Confused by nonlocal models and relativity

[Tendex]

Slitely off topic, but can you explain the difference between Einstein causality and signal causality.

Ha, that's a good one. Let me try and untie a little this cute knot. Signal causality is what's left when you take SR usual causality structure(i.e. Einstein causality or regular EPR local realism ) and in order to make it fit into a quantum theory with no classically predetermined measure values you only leave operational "no faster than light" signaling.
But, you do it by appealing to an essential element of Einstein causality of SR(in what is called "(anti)commutation of spacelike separated operators), i.e behaviour outside the light cone, which of course implies there must be an inside the SR light cone structure somewhere but chose to ignore it for the moment to the effect of rejecting causal realism as QFT is part of quantum theory and its experiments violate Bell's inequalities constructed according to causal realism.
And then of course people has a bit of confusion with all these terms like "real", " causal" and "local" which is not surprising since they usually are not even aware of all this mess.

 

[vanhees71]

Standard textbook QM is nonlocal if the quantum state is real: https://arxiv.org/abs/0706.2661

Whether the quantum state is real is a matter of interpretation, so one cannot say that 100% for sure QM is not nonlocal.

It depends on, what you mean by "nonlocal". Non-relativistic quantum mechanics is of course non-local since there are interactions at a distance as a rule rather than an exception. Relativistic quantum field theory by construction has only local interactions, i.e., the Hamiltonian density commutes with any local observable at space-like distances of their arguments ("microcausality").

All QT is "nonlocal" in the sense that there is entanglement which describes correlations which can be between far-distant parts of a quantum system. Einstein invented for this kind of "nonlocality" the notion of "inseparability". That's much more concise than to call that "nonlocality".

I'd plead to always say what one means by "local" or "nonlocal" (usually one shortly says relativistic QFT is "local" because of the locality of interactions) or better avoid this expression if not clearly specified.

 

[atyy]

Are you saying there is a single objective cut defined over all of space? Different observers aren't permitted to have different cuts?

Yes, different observers cannot have different cuts. So there is a single cut defined over all of space. Whether the cut is objective or subjective is not relevant. At this level, there is a single user or single pair of users (Alice and Bob). To them the quantum formalism will be consistent, and also predict the violation of the Bell inequalities. If they take the wave function to be real, then the formalism is nonlocal to them.

 

[Tendex]

Probability is simply part of logic, the logic of plausible reasoning. Such laws of reasoning are the base for evaluating scientific theories, thus, they have to hold in scientific theories, and if they don't, the theory should be rejected as logically inconsistent.

I agree with this provided one leaves room for the possibility that the current formal mathematical logic can be extended to a more general one for physical theories if at some point experiments lend themselves to a more comprehensive form of scientific theory. This is of course not my idea but was contemplated by people like Poincaré, Weyl and Brouwer.

 

[martinbn]

Yes, different observers cannot have different cuts. So there is a single cut defined over all of space. Whether the cut is objective or subjective is not relevant. At this level, there is a single user or single pair of users (Alice and Bob). To them the quantum formalism will be consistent, and also predict the violation of the Bell inequalities. If they take the wave function to be real, then the formalism is nonlocal to them.

So, what does it mean to take the wave function to be real!?

 

[Elias1960]

I think the author of the bible on QFT, Weinberg, might not agree with you on that.

Maybe. But if there is not even agreement about what is real in QM, and the minimal interpretation of QM remains silent about this, how can one expect agreement in QFT, which is, last but not least, the same QM applied to fields? The minimal interpretation of QFT will be even more minimal because it will exclude the Schroedinger picture too (given that it does not look Lorentz-covariant, and everything which is not Lorentz-covariant has to be hidden). Moreover, there is some additional disagreement about the ontology, namely particles vs. fields. There will be hardly an agreement about this.

This is a claim you've made before, that rejecting Kolmogorov's axioms is "logically impossible" or something like that.
We have now constructed generalized probability theories, not just QM such as PR box probabilities, so this just is not true. Your contention that classical probability is the only logically possible such theory is contradicted by the literature.
Other probability theories simply reject some of Cox's assumptions.

How many times I have to repeat the same trivialities? That you can invent mathematical structures that violate some of the rules of logic or probability theory is not questioned at all. What is questioned is that they are applicable as the laws of logic, including the logic of plausible reasoning. You cannot simply change the laws of reasoning.

(You can, of course, doubt some laws of reasoning, and refuse to apply them yourself. This will weaken your ability to argue reasonably, thus, will be quite stupid. But to invent new, different laws of reasoning will simply transform your reasoning into nonsense.)

Thank you that by adding "or something like that" you have at least admitted that I have not written the nonsense you have invented out of thin air.

EPR realism is ruled out by the highly accurate observations of the violation of Bell's inequality, while Einstein causality is very well confirmed by all observations.

Wrong. EPR realism is as well very well confirmed by all observations. What is falsified by the violation of Bell's inequality is the combination of EPR realism with Einstein causality, as well as the combination of a meaningful notion of causality at all (which has to include Reichenbach's common cause principle) with Einstein causality.

So, preserving Einstein causality given the violation of Bell's inequality requires one to give up realism in the extremely weak form of the EPR criterion of reality and also to reduce causality to positivistic signal causality, rejecting Reichenbach's principle of common cause. The tobacco industry will be happy with rejecting this horrible principle which forces them to find causal explanations for the correlations between smoking and lung cancer. Everybody else will simply ignore this, so that after this we have two scientific methods, one applicable for essentially everything, which considers it a necessity to find causal explanations of observable correlations, and a special exception for the violation of the Bell inequality, where one can simply ignore them, given that every causal explanation would violate Einstein causality, thus, would be anathema.

To read

... nor has it been convincingly extended to relativistic QT.

in the answer to a post where the standard reference to this extension has been given makes me wonder if giving such references makes sense at all.

Slitely off topic, but can you explain the difference between Einstein causality and signal causality.

Signal causality means if $A \to B$ then you can also send a signal from A to B. That means, doing something at A changes some probability at B. The violations of the Bell inequalities do not allow such signaling, because they allow for two causal explanations, either $A \to B$ or $B \to A$. If this could be used to send a signal from A to B, the explanation $B \to A$ would be impossible.

 

[martinbn]

Signal causality means if $A \to B$ then you can also send a signal from A to B. That means, doing something at A changes some probability at B. The violations of the Bell inequalities do not allow such signaling, because they allow for two causal explanations, either $A \to B$ or $B \to A$. If this could be used to send a signal from A to B, the explanation $B \to A$ would be impossible.

But my question was about the difference between this and Einstein causality. They still seem the same to me. Einstein causality means that if $A \to B$, the A is in tha past lightcone of B, so you can send a signal from A to B. If you can send such a signal, then A is in the past of B. So, what is the difference?

 

[DarMM]

Yes, different observers cannot have different cuts. So there is a single cut defined over all of space. Whether the cut is objective or subjective is not relevant. At this level, there is a single user or single pair of users (Alice and Bob). To them the quantum formalism will be consistent, and also predict the violation of the Bell inequalities. If they take the wave function to be real, then the formalism is nonlocal to them.

I think it is relevant. The usual understanding is that if the cut is subjective then you get contradictions from extended Wigner's Friend scenarios. If the cut is objective then you have Many-Worlds, just we also have a non-relativistic covariant degree of freedom defined over all of space which specifies where the worlds "stop". So it's either inconsistent or MWI + nonlocal degree of freedom.

There are other problems with textbook QM + $\psi$ directly describing a physical degree of freedom (Dirac-Von Neumann) unlike textbook QM + $\psi$ as a book keeping device for probabilities (Copenhagen). It's not really an interpretation anybody in Foundations holds to anymore.

 

[DarMM]

So, what does it mean to take the wave function to be real!?

Essentially that $\psi$ directly describes a physical degree of freedom in the system. It's a shorthand common in physics.

 

[DarMM]

How many times I have to repeat the same trivialities? That you can invent mathematical structures that violate some of the rules of logic or probability theory is not questioned at all. What is questioned is that they are applicable as the laws of logic, including the logic of plausible reasoning. You cannot simply change the laws of reasoning.

You can change the laws of probability, as evidenced by Quantum Probability. A probability theory that doesn't obey the Kolmogorov axioms or obey Cox's assumptions.

 

[Elias1960]

But my question was about the difference between this and Einstein causality. They still seem the same to me. Einstein causality means that if $A \to B$, the A is in tha past lightcone of B, so you can send a signal from A to B. If you can send such a signal, then A is in the past of B. So, what is the difference?

The difference is that if you have a 100% correlation between A and B spacelike separated, then in normal, classical causality this requires an explanation, either by some direct causal influence $A \to B$ or $B\to A$ or by some common cause $C\to A, C\to B$.

This requirement for a causal explanation of observable correlations has to be given up if one follows those who reject all explanations which use a preferred frame. But once no other explanations exist, one has to give up causality.

The difference is that if you require a causal explanation, then you have only two choices for spacelike separated events A and B: Either $A \to B$ or $B\to A$. Both explanations would violate Einstein causality, thus, Einstein causality would be falsified if this principle would not be given up. A common cause explanation $C\to A, C\to B$ is what the violation of the Bell inequality excludes.

 

[martinbn]

Essentially that $\psi$ directly describes a physical degree of freedom in the system. It's a shorthand common in physics.

That is still unclear to me. How could it be a degree of freedom! Anyway, since I don't know it I will stay out of it.

 

[martinbn]

The difference is that if you have a 100% correlation between A and B spacelike separated, then in normal, classical causality this requires an explanation, either by some direct causal influence $A \to B$ or $B\to A$ or by some common cause $C\to A, C\to B$.

This requirement for a causal explanation of observable correlations has to be given up if one follows those who reject all explanations which use a preferred frame. But once no other explanations exist, one has to give up causality.

The difference is that if you require a causal explanation, then you have only two choices for spacelike separated events A and B: Either $A \to B$ or $B\to A$. Both explanations would violate Einstein causality, thus, Einstein causality would be falsified if this principle would not be given up. A common cause explanation $C\to A, C\to B$ is what the violation of the Bell inequality excludes.

I think I am not asking my question clearly. I just want to know the definitions of the two different causalities that you use. Einstein causality and signal causality. What has to be concluded by Bell's theorem is a separate question.

 

[DarMM]

That is still unclear to me. How could it be a degree of freedom! Anyway, since I don't know it I will stay out of it.

Well how can anything describe a physical degree of freedom? I assume you understand the sense in which the electric field $E$ describes a physical degree of freedom?

Or do you mean it's more natural to take it as describing a probability distribution rather than a physical degree of freedom.

 

[martinbn]

Well how can anything describe a physical degree of freedom? I assume you understand the sense in which the electric field $E$ describes a physical degree of freedom?

Or do you mean it's more natural to take it as describing a probability distribution rather than a physical degree of freedom.

I am guessing I don't know what degree of freedom is. It is intuitive in mechanics. A point in the plane has two degrees of freedom. In electrodynamics the electric field has infinitely many degrees of freedom. But wouldn't have thought that the field itself is called a degree of freedom.

 

[DarMM]

Well let's make it simpler. You understand the sense in which $E$ is taken to describe an actual propagating physical system?

 

[martinbn]

Well let's make it simpler. You understand the sense in which $E$ is taken to describe an actual propagating physical system?

Yes.

 

[DarMM]

Then saying $\psi$ is real is a shorthand for saying that $\psi$ is similarly describing an actual propagating physical system.

As opposed to $\psi$ just describing the probabilities of various outcomes.

 

[martinbn]

Then saying $\psi$ is real is a shorthand for saying that $\psi$ is similarly describing an actual propagating physical system.

As opposed to $\psi$ just describing the probabilities of various outcomes.

That's is what I thought initially. But it makes no sense to me to say that $\psi$ an actual propagating physical system. Because the domain on which it is defined is not space-time. If you have more than two particles how do you makes sense of it?

 

[DarMM]

That's is what I thought initially. But it makes no sense to me to say that $\psi$ an actual propagating physical system. Because the domain on which it is defined is not space-time. If you have more than two particles how do you makes sense of it?

Oh I see. Yes, you run into problems like this once you say $\psi$ is real. It depends on the interpretation how exactly you get out of this. Some versions of MWI for example say that spacetime is simply an illusion caused by decoherence and only $\psi$ exists. I'm sure proponents of the various views can give details.

I myself have difficulty with $\psi$ being real. It literally is a mathematical generalization of a probability distribution, obeys theorems corresponding to a probability distribution, collapse is a generalization of Bayesian updating mathematically. It even obeys a de Finetti type theorem. Why a physical "wave" would obey de Finetti's theorem I find hard to understand.

 

[atyy]

I think it is relevant. The usual understanding is that if the cut is subjective then you get contradictions from extended Wigner's Friend scenarios. If the cut is objective then you have Many-Worlds, just we also have a non-relativistic covariant degree of freedom defined over all of space which specifies where the worlds "stop". So it's either inconsistent or MWI + nonlocal degree of freedom.

Well, but what if Wigner's Friend scenarios are impossible, because you never shift the cut once you make it.

There are other problems with textbook QM + $\psi$ directly describing a physical degree of freedom (Dirac-Von Neumann) unlike textbook QM + $\psi$ as a book keeping device for probabilities (Copenhagen). It's not really an interpretation anybody in Foundations holds to anymore.

But it is consistent if naive, and it is discussed in the paper https://arxiv.org/abs/0706.2661 by Harrigan and Spekkens that proposed the psi-ontic/psi-epistemic definitions that have been widely used. There he calls it ψ-complete.

"It will be useful for us to contrast hidden variable models with the interpretation that takes the quantum state alone to be a complete description of reality. We call the latter the ψ-complete view, although it is sometimes referred to as the orthodox interpretation."

"Specifically, we show that for models wherein the quantum state has the status of something real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. "

"Nonetheless, even if one ignores the non-separability of entangled quantum states, it is straightforward to showthat the manner in which such states are updated after local measurements implies a failure of local causality if one adopts a ψ-complete model "

 

[DarMM]

Oh certainly it is discussed in Spekkens paper there, I don't deny that. However those papers pre-date the extended Wigner's friend cases. It's also not an interpretation that is really held by anybody.

I'll say more soon, just want to look at some notes.

 

[Elias1960]

I think I am not asking my question clearly. I just want to know the definitions of the two different causalities that you use. Einstein causality and signal causality. What has to be concluded by Bell's theorem is a separate question.

To define causality seems too complex for a forum post. All you can expect is some short, sloppy description of the most important aspects.

Signal causality: You have something a human being can influence by his free decision, let's denote this with A. We have something which can be observed, denoted by B. If $P(B|A) \neq P(B)$, then we are able to send a signal from A to B. This concept of causality is completely positivistic, that there exist some causal connection $A\to B$ can, at least in principle, be proven by explicitly sending a signal. Moreover, to name this "causality" is essentially misleading because it is well-defined in a world without any causality in the usual meaning of the word, all that is used here is probability theory (if a signal-causal relation exists follows from some fact about observable probabilities). Ok, some notion of free will or at least some independence of the decisions of experimenters, to define the difference between what the experimenter is doing at A (making a decision what to send to B) and what the observer is doing at B (observing what should contain A's message).

Classical causality: I have no simple list of axioms, simply because I have not tried to find one. The conceptual difference is quite big, causality is a different structure of the world which is not directly observable, but what is supposed to explain what we observe in a non-mystical way. Causal explanations are always theoretical, hypothetical explanations, and go beyond what they are supposed to explain. "Signal causality" is, in this sense, no causality at all, but, instead, simply a description of some observation.

Important properties of causal explanations are: 1.) There is no causal influence into the past. As a consequence (and in situations where "the past" is not well-defined by observation) there are no causal loops. 2.) The axioms of a partial order relation. 3.) Free will decisions or at least variants of them (combining dice throwing, a free-will decision, a deterministic random number generator, some random quantum effect, and characteristics of light coming from different ends of the universe by some well-defined algorithm into a single decision) do not have a cause (no superdeterminism). 4.) And, of course, particular causal influences should correspond to what follows from the scientific laws, in particular, the laws of physics, forbidding most of the astrological "causal" influences (but not all, the date of birth has a lot of influence on weather and so on in the first year of life, which can have, at least in principle, a lot of influence on character-forming and so on).

But the most important thing is what makes a causal explanation a necessity, or a missing causal explanation an open scientific problem. Without such a necessity, causality could be simply ignored. The "but this has no explanation" could be answered by a simple "so what?". And here the key principle is Reichenbach's principle of common cause. Every nontrivial correlation requires a causal explanation.

Einstein causality I have not even mentioned, because it comes in those two variants too.

 

[atyy]

Oh certainly it is discussed in Spekkens paper there, I don't deny that. However those papers pre-date the extended Wigner's friend cases. It's also not an interpretation that is really held by anybody.

Well, what to me seems unintuitive about what you are saying is that you seem to be saying that the minimal interpretation requires one to believe that psi is not real, whereas typically we say that the minimal interpretation is agnostic.

Also, I don't understand why the extended Wigner friend scenarios have anything to do with it. I thought we agreed the papers don't make sense from a Copenhagen viewpoint. And from a Copenhagen viewpoint, it doesn't seem as if one has to to be committed anyway to the reality or non-reality of the wave function. Maybe just think of it as two levels of reality. There is the outer reality, and the inner reality which is just a tool to compute the probabilities of measurement outcomes, the latter part being exactly the same as a version of Copenhagen where the wave function is not real.

I suspect that what you are saying is that the minimal interpretation cannot emerge from Bohmian Mechanics by definition.

 

[DarMM]

Well, what to me seems unintuitive about what you are saying is that you seem to be saying that the minimal interpretation requires one to believe that psi is not real, whereas typically we say that the minimal interpretation is agnostic.

No, for example you might leave open that Bohmian Mechanics and MWI are true.

Also, I don't understand why the extended Wigner friend scenarios have anything to do with it. I thought we agreed the papers don't make sense from a Copenhagen viewpoint. And from a Copenhagen viewpoint, it doesn't seem as if one has to to be committed anyway to the reality or non-reality of the wave function

They make sense, they just don't invalidate Copenhagen. $\psi$ not being real is a fundamental part of the Copenhagen view in any exposition of it I've read. Either in the writings of Bohr, Heisenberg, Pauli, Peierls or more modern papers or courses like Matt Leifer's.

 

[Elias1960]

I agree with this provided one leaves room for the possibility that the current formal mathematical logic can be extended to a more general one for physical theories if at some point experiments lend themselves to a more comprehensive form of scientific theory. This is of course not my idea but was contemplated by people like Poincaré, Weyl and Brouwer.

In principle, indeed, I would not completely exclude the possibility that some empirical facts could influence the laws of reasoning.

Last but not least, our ability of reasoning is the result of evolution, which means, it is useful for our survival, but this usefulness is based on the quite small domain of circumstances that are important for the survival of humans and their predecessors. This classical argument is not without any value. At least in principle one cannot exclude the possibility that some completely strange observations made in a domain inaccessible to our predecessors could force us to modify even our laws of reasoning.

But this would require much more otherwise unexplainable evidence. In the actual world of modern physics, there is essentially nothing in contradiction with common sense at all, at least if one allows common sense compatible explanations of modern physics to be discussed (thus, not in this forum, which forbids to discuss even the original interpretation of SR). Everything in modern physics is compatible even with Newtonian/Kantian understanding of absolute space and time (with relativistic effects explained by a distortion of clocks and ruler by the gravitational field) and a completely classical ontology where the wave function described only our incomplete knowledge about the real configuration of the quantum system and the world around it. If even a rejection of classical spacetime theory and classical ideas about the ontology depend on heavy metaphysical decisions against common sense, there is no base for even questioning the laws of reasoning, which are yet on another, higher level.

Indeed, common sense would have no problem with accepting that Newtonian absolute space and time are simply wrong. Not even this is certainly falsified by any empirical evidence, theories which accept a Newtonian spacetime and explain relativistic effects by distortions of rulers and clocks exist, they even give the Einstein equations in some natural limit, thus, are quite plausibly viable.

Similarly for the quantum part. Common sense has no problem with accepting that beyond the classical configuration there is something more, say, some wave function, which really exists. But not even this follows from any experimental evidence. Caticha's entropic dynamics gives a completely classical picture of what really exists (the same as in a classical Lagrangian picture, a trajectory in the configuration space), and the wave function is, together with the Schroedinger equation, derived from entropic inference, as a function which describes simply our incomplete knowledge about the quantum system.

So, modern physics has viable interpretations which are compatible not only with the classical laws of reasoning (in their modern understanding, classical logic together with the logic of plausible reasoning) but even with old Newtonian absolute space and time as well as a completely classical ontology as used in the Lagrange formalism.

That means, while in principle I would acknowledge that a world where empirical evidence forces us to modify even the laws of reasoning, the actual world of modern physics is far away from such a situation.

Spoiler: What is missed in physics is the counterrevolution after the revolution

All revolutions in the political area have, after some time, experienced some sort of counterrevolution, sometimes explicitly, sometimes simply by the new elites silently reviving useful and necessary things which have been destroyed during the revolution.

Kuhn describes science in terms of having revolutions too. The characteristic of those scientific revolutions is that everything can be questioned. But, once the revolution was successful and a new theory found, what about all the things which have been unjustly questioned and rejected? If science would be a rational endeavor, it would be reasonable to try to revive them. Thus, given that it is unknown what has been rejected without justification, one should have tried to revive everything that has been rejected during the revolution. There would have been ideas that could be revived, and ideas where the revival appeared to be impossible. But this was not done.

To question everything during a scientific revolution is fine, adequate, justified. But once the revolution was successful, and the new theory was found, a similar counterrevolution has to start, which tries to make everything which has been rejected during the revolution compatible with the new theory. And this counterrevolution is completely missed. The scientific analogon of Jacobin terror continues up to now.

 

[atyy]

No, for example you might leave open that Bohmian Mechanics and MWI are true.

Well, perhaps we use the word interpretation in a different way. How about minimal formalism instead of minimal interpretation. To me, I would say that the minimal formalism emerges from BM, and that within the minimal formalism it is consistent to treat the wave function as real, but complete within the formalism - although it is incomplete from the point of BM. It is complete in the sense of completeness for the purpose of making predictions of the probabilities of measurement outcomes.

They make sense, they just don't invalidate Copenhagen. $\psi$ not being real is a fundamental part of the Copenhagen view in any exposition of it I've read. Either in the writings of Bohr, Heisenberg, Pauli, Peierls or more modern papers or courses like Matt Leifer's.

I would more typically say that there is neither a commitment to the wave function being real or not real. I think it is the meta-rule of Copenhagen that you must first set up the cut before you do anything else, and that you don't move the cut, that makes the extended Wigner friend scenarios irrelevant for Copenhagen. As long as you don't move the cut, it doesn't matter whether you think the wave function is real or not.

 

[DarMM]

Well, perhaps we use the word interpretation in a different way. How about minimal formalism instead of minimal interpretation. To me, I would say that the minimal formalism emerges from BM, and that within the minimal formalism it is consistent to treat the wave function as real, but complete within the formalism - although it is incomplete from the point of BM. It is complete in the sense of completeness for the purpose of making predictions of the probabilities of measurement outcomes.

I'm not sure I fully follow, but the wavefunction of minimal QM comes from the Bohmian epistemic $\psi$ not the fundamental $\Psi$ of Bohmian Mechanics. This distinction matters in Frauchiger-Renner type scenarios.

I would more typically say that there is neither a commitment to the wave function being real or not real

I don't agree with this. Bohr, Heisenberg, Pauli, Peierls and many others explicitly state the wave-function is just a catalogue of probabilities. More modern Copenhagen people say this. It's what's said about Copenhagen in the Perimeter Institute's Quantum Foundations courses no matter who teaches it each year.

See slide 11 here:
http://mattleifer.info/wordpress/wp-content/uploads/2018/05/Lecture26.pdf

 

[vanhees71]

I am guessing I don't know what degree of freedom is. It is intuitive in mechanics. A point in the plane has two degrees of freedom. In electrodynamics the electric field has infinitely many degrees of freedom. But wouldn't have thought that the field itself is called a degree of freedom.

I'm also lost. Now the apparently clear notion of "physical degree of freedom" is also blurred by philsophical unclear redefinitions.

In physics there are two kinds of "degrees of freedom", applying either to point-particle mechanics, where it is described as a finite number of independent configuration-space variables $q^k$ with $k \in \{1,\ldots f \}$, or field-degrees of freedom. E.g., the electromagnetic field is described by 6 field-degrees of freedom, e.g., the 3 electric and the 3 magnetic field components.

Now they start to claim that the quantum mechanical wave function $\psi$ is a "degree of freedom". What should that mean? The important point is that first of all this applies only to non-relativistic QM, where a single wave function makes sense as the representative of a $N$-paticle system with a fixed number of particles. It's a function of time and configuration space $(\vec{x}_1,\ldots,\vec{x}_N)$ and not a field to begin with. Also its meaning is very clear: $|\psi|^2$ is the probability density for the $N$-particle system to be in configuration space at $(\vec{x}_1,\ldots,\vec{x}_N)$ when measured at time, $t$. I don't see any way, which sense it should make to declare the wave function to be a "degree of freedom". This was a very early idea by Schrödinger, which was almost immediately given up, which made Schrödinger in fact very upset, but he had to give in finally, since there's no sensible way to define the wave function as some kind of field.

In relativistic physics since the particle number is not conserved (which is an empirical fact) you cannot use wave functions to begin with but have to introduce quantized fields. So far the only successful kind of QFTs are those with local interactions, i.e., using realizations of the proper orthochronous Poincare group using field operators which admit to define local observables which obey the microcausality constraints, which is realized by using fundamental field operators obeying either canonical equal-time commutator (bosons) or anticommutator (fermions) relations. Sometimes one talks sloppily about these quantized fields in terms of "field-degrees of freedom" though this is kind of misleading either. At least fermion fields are hard to interpret classically at all. From the path-integral formalism we know that such fields should be Grassmann valued functions, but what this should physically mean as a classical field theory, i.e., not as a calculational tool to evaluate fermionic path integrals, is not clear at all.

 

[akvadrako]

Why a physical "wave" would obey de Finetti's theorem I find hard to understand.

I don't know about Finetti's theorem specifically, but I suspect the connection between objective reality and probability theory is something like this: breaking the rules of probability is forbidden logically because it would lead to an inconsistency, i.e. probability should include everything possible. Perhaps the universe has the same set of restrictions. I was reminded of this idea in a recent paper by Adán Cabello, The problem of quantum correlations and the totalitarian principle.

This result fits with Wheeler’s thesis that the universe lacks of laws for the outcomes of some experiments and with Born’s intuition that quantum theory is a consequence of the non-existence of ‘conditions for a causal evolution’