Definition of strong Einstein causality

For the proof of Bell's inequality we need Einstein causality, in a variant we have named here strong Einstein causality. For the consideration of various alternatives we need, as well, some other, weaker notions of causality, namely weak Einstein-causality as well as strong and weak classical causality.

Strong and weak notions of causality

We have defined two classes of theories — statistical and realistic theories. There are two notions of causality, which correspond to these two types of theories. We are interested here only in one very special, but central question related with causality: Does the decision of the experimenter to use some value a as the input have some influence on the result y of the observation or not?

Statistically provable causal influence

In a statistical theory we have only probability distributions ρ(y,a). But the formula for the measurement result

E(f,a) = ∫ f(y) ρ(y,a) dy

allows in many cases a positive answer: If ρ(y,a) depends on a, then we have some causal influence of the choice of the experimenter a on the measurement result y. This can be named a statistically provable causal influence. Such a provable causal influence can be used for information transfer: The choice of the input parameter a by the experimenter is sending the signal, the measurement of y is receiving the signal. (If ρ(y,a) depends only weakly on a, one has to repeat many such measurements with the same input a to identify a from y with sufficient certainty. But in principle such an application is always possible, if ρ(y,a) depends on a.)

This notion of causality we name weak causality.

Real causal dependence

If there is no such weak causal dependence, it does not follow at all that there is no causal influence of a on the measurement result y. A simple example proves this: Assume somebody tells us numbers between one and six. He is using, for this purpose, a dice. Now, for a = 1, he tells us the value x shown by the dice. But for a = 0, he tells, instead, the value 7-x. The value of a obviously has a causal influence on the result of the measurement.

But, using only the probability distribution ρ(x) = 1⁄6 (if the dice is fair) we cannot prove the existence of this causal influence. It doesn't matter if y = x or y = 7-x, we have always p = 1⁄6, thus, no dependence of ρ(y,a) on a.

Thus, to describe this type of causal influence, we need another notion of causality. This notion follows in a natural way from the realistic explanation:

There exists a causal influence of a on y, if the function y(x,a) depends on a. This notion of causal influence we name real causal influence or strong causality.

Thus, we have two different notions of causality — a weaker notion of causality, where only causal influences which can be proven statistically, or used for information transfer, can be identified, and a stronger notion of causality, which is based on a realistic explanation.

The notion of causality used in Bell's inequality is the second, strong notion of causality.

Einstein-causality vs. classical causality

Above versions of definition of causality have two variants: First, there is the stronger version of Einstein causality, with the speed of light in vacuum as an absolute upper bound for causal influences. The alternative is classical causality in a preferred frame, which forbids only causal influences into the past (relative to the preferred absolute time t).

As a consequence, we have to distinguish four notions of causality: Classical resp. Einstein causality, each in their weak (statistically provable) and strong (based on a realistic explanation) variant.

Which notion of causality has to be given up to preserve realism?

We have to give up only those notions of causality, which, together with realism, allow to prove Bell's inequality for space-like separated events. This is because this inequality is violated by quantum theory and, it seems, in reality as well.

But in the proof of Bell's inequality we need strong Einstein causality, that means, the strongest of the four notions of causality. Nor weak Einstein causality, nor strong classical causality allow to prove Bell's inequalities for space-like separated events. Therefore, they don't have to be given up.

That means, for the preservation of realism we don't have to give up anything: Because the notion of causality which we have to give up — the strong, realistic version of Einstein-causality — becomes meaningless if we give up realism, thus, has to be given up anyway.

So, the conflict between realism and Einstein causality has a simple solution: We don't have to give up realism, because it does not allow us to preserve the strong, realistic notion of Einstein causality anyway, and the weak, statistical notion of Einstein causality is compatible with realism. Thus, giving up realism we would not gain anything.

Conflict with the principle of manifest relativistc symmetry

To avoid arguments of this type, there is some possibility which is worth to be considered: We can, last but not least, derive strong Einstein causality from some other principles, especially some strong version of relativistic symmetry. In this case, the conflict between realism and Einstein causality becomes a conflict between realism and this strong version of relativistic symmetry.

To avoid a similar argumentation, the symmetry principle would have to be formulated in such a way, that, after giving up realism, it does not automatically reduce to symmetries of observables only. Else, we would be in the same situation: Symmetries of observables only remain compatible with realism. Thus, we would not gain anything by giving up realism.

Such a possibility exists. It is the principle of manifest relativistic symmetry — a very general and obviously highly metaphysical principle.