How do entanglement experiments benefit from QFT (over QM)?

[DarMM]

What is the "canonical description of QM"? There is the minimal interpretation, which remains silent over such questions as if the $C^∗$ structure is something fundamentally important. Those who claim that the $C^∗$ structure is something fundamentally important support an interpretation beyond the minimal one and involve their personal QM philosophy. And naming this "canonical description of QM" (instead of giving it the appropriate name, like "Copenhagen interpretation" or whatever) could be even suspected to be an attempt to hide this.

This is daft though. QM has a C*-algebra structure, that's just the mathematical formalism. If we can't say the mathematical formalism has properties because there might ultimately be another deeper theory one can basically say nothing about any theory.

"Does General Relativity have differentiable manifolds? Who knows there might be a deeper theory."

Statements like these mix up mathematical facts of the formalism with claims about ontology. I'm not interested in the latter. I'm saying the actual formalism that is used by most physicists, the actual C*-algebra set up of QM, has multiple sample spaces.

 

[DarMM]

BM interpretation making the same predictions (otherwise it wouldn't be a QM interpretation)

It possibly makes the same predictions as non-relativistic QM (there is no fully general proof that it does) ignoring Wigner's friend scenarios.

It simply doesn't make the same predictions with regard to QFT.

also has multiple sample spaces for its predictions and that doesn't prevent it from having one sample space for the much bigger set of all possible outcomes

It doesn't. See the random variables/response functions I mentioned to @atyy above.

 

[Elias1960]

The formalism itself does not have a single sample since it has a non-commutative $C^*$-algebra. This isn't interpretational. The number of sample spaces is a mathematical fact.

Bohmian Mechanics is an alternate formalism and ultimately a different theory. As a different formalism different mathematical statements are true about it. It has a single sample space of the form:
$$\mathcal{H}\times Q^{\otimes n}$$
where $Q$ is some manifold (generally orbifold) of particle positions.

This simply is mathematically a very different set up. QM itself has no such Hilbert-Orbifold state space.

While I agree that BM is ultimately a different theory (simply because the QM states are only a small subset of states of BM theory, namely quantum equilibrium states), what distinguishes different theories are different empirical predictions, not a different mathematical apparatus. In fact, one can use the mathematical apparatus of non-commutative $C^*$-algebra in BM too if one likes, the math formalism does not have patents or require licenses. So, BM restricted to quantum equilibrium states is the same theory as Schrödinger theory, given that it makes the same empirical predictions.

Of course, for particular quantum experiments, one can use the correspondingly reduced sample spaces restricted to the particular experiment. So the existence of these reduced versions is a triviality which proves nothing. And, given that the combination of all this into a single big sample space is a quite straightforward and trivial exercise if one recognized how it has to be done appropriately, and this construction is essentially theory-independent and works even for metatheoretical considerations, the point that such a common sample space exists is not even part of a theory, it is part of logical reasoning about such theories.

The choice of what defines the ontology is clearly interpretational. It exists only in realistic interpretations, and not in the mathematical apparatus itself.

 

[Tendex]

It possibly makes the same predictions as non-relativistic QM (there is no fully general proof that it does) ignoring Wigner's friend scenarios.

It simply doesn't make the same predictions with regard to QFT.

This is irrelevant for your own point, it is enough with the predictions of NRQM, the noncommutative algebra applies to them.

It doesn't. See the random variables/response functions I mentioned to @atyy above.

See the construction of random outcomes by Kochen and Specker quoted above and valid for any theory.

 

[DarMM]

In fact, one can use the mathematical apparatus of non-commutative $C^*$-algebra in BM too if one likes, the math formalism does not have patents or require licenses. So, BM restricted to quantum equilibrium states is the same theory as Schrödinger theory, given that it makes the same empirical predictions.

It is a conjecture that it is the same theory for a particular subset of non-rel QM. It's completely proven. As I've said it's acknowledged by Bohmians that it's not the same in Wigner's friend scenarios.

QM is distinguished from Bohmian Mechanics because it only has the C*-algebra strucutre and states upon it. Thus QM has multiple sample spaces.

 

[DarMM]

This is irrelevant for your own point

It is not. The most general type of algebraic structures we see even in non-relativistic physics are type-$III$ C*-algebra factors which Bohmian Mechanics cannot replicate. It is a conjecture that it replicates a Wigner Friend excluding subset of non-relativistic finite degree of freedom Quantum mechanics. This is what I mean, it's an alternate formalism that has a lot of work to do to replicate QM and we know it won't replicate QM in Wigner's friend scenarios. Thus it is a different theory, so why are we using it to object to facts about the mathematical structure of QM?

See the construction of random outcomes by Kochen and Specker quoted above and valid for any theory.

This is a complete non-sequitur. That construction shows what I am saying, an alternate formalism with infinitely many contextual variables for each variable in QM, all defined on a single sample space. It doesn't show Bohmian Mechanics also has multiple sample spaces.

 

[Tendex]

It is not. The most general type of algebraic structures we see even in non-relativistic physics are type-$III$ C*-algebra factors which Bohmian Mechanics cannot replicate. It is a conjecture that it replicates a Wigner Friend excluding subset of non-relativistic finite degree of freedom Quantum mechanics. This is what I mean, it's an alternate formalism that has a lot of work to do to replicate QM and we know it won't replicate QM in Wigner's friend scenarios. Thus it is a different theory, so why are we using it to object to facts about the mathematical structure of QM?This is a complete non-sequitur. That construction shows what I am saying, an alternate formalism with infinitely many contextual variables for each variable in QM, all defined on a single sample space. It doesn't show Bohmian Mechanics also has multiple sample spaces.

By definition of QM interpretation, (you are referring to an alternative Bohmian theory, not to the QM interpretation then) all QM interpretations make the same predictions and they can apply the noncommutative C*-algebra to them, so they have multiple sample spaces for their predictions.
Additionally, as Kochen&Specker assert explicitly, their single sample space can be applied to ANY theory(as long as it follows mathematical logic and deals with probabilities of course)

 

[atyy]

I don't agree with this. The formalism itself does not have a single sample since it has a non-commutative C*-algebra. This isn't interpretational. The number of sample spaces is a mathematical fact.

Bohmian Mechanics is an alternate formalism and ultimately a different theory. As a different formalism different mathematical statements are true about it. It has a single sample space of the form:
$$\mathcal{H}\times Q^{\otimes n}$$
where $Q$ is some manifold (generally orbifold) of particle positions.

This simply is mathematically a very different set up. QM itself has no such Hilbert-Orbifold state space.

One can be neutral on which formalism is physically correct, but not on the mathematical facts of each formalism. And it is a mathematical fact that QM has multiple sample spaces. Nobody in the actual literature phrases this as or considers it as interpretational.

It's just a matter of language, so I won't argue with that.

But consider orthodox QM with either wave function and collapse to be real, or not real. When the wave function is not real, the mathematics of orthodox QM is a non-commutative C*-algebra. Now take the wave function and collapse to be real, is the language of naive QM no longer a non-commutative C*-algebra?

 

[Elias1960]

It possibly makes the same predictions as non-relativistic QM (there is no fully general proof that it does) ignoring Wigner's friend scenarios.

The simple straightforward proof is sufficient. Wigner's friend scenarios are artificial constructions which are irrelevant for the comparison of empirical predictions.

It simply doesn't make the same predictions with regard to QFT.

Wrong. Once again the reference

Bohm.D., Hiley, B.J., Kaloyerou, P.N. (1987). An ontological basis for the quantum theory, Phys. Reports 144(6), 321-375

This is daft though. QM has a C*-algebra structure, that's just the mathematical formalism. If we can't say the mathematical formalism has properties because there might ultimately be another deeper theory one can basically say nothing about any theory.

You are free to say whatever you like about the mathematical formalism. Of course, a single sample space for the whole theory can be reduced to much smaller subspaces for particular experiments, so, there is nothing strange with multiple sample spaces. The nonsensical part is the claim that there does not exist a single one, if it has been explicitly constructed and presented.

You can also say that there exist no single sample space with particular additional properties related with that C*-algebra structure. No problem. But in this case, the objection that the C*-algebra structure may be of no fundamental interest is a reasonable objection. The single sample space which does not care about the C*-algebra structure has been explicitly presented. Moreover, the construction is essentially part of logic, not of the particular theory (it is the Stone space of the Boolean algebra of the meaningful propositions of the theory).

Why the non-existence of such a sample space with some additional properties, like some compatibility with some C*-algebra structure, is something worth to be mentioned, is something which has to be explained by the one who makes this claim.

I'm saying the actual formalism that is used by most physicists, the actual C*-algebra set up of QM, has multiple sample spaces.

And this information is quite irrelevant because for a given particular experiment one can always reduce the single sample space to various much smaller ones which fix all the information about that particular experiment. The information which would be problematic would be that no single one exists. The information that no single one with particular additional properties exists would be uninteresting too, if presented without any arguments that the additional structure is somehow fundamentally important.

 

[DarMM]

By definition of QM interpretation, (you are referring to an alternative Bohmian theory, not to the QM interpretation then

I'm referring to what Bohmian Mechanics is actually like. The meaning of the English word "interpretation" doesn't alter the mathematical facts. One could argue if "interpretation" is the correct word, but that's a separate issue.

Additionally, as Kochen&Specker assert explicitly, their single sample space can be applied to ANY theory

The resulting construction is not part of QM. One can use the machinery of the generalized Nash's embedding theorem to construct a 231-D Minkowski space in which to embed any spacetime from General Relativity, but the resulting Minkowski space is not part of GR.

Sure you can apply the construction, one can imagine building structures out of almost anything in QM by applying some construction to them. If the resulting object isn't part of the theory though, why does it matter?

 

[DarMM]

Wrong. Once again the reference

That doesn't replicate QFT. Even Bohmians say they cannot replicate QFT. Show me that being used to compute a weak force cross section.

And this information is quite irrelevant

The actual mathematical structure of the commonly used formalism is irrelevant? Well there is little I can say to that.

 

[DarMM]

But consider orthodox QM with either wave function and collapse to be real, or not real. When the wave function is not real, the mathematics of orthodox QM is a non-commutative C*-algebra. Now take the wave function and collapse to be real, is the language of naive QM no longer a non-commutative C*-algebra?

We know what you call "orthodox QM with wave function collapse as real" is self-contradictory.

Non-self contradictory approaches that have the wave-function as real require supplementation with additional variables and thus are different theories.

It's just a matter of language, so I won't argue with that.

I don't think it is. A formalism either has a mathematical property or it does not. Something being a mathematical property of a formalism is not a matter of language I would have said.

 

[DarMM]

I think at this point everything possible has been said.

I think it should be clear that QM the formalism itself is a generalized probability theory with a non-commutative C*-algebra.
(Opinions here that such a generalization is not sensible are crankish internet opinions not found in the literature)

This then causes there to be multiple sample spaces, which prevent decorrelating explanations and thus allows violations of Bell's inequality.

I'll close with that.

 

[Tendex]

I'm referring to what Bohmian Mechanics is actually like. The meaning of the English word "interpretation" doesn't alter the mathematical facts. One could argue if "interpretation" is the correct word, but that's a separate issue.

The mathematical fact I was pinpointing in the usual way is that all theories making the same predictions as QM regardless of their ontology can have multiple sample spaces, you got a problem with this?

The resulting construction is not part of QM. One can use the machinery of the generalized Nash's embedding theorem to construct a 231-D Minkowski space in which to embed any spacetime from General Relativity, but the resulting Minkowski space is not part of GR.

Sure you can apply the construction, one can imagine building structures out of almost anything in QM by applying some construction to them. If the resulting object isn't part of the theory though, why does it matter?

GR is(as I already told you) so far about 4-dimensional spacetimes so I guess you can leave out a 231-dimensional space but last I checked Hilbert spaces such as the used by Kochen&Specker with quantum observables and quantum states and explicitly claiming is a QT construction might have something to do with QM and be part of the theory, but who knows?

 

[Elias1960]

That doesn't replicate QFT. Even Bohmians say they cannot replicate QFT. Show me that being used to compute a weak force cross section.

Why would that be a problem? You somehow seem to think that Bohmians are not allowed all the mathematical apparatus of standard QFT. (It seems to be a property of many of your claims that you confuse interpretational questions with mathematical formalism which has nothing to do with the interpretation.)

The actual mathematical structure of the commonly used formalism is irrelevant? Well there is little I can say to that.

No, not the actual mathematical structure, but the claim that there exist multiple sample spaces. It is irrelevant because it is a triviality too. One can always reduce the single sample space for multiple particular experiments to multiple particular sample spaces. So, please read what I write, and don't cut necessary context in your quotes which already answers your polemical rhetorical question.

 

[Morbert]

Yes. The information you have is insufficient to identify the truth value. So what? It does not mean that it does not have a truth value.

A Boolean logic is an event algebra that follows from a sample space of elementary event propositions. If you construct a Boolean logic around a sample space appropriate for the $S_x$ measurement you made, you cannot include propositions like $S_z = \uparrow$, as you cannot construct a sample space that contains all the necessary elementary propositions about $S_x$ and $S_z$. I.e. It's not just that the truth of the proposition is unknown. The proposition itself cannot be made.

You can, of course, build an alternative Boolean logic around an alternative sample space that contains elementary propositions like $S_z = \uparrow$, but this logic necessarily excludes propositions logically equivalent to the measurement outcome you just observed, and so cannot be used in the context of the measurement result you are analysing.

 

[Jimster41]

The information you have is insufficient to identify the truth value. So what? It does not mean that it does not have a truth value.

I thought it was more like @Morbert is saying. Re the the cat being either dead, alive or dead and alive - There are a incompatible clauses in Boolean algebra else why the XOr and it’s not the same as the t-shirt being red or green or red and green.

Additionally, as Kochen&Specker assert explicitly, their single sample space can be applied to ANY theory(as long as it follows mathematical logic and deals with probabilities of course)

That’s what I thought experiments in QM show (and our empirical reality reinforces) there are justifications for taking a set of proposable propositions and carving them down to plausible ones via formalism. Your positions feels like it’s advocating for The supremacy of the first important Cantor-like metafying step - the collection of all proposals we can propose (including “the perp was a cat that was entirely red and entirely green”). Sure it’s an important step but so is the sorting out of what the algebra of empirical reality supports.

The simple straightforward proof is sufficient. Wigner's friend scenarios are artificial constructions which are irrelevant for the comparison of empirical predictions

I’m a bit surprised to hear you say this. To me it seems like you’ve been taking the exactly opposite position w/respect to dinosaur guy the whole time. He’s saying a sharp description of QM formalism w/respect to the algebra type empirical observations support is key to discourse about it and you’ve been saying general propositional set building can be done on the propositions about observables to embed whatever distinction that process makes into a common and indistinguishable set.

Sometime I wish there was like a supermoderator who could come along and when closing these great (albeit contentious) threads do a bit of summarizing for the listeners - maybe some sound healing, a little deep breathing.

 

[DrClaude]

This thread has ran its course. Time to close.

Thanks to all that have participated.

 

[DrClaude]

As a closing note, @*now* would like to point out the following paper:

https://arxiv.org/pdf/1806.08150.pdf

The notion of locality in relational quantum mechanics
P. Martin-Dussaud, C. Rovelli, and F. Zalamea

The term ‘locality’ is used in different contexts with different meanings. There have been claims that relational quantum mechanics is local, but it is not clear then how it accounts for the effects that go under the usual name of quantum non-locality. The present article shows that the failure of ‘locality’ in the sense of Bell, once interpreted in the relational framework, reduces to the existence of a common cause in an indeterministic context. In particular, there is no need to appeal to a mysterious space-like influence to understand it.