Correlations vs. locality:
Can you hear the shape of a drum?
Is Nature local or non-local? Those are battle lines between the epistemic and ontic camps. But can we approach the problem from a different angle? I don't remember the quote exactly, but Bell once stated something along the following line: quantum correlations cry for an explanation. Or do they? I will attempt to make the case for the contrary.
If you think quantum correlations (which go above Bell limit) are in need of an explanation, then very likely you are in the ontic, beable, non-local camp. Personally I am not in this camp, and I am not in anyone's camp. What I am trying to do is reconstruct quantum mechanics from physical principles and then mathematically arrive at the correct quantum mechanics interpretation.
So what do I know for now? Quantum mechanics is locality-independent. This means that considerations of locality have no role whatsoever in deriving quantum mechanics. Quantum correlations which go above Bell's limit are a mathematical consequence of the quantum formalism. Do I find correlations above the Bell limit strange? Indeed I do, because as a living organism I am the result of a long natural selection process which favors classical intuition as a necessary tool for survival. However, as a physicist, it is not the correlations alone which are troublesome to me, but correlations over spatially separated experiments. And if quantum correlations are natural mathematical consequences of the quantum formalism, what is in deep need of an explanation is the very idea of distance. This is a different paradigm from the one put forward by Bell.
We tend to take the idea of space or space-time, or locality, or neighborhood for granted because this how nature is and physics is an experimental science, but if everything is quantum mechanical at core, and if locality plays no role in deriving quantum mechanics, where does the metric tensor comes from? (I will attempt to show that this is the deep mystery, and not the correlations) The funny thing is that the answer is known and was arrived at by a completely unexpected route starting with a strange question: Can you hear the shape of a drum? Moreover, although there are exceptions, the answer is not well known to neither the quantum foundations community, nor to the string theory/high energy physics, and paradoxically, it is well known to mathematicians who uncovered an extremely rich mathematical domain: noncommutative geometry. I will start exploring this area in this post and continue the topic in subsequent ones.
So what is the shape of a drum question about? When you go to a symphonic concert, you can clearly identify pianos from trumpets, trumpets from drums, etc. Why? Because they sound different even when they play the very same note.
But now let's make the problem harder and pick the same type of instrument. The shape of the instrument determines the spectral characteristics of the sound. Can we solve the inverse problem? Do the eigenfrequencies uniquely determine the shape of the instrument? The answer is negative as counterexamples show. However we are onto something interesting here. Eigenvalues and eigenvectors naturally occur in quantum mechanics. Also although the answer is negative, we can tell an instrument from another, and therefore we must be missing just a little bit of information to solve the inverse problem. And if we are able to solve the inverse problem, we have succeeded into recovering the metric tensor information in a very different language and moreover this language is common to quantum mechanics as well.
Please stay tuned for the continuation next week.
If you think quantum correlations (which go above Bell limit) are in need of an explanation, then very likely you are in the ontic, beable, non-local camp. Personally I am not in this camp, and I am not in anyone's camp. What I am trying to do is reconstruct quantum mechanics from physical principles and then mathematically arrive at the correct quantum mechanics interpretation.
So what do I know for now? Quantum mechanics is locality-independent. This means that considerations of locality have no role whatsoever in deriving quantum mechanics. Quantum correlations which go above Bell's limit are a mathematical consequence of the quantum formalism. Do I find correlations above the Bell limit strange? Indeed I do, because as a living organism I am the result of a long natural selection process which favors classical intuition as a necessary tool for survival. However, as a physicist, it is not the correlations alone which are troublesome to me, but correlations over spatially separated experiments. And if quantum correlations are natural mathematical consequences of the quantum formalism, what is in deep need of an explanation is the very idea of distance. This is a different paradigm from the one put forward by Bell.
We tend to take the idea of space or space-time, or locality, or neighborhood for granted because this how nature is and physics is an experimental science, but if everything is quantum mechanical at core, and if locality plays no role in deriving quantum mechanics, where does the metric tensor comes from? (I will attempt to show that this is the deep mystery, and not the correlations) The funny thing is that the answer is known and was arrived at by a completely unexpected route starting with a strange question: Can you hear the shape of a drum? Moreover, although there are exceptions, the answer is not well known to neither the quantum foundations community, nor to the string theory/high energy physics, and paradoxically, it is well known to mathematicians who uncovered an extremely rich mathematical domain: noncommutative geometry. I will start exploring this area in this post and continue the topic in subsequent ones.
So what is the shape of a drum question about? When you go to a symphonic concert, you can clearly identify pianos from trumpets, trumpets from drums, etc. Why? Because they sound different even when they play the very same note.
But now let's make the problem harder and pick the same type of instrument. The shape of the instrument determines the spectral characteristics of the sound. Can we solve the inverse problem? Do the eigenfrequencies uniquely determine the shape of the instrument? The answer is negative as counterexamples show. However we are onto something interesting here. Eigenvalues and eigenvectors naturally occur in quantum mechanics. Also although the answer is negative, we can tell an instrument from another, and therefore we must be missing just a little bit of information to solve the inverse problem. And if we are able to solve the inverse problem, we have succeeded into recovering the metric tensor information in a very different language and moreover this language is common to quantum mechanics as well.
Please stay tuned for the continuation next week.
Hi Florin,
ReplyDeleteAs I understand the eigenvectors and eigenvalues give fantastic agreement of quantum theory with experiments. So unless you are talking about gravity, what metric tensor is necessary?
Gravity will enter the picture as well but not at the string theory level but at the Standard Model energies; it is quite an amazing thing how the story will turn out to be.
DeleteWhat we are after is an equivalent description of the information contained in the metric tensor, and we want to do it in a spectral way compatible with quantum mechanics.
Florin,
Delete"If you think quantum correlations (which go above Bell limit) are in need of an explanation, then very likely you are in the ontic, beable, non-local camp."
During discussing the subject 'GHZ for die-hard local realists" I've made the folowing points:
A
"Is it possible to have a distribution of charged particles satisfying the two propositions bellow?
1. Some charges/groups of charges evolve independently of other charges/groups of charges.
2. all charges move in agreement with Maxwell's theory.
If you think this is possible please exemplify. You can place the charges at any distance you want, in any configuration you want."
B
OK, so let’s only discuss the experiment from the subjective point of view of Alice. She measures her particle, she gets +1/2, she knows Bob’s particle must be -1/2. Do you agree that the spin of Bob’s particle, from the subjective point of view of Alice is well defined (-1/2)? Does she need to wait for Bob to perform his measurement and confirm it to her before calling the -1/2 value well-defined?"
First point, if not refuted proves that Bell's theorem does not apply to classical electromagnetism. The second (I have found that this experiment is Bohm's version of EPR) proves that locality and non-realism are incompatible.
So, the correct camps are:
locality+realism
non-locality + non-realism
"Do I find correlations above the Bell limit strange? Indeed I do, because as a living organism I am the result of a long natural selection process which favors classical intuition as a necessary tool for survival."
But the point is that the Bell correlations are not opposed to classical determinism. On the contrary, in a deterministic theory like classical electromagnetism everything is correlated with everything else so why be surprised? The wrong intuition is in neglecting modern classical field theories and thinking in terms of pre-field physics.
Andrei
Dear Andrei,
DeleteHow you implement correlations for Bell limit is irrelevant, and you can write actual computer programs any way you like but subject to Bell's demands and you will not be able to obtain correlations above his limit. This is because Bell's theorem is a provable mathematical statement regardless of the physics involved.
There are non-local realistic models able to beat Bell's limit and achieve quantum correlations, so your dichotomy: "locality+realism
non-locality + non-realism"
is incorrect.
Based on the very definition of "locality", one can argue QM is non-local, but this is semantics.
Dear Florin,
DeleteThe proof of Bell's theorem requires some assumptions. One of them is that the detector settings and the hidden variable are independent parameters (independence or free-will assumption). This assumption is false for classical electromagnetism as my argument A above shows. I will write it here again:
"Is it possible to have a distribution of charged particles satisfying the two propositions bellow?
1. Some charges/groups of charges evolve independently of other charges/groups of charges.
2. all charges move in agreement with Maxwell's theory.
If you think this is possible please exemplify. You can place the charges at any distance you want, in any configuration you want."
Now, if you cannot refute this argument you have to accept that classical electromagnetism is not within the scope of Bell's theorem so it can potentialy explain the experimentaly observed correlations.
In fact, even a clockwork mechanism could be built so that it produces any correlations you want above Bell's limit. The key for its success is that the source and the detectors cannot evolve independently for they are part of the same mechanism.
To make an analogy, the proposition that the sum of the angles of a triangle is 180deg is a proven mathematicasl statement, but it depends on the assumption that the triangle is drown on a plane surface. If you have a sphere the sum can be lower, if you have a saddle, the sum is higher. You need to be careful to only apply Bell's theorem on those physical theories that fall within its scope.
"There are non-local realistic models able to beat Bell's limit and achieve quantum correlations, so your dichotomy: "locality+realism
non-locality + non-realism"
is incorrect."
Sure, you are right. Given that local-realistic theories are possible and all non-local theories are possible it follows that all non-local realistic theories are possible. But my main point is that non-realistic local theories are impossible (My argument B above).
"Based on the very definition of "locality", one can argue QM is non-local, but this is semantics."
In my oppinion it doesn't make much sense to speak about locality in the absence of a space-time description of the phenomena described by the theory. QM predicts correlations between distant places but this fact alone is not enough to conclude that it is non-local because local theories like classical electromagnetism or general relativity also predict such correlations.
Andrei
I would say the question is this. Suppose you have a group G that describes the eigenvectors of the system. If these vectors are entangled states, then we have knowledge of the whole system in a sense “modulo” the individual quantum systems in the entanglement. Entanglement is a case where the whole is greater than the parts, where there are actually no ontological meaning to the existence of the parts.
ReplyDeleteI have been looking into how it is that spacetime is quantum mechanics in disguise. There is a lot about the conservation of qubits in the literature today. I prefer to think according to the symmetries of entanglement that in a Noetherian sense is equivalent to the conservation of quantum phase. A consequence of this is that quantum information is conserved.
This has a way of then looking at locality and nonlocality. The local properties of a quantum system are identical to the local properties of a classical system. The Schrodinger equation does describe the propagation of a wave or field through spacetime, and this is causal structure of spacetime for the development of local physical properties. Quantum mechanics on the other hand is nonlocal, and casts this system into the language of nonlocal properties. However, I think most of us have the sense that classical structures are an approximation, and this holds for spacetime as well. Spacetime is then ultimately built up from nonlocal entanglements. In this way the properties to any system that we describe a local is then really an approximation or maybe a sort of illusion.
If this or something similar works it would mean that spacetime as a metric geometry with local properties is a coarsed grained approximation. If I were to encapsulate this I would say
quantum gravity (global and nonlocal symmetries) = G/local symmetries,
for G the general group of symmetries.
Lawrence,
DeleteI am not an expert in quantum gravity so I cannot comment. What I will arrive will be the Standard Model weakly coupled with gravity (general relativity). Here gravity is not quantized, but QM is "geometrized". This work arose out of the classification of von Neumann algebras and advanced operator algebra theory.
Spacetime is built up from quantum entanglement. I would say that large N quantum entangled systems ---> geometry. In the end maybe everything is quantum mechanics, and frankly I am thinking it really is just good old fashioned QM for the most part. All the strings, things, branes etc are just "decorations."
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DeleteSorry guys, I was very busy and did not have a chance to respond.
ReplyDelete