# Violation of Bell inequalities for classical fields?

In the model above post # 45 the elapsed time/space like separation between measurements with any relative angle does not apply. For every angle Alice sets at detector A there is a pre existing / hidden variable correlated outcome relative to any angle at detector B.
That sounds like superdeterminism which can never be refuted experimentally, hence is not a very useful model to have.

zonde
Gold Member
I am not claiming that it does happen; i am just claiming that the assumptions used to derive Bell-type inequalities are not satisfied by classical fields.
There are different derivations of Bell-type inequalities that use different assumptions. Your claim does not sound serious if it seems like you are unaware of these other ways of deriving Bell-type inequalities.
Hence Bell-type argument and the experimental verification of the inequalities rules out a theory satisfying the Bell locality assumption, But not a classical field theory that is local in the causal sense.
What is "local in the causal sense"? Is it this:
in a classical relativistic field theory the value of a field at a position x at time t (in a fixed foliation defining observer time) depends on the values of the field at all points at position in the past light cone of x at any fixed earlier time. This allows Bell nonlocal behavior in a causally local field theory.

A. Neumaier
2019 Award
What is "local in the causal sense"?
Causal means respecting the causality relation induced by the Minkowski metric. For a classical field theory,
it means a family of Poincare invariant hyperbolic differential equations, leading to an upper bound of c for the signal velocity.
Hyperbolicity and Poincare invariance imply the statement you had highlighted.

unaware of these other ways of deriving Bell-type inequalities
I probably have seen all until about 2007, when I started to lose interest since it didn't cover field theories. Maybe you can point me to a more recent paper where I can find a set of assumptions that covers causal fields.

Professor Neumaier, in the example in your slides, why can't I use a classical field as a hidden variable?

I'm perplexed, and I want to understand this, because you said basically the same thing Bell said in La nouvelle cuisine.
in a classical relativistic field theory the value of a field at a position x at time t (in a fixed foliation defining observer time) depends on the values of the field at all points at position in the past light cone of x at any fixed earlier time.
This is what Bell said as well. But he then went on to say that, given all the information in the past light cone of x, any information in the region Y space-like separated from x should be irrelevant for any prediction that could be made about a measurement at x. This allows Bell to factorize the probability. But this is not the case in quantum mechanics because a measurement result in Y can provide information that is not available elsewhere that predicts a future outcome at x. So Bell violation should also rule out hidden variables that are classical fields.

A. Neumaier
2019 Award
given all the information in the past light cone of x, any information in the region Y space-like separated from x should be irrelevant for any prediction that could be made about a measurement at x.
Given all the information in the past light cone of x, any other information is irrelevant, no matter what the separation.

So Bell violation should also rule out hidden variables that are classical fields.
But the past light cone of x will intersect the past light cone of y in a very big region (as the experiment started with a joint preparation),
whereas the usual arguments silently assume that once the trajectories of the particles are separate (i.e., already after an infinitesimal time), their dynamics is independent. This is the case for point particles but not for extended objects such as waves.

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Nugatory
Thank you for your quick reply. Yes, a complete information from any slice of time t in the past light cone completely specifies what happens at x. But the dynamics from Maxwell's equations is local. So can you clarify what the problem with intersecting past light cones is? I fail to see how classical fields can behave in any way like the situation with entangled states that I described above.

A. Neumaier
2019 Award
I fail to see how classical fields can behave in any way like the situation with entangled states that I described above.
I don't see how they can behave like that either.

But I see that all arguments I know to derive testable inequalities (I studied this thoroughly until about 2007) make use of assumptions that are inspired by a point particle picture and do not apply to waves.

This just means that in order to rule out classical local hidden field theories one needs more careful arguments; maybe these exist but I haven't seen any. The situation is complicated since the analysis must be dynamical, unlike the simple arguments used for Bell-type inequalities.

Truecrimson
stevendaryl
Staff Emeritus
That sounds like superdeterminism which can never be refuted experimentally, hence is not a very useful model to have.
I don't know why people keep saying that. You can't refute the claim "Everything is determined ahead of time". But you can certainly refute a superdeterministic theory that makes definite predictions. If your superdeterministic theory predicts that "Bob and Alice will always choose the same detector setting", and they DON'T choose the same detector setting, then that particular superdeterministic model is refuted.

But I see that all arguments I know to derive testable inequalities (I studied this thoroughly until about 2007) make use of assumptions that are inspired by a point particle picture and do not apply to waves.
Wrong. There is nothing in the hidden variables which makes any assumption somehow "inspired" by point particles. The hidden variables may be anything, they may even live in 26 dimensional spaces or somewhere else, the other ingredients are the decisions of the experimenters what to measure and the results of the measurements, above are, obviously, macroscopic items, thus, in no way assume anything about microscopic theory.

A. Neumaier
2019 Award
The hidden variables may be anything
Of course, but in addition to the hidden variables there are the point-like trajectories. I am talking about the latter - these influence the arguments.

No, there are no point-like trajectories in the argument. There are two decisions, a and b, and two measurement results, A and B, and there is the assumption that a does not influence B and b does not influence A. No point-like trajectories necessary. Only the hidden variable, which also influences the measurement outcomes, $A(a,\lambda), B(b,\lambda)$.

A. Neumaier
2019 Award
there are no point-like trajectories in the argument.
These appear silently, in the arguments connecting the formal claims. What goes in there is rarely spelled out completely.
There must be some causal connection between what was prepared and what is measured, and this causal connection is usually described rather informally, using point particle intuition. For waves, the causal connection is complicated, hence there don't seem to be simple arguments.

All what one needs is that the general dependence $A(a,b,\lambda)$ can be reduced to $A(a,\lambda)$, because one assumes that the free decision b what to measure at B cannot causally influence the measurement result A. There is nothing implicit here, this is, completely explicit, Einstein causality. Once this assumption about the non-existence of a causal influence is made, we can continue and prove Bell's inequality.

wle
The propagation of fields violates the basic assumption of Bell-type arguments that systems in disjoint regions propagate independently once they are separated. in a classical relativistic field theory the value of a field at a position x at time t (in a fixed foliation defining observer time) depends on the values of the field at all points at position in the past light cone of x at any fixed earlier time. This allows Bell nonlocal behavior in a causally local field theory.
How so? What you say about classical relativistic fields here fits in exactly with the locality assumption that Bell was working with. The exact statement in La nouvelle cuisine for instance is:
J. S. Bell said:
A theory will be said to be locally causal if the probabilities attached to values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a space-time region 3 (Fig. 4).
(Fig. 4 is this Minkowski diagram, with the caption "Full specification of what happens in 3 makes events in 2 irrelevant for predictions about 1 in a locally causal theory.") Bell goes on to explain how this (together with the standard "no superdeterminism/retrocausality" assumption) implies that correlations observed in a Bell-type experiment should admit a factorisation of the form $$P(ab \mid xy) = \int \rho(\lambda) P_{\mathrm{A}}(a \mid x; \lambda) P_{\mathrm{B}}(b \mid y; \lambda) \,, \qquad (*)$$ which in turn implies the Bell inequalities.

Intuitively, (*) roughly expresses the idea that correlations between the outcomes ##a## and ##b## should be explicable in terms of initial conditions ##\lambda## somewhere in the union of their past light cones. This assumes nothing about the underlying dynamics, so I see no reason that a classical relativistic field theory like electromagnetism should be an exception.

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zonde
Gold Member
I probably have seen all until about 2007, when I started to lose interest since it didn't cover field theories. Maybe you can point me to a more recent paper where I can find a set of assumptions that covers causal fields.
No, I can't do that. But you are arguing that there is important difference between "locality" in Bell sense and "causal locality" of field model. Here:
Thus causal locality and Bell locality are two essentially different concepts.
So I can point you to older Eberhard paper that is using different locality assumption for derivation of inequality. But this derivation of inequality relies on detection efficiency model that is particle based so it can't be applied to field model straight away.

stevendaryl
Staff Emeritus
No, I can't do that. But you are arguing that there is important difference between "locality" in Bell sense and "causal locality" of field model.
As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows:

$P(A \wedge B | \lambda, \alpha, \beta) = P(A|\lambda, \alpha) P(B|\lambda, \beta)$

where $A$ is some yes/no measurement at one location, $B$ is some yes/no measurement at another location, $\alpha$ is the state, or situation, at the first location, $\beta$ is the state, or situation at the second location, and $\lambda$ is state information shared by both measurements. This factorizability assumption isn't the same thing as Einstein causality (or signal locality), because QM is not factorable in Bell's sense, but satisfies Einstein causality (no FTL signals).

wle
As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows:

$P(A \wedge B | \lambda, \alpha, \beta) = P(A|\lambda, \alpha) P(B|\lambda, \beta)$

where $A$ is some yes/no measurement at one location, $B$ is some yes/no measurement at another location, $\alpha$ is the state, or situation, at the first location, $\beta$ is the state, or situation at the second location, and $\lambda$ is state information shared by both measurements.
If you're following Bell then, strictly speaking, the definition of locality he works with is what I quoted in my previous post. In your notation: that, given sufficiently specified information ##\lambda## in a suitable part of the past light cone of (e.g.) ##A##, knowledge of ##B## and ##\beta## should be redundant for making predictions about ##A##, or $$P(A \mid B, \alpha, \beta, \lambda) = P(A \mid \alpha, \lambda) \,.$$ The factorisation condition you wrote follows from this and the definition of conditional probability: $$\begin{eqnarray*} P(A, B \mid \alpha, \beta, \lambda) &=& P(A \mid B, \alpha, \beta, \lambda) P(B \mid \alpha, \beta, \lambda) \\ &=& P(A \mid \alpha, \lambda) P(B \mid \beta, \lambda) \,. \end{eqnarray*}$$

zonde
Gold Member
As I said, I think Bell's notion of locality is the assumption that probability distributions factor as follows
Yes (also wle made a good point)
This factorizability assumption isn't the same thing as Einstein causality (or signal locality), because QM is not factorable in Bell's sense, but satisfies Einstein causality (no FTL signals).
There is distinction between "can't be used to produce FTL signals" and "do not need FTL signals to explain phenomena".
As I understand you define "Einstein causality" as the former. But I don't see much point in such a utilitarian definition.

stevendaryl
Staff Emeritus
Yes (also wle made a good point)

There is distinction between "can't be used to produce FTL signals" and "do not need FTL signals to explain phenomena".
As I understand you define "Einstein causality" as the former. But I don't see much point in such a utilitarian definition.
Well, to me the fact that QM nonlocality can't be used to send signals is pretty important. I'm not exactly sure what it means, but my feeling is that if there were truly something faster-than-light going on, there should be a way to expose it to experiment.

zonde
Gold Member
Well, to me the fact that QM nonlocality can't be used to send signals is pretty important. I'm not exactly sure what it means, but my feeling is that if there were truly something faster-than-light going on, there should be a way to expose it to experiment.
Being skeptical is important part of doing science.

Well, to me the fact that QM nonlocality can't be used to send signals is pretty important. I'm not exactly sure what it means, but my feeling is that if there were truly something faster-than-light going on, there should be a way to expose it to experiment.
The point why it is impossible is quite simple, it follows from the logic of the proof. We have Reichenbach's principle of common cause: A correlation requires causal explanation, and there are three possibilities: A causes B, B causes A, and a common cause C for A and B. Then, the whole point of Bell's inequality is to rule out the common cause explanation. That means, two explanations remain, A causes B, or B causes A.

Given that above explanations kill Einstein causality, it is clearly not a defense for Einstein causality that we are, yet, unable to identify with certainty which of the two explanations is correct. Let's not forget that there is only one plausible culprit (what happens earlier in the CMBR frame is the cause), and there is a natural explanation why it is so difficult to identify the culprit: Even if microscopically there is a preferred frame, the symmetry group of the large distance wave equation hides this preferred frame. The simplest example is usual atomic matter, which gives the standard wave equation for acoustic waves in the large distance limit, and the symmetry group for the standard wave equation $(\partial_t^2 - \Delta)\varphi(x,t)=0$ is the Poincare group, which is not the fundamental symmetry group of atomic matter, but only of its large distance approximation.