2.2. The Additional Hypothesis
All real measurements occur successively in time. For example,
represents the following process: set the analyzers’ angles to {
α,
β} during the time interval [
θ,
θ + Δ
T], sum up the number of coincidences detected after each analyzer’s outputs, and calculate the ratio in Equation (3). Note that
E(
α,
β) in Equation (4) is an average over the space of the hidden variable
λ, while
E(
α,
β)
measured in Equation (7) is an average over time. In practice, these two averages are
implicitly assumed to be equal, but they may not be. The assumption stating that these two averages are equal is the well-known ergodic hypothesis. In other words, Bell’s inequalities are
derived from assuming LR, but they can be
used in the real world only if ergodicity (or a similar hypothesis, see later) is also assumed. The idea that non-ergodicity may be related with the solution of the QM vs. LR controversy is not new [
13,
14,
15]. Yet, it must be warned that the meaning given to “non-ergodicity” in those early approaches was influenced by the proposal of an LR model based on a system of coupled oscillators (thus similar to the best known example of non-ergodicity: the Fermi–Pasta–Ulam system) [
16]. This proposal was eventually shown to be reducible to a form of the time-coincidence loophole [
17]. Instead, the meaning given to “non-ergodicity” here is more general.
The historical connection between the terms “non-ergodicity” and “time-coincidence loophole” may lead to confusion. I stress that “non-ergodicity” here has no relationship with the time-coincidence loophole. This loophole is based on a “conspiratorial” modification of photons’ detection times [
17]. Somebody might still think that the recent loophole-free experiments are also free from the need of the, as it were, “ergodicity hypothesis,” since they only take into account an outcome for the whole
time interval that the measurement lasts, and that fine detail of what happens inside that time interval is ignored. This approach may close the time-coincidence loophole, but it does not avoid the “ergodicity hypothesis.” Note that, at a given time or, if preferred, at the time interval (say) #1234, only one analyzers’ setting is fixed (say {
α,
β’}, not two or more of the four possible settings, see
Figure 1). The result of the observation in time interval #1234 is then saved in only one “box” (say, {−,+,
α,
β’}, not in 2or more of the 16 possible boxes). When the experimental run ends after summing up the results of many time intervals (say, from #1 to #10
6), the numbers in the boxes are fatally averages
over time, while the terms in the (theoretically derived) Bell’s inequality are averages
over the ensemble of states of the hidden variable. Simply, the two averages are not necessarily equal. We cannot insert the numbers in the boxes into the derived Bell’s inequality (and to expect the result to be logically linked to the premises for deriving that inequality) unless we suppose the two averages (time and ensemble) are equal, which is the
usual meaning of “ergodicity”.
Now, let us explore the conditions that are necessary to apply Bell’s inequalities as we are used to. We will find that “ergodicity” is only one of the possible hypotheses in a set.
If measurement independence is assumed valid (what is a consequence of assuming locality), then the distribution of the time intervals among the angle settings (that is, if it is a single continuous interval or many separated small intervals, as in the experiments with random varying analyzers [
7,
8,
9,
10]) is irrelevant. Let us assume then, for simplicity, that the angle setting at Station A is
α from
t = 0 to
T/2 and
α’ from
T/2 to
T, and at Station B the setting is
β’ from
t = 0 to
T/4 and from
t = 3
T/4 to
T, and
β from
t =
T/4 to 3
T/4. A different distribution requires a more involved notation of the integration intervals, but the result is the same. Now the problem in the usual derivation of the CHSH inequality appears clearly in the passage from the first to the second equality in Equation (5), which now becomes
The rhs in the first line of Equation (8) is what is actually measured, while the integral in the second line is the expression that leads to the CHSH inequality. The expressions are different simply because the integration intervals are different. The same happens with the integrals for the term added and subtracted to derive Equation (6) from Equation (5). In consequence, the measured values may violate the CHSH inequality Equation (1) or not but, at this point, this result implies nothing about the validity of LR. This is because the CHSH inequality is derived from the second line of Equation (8), which is different from the measured expression (the first line in Equation (8)). In a few words: the logical link is broken. To restore the logical link it would be necessary to sum all the observables under the same interval of integration. However, going back in time to measure with a different setting to complete the integral is impossible. This is an impossibility from which there is no escape, no matter the inequality or the setup used. One may say, “Well, this is sad, we have worked for so many years for nothing, let’s forget about all this” or, instead, to explore under what hypotheses (in addition to LR) the usual Bell’s inequalities maintain their significance. The key is to bridge the gap between the first and the second lines in Equation (8).
In order to do this, the integrals in the first line in Equation (8) must be completed with expectation values obtained under conditions that did not occur or counterfactual values. This step does not require additional assumptions, since realism, which is already assumed in the derivation of Bell’s inequalities, does allow counterfactual reasoning. The first line of Equation (8) completed in that way is now equal to the second line:
where the underlined terms are the sum of three counterfactual time averages:
For example, the factor
AB(
α,
β,
t) indicates the result of a measurement performed at a time value when it actually was
B ≠
β, that is, a counterfactual result. By definition, its numerical value is
which is unknown, because
all the C
ij = 0. It is a zero-over-zero indeterminacy. Yet, counterfactual definiteness ensures that the rhs of Equation (12)
does take some value (see later).
The derivation of the CHSH inequality continues as usual, and the final expression is
I stress that it is this Equation (13), not Equation (1), the inequality that is derived by assuming LR only. Equation (13) is not as trivial as it may appear, for the (underlined) counterfactual terms span over three time intervals instead of one (see Equations (10) and (11)), and hence they can be, in principle, as much as thrice as large as the corresponding (non-underlined) factual ones.
As Bell’s inequalities are derived within LR, counterfactual reasoning is acceptable and no additional hypotheses are necessary to derive Equation (13). However, there is now the problem of giving numerical values to the counterfactual terms. In order to solve this problem, a “possible world” must be defined to ensure logical consistency [
18]. There is no mystery in this situation, just a lack of information. Let us consider an example from everyday life: Let us suppose that, when I go to the cafeteria and find my friend Alice there, the result of the observable
A≡ “I find Alice in the cafeteria” is 1, and 0 when I do not find her there. After many visits to the cafeteria, I measure the expectation value 〈
A〉 = 0.3. This is the available information. Now, let us consider the question, “What is the expectation value of
A when I don’t go to the cafeteria?” Assuming that Alice and the cafeteria have a well defined existence even when I do not go there (roughly speaking, if realism or counterfactual definiteness is assumed), and then slightly changing the definition of
A, from “I find” to “To find,” the expectation value of
A is some well defined number, say,
q. However, the value of
q cannot be known with the information available at this point. More information is needed regarding the behavior of Alice and the properties of the cafeteria when I am not observing them (that is, a “possible world” must be defined) to assign a numerical value to
q. Defining
this missing information means a hypothesis
in addition to LR. If this additional hypothesis is not made (that is, if this missing information is not provided), then the values of the counterfactual terms in Equation (13) (which
cannot be measured) remain undefined, and it is therefore impossible to know whether or not the results of an experiment (which measures only the factual terms) violate the inequality.
Once a possible world is defined, the counterfactual terms can be calculated. Depending on the possible world chosen, Equation (1) is retrieved from Equation (13), or not [
11]. I stress that the important point here is this:
the definition of a possible world unavoidably entails one assumption in addition to LR. This weakens the consequences of the violation of Bell’s inequalities reported in the experiments, for the violation can be interpreted as a refutation of the additional assumption, not necessarily of LR. Note that this weakening is not due to an experimental imperfection, as in the case of the loopholes. The setup in
Figure 1 is assumed ideally perfect. The weakening is a consequence of the fact that real measurements are performed in time, and that it is impossible to measure with two different angle settings at the same time. It seems to be related to a weak point in the usual way the theory of probability is applied to the results of observations. In the usual way, events are thought to occur in abstract, independent parallel worlds. The average over an ensemble of these parallel worlds allows the simple calculation of probabilities. In any actual observation instead, events occur (and averages are obtained) successively in time,
in the only available real world. The difference between these two situations is at the core of the recent resolution of a paradox in gambling theory [
19,
20].
2.3. The Conditions for Retrieving the Validity of Bell’s Inequalities
The simplest way to retrieve Equation (1) from Equation (13) is to suppose a possible world where the factual and counterfactual observables are equal:
AB(
α,
β,
t) =
AB(
α,
β,
t) (and the same for all the other counterfactuals). In the example of the cafeteria, this possible world means that, each day, Alice would have been there, or not, regardless of whether I went to the cafeteria or not. This choice has some technical drawbacks [
11]. A less restrictive and, in my opinion, more appealing alternative is to suppose that the factual and counterfactual
time averaged expectation values are equal (in the example of the cafeteria, this means that
q = 〈
A〉 = 0.3). Each of the three counterfactual terms in the rhs in Equations (10) and (11) is then equal to the factual term; hence
which is the same as the other counterfactuals, and Equation (1) is retrieved from Equation (13). This alternative defines what I call the “homogeneous dynamics assumption” (HDA). The logic inference is then:
On the other hand, the ergodic hypothesis means that the time averages are equal to the ensemble average. As the latter is unique, it implies that all the time averages (both factual and counterfactual) are equal. Therefore, ergodicity ⇒ HDA. Yet, the HDA is different from ergodicity, for it is conceivable that the factual and the counterfactual time averages are equal among them, but that they are still different from the ensemble average. It is also conceivable that Equation (14) holds even if the HDA is not valid. The three counterfactual integrals in Equations (10) and (11) may all be different from each other, and still their sum may be equal to thrice the factual one. These different logical implications involved are summarized in
Figure 2. The set of hypotheses painted in gray retrieve the usual meaning of Bell’s inequalities. In the “other possible worlds” (outside the gray set), there is no logical link between the observed violation of Bell’s inequalities and the validity of LR.
As can be seen in
Figure 2, the HDA does not define the only possible world where Bell’s inequalities are retrieved. Yet, I believe the HDA to be the most plausible one in physical terms. It indicates that the time average of a dynamical variable recorded during the interval [
t1,
t1 +
T] (say, when the variable is being observed) is equal to the time average recorded during any other interval [
t2,
t2 +
T] (say, when the variable is
not being observed) provided, of course, that
T is sufficiently long. The relationship between the validity of the HDA and of a Markovian underlying dynamics seems intuitive, and deserves to be examined in detail elsewhere.
In fact, the HDA seems so plausible that doubts were raised on the existence of a physical situation where it does not hold. In other words, there is no doubt that possible worlds can be
defined to violate the HDA (see [
11]); the question is whether some physically reasonable model exists that, as a
consequence of its evolution, violates the HDA. It is desirable that such model also violates Bell’s inequalities, and that it does so for some (not all) values of its parameters. In this way, the model is useful to study the relationship between the validity of the HDA and that of Bell’s inequalities. In the next section, one such model is proposed.