An argument against Bohmian mechanics?

[Demystifier]

From time to time some discussions on PF give me the impression that there is something about statistical mechanics that I simply don't see. Maybe its because I'm just a master's student who is just learning it and I need time to see it the way you guys see it but I'm just curious what's going on. You know what I mean? If yes, can you provide a reference that can shed some light on these things you guys sometimes talk about?

See the links in post #12 of this thread.

 

[PeterDonis]

I don't think there is any content in saying that it is nonlocal

Sure there is; "nonlocal" means not factorizable, "local" means factorizable. Or, equivalently, nonlocal means "violates Bell inequalities", local means "satisfies Bell inequalities". It's easy to test. We've been over this.

 

[PeterDonis]

If all the uncertainty in the electron's trajectory depends only on it's initial position then Bell inequalities apply.

Nope. Bohmian mechanics is an explicit counterexample: each individual trajectory is deterministic, but its equation of motion includes the quantum potential, which is what allows it to violate the Bell inequalities (because the quantum potential leads to the QM prediction for joint probabilities of measurement results, which is not factorizable). Bell explicitly discussed this exact point.

 

[rubi]

Let's analyze the interplay between classical logic and probability theory.

In (a model of) classical propositional logic, we have n-ary predicates $P(x_1,\ldots,x_n)$ that (essentially, I don't want to introduce $\models$) map the elements of a set $X$ into the truth values $\top$, $\bot$. These predicates (applied to elements) constitute statements. We can build new statements by using logical connectives like $\neg$, $\wedge$, $\vee$ in the usual way and these statements will also attain truth values.

Now, we can represent this algebra of statements by subsets of $X$ in the following way: To each unary predicate $P(x)$, we associate a subset $U_P = \{x : P(x)\}$. Using some basic set theory, we can check that $U_{P\wedge Q} = U_P \cap U_Q$, $U_{P\vee Q} = U_P \cup U_Q$ and $U_{\neg P} = X \setminus U_P$.

Let's say, we are interested in a list of unary predicates about the set $X$. We can form the algebra of predicates, obtained from applying arbitrary logical connectives to that list. We can represent this algebra of predicates by a set algebra $\Sigma$ of subsets of $X$. This algebra can equivalently be obtained by taking arbitrary (finite) unions and intersections of the subsets $U_P$ associated to the predicates we're interested in. In order to make contact with probability theory, we need to allow for countably many unions and intersections and we need $X\in\Sigma$.

Associating probabilities to statements like $P(x)$ means specifying a probability measure on $\Sigma$, i.e. a function $p:\Sigma\rightarrow [0,1]$ that satisfies some basic axioms like $p(X)=1$. The triple $(X,\Sigma,p)$ forms a probability space.

We can define random variables on $X$ as (measurable) functions $A:X\rightarrow\mathbb R$. We will later be interested in the random variables
$$\chi_P(x) = \begin{cases}1 & P(x) \\ 0 & \neg P(x)\end{cases} \text{.}$$
Using these random variables, the statement $P(x)$ is equivalent to $\chi_P(x) = 1$. We can also check that $\chi_P(x)^2 = \chi_P(x)$.

On the set $\mathcal R$ of random variables, we can introduce the so called valuation maps $\nu_x : \mathcal R \rightarrow \mathbb R$, which just evaluate a random variable at $x$: $\nu_x(A)=A(x)$. If $X$ is not empty, then obviously at least one such $\nu_x$ exists. We can easily check that $\nu_x(AB)=\nu_x(A)\nu_x(B)$ and $\nu_x(A+B)=\nu_x(A)+\nu_x(B)$. If we have $\nu_x(A^2)=\nu_x(A)$, then it follows that $A(x)=1$ or $A(x)=0$, i.e. $A(x)$ is a function of the type $\chi_P(x)$.

We now want to know, whether we can represent all propositions about quantum systems using classical logic. So let's first define what we mean by a proposition about a quantum system. Given a Hilbert space $\mathcal H$, a quantum proposition $\hat P\in\mathcal P$ is just an orthogonal projector $\hat P:\mathcal H\rightarrow \mathcal H$ (i.e. $\hat P^\dagger = \hat P$ and $\hat P^2=\hat P$).

If quantum propositions $\hat P$ could be represented using classical logic, then there would be a map $r:\mathcal P\rightarrow \mathcal R$ of the quantum propositions into the random variables on some probability space $(X,\Sigma,p)$. On a classical probability space, there exists at least one valuation map $\nu_x$. If a such a map $r$ did exist, then we could define a valuation map on the set of quantum propositions by $\nu=\nu_x\circ r$. Since commuting quantum propositions behave like propositions in classical logic, we would require $\nu$ to behave like a classical valuation map at least for commuting observables, i.e. we would require $\nu(\hat P+\hat Q)=\nu(\hat P)+\nu(\hat Q)$ and $\nu(\hat P\hat Q)=\nu(\hat P)\nu(\hat Q)$. From that, we can also be sure that quantum propositions would be mapped to classical propositions, because $\nu(\hat P^2)=\nu(\hat P)$ implies that $r(\hat P)=\chi_P$ for some $P$.

Now here comes the surprise: A well-known no-go theorem (Kochen-Specker) says that for $dim(\mathcal H)>2$, no such valuation map $\nu$ with these properties exists! Not all propositions about quantum systems can be embedded into classical logic! If quantum mechanics is correct, we cannot reason about properties of quantum systems with only the laws of classical logic. This is independent of the interpretation and thus also applies to Bohmian mechanics.

Such a probability cannot be assigned in standard QM, but it does not imply that it cannot be assigned in any theory, perhaps some more fundamental theory (yet unknown) for which QM is only an approximation.

The above considerations show that we cannot hope for this unless quantum mechanics is incorrect.

There should exist general logical rules which can be applied to any theory of nature, not only to a particular theory (such as QM) which, after all, may not be final theory of everything. In such a general logical framework, the statement $S_x=+1\wedge S_y=-1$ must not be forbidden.

Such a theory cannot make the same predictions as quantum mechanics.

What can such a theory look like? Well, the minimal Bohmian mechanics is not such a theory, because minimal BM cannot associate a probability with $S_x=+1\wedge S_y=-1$. This is because spin is not ontological in minimal BM. However, there is a non-minimal version of BM (see the book by Holland) in which spin is ontological and $S_x=+1\wedge S_y=-1$ makes perfect sense. Nevertheless, this theory makes the same measurable predictions as standard QM.

This must necessarily use the $d=2$ loophole and can't be extended to all quantum systems.

I am not saying that this non-minimal version of BM is how nature really works. Personally, I think it isn't. What I am saying is that there is a logical possibility that nature might work that way. Therefore it is not very wise to restrict logical rules to a form which forbids you to even think about such alternative theories.

You can of course leave this possibility open, but then you must propose a theory that disagrees with quantum mechanical predictions.

 

[RockyMarciano]

Sure there is; "nonlocal" means not factorizable, "local" means factorizable. Or, equivalently, nonlocal means "violates Bell inequalities", local means "satisfies Bell inequalities".

Ok, I see where the confusion comes from, according to your definitions you are using the word "local" for what I'm calling "classical realistic", and is represented mathematically by the Bell inequalities. And you are calling "nonlocal" what I define as "nonrealist". I think my terminolgy is more adequate to the mainstream use, as used for example in the recent Nature paper on Bell violation by Hansen. Although I've seen the terminology "nonlocal" for theories violating the Bell inequalities in pop-science literature, I prefer to keep the term nonlocal for theorias that allow FTL signaling, like in the Nature paper.
So when I say nonlocal doesn't add anything meaninful to realist I really mean that "nonrealist realist" theories don't make much sense.

 

[PeterDonis]

I see where the confusion comes from

Yes, it comes from trying to use ordinary language instead of math. As I've already pointed out.

I think my terminolgy is more adequate to the mainstream use

The definition I gave for "nonlocal" is the one I've seen in every paper I've read on the Bell inequalities (not surprising since it was Bell's definition). None of them have equated "nonlocal" with "allows FTL signaling". I haven't read the recent paper by Hansen that you mention; do you have a link?

As far as usefulness, defining "nonlocal" as "allows FTL signaling" doesn't seem very useful to me, since by that definition we have no nonlocal theories that are not ruled out by experiment.

 

[RockyMarciano]

Yes, it comes from trying to use ordinary language instead of math. As I've already pointed out.

The math content here is in the Bell inequalities and it is clear enough.

The definition I gave for "nonlocal" is the one I've seen in every paper I've read on the Bell inequalities (not surprising since it was Bell's definition). None of them have equated "nonlocal" with "allows FTL signaling". I haven't read the recent paper by Hansen that you mention; do you have a link?

As far as usefulness, defining "nonlocal" as "allows FTL signaling" doesn't seem very useful to me, since by that definition we have no nonlocal theories that are not ruled out by experiment.

I linked it in #230. There "local" is defined as "not allowing FTL signaling". I'm not sure they define "nonlocal" explicitly, I'm defining it by oppostion to "local".

 

[RockyMarciano]

Nope. Bohmian mechanics is an explicit counterexample: each individual trajectory is deterministic, but its equation of motion includes the quantum potential, which is what allows it to violate the Bell inequalities (because the quantum potential leads to the QM prediction for joint probabilities of measurement results, which is not factorizable). Bell explicitly discussed this exact point.

But that is because Bohmian mechanics has an equilibrium hypothesis that allows the Born rule(not a postulate in BM but derived with this hypothesis, which directly contradicts its deterministic part in practice but that allows it to claim all the predictions of QM coincide with BM observable predictions.

 

[PeterDonis]

The math content here is in the Bell inequalities and it is clear enough.

Yes, but that's not the math content you're using to define "local" and "nonlocal".

There "local" is defined as "not allowing FTL signaling".

Yes, I see. So by their definition, QM is "local" but not "realistic".

 

[PeterDonis]

that is because Bohmian mechanics has an equilibrium hypothesis that allows the Born rule...

This is true but it doesn't change the fact that BM is a counterexample to the claim zonde was making.

 

[RockyMarciano]

Yes, but that's not the math content you're using to define "local" and "nonlocal".

I'm using the purely physical i.e. observable content of those terms.

Yes, I see. So by their definition, QM is "local" but not "realistic".

Well, yes,but as you say by that definition of local there are no physical theories that are nonlocal.

 

[RockyMarciano]

This is true but it doesn't change the fact that BM is a counterexample to the claim zonde was making.

Yes, in ordinary language, and it also confirms there is no physical content to calling a realist theory local or nonlocal, as those additions have no observable consequences even in principle for such theories.

 

[PeterDonis]

I'm using the purely physical i.e. observable content of those terms.

In other words, you're agreeing with me that you're not using the math content that you said was clear. You're using other math content--"no FTL signaling" can also be described precisely using math (and in fact it should be, since otherwise it's easy to misunderstand what it means in a curved spacetime, as numerous threads in these forums have shown).

 

[RockyMarciano]

In other words, you're agreeing with me that you're not using the math content that you said was clear. You're using other math content--"no FTL signaling" can also be described precisely using math (and in fact it should be, since otherwise it's easy to misunderstand what it means in a curved spacetime, as numerous threads in these forums have shown).

Do you mean the physics don't have math content? That's an odd assertion in a physics forum.

 

[RockyMarciano]

I explained in #368 how the math content of realism as reflected in the Bell inequalities bears on the observability of FTL, there is no observable content regardless of if it is claimed that a realist theory allows FTL(the Bohmian case) or it doesn't allow FTL(the "local realist case, because an experiment showing it would imply the use of FTL signaling).

 

[PeterDonis]

Do you mean the physics don't have math content?

No. I'm just trying to figure out which math content you are referring to when you talk about "no FTL signaling".

I explained in #368 how the math content of realism as reflected in the Bell inequalities bears on the observability of FTL

As far as I can tell, you are just pasting the labels "local", "nonlocal", "realist", "nonrealist" in various places but never actually saying what, physically, you think is observable or not observable.

Testing for FTL signaling is simple: send the same information in an ordinary light signal and in whatever other kind of signal you are trying to test for FTL signaling. If the information can be extracted from the latter signal at the receiver before the ordinary light signal arrives, you have FTL signaling; if not, not. EPR-type experiments test negative for FTL signaling by this criterion, because the full information contained in the correlations between measurement results is not available at either location until the result from the other location has been communicated by an ordinary light signal (or, in practice, something slower).

Testing for violations of the Bell inequalities is also simple; I don't think I need to elaborate on that here.

What else is there to test?

 

[zonde]

This was also my point in #283. The distinction local versus nonlocal has no empirical or operational meaning for classical realistic(i.e. hidden variables) theories because one can either consider they have nonlocal implicit influence(but no way to observe FTL signaling) as you say or if one decides the theory is local by the independent construction of the hidden variables there is no way to design an experiment to show this as it would be require the use of an instantaneous measure of spacelike separated events(i.e. FTL signaling) to do it, which is a strange way for a theory to be local .

Why do you think that experiment would need "instantaneous measure of spacelike separated events" to tell apart nonlocal implicit influence and local hidden variables? The only reason I could imagine would be that you take measurements as non factual (the MWI way).
And just to make sure that we are on the same page I would like to ask if you are familiar with this very simple counterexample type of Bell inequality: https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138

 

[Demystifier]

The above considerations show that we cannot hope for this unless quantum mechanics is incorrect.

Such a theory cannot make the same predictions as quantum mechanics.

You can of course leave this possibility open, but then you must propose a theory that disagrees with quantum mechanical predictions.

Well, QM and the theory I am talking about have the "same" predictions only in the FAPP (For All Practical Purposes) sense. In principle they differ, but in practice they do not.

Loosely speaking, this is like comparing thermodynamics with statistical physics. In thermodynamics the second law $dS\ge 0$ is an exact law. In statistical physics a deviation from $dS\ge 0$ is possible, but the probability for such a deviation is so small that it can be neglected FAPP. Bohmian mechanics (either minimal or non-minimal) is for QM what Boltzmann classical statistical mechanics was for thermodynamics at the end of 19th century. At that time atoms were "hidden variables" and Mach criticized Boltzmann's statistical mechanics for basing the theory on unobserved atoms. Boltzmann was so depressed by the fact that the mainstream physicists did not accept his theory that he eventually committed suicide. I think that's a very important lesson from the history of science.

 

[RockyMarciano]

As far as I can tell, you are just pasting the labels "local", "nonlocal", "realist", "nonrealist" in various places but never actually saying what, physically, you think is observable or not observable.

Then why do you avoid quoting the next sentences of my post where I actually explain what, physically, I say is not observable(I only spoke about non observability, that is about claims without physical content). I'll repeat it, the first case(non observability of FTL signaling in realist theories that claim FTL influence is explained by Demystifier in #141 so I guess is clear enough. The second case I explained is the non empirical content of claiming that a realist theory doesn't allow FTL influence. The reason that is claimed for a so called "local hidden variables" theory is that the assumption is made that a realist theory has predetermined values for all possible measurement outcomes, but this assumption could only be validated in practice by designing an experiment that performed simultaneous measurements in spacelike separated points. and such an experiment requires FTL signaling.
I guess if you quote this post cleverly enough again you can claim once more that I never actually explained what I think is not observable for realist theories, but again why would you do it?

 

[stevendaryl]

Sure, they just measure the x component of the spin. What I described, is what's usually called "collapse", i.e., the preparation of an observable to have a determined value by filtering, which indeed means just to block all the unwanted states.

The whole point of EPR was to distinguish between the quantum collapse from various of the unproblematic variants.

 

[RockyMarciano]

Why do you think that experiment would need "instantaneous measure of spacelike separated events" to tell apart nonlocal implicit influence and local hidden variables?

No, not to tell apart, there is no experiments to tell apart the claim of "FTL influence" from "non FTL influence" for hidden variables theory if in those theories those claims have no empirical content.
I said that the claim of "no FTL influence" in a hidden variables theory would need "instantaneous measure of spacelike separated events" to be substantiated, because the claim is made based on the assumption of independence of the hidden variables at each spacelike separated point, in other words that the values of all possible measurement results are predetermined, and the only way to test this predetermination in principle is by an experiment that measures spacelike separated events instantaneously(so one can verify the predetermination of results for all possible measurements at any point) and that would require FTL signaling.
I really don't think this is so difficult to see, one just have to acknowledge that hidden variable theories assume the classical galilean relativity with infinite speed of propagation of information and if you think about it that is the only way they can ensure a claim about non influence of one measurement on the rest of measurements separated spatially necessary for the hidden variables to determine measurement outcomes.

The only reason I could imagine would be that you take measurements as non factual (the MWI way).

Nothing to do with this. See above.

And just to make sure that we are on the same page I would like to ask if you are familiar with this very simple counterexample type of Bell inequality: https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138

Sure but I'm afraid they are using the term nonlocal there to mean what in the Nature paper I linked called local nonrealist.

 

[stevendaryl]

I can't quite follow this,( but I'm trying my best). If the electron were in a superposition of u and d states wouldn't it have needed to have been prepared in a definite state of l or r to be guaranteed to produce a random result in the u/d direction? If the up coefficient were bigger than the down coefficient (so to speak) would it not be expected to migrate one way rather than the other?

Sorry for not responding to this sooner. In the EPR experiment, Alice and Bob each have probability exactly 1/2 of measuring spin-up along whatever direction they choose to measure spins. The mathematical explanation is that neither Alice's nor Bob's particle is in a superposition, but the two-particle composite system is in the state $\frac{1}{\sqrt{2}} (|u\rangle |d\rangle - |d\rangle |u\rangle)$.

I'm not sure if that answers your question...

 

[zonde]

I said that the claim of "no FTL influence" in a hidden variables theory would need "instantaneous measure of spacelike separated events" to be substantiated, because the claim is made based on the assumption of independence of the hidden variables at each spacelike separated point, in other words that the values of all possible measurement results are predetermined, and the only way to test this predetermination in principle is by an experiment that measures spacelike separated events instantaneously(so one can verify the predetermination of results for all possible measurements at any point) and that would require FTL signaling.

You are mixing theory with reality. A theory is what we put into it. There is no hidden content. If a theory says that such and such phenomena is explained by such and such mathematical model (plus correspondence rules) then that's all about it. And we test the theory by verifying it's predictions. We can't prove that some theory is universally valid for any conceivable measurement.

 

[Demystifier]

@stevendaryl are you now satisfied with #371?

 

[stevendaryl]

@stevendaryl are you now satisfied with #371?

I'm still a little confused about Bohmian mechanics and probabilities, in the case of spin-1/2.

Empirically, we have the following situation:

1. You prepare a beam of electrons and filter out only those that are spin-up in the direction $\hat{a}$ (by sending them through a Stern-Gerlach device and only considering those that deflect in a particular direction).
   
2. You perform a second filtering, and filter out those that are spin-up in the direction $\hat{b}$.
3. The fraction of those who pass both filters is given by: $cos^2(\frac{\theta}{2})$ where $\theta$ is the angle between $\hat{a}$ and $\hat{b}$. This can also be written, using a trigonometric identity as: $\frac{1}{2}(1+cos(\theta)) = \frac{1}{2}(1+ \hat{a} \cdot \hat{b})$, if $\hat{a}$ and $\hat{b}$ are unit vectors.

Now, we try to explain this using a local deterministic hidden-variable theory (we'll get to the nonlocal version in a moment). That means we assume that there is a variable $\lambda$ attached to each electron, and it's value is distributed according to some probability distribution $P(\lambda)$, and this variable determines whether the electron passes the second filter. Then to agree with predictions, the set $V_{\hat{b}}$ of all values of $\lambda$ such that the electron will pass the second filter must have a measure equal to $\frac{1}{2}(1+ \hat{a} \cdot \hat{b})$. We can go through a Bell-type argument to show that there is no probability distribution $P(\lambda)$ that can simultaneously give this measure to each set $V_{\hat{b}}$ for all possible values of $\hat{b}$. (We can actually do better than that, and find three values of $\hat{b}$, and prove that there is no probability distribution that works for those three values.)

So my question is: how does allowing nonlocal interactions change this story? I think I just need to work this out myself. Presumably, allowing nonlocal interactions makes the problems of the impossible probability distribution disappear, but at the moment, I'm not sure I can put together why.

 

[Demystifier]

I'm still a little confused about Bohmian mechanics and probabilities, in the case of spin-1/2.

Empirically, we have the following situation:

1. You prepare a beam of electrons and filter out only those that are spin-up in the direction $\hat{a}$ (by sending them through a Stern-Gerlach device and only considering those that deflect in a particular direction).
   
2. You perform a second filtering, and filter out those that are spin-up in the direction $\hat{b}$.
3. The fraction of those who pass both filters is given by: $cos^2(\frac{\theta}{2})$ where $\theta$ is the angle between $\hat{a}$ and $\hat{b}$. This can also be written, using a trigonometric identity as: $\frac{1}{2}(1+cos(\theta)) = \frac{1}{2}(1+ \hat{a} \cdot \hat{b})$, if $\hat{a}$ and $\hat{b}$ are unit vectors.

Now, we try to explain this using a local deterministic hidden-variable theory (we'll get to the nonlocal version in a moment). That means we assume that there is a variable $\lambda$ attached to each electron, and it's value is distributed according to some probability distribution $P(\lambda)$, and this variable determines whether the electron passes the second filter. Then to agree with predictions, the set $V_{\hat{b}}$ of all values of $\lambda$ such that the electron will pass the second filter must have a measure equal to $\frac{1}{2}(1+ \hat{a} \cdot \hat{b})$. We can go through a Bell-type argument to show that there is no probability distribution $P(\lambda)$ that can simultaneously give this measure to each set $V_{\hat{b}}$ for all possible values of $\hat{b}$. (We can actually do better than that, and find three values of $\hat{b}$, and prove that there is no probability distribution that works for those three values.)

So my question is: how does allowing nonlocal interactions change this story? I think I just need to work this out myself. Presumably, allowing nonlocal interactions makes the problems of the impossible probability distribution disappear, but at the moment, I'm not sure I can put together why.

In this example there is no entanglement and therefore there is no need for non-local hidden variables. Indeed, the Bohmian description of this case is local. What you call "Bell-type argument" above, should really be a Kochen-Specker type argument. In other words, the non-trivial property of hidden variables we need here is contextuality, not non-locality. The notion of Bell non-locality includes also contextuality, but the converse is not true; contextuality does not necessarily include Bell non-locality. Contextuality means that your $P(\lambda)$ changes when the experimental setup changes. But it is local, in the sense that this change can be described by a local hidden variable theory.

 

[zonde]

Nope. Bohmian mechanics is an explicit counterexample: each individual trajectory is deterministic, but its equation of motion includes the quantum potential, which is what allows it to violate the Bell inequalities (because the quantum potential leads to the QM prediction for joint probabilities of measurement results, which is not factorizable).

I suppose I could get some understanding by comparing two conditional wavefunctions of electron form entangled pair where the spin of other electron is already measured along $\vec a_y$ or $\vec a_z$. Simpler explanations (like wikipedia or this Goldstein's article) of BM give only explanations for spinless conditional wavefunctions. Anyways measurement of remote entangled electron shouldn't change configuration (position) of the first electron directly. Any change is only trough (conditional) wavefunction, as it seems to me.

Bell explicitly discussed this exact point.

Is this Bell's explanation in book? Or maybe there is some online link?

 

[stevendaryl]

In this example there is no entanglement and therefore there is no need for non-local hidden variables. Indeed, the Bohmian description of this case is local. What you call "Bell-type argument" above, should really be a Kochen-Specker type argument.

Well, the Bell argument "factors" into two different arguments:

1. The perfect correlations/anti-correlations between distant pairs implies (under the assumption of local realism) that there is a variable $\lambda$ that deterministically decides the outcome of every possible spin measurement.
2. Given #1, the problem becomes to come up with a function $A_{\alpha, \lambda}$ that returns $\pm 1$ for every value of $\lambda$ and which has the correct statistics.

The second problem is no different, depending on whether you're dealing with entangled pairs or just trying to explain the result of measurements of a single electron.

In other words, the non-trivial property of hidden variables we need here is contextuality, not non-locality. The notion of Bell non-locality includes also contextuality, but the converse is not true; contextuality does not necessarily include Bell non-locality. Contextuality means that your $P(\lambda)$ changes when the experimental setup changes. But it is local, in the sense that this change can be described by a local hidden variable theory.

Yes, a single experiment can easily be explained using hidden variables, but we are only interested in those explanations that can be extended to the twin-pair problem. It's no use to come up with an explanation that we know (from other information) is false.

But what I'm asking for is how allowing contextuality changes the mathematics. What's an example of a contextual explanation?

 

[Demystifier]

What's an example of a contextual explanation?

Bohmian mechanics. To show that BM leads to the same measurable predictions as standard QM, BM needs to take into account the wave function of the measuring apparatus (WFMA). (Standard QM can also take into account the same WFMA, but with some Copenhagen-like doctrine the predictions of standard QM can be obtained even without WFMA.) This WFMA depends on the measurement setup, which makes it contextual. The motion of Bohmian particles also depends on this wave function, so contextuality at the level of wave functions translates into contextuality of particle trajectories.

 

[stevendaryl]

Bohmian mechanics.

I guess what I was asking for was something more abstract. In terms of the hidden-variables story that I gave, how do things get modified by allowing contextual or nonlocal measurements? Presumably, there is still a $\lambda$, and there is still a probability distribution on $\lambda$. So what gets modified?

 

[Demystifier]

I guess what I was asking for was something more abstract. In terms of the hidden-variables story that I gave, how do things get modified by allowing contextual or nonlocal measurements? Presumably, there is still a $\lambda$, and there is still a probability distribution on $\lambda$. So what gets modified?

When the measurement setup is changed, then it is $P(\lambda)$ that gets modified.

 

[stevendaryl]

When the measurement setup is changed, then it is $P(\lambda)$ that gets modified.

That doesn't seem right. $\lambda$ is a variable permanently associated with each electron (in Bohmian mechanics, it would be the position of the electron at some reference time, $t_0$). So the values of $\lambda$ are chosen before the choice of the detector setting. Maybe the sets $V_{\hat{b}}$ are changed when you change the detector settings?

 

[RockyMarciano]

... We can't prove that some theory is universally valid for any conceivable measurement.

I'm not sure I understand what you posted but it seems you missed my point, I'm saying that there should be falsifiable content in a theory, in other word the physical claims of a theory should be testable in principle, if a hidden variables theory claims to allow or not allow FTL influences that should be testable or it has no meaning in addition to having hidden variables determining the measurement outcomes. This has nothing to do with proving anything, physical theories are not proven, just confirmed until empirical info discards them, but first it must have empirically testable meaning for its claims at least in principle, or else those claims are scientifically meaningless.

 

[RockyMarciano]

I guess what I was asking for was something more abstract. In terms of the hidden-variables story that I gave, how do things get modified by allowing contextual or nonlocal measurements? Presumably, there is still a $\lambda$, and there is still a probability distribution on $\lambda$. So what gets modified?

I think my #393 could help here, quantum contextuality is basically the way some hidden variables interpretations of QM get around the contradiction between their deterministic nature and the non-deterministic nature imposed on QM by the Born rule, otherwise they wouldn't have the same results and couldn't claim to be just an interpretation.
To answer your question nothing gets modified mathematically by contextuality, just the end result, the classical probabilities of deterministic theories are switched in an ad hoc way by the non-deterministic probabilities produced by the Born rule(contradicitng all the mathematical arguments associated to hidden variables and the deterministic narrative). I would say for anyone except the proponents of such interpretations the arbitrariness should be more than evident.
It is just a way to circunvent the Kochen-Specker theorem by introducing the caveat that the hidden variables can behave for predictions as dictated by QM's Born rule postulate

 

[RockyMarciano]

That doesn't seem right. $\lambda$ is a variable permanently associated with each electron (in Bohmian mechanics, it would be the position of the electron at some reference time, $t_0$). So the values of $\lambda$ are chosen before the choice of the detector setting. Maybe the sets $V_{\hat{b}}$ are changed when you change the detector settings?

But note that when they are allowed to change depending on context(detector settings), that is no longer a hidden variables model.