Violation of Bell inequalities for classical fields?

[Ilja]

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]

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.

 

[Ilja]

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:

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.

 

[zonde]

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]

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]

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]

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]

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.

 

[Ilja]

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.