed of parts that interacted in the past has the same property. You simply don't have the slightest clue what you're talking about.

6. This argument is classical. Every classical system composed of parts ... . Sure, but in the end your objection to nonlocality is you equate it with some action at a distance or nonlocal signalling.

LC

7. Lubos,

I can still argue with you on noncomutativity: I understand what you state, but I contend the math description goes much deeper. However I will leave this to another day.

I want to concentrate instead on "Show me a non-factorizable state in CM"

I did not think I would (ever) say this, but on this I was wrong and you were right. I guess I was too arrogant asking for an example which I should have easily discovered myself. Touche, you got me here.

8. Thanks for the progress.

Otherwise all of quantum physics is hidden in the algebra of observables.

3. Well there is always more mathematical story but whatever mathematical value the phase space formulation of QM has it isn't probability theory and it doesn't really help (me) to understand and interpret the physical theory.

The idea that QM is 'just' probability theory applied to mechanics when noncommutativity (as well as uncertainty) can't be ignored does help. It fits in very nicely with Max Born's insights (even better in light of a more modern, Cox-Jeffreys-Jaynesian, view of probability I think) and with other insights. You'd have to show me that this view is untenable, not just that there is more mathematical story - especially not any 'psiontology-esque' mathematical story - to QM, before I'd let go of it.

4. Florin's blog entries are a basic introduction to these nonlocal properties of QM. If one wants to deny or ignore these they are certainly free to do so. However, entanglement symmetries are becoming a large part of quantum black hole physics. In effect by denying nonlocality you foreclose your understanding of new developments.

If Florin wants to elaborate on this that is his choice, but I don't really have the time for that. One can do QFT with locality of fields or amplitudes and "shut up and calculate." However, standard QFT may be reaching its limits for it predicts a vast set of degrees of freedom or Hilbert spaces that may be "too large."

In science I think one needs both perspectives that are liberal and conservative. One needs some conservative basis in order to ask questions about developments, but one should also have a liberal sense in order to think in adventurous ways. Going too far with either is probably not advised; think of it as a scientific form of the Buddhist "middle way."

Lubos is an arch-conservative; in fact he is so conservative that he is politically nearly fascist and he also rejects a lot of science. He called AMPS nonsense, when that has turned into a very motivating problem. He denies CO_2 climate change, when the data in the last 10 years has overwhelmingly gone against his stance.

5. Florin,
I am interested in following:
"Show me a non-factorizable state in CM"
I understand both you and Lubos agree that this is possible. Can you give an example?

1. I have been pondering this as well. I found this fairly recent paper on the topic https://arxiv.org/pdf/1209.4036v3.pdf

LC

2. Kashyap,

He just did above and he was right. Please read my next post where I'll explain it in detail.

Lost the battle, but not the war :)

3. This comment has been removed by the author.

4. Florin,
My guess(!) is that many body system (with may be three body forces etc) may be non-factorizable.Surely you can construct model for solar system by adding two by two forces and solve perturbatively; NASA does it everyday, but I am talking about solving exactly with Fadeev type analysis etc. Also I am not sure in General relativity with each mass distorting space-time, the system is factorizable. What do you say?

6. You appear to be making nice progress. You now are back to a simple source, beamsplitter, and observer and are still arguing about many events. Now just argue about one photon event.

If an observer measures a photon at path A, can that observer conclude that the photon only then traveled on path A? Classically, yes. Quantum says that it is impossible to say which path the photon traveled since that one photon was a superposition state between two paths.

Now you can argue endlessly about whether a photon path is predictable and therefore classically determinate and so there is no free will or whether the photon path is fundamentally unknowable and therefore uncertain and there is free will.

Finally, by pairing each photon with its entangled sibling at the CMB creation, you can then get a pilot wave type of result for the biphoton and find that biphoton gravity does act determinate.

In quantum, you can always create photon pairs as commuting observables and therefore represent kind of a Bohmian hidden variable, but really the underlying universe is still quantum.

While a single photon has an uncertain future, an entangled biphoton has a determinate future since whatever happens to one simultaneously happens to the other in a complementary way.

In any algebra, you can use noncommuting primitives as entangled products to form commuting observables.

7. You appear to be making nice progress. You now are back to a simple source, beamsplitter, and observer and are still arguing about many events. Now just argue about one photon event.

If an observer measures a photon at path A, can that observer conclude that the photon only then traveled on path A? Classically, yes. Quantum says that it is impossible to say which path the photon traveled since that one photon was a superposition state between two paths.

Now you can argue endlessly about whether a photon path is predictable and therefore classically determinate and so there is no free will or whether the photon path is fundamentally unknowable and therefore uncertain and there is free will.

Finally, by pairing each photon with its entangled sibling at the CMB creation, you can then get a pilot wave type of result for the biphoton and find that biphoton gravity does act determinate.

In quantum, you can always create photon pairs as commuting observables and therefore represent kind of a Bohmian hidden variable, but really the underlying universe is still quantum.

While a single photon has an uncertain future, an entangled biphoton has a determinate future since whatever happens to one simultaneously happens to the other in a complementary way.

In any algebra, you can use noncommuting primitives as entangled products to form commuting observables.