Are there useful generalizations of Bell's criterion of realism?

One can, of course, give some nonsensical definition of "realism" and then use this definition for name-calling: Whatever does not fit into this definition of "realism" is mysticism, not realistic, simply because it does not fit into our definition of "realism".

So, maybe the definition of realism we have given is nonsensical? This is a possibility we cannot exclude completely, a line of argumentation which is always open. But those who want to follow this line of argumentation have to do some homework: They have to present an alternative definition of realism.So it is quite reasonable to argue that

Only if they present such an alternative definition, it is possible to compare the two competing notions of realism and to decide which of the two notions deserves to be named "realism", and which deserves a different label.

Now, there is a large class of possible definitions of realism which can be excluded from the discussion as irrelevant: All those notions of realism which are stronger, which require more than our definition.

Note that our notion of realism is already extremely weak: All the classical religions, which postulate various ghosts, gods and so on as really existing entities, which causally influence our world, are accepted as realistic theories. So it is quite reasonable to argue that our notion of realism is much too weak, and the correct notion of realism should be much stronger, so that religious explanations are excluded. I do not argue about this – in fact, there is at least one additional condition for a realistic theory which is necessary but not covered by our criterion: The consistency of the realistic explanations of different experiments. The point is that whatever the stronger notion of realism, it is also in conflict with relativistic symmetry and Einstein causality, for the same reasons. Therefore, any stronger notion of realism certainly does not save nor Einstein causality, nor realistic versions of relativistic symmetry. To save relativity, one needs a weaker version of realism.

Are there such weaker notions of realism? Let's consider here some possibilities:

The empty notion of realism

There is, obviously, the trivial, empty version of realism: A statistical theory is already a realistic theory. There are no additional requirements which have to be fulfilled by a realistic theory.

Now, first, there is no good reason to name such a theory realistic. Once there is no difference between a realistic theory and a statistical theory, there is no reason to use different names for them.

Even more, the empty notion of realism contains no trace of the central idea of realism – an observer-independent reality. The statistical theory defines the probability measure ρ(y,a) for the results of our observations, and it depends in a clear and obvious way from the parameters controlled by the observer. So there is nothing observer-independent in this notion.

Then there is the methodological point: This notion of realism means giving up the search for realistic explanations. We don't have to do anything to find a "realistic explanation" in this weakest sense: We observe something, transform the observed frequencies into some probability measure ρ(y,a), and that's all – we have already found a "realistic explanation", no need to search for a better one.

In this sense, the only difference between giving up realism and using this "weak" definition of realism is a rhetorical one. In above cases we have to give up the content of realism: We no longer have to search for realistic explanations. The difference is only in the description: In the first case, we openly give up realism. In the second case, we do not admit that, everything seems fine, we only use a "weak" notion of realism instead of the "strong, classical" notion.

In other words, the first notion openly admits that something very serious has changed: Realism, one of the foundations of the scientific method itself, has been rejected. The second version simply hides this, but the effect is the same: We no longer search for realistic explanations. The non-existence of a realistic explanation is no longer an open scientific problem, and to find such an explanation is no longer a scientific progress.

This seems sufficient to reject this "weak" notion of realism: To name the rejection of realism a version of realism only causes confusion.

Allowing for a dependence on the parameters a

What about allowing for a dependence of ρ(λ) in

∫ f(y) ρ(y,a) dy = ∫ f(y(λ,a)) ρ(λ) dλ

on the parameters a? We would obtain

∫ f(y) ρ(y,a) dy = ∫ f(y(λ,a)) ρ(λ,a) dλ.

But this is exactly the trivial notion of realism: We can use simply y = λ, ρ(y,a) dy = ρ(λ,a) dλ. No need to invent something nonetrivial – whatever we observe, a "realistic" explanation always exists, and it is trivial to find it. So, we have to reject this proposal.

Moreover, it clearly contradicts the basic idea of realism – that reality is observer-independent.

Allowing stochastic functions y(λ,a)

Our definition of realism seems to contain an element of determinism: The function y(λ,a) in our definition is a deterministic function. What if we replace it with a stochastic function?

If we use classical, Kolmogorovian stochastics, we do not obtain anything new. A stochastic function is defined as a deterministic function on some unspecified space of elementary events Ω. So, we replace the deterministic function y(λ,a) by another deterministic function y(λ,a,ω), and instead of a single probability measure on Λ we have now also a probability measure on Ω.

We also have to specify the properties of the probability measure on Ω In particular, the general definition of a stochastic function does not tell us much about the dependence of the measure on the parameters a. But there is not much choice, and we can consider above alternatives:

Thus, to replace the deterministic function y(λ,a) by some stochastic function y(λ,ω,a) does not lead to anything new.

Non-Kolmogorovian probability theory?

But maybe it makes sense to use some non-Kolmogorovian probability theory?

Sorry, but I don't know what this means. The only interpretation of this notion which makes sense to me is the idea that one should not use the space of events Ω and the probability measure ρ(ω) on it but to work only with probabilities of statements about observables like Ρ(A(y)|a) that a statement A(y) is true.

This idea may be attractive for positivists, who prefer to consider only statements about observables instead of statements which contain unobservables like the events ω ∈ Ω. It seems not very interesting for realists. Except, maybe, for the sake of the argument if arguing with positivists.

But, of course, for the sake of the argument one can consider this approach too.

What do we have to change? First, we need a replacement for the notion of a realistic explanation in terms of pure probabilities. The following notion seems appropriate: Assume we observe if two statements A(y), B(y) are independent or not. Independence means

Ρ(A & B | a, b) = Ρ(A | a, b) Ρ(B | a, b).

If they are not independent, we would like to have some realistic explanation for this. The realistic explanation may be given in the following way: There are some common causes λ ∈ Λ. If these common causes are known, then the correlation should disappear if we take into account these common causes. That means, the conditional probabilities should be already independent:

Ρ(A & B | λ, a, b) = Ρ(A | λ, a, b) Ρ(B | λ, a, b).

This condition is named in the literature output independence. It is not a nice choice, and causes a lot of confusion: Some people think that it makes sense to consider theories which do not fulfill this condition. But it does not make sense: A theory which does not fulfill the condition of output independence is simply a theory which does not give a realistic explanation of the observed correlations. Thus, for a realistic theory output independence is a basic requirement. A much better name is completeness: A theory which does not fulfill this condition is simply incomplete, because it does not explain the observable correlations.

In this context, realism is, one the one hand, the thesis that there exist a set of parameters λ ∈ Λ which makes the theory complete. On the other hand, it is the methodological principle that, if no such complete theory exists, we have to search for such a theory, and, in case we have found such a complete theory, that we have to prefer it in comparison with incomplete competitors.

Once this condition is fulfilled, we can formulate now the condition of Einstein causality as

Ρ(A | λ, a, b) = Ρ(A | λ, a);     Ρ(B | λ, a, b) = Ρ(B | λ, b);

But this appears already sufficient to prove Bell's inequality. One needs only the following elementary lemma of probability theory:

If two independent random variables Z1, Z2 satisfy Ρ(Z1=Z2)=1 then there exists a constant c such that Ρ(Z1=c)=1 and Ρ(Z2=c)=1.

Now, in the situation of the EPR-Bohm experiment, the consideration of the case a=b, which gives A=B, gives a constant (which possibly depends on λ as well as the value a=b) such that

Ρ(A=c(λ,a) | λ, a) = Ρ(B=c(λ,b) | λ, b) = 1.

which finishes the EPR part of the proof of Bell's inequality.

As a consequence, this generalization does not change anything: Taken together with Einstein causality (which gives the parameter independence), the requirement of completeness (named "output independence") allows to derive Bell's inequality. Thus, one of the assumptions has to be false. Or realism, or Einstein causality has to be rejected.

Thus, the question if this is really a generalization of the notion of realism becomes irrelevant for the main point of these pages, which is the consideration of the conflict between realism and relativity. The conflict remains. And all the arguments we have given in favour of realism remain valid too.

Something else?

But, maybe there is some other, weaker notion of realism? A notion of realism which is not in conflict with relativistic symmetry but is, on the other hand, nontrivial, and allows to distinguish a reasonable notion of realistic explanation from the pure description of the measurement results?

One can, of course, never know what some future researchers may propose. But because we cannot evaluate and criticize proposals which have not been made, they do not count in science.

But assume somebody makes such a proposal: Some principle of, say, "quantum realism", stronger than pure description of the measured probability distributions, but weaker than realism. What would follow?

First, whatever this new notion, the scientific way to distinguish them is to give them different names. A natural choice for these names would be to name the new, weaker one "quantum realism", and the notion considered here "classical realism".

Or simply realism: Last but not least, our notion realism is an established scientific notion, used by EPR and in Bell's theorem, and more than 70 years after the publication of the EPR criterion of reality no weaker notion of realism has been proposed. So, please use a new name for the new notion, and leave the old name for the old, established notion.

After this, there would be no reason to change anything in our argumentation: There would be a conflict between classical realism and relativistic symmetry, and we would have to make a choice between them. The principles we have proposed for making such choices remain unchanged. There may be a little shift in some of the evaluations: Where today the relativistic alternative requires to give up realism completely, the loss may be not that strong if there is a nontrivial notion of quantum realism compatible with relativistic symmetry. -----------------------------------------

Observed correlations cry for realistic explanations. There should be some common causes – some elements of reality which causally influence the results we observe.

But what counts as a realistic explanation of our observations? One is confronted often enough with pseudo-explanations: descriptions of the observed facts in different words, circular explanations and so on. The definition of the meaning of "realistic explanation" is what we consider to be the main point of the definition of realism. Realism is then nothing more than the hypothesis that for every correlation exists a realistic explanation.

Empirical theories

It is important to note that empirical science and realism are different notions: An empirical theory has to make testable predictions: If we do some experiment, where the experimenter has a free choice of some control parameters a ∈ A and observes some measurement result y ∈ Y, the empirical theory has to predict a probability distribution ρ(y | a)dy. This probability distribution gives then everything else. For example, for every function f(y) of the measurement results we can compute the expectation value E(f | a)

E(f | a) = ∫ f(y) ρ(y | a) dy,

and for every statement A ⊂ Y about the measurement results we can compute the probability Ρ(A | a) that the statement is true by

Ρ(A | a) = ∫A f(y) ρ(y | a) dy.

The empiricist point of view is that this is all science can give: Some sort of black box which predicts the observable probability distributions ρ(y | a)dy. The word "explanation" does not have any well-defined meaning. The rules which allow to compute the probability distribution ρ(y | a)dy are all what science can give, thus, is already a sufficient explanation. To ask for more explanations is metaphysics, anathema, classial prejudice.

Realistic explanations

Assume we have two statements A and B. Then, these statement are independent if

Ρ(A and B | a) = Ρ(A | a) Ρ(B | a).

If they are not independent, they are correlated, and we have observed a non-trivial correlation. Realism requires that such a correlation has a realistic, causal explanation: There should be some causes λ. These causes explain the observed correlation if the conditional probabilities are already independent:

Ρ(A and B | λ, a) = Ρ(A | λ, a) Ρ(B | λ, a).

What we observe in repeated experiments are frequencies. A physical theory has to in particular descriptions This has to be clarified, if we want to be able to distinguish "explanations" which are not more than a description of the observed facts in other words