Reality Dosen't Shine, It Twinkles

1. Introduction

In recent years many physicists have written popular books in which they make grand assertions such as reality does not exist, time does not exist, reality is non-local, as if all of these were proven facts [1,2]. This paper is part of a series of papers [3,4,5,6,7,8] which add to a growing literature devoted to challenging these assertions [9,10,11,12,13,14,15,16] and returning a semblence of credulity to our understanding of reality. I have argued [5] that the interpretational problems of quantum mechanics stem from an implicit attachment to a viewpoint which is founded upon the concept of object. While classical entities share many characteristics in keeping with the concept of object, biological entities do not, and many quantum systems share more features in common with biological entities than they do with objects. Mathematics entities, on the other hand, are the epitome of objects, and it is not all that surprising that inaminate matter, which possesses characteristics closely resembling those of objects, should be describable in terms of mathematical entities. Efforts to do the same in the case of quantum mechanical entities, while quite successful from a computational perspective, have resulted in endless confusion regarding how the theory should be interpreted.

The advances in experimental methodology which now allow for many questions in quantum foundations to be put to empirical observation, and the awarding recently of the Nobel prize for work in this area, has moved the study of quantum foundations from the backwater of philosopause and into mainstream physics. The vacuous advice to "shut up and calculate" is no longer sustainable, and the question of effective intepretations of quantum mechanics has become ever more pressing.

The quantum realm is frequently described as being fundamentally distinct from the classical realm, even though the classical realm is supposed to be founded upon the quantum realm. This has led to discussions of a so-called boundary between the classical and quantum realms [7]. Entanglement is frequently mentiond as one situation which distinguishes the quantum from the classical realm.

2. Bell Inequalities Revisited

In "On the Einstein-Podolsky-Rosen paradox" [17](Chapter 2), Bell purported to show that local hidden variables as proposed by Einstein, Podolsky and Rosen [18] could not reproduce certain results as predicted by quantum mechanics. In particular, he imagined a pair of spin $\frac{1}{2}$ particles, $1,2$, being created as an entangled pair and moving in opposite directions to a pair of detectors operated by two independent observers, A and B. Each is free to measure the spin component along an angle of their choice, independent from one another. Thus observer A measures ${\sigma }_1·\mathbf{a}$ while observer B measures ${\sigma }_2·\mathbf{b}$. Bell then assumed the existence of a very general form of hidden variable, which he represented as $\lambda$, which could represent a single variable or multiple variables, functions of one or more variables, deterministic or stochastic. Bell assumed that the measurement obtained by observer A of ${\sigma }_1·\mathbf{a}$ is a function of $\mathbf{a}$ and $\lambda$, denoted $A(\mathbf{a},\lambda )$ and likewise for observer B measuring ${\sigma }_2·\mathbf{b}$, denoted $B(\mathbf{b},\lambda )$. Bell further assumed that the result A is independent of $\mathbf{b}$ and that of B is indepdendent of $\mathbf{a}$. Since we are dealing with spin measurements,

$$A(\mathbf{a},\lambda )=\pm 1,B(\mathbf{b},\lambda )=\pm 1$$

and since these particles are entangled, it follows that $A(\mathbf{a},\lambda )=-B(\mathbf{a},\lambda )$.

Assume that the hidden variable $\lambda$ is an element of some measure space $(\Lambda ,\Omega ,\mu (\lambda \left)\right)$ where $\Lambda$ is the set of variables, $\Omega$ is a set of measurable subsets of $\Lambda$ and $\mu \left(\lambda \right)$ is a measure on $\Omega$. The expectation value of the product of the two measurements $E(\mathbf{a},\mathbf{b})$ should then take the value

$$E(\mathbf{a},\mathbf{b})=\int d\mu \left(\lambda \right)A(\mathbf{a},\lambda )B(\mathbf{b},\lambda )=<({\sigma }_1·\mathbf{a})({\sigma }_2·\mathbf{b})>=-\mathbf{a}·\mathbf{b}$$

according to quantum mechanics.

However, there is a problem with this presumption of hidden variables since it is possible to form an inequality (for example the CHSH inequality), which it is argued, must be satisfied by any form of hidden variable but which is violated by quantum mechanics. A standard argument for the CHSH inequality is to consider the function $chsh$ of four variables $a,a^{\prime },b,b^{\prime }$

$$chsh(a,a,b,b^{\prime })=ab+a^{\prime }b+a{b}^{\prime }-a^{\prime }{b}^{\prime }$$

where the range of values for each variable is $[-1,1]$. In [19], Shimony presents an argument derived from Mermin showing that this function must take values within $[-2,2]$. Clearly this is a linear function defined on the simplicial region ${[-1,1]}^4$. Being linear, it must take its extreme values on the boundary of the simplex, in particular, at the corners $(\pm 1,\pm 1,\pm 1,\pm 1)$. The formula may be rewritten in the form $(a+a^{\prime })(b+b^{\prime })-2a^{\prime }{b}^{\prime }$. On the corners, the value of the $(a+a^{\prime })(b+b^{\prime })$ must be either 0 or $\pm 2$ and a simple check shows that the value of $2a^{\prime }{b}^{\prime }$ must be $\pm 2$ respectively. Hence the maximum value must be $\pm 2$.

Therefore if there exists a probability measure $\mu$ on the simplex ${[-1,1]}^4$, then

$$|\int chsh(a,a^{\prime },b,b^{\prime })d\mu |\le \int |chsh(a,a^{\prime },b,b^{\prime })|d\mu \le 2\int d\mu \le 2$$

Now the CHSH formula can be written as

$$CHSH(\mathbf{a},{\mathbf{a}}^{\prime },\mathbf{b},{\mathbf{b}}^{\prime };\lambda )=$$

$$\int [A(\mathbf{a},\lambda )B(\mathbf{b},\lambda )+A({\mathbf{a}}^{\prime },\lambda )B(\mathbf{b},\lambda )+A(\mathbf{a},\lambda )B({\mathbf{b}}^{\prime },\lambda )-A({\mathbf{a}}^{\prime },\lambda )B({\mathbf{b}}^{\prime },\lambda )]d\mu =$$

$$\int [E(\mathbf{a},\mathbf{b},\lambda )+E({\mathbf{a}}^{\prime },\mathbf{b},\lambda )+E(\mathbf{a},{\mathbf{b}}^{\prime },\lambda )-E({\mathbf{a}}^{\prime },{\mathbf{b}}^{\prime },\lambda )]d\mu$$

$$\int chsh(A\left(\mathbf{a}\right),A\left({\mathbf{a}}^{\prime }\right),B\left(\mathbf{b}\right),B\left({\mathbf{b}}^{\prime }\right),\lambda )d\mu$$

and it follows immediately that $\left|CHSH\right(\mathbf{a},{\mathbf{a}}^{\prime },\mathbf{b},{\mathbf{b}}^{\prime }\left)\right|\le 2$

It is well known that in the setting of quantum mechanics this inequality is in fact violated. So how can this happen? The mathematics is quite clear, and straightforward. It is simply the case that if one has a set of variables on the 4-dimensional simplex ${[-1,1]}^4$, and forms the function chsh, and there exists a suitable measure $\mu$, then the value of the integral will be bound to the interval $[-2,2]$. Since the form of the function chsh is fixed, the only assumption which can be challenged is the existence of the measure $\mu$. Note that in the derivation above, there is no mention of causation, contextuality, non-commutativity, or of locality. The derivation is purely mathematical. The limitiation arises because of the form of the function chsh, where the individual terms associate the variables.

If the individual terms are allowed to disassociate, to be independent of one another, then the inequality can be violated. Consider the function $ichsh=a+b+c-d$ on ${[-1,1]}^4$ where the variables $a,b,c,d$ are independent of one another. Then this function clearly takes values in the range $[-4,4]$. Suppose that we attempt to force $chsh$ to take a value $\pm 4$. This will obviously occur if the first three terms are positive and the fourth term ($a^{\prime }{b}^{\prime }$) is negative. Then sign(a)=sign(b), sign(b)=sign($a^{\prime }$), sign(a)=sign($b^{\prime }$) from the first three terms, which implies that sign($a^{\prime }$)=sign($b^{\prime }$), so that the fourth term cannot be negative. Likewise the smallest possible value occurs if $ab=b{a}^{\prime }=a^{\prime }{b}^{\prime }=-1$ and $a{b}^{\prime }=1$, yielding a value of -4 for the function. Thus function will take this minimal value if the first three terms are negative and the fourth is positive. In that case sign(a)=-sign(b), sign(b)=-sign($a^{\prime }$), sign(a)=-sign($b^{\prime }$) from the first three terms, which implies that sign(a)=-sign(b)=sign($a^{\prime }$)=-sign($b^{\prime }$), so that the fourth term cannot be positive. Thus the $chsh$ cannot take these extremal values $\{-4,4\}$. The problem is that each term in chsh is a product of two variables, effectively correlating or entangling them.

The only way to disentangle them is by introducing additional factors, and since we are interested in the integral of chsh over the simplex, one approach is to introduce different measures for different terms. This changes the identity of the random variables associated with the terms, echoing a point made by Dzhafarov [20] that contextuality is about the identity of random variables. The assumption that classical random variables should always possess a joint distribution is just that - an assumption. It is not true generally. It has been known since the time of Kolmogorov that such assumptions are not universally valid [21]. Vorob’ev made this quite explicit in 1962 [22] when he determined the exact conditions which must be met in order that a joint probability measure exist for a collection of random variables. His arguments are entirely classical and lie within the framework of Kolmogorov probability theory. No appeal to non-Kolmogorov, quantum mechanical probabilities is necessary. It is simply a fact of both the classical and quantum worlds that joint probabilities need not exist for an arbitrary collection of random variables.

Dzhafarov has extended the usual Kolmogorov framework to more formally take into account the effect that context has on random variables. In the Contxtuality by Default approach of Dhzafarov and Kujala [23], each random variable is identified by the property q that it measures and the context a within which it is measured, and so each random variable is denoted as $R_q^a$ and a collection of random variables can be organized in the form of a matrix. The CHSH situation can be viewed a a cyclic 4 system, that is a system of 4 random variables which can be arranged in an array having the following form:

$$\begin{array}{ccccc}{q}_1& q_2& q_3& q_4& \\ R_{q_1}^{a_1}& & & R_{q_4}^{a_1}& a_1\\ R_{q_1}^{a_2}& R_{q_2}^{a_2}& & & a_2\\ & R_{q_2}^{a_3}& R_{q_3}^{a_3}& & a_3\\ & & R_{q_3}^{a_4}& R_{q_4}^{a_4}& a_4\\ \end{array}$$

Dzhafarov calls each row in the matrix a bunch, and each column a connection. The random variables in each bunch have the same context and so can be presumed to possess a joint distribution. The random variables in each conenction have different contexts, and it cannot be assumed, a priori, that they possess a joint distribution. In most cases they will not. If they do, they are called consistently connected. If they do not, they are called inconsistently connected. If we assume that the random variables lie in the range $[-1,1]$, we may determine expectation values within each context, that is

$$E(i,j;n)=\int R_{q_i}^{a_n}{R}_{q_j}^{a_n}d\mu$$

where $\mu$ is the probability measure for the joint probability distribution for the random variables.

Consider the formula

$$R_{q_1}^{a_2}{R}_{q_2}^{a_2}+R_{q_2}^{a_3}{R}_{q_3}^{a_3}+R_{q_3}^{a_4}{R}_{q_4}^{a_4}-R_{q_1}^{a_1}{R}_{q_4}^{a_1}$$

If we ignore the context, we have

$$R_{q_1}{R}_{q_2}+R_{q_2}{R}_{q_3}+R_{q_3}{R}_{q_4}-R_{q_1}{R}_{q_4}=chsh(R_{q_1},R_{q_2},R_{q_3},R_{q_4})$$

so this formula appears to suitably generalize the chsh formula to include context.

If there is a joint probabilty measure $\mu$ for all of the random variables in the formula, which effectively means that context may be ignored, then the analysis for the chsh formula applies to this formula as well (since they become equivalent as seen above), and so it follows that

$$\begin{array}{c}\hfill |\int [E(1,2,2)+E(2,3,3)+E(3,4,4)-E(1,4,1)\left]d\mu \right|\le \end{array}$$

(1)

$$\begin{array}{c}\hfill \int \left|\right[E(1,2,2)+E(2,3,3)+E(3,4,4)-E(1,4,1)\left]\right|d\mu \le 2\int d\mu =2\end{array}$$

(2)

In the event that a joint probability measure such as $\mu$ does not exist, then it is incorrect to integrate over the function $E(1,2,2)+E(2,3,3)+E(3,4,4)-E(1,4,1)$ on the whole. Instead we must integrate each term separately, using the joint probability distribution appropriate for each pair of random variables. In this case we obtain

$$\int E(1,2,2)d{\mu }_{1,2,2}+\int E(2,3,3)d{\mu }_{2,3,3}+\int E(3,4,4)d{\mu }_{3,4,4}-\int E(1,4,1)d{\mu }_{1,4,1}$$

This is the same as for the function ichsh since each integral extremizes to $\{-1,1\}$, so that

$$\begin{array}{c}\hfill |\int E(1,2,2)d{\mu }_{1,2,2}+\int E(2,3,3)d{\mu }_{2,3,3}+\int E(3,4,4)d{\mu }_{3,4,4}-\int E(1,4,1)d{\mu }_{1,4,1}|\le \end{array}$$

(3)

$$\begin{array}{c}\hfill |\int E(1,2,2)d{\mu }_{1,2,2}|+|\int E(2,3,3)d{\mu }_{2,3,3}|+|\int E(3,4,4)d{\mu }_{3,4,4}|+|\int E(1,4,1)d{\mu }_{1,4,1}|\le 4\end{array}$$

(4)

.

Let me emphasize again that there is no requirement for non-local influences or for quantum mechanical probabilities for this to be true. The only requirement is that the joint probabilities be contextual.

In the application of the CHSH inequality to the case of entangled particles, the random variables involved are consistently connected, whereas in most classical settings they are inconsistently connected. One might assume, therefore, that consistently connected random variables, having the same probability measures across contexts, should therefore admit joint distributions across those contexts. In the next section I shall present a simple example in a classical setting in which the random variable are consistently connected but the system is still contextual and a CHSH inequality is maximally violated. The case of consistent connectedness appears to violate our intuitions because having the same appearing random variables makes one assume that the same conditions underlying their generation pertain across contexts, and so a joint distribution formed of simple products holds across all contexts. This failure of intuition is not due to the presence of non-local influences but due to a failure to know the actual mechanism underyling the generation of the random variables. In such a case we say that the marignal probabilites are degenerate, since they do not specify a unique joint distribution but instead may arise from multiple, distinct joint distributions. If we do not know beforehand the correct joint distribution for the system under study, then we must correct for our ignorance.

In the case of a Bell situation, we have an entangled pair of particles described by a wave function of the form

$$\frac{1}{\sqrt{2}}\left(\right|-1{>}_1|1{>}_2+|1{>}_1|-1{>}_2)$$

From the vantage point of Observer 1, the marginal probabilities are determined by projecting onto each possible state, for example for state $-1$

$$|-1{>}_1{<-1|}_1\frac{1}{\sqrt{2}}\left(\right|-1{>}_1|1{>}_2+|1{>}_1|-1{>}_2)=\frac{1}{\sqrt{2}}\left(\right|-1{>}_1\left|1{>}_2\right)$$

yielding a probability of 1/2 and likewise for the 1 state. Thus the probability distribution for Observer 1 will be (1/2,1/2) and similarly for Observer two. The joint probability is given by

$$\begin{array}{ccc}& -1& 1\\ -1& 0& 1/2\\ 1& 1/2& 0\end{array}$$

These marginal probabilites are the same for the situation of two free particles whose wave function is

$$\frac{1}{2}\left(\right|-1{>}_1+|1{>}_1\left)\right(|-1{>}_2+\left|1{>}_2\right)$$

however, the joint probability in this case is

$$\begin{array}{ccc}& -1& 1\\ -1& 1/4& 1/4\\ 1& 1/4& 1/4\end{array}$$

so that in the case of two entangled particles, the marginal probabilities associated with each particle, for each Observer, are degenerate - they simply do not convey enough information about the mechanism underlying their generation.

When arbitrary joint distributions do not exist we can say that the system exhibits Type I contextuality or what Dzhafarov has termed contextuality by default. Type II contexuality (what Dzhafarov has termed "true contextuality") can be detected using a generalization of the CHSH inequality. That inequality suffices when the random variables are consistently connected. When the random variables are inconsistently connected, then this Type I contextuality must be compensated for. Dzhafarov, Zhang and Kujala [23] aruged for a more general inequality, given here for cyclic systems of order n. This inequality is $\Delta C=$

$$\mathbf{s}(\pm <R_{q_1}^{a_2}{R}_{q_2}^{a_2}>\pm <R_{q_2}^{a_3}{R}_{q_3}^{a_3}>\cdots \pm <R_{q_{n-1}}^{a_n}{R}_{q_n}^{a_n}>)-(n-2)-\sum _1^n|<R_{q_i}^{a_i}>-<R_{q_i}^{a_{i+1}}>|\le 0$$

where $\mathbf{s}$ means the maximum taken over all combinations of terms such that the number of minus terms is always odd. The first term is the CHSH term, the second compensates for the degree of cyclicity, while the third term compensates for inconsistent connectedness.

It is often asserted that Type II or true contextuality is unique to quantum mechanics. However, Dzhafarov and colleagues have demonstrated the existence of True Contextuality in two experiments [24,25,26] as have other authors [27,28,29]. In addition a simple thought experiment involving ice cream preferences shows violation of even the Tsirel’son bound under ideal conditions [6]. In none of these cases are superluminal influences involved - they result from contextual effects. Several authors, most notably Khrennikov [13,14] and Dzhafarov [15,20,30] have argued that the issue is not the presence of non-locality, but rather the presence of contextuality. As mentioned above, Dzhafarov has argued that the problem lies with the identity of random variables, and their tendency to change in the presence of different contexts [20]. Unfortunately their arguments seem to be lost on mainstream physicists. The argument above is presented in the hopes that its transparency might put an end to the debate about non-locality and show that the issue is one of the presence or absence of contextuality, which is not unique to quantum mechanics but which can occur classically as well. It is about the dynamics of the system, not its scale or some other arbitrary classical-quantum distinction [7].

3. The Problem of Worldviews

In [5] I argued that the failure to account for the presence of (implicit) biases derived from shared worldviews led to assertions concerning the non-existence of time. The main arguments in support of this position all suffer from one of three logical failures: begging the question, the fallacy of misplaced concreteness or the fallacy of misplaced omniscience. The tension in interpreting quantum mechanics arises, in part, from the tension between the more classical, Objectivist Worldview, whose central entities are objects, and ideas of emergence, contextuality, non-commutativity, non-separateness, which are more in keeping with a Processist worldview, whose central entities are processes (in the sense of Whitehead [31]). In the Processist worldview, entities are generated; they happen, they become. They exist for a time, then fade away. Objects, on the other hand, at last in ideal form, simply exist; they are eternal, as are their properties, at least until some interaction results in a change. Objects are ideal for study using mathematics and propositional logic, since their entities of study are all ideal objects.

The core attributes of an object are:

• It exists independent from any other entity -it can be isolated and treated as a whole unto itself
• It is eternal - it does not become, it merely is
• It is passive - it reacts, it does not act
• Its properties are intrinsic and non-contextual - they are fixed, complete and independent of the actions of any other entity
• Its motion is determined by fixed laws - which may be deterministic or stochastic (usually explained away as due to ignorance on the part of the observer)
• Its motion is often attributed to variational principles - optimality, minimal, maximal - always extremized in some direction
• Its interactions with other objects are always local
• History is irrelevant - the future motion of an object depends only on its present state (and sometimes not even that in the case of stochastic objects)

Processes are wholly unlike objects, although they may give rise to objects in particular circumstances. Whitehead considered process to be the generator of reality [31]. Unlike in the Objectivist worldview, which considers the entities of reality to simply exist, Whitehead considered becoming to be logically prior to being. In other words, the elements of reality do not simply exist, instead, they must come into existence through a process of becoming. Prior to becoming, they have no existence. Subsequent to becoming, they exist briefly, following which they again fade from existence. Reality is a continual succession of becoming, being, and fading away. The basic elements of reality which are generated by processes are termed actual occasions. These occasions have several characteristics, many of which are shared by organisms. They are:

• Actual occasions are both ontological and epistemological (informational) in character.
• Actual occasions are transient in nature. They arise, linger just long enough to pass their information on to the next generation of actual occasions, and then fade away.
• Process theory posits the existence of a transient now structured as a compound present: current generation of actual occasions, generating process, and next generation of actual occasions.
• Actual occasions are holistic, discrete, finite, possessing a “fuzzy” extensionality.
• Actual occasions are not directly observable. Only interactions among processes are discernable.
• Observable physical entities are emergent upon actual occasions
• Information propagates causally from prior to nascent actual occasions as a discrete wave.
• Information form prior actual occasions is incorporated into nascent actual occasions through the act of prehension

The entities generated by processes are contingent, and thus susceptible to contextual effects, as a fundamental characteristic. Entangled systems are difficult to conceptualize within an Objectivist worldview, requiring leaps of faith or mental gymnastics such as postulating the existence of undetectable instantaneous non-signalling influences, which nevertheless somehow send signals between particles, just not between observers. From a Processist worldview, entangled systems are systems which are generated by a single process - they have a common cause as it were, and that suffices [3,16]. Understanding the origin of contextuality in any given system is complex - there are no universal features given our current understanding. In particular though, non-locality is not a prerequisite for contextuality.

The contingent nature of process suggests the need for a change in the logical system to be used to reason about such systems. Multi-valued, modal, fuzzy, and intuitionistic logics seem particularly promising in this regard and Gisin has already carried out some work in this area [33,34,35]. Notions of reality such as counterfactual determinism all harken back to Objectivist sensibilities at least - the idea that something can be real only if it manifests all of its properties all of the time. This is similar to the notion of realism proposed in the 1935 EPR paper [18]: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”. In [3] I suggested a weaker form of realism, namely that an entity could be said to be real if it can be shown to make a difference. An entity can thus be a generator of a property, or dispose a set of possible values of a property, without necessarily fixing the value of the property for all time, and still be considered real. This allows processes to be real, not merely objects.

Let us re-examine the CHSH inequality. In the EPR setting it states that

$$|\int [E(\mathbf{a},\mathbf{b},\lambda )+E({\mathbf{a}}^{\prime },\mathbf{b},\lambda )+E(\mathbf{a},{\mathbf{b}}^{\prime },\lambda )-E({\mathbf{a}}^{\prime },{\mathbf{b}}^{\prime },\lambda )\left]d\mu \right|\le 2$$

and in the setting of arbitray random variables takes the form

$$|\int [R_{q_1}{R}_{q_2}+R_{q_2}{R}_{q_3}+R_{q_3}{R}_{q_4}-R_{q_1}{R}_{q_4}\left]d\mu \right|\le 2$$

Note that time plays no role in the above expressions. In the parlance of logic, these are propositional statements, which hold eternally, absolutely. This is a problem in both quantum and classical mechanics, because observations of the values of random variables must take place over time, and it may be impossible to evaluate certain sets of random variables simultaneously. Non-commutativity of observables is an issue in classical just as in quantum mechanics. It has little to do with the mathematics of Hilbert spaces, it has to do with the complexity of entities (for example, try dissecting a cat first and exercising it later!). Observations, let alone measurements, may not be made together or in certain orders, and thus it is not at all cear in the above formula that these random variables refer to the same entities. They may refer to the same properties, but the nature of a random variable is determined by the probability measure associated with it, not merely but the set of its possible observed values. Declaring a fromula such as those above requires making the assertion, or the assumption, that these random variables are the same random variables in each instance in the formula, and there that a joint probability distribution for the whole collection exists. Time and order thus serve as fundamental markers of context in any real situation, and it must be shown in advance that these make no difference in the determination of the random variables, and thus that context may be ignored. Otherwise, time and order must be explicitly noted.

If time and order must be noted then they become part of the context which must be associated with the random variable. Since some quantum mechanical and some classical systems violate the inequality, it follows that context must be taken into account when evaluating the inequality. Thus we cannot use the above form of the inequality, we must at the very least use the form

$$|\int E(1,2,2)d{\mu }_{1,2,2}+\int E(2,3,3)d{\mu }_{2,3,3}+\int E(3,4,4)d{\mu }_{3,4,4}-\int E(1,4,1)d{\mu }_{1,4,1}|\le 4$$

.

These different contexts cannot, in general, be applied simultaneously, so that time must be an implicit component of these contexts. Making the time component explicit, as in

$$|\int E(1,2,t_2)d{\mu }_{1,2,t_2}+\int E(2,3,t_3)d{\mu }_{2,3,t_3}+\int E(3,4,t_4)d{\mu }_{3,4,t_4}-\int E(1,4,t_1)d{\mu }_{1,4,t_1}|\le 4$$

.

This in turn shows that if the CHSH inequality is violated, then the system must manifest different random variables, which implies different distributions of values or simply different values, at different times. This is not compatible with an Objectivist worldview in which properties as given a propositional quality, and thus are expected to be enduring in the absence of interaction. This is compatible, however, with a Processist worldview, in which a system merely disposes a system to express particular values for a property. The actual value expressed at any given moment in thus a product of the process and the conditions which pertain at the time of determination. While this might be an unusual phenomenon in the world of inanimate entities, it is commonplace in the world of biological and psychological entities.

This shows that in situations in which the CHSH, or other such inequalities are violated, then entities must behave in a temporally transient manner, with properties exhibiting different values or distributions at different times. Thus properties cannot be enduring, they must instead be fleeting, or at least transient. Thus at least some of the entities that form what we call reality cannot be enduring, i.e. they do not "shine", instead they must vary in some manner from duration to duration, i.e. they "twinkle".

4. Classical Consistent Connectedness

In applying the CHSH inequality to the EPR setting, much is often made of the fact that the random variables as observed by the two observers take the same form, regardless of whether they are observing two free particles or an entangled pair. These observables are consistently connected, in the language of Dzhafarov, which means that in applying his generalized inequality to the EPR situation, the third term correcting for inconsistent connectedness vanishes and one is left with the original CHSH inequality. The appearance of contextuality in the setting of consistent connectedness might be viewed as being unique to the quantum mechanical situation, since at first glance it would seem to suggest, in the classical case, the presence of a joint distribution for the random variables. After all, their form is independent of context so intutitively one might think that a joint distribution should exist. However, in the EPR case one sees, as noted in the discussion of a previous section, that the distributions of states in the two cases are decidedly different. This is only made worse when spin measurements are taken into account.

In this section I wish to present two examples of a classical situation, in which the relevant variables are consistently connected, there are no non-local influences, yet the situation nevertheless still exhibits true, Type II contextuality, and thus exhibits a strong form of contextuality in spite of being classical. It is hoped that by presenting such an example it will dispel some of the mysticism that all too often creeps into discussions of quantum mechanical phenomena, and drives home the point that a Processist, or generative view of real entities, is compatible with both classical and quantum realms. The mysteries of quantum mechanics are more an expression of an implicit bias due to a limited worldview, rather than expressing a fundamental break from our usual experienced reality.

The toy model to be considered is a two player co-operative game. It is physically realizable. Each player is given a set of pieces which can be combined with a suitable piece from the other player to form a small structure similar to a miniature Eiffel tower. These pieces are stacked in various ways to form a larger structure, but for the current purpose the details of that precedure are not important. The pieces of Player I are semi-transparent, red in colour. Each represents one half of a tower, with equal numbers of left and right half pieces. Within each piece is a rotatable disk with a single hole near the circumference. The pieces of Player II are similar, except being blue in colour and instead of a hole there is a small pin at a matching location on its disk. There is also a base piece, shaped like

$$\begin{array}{ccc}--& --& \\ \searrow & & \searrow \\ \nearrow & & \nearrow \\ --& --& \end{array}$$

and oriented left or right depending on the direction of the arrows.

Players alternate moves, and during each move, select a particular attribute: orientation of piece P, orientation of base B, orientation of disk D, such that at the end of play, the two half pieces fit together to form a single tower, the disk is oriented with the hole/pin along a vertical plane such that the pin enters the hole (holding the pieces together), and the combined pieces are placed along the long axis of the base with a left piece on the left, right on the right.

There are two observers. Observer I examines the contributions of Player I, while Observer II examines those of player II. The pieces are semitransparent so that it is possible to examine the orientation of the internal disks without disrupting the structure itself. In this way, measurements may be carried out without disturbance, showing that the concept of "disturbance" is not germane to the issue of contextuality. Moreover, observations may be carried out independent of one another, and in any order, so that non-commutativity of observations is irrelevant. However, there is a subtlety here as regards the relationship between observation and game play which will be discussed below.

Each player is free to choose which attribute to select during a move. At the end of play, if the attributes do not suitably align, then the pieces are removed, since they will not form a stable unit. Each player is presumed to adopt a particular strategy towards play, which helps to determine what move they should make in response to previous moves by both players, so as to form a stable unit at the end of play. Winning strategies are those which guarantee success and I will only consider winning strategies here. Moves are conditional upon previous moves. While individual strategies are possible, a simpler approach is for each player to adopt a similar set of responses to prior game play. This makes describing these strategies much simpler. The following table spells out the move that either player should make based upon what has previously been chosen (by either player).

The columns refer to which attributes have already been chosen, by either player. The rows show the responses of the current player to those existing attributes. Each player is free to choose which move to make but since the initial moves are most important, games will be labeled by the first attribute choice of each player, $choic{e}_1,choic{e}_2$ where $choic{e}_1$ can be one of $P_1,B,D_1$ and $choic{e}_2$ can be one of $P_2,B,D_2$. We assign values as follows: Left piece (1), right piece (-1), left direction of block (1), right direction of block (-1), disk hole up/pin up (1), disk hole down/pin down (-1).

The initial moves are laid out in Table 1. The columns refer to the initial move of PLayer I, while the rows refer to the subsequent initial move of Player II. The subsequent moves depend upon those initial moves and as described in Table 2 and Table 3. The rules asosciated with the choice of block orientation are more complicated and depend upon which player plays the block first. In the tables, B1 means that Player I played the block first, and conversely for B2.

The subsequent play depends upon which attributes have already been chosen and hold similarly for both Players, thus in Table 2 only the attribute type is mentioned as the Player designation is irrelevant. For which attribute is selected, reference is made to any attributes previously selected by either player. The presence of an attribute of the same type takes precedence over the other attributes, marked in bold emphasis. Since one cannot make two base choices, P takes precedence over D for B. The rules involving block play are more complicated as they depend upon which player played the block. They are presented in Table 3.

As an example, suppose play is $(P,B)$. Player I plays first and chooses their piece at random, say left. Player II then chooses the opposite orientation of the block, right. This ensures that when Player II chooses its piece, it is aligned with its block. Player I has only D to play. Since its P is congruent, it aligns with P1 and so is up. That forces Player II to pick up for its disk and right for its piece, since Player I has no plays left.

Now let there be two Observers, I and II. Each observer is free to observe the orientation of any component throughout the structure. Observer I examines components of Player I, Observer II examines components of Player II. Observer I’s observable as thus $P1,B,D1$ while those of Observer II are $P2,B,D2$.

Using the above observables we may form a 4-cyclic system as follows:

$$\begin{array}{ccccc}{P}_1& D_2& B& P_2& \\ R_{P_1}^{a_1}& & & R_{P_2}^{a_1}& a_1=(P1,P2)\\ R_{P_1}^{a_2}& R_{D_2}^{a_2}& & & a_2=(P1,D2)\\ & R_{D_2}^{a_3}& R_B^{a_3}& & a_3=(D2,B)\\ & & R_B^{a_4}& R_{P_2}^{a_4}& a_4=(B,P2)\\ \end{array}$$

The context is chosen by analogy with the CHSH situation. In the case of measurements by the two observers in a CHSH experiment, the two observers fix their choices of measurements. This forces an interaction with the entangled pair, formally through projection operators onto the observable vectors. This in essence forces a projection of the wave functions and hence forces a particular dynamical evolution which is compatible with the observables being measured. Here, the choice of observables involves setting the initial choices of the players to be the observables in question, which ensures that they are manifested in the evolution of the game.

The probabilities for the various possibilities for the observables are

$$\begin{array}{cccc}& & P1& \\ & & -1& 1\\ P2& -1& 0& 1/2\\ & 1& 1/2& 0\end{array}$$

$$\begin{array}{cccc}& & P1& \\ & & -1& 1\\ D2& -1& 1/2& 0\\ & 1& 0& 1/2\end{array}$$

$$\begin{array}{cccc}& & B& \\ & & -1& 1\\ D2& -1& 1/2& 0\\ & 1& 0& 1/2\end{array}$$

$$\begin{array}{cccc}& & B& \\ & & -1& 1\\ P2& -1& 1/2& 0\\ & 1& 0& 1/2\end{array}$$

The marignal probabilities for each of the random variables may be determined from the above probability distributions and it is obvious that

$$\begin{array}{c}\hfill R_{P_1}^{a_1}=R_{P_1}^{a_2}=(1/2,1/2)\end{array}$$

(5)

$$\begin{array}{c}\hfill R_{P_2}^{a_1}=R_{P_2}^{a_4}=(1/2,1/2)\end{array}$$

(6)

$$\begin{array}{c}\hfill R_{D_2}^{a_2}=R_{D_2}^{a_3}=(1/2,1/2)\end{array}$$

(7)

$$\begin{array}{c}\hfill R_B^{a_3}=R_B^{a_4}=(1/2,1/2)\end{array}$$

(8)

so that the cyclic 4 system is consistently connected. This means that the Dzhafarov inequality reduces to the usual CHSH inequality. A quick check shows that $\Delta C=$

$$\mathbf{s}(\pm <R_{P_1}^{a_2}{R}_{D_2}^{a_2}>\pm <R_{D_2}^{a_3}{R}_B^{a_3}>\cdots \pm <R_B^{a_4}{R}_{P_2}^{a_4}>)-(4-2)-\sum _1^n|<R_{q_i}^{a_i}>-<R_{q_i}^{a_{i+1}}|=$$

$$\mathbf{s}(\pm <R_{P_1}^{a_2}{R}_{D_2}^{a_2}>\pm <R_{D_2}^{a_3}{R}_B^{a_3}>\cdots \pm <R_B^{a_4}{R}_{P_2}^{a_4}>)-2\le$$

$$<R_{P_1}^{a_2}{R}_{D_2}^{a_2}>+<R_{D_2}^{a_3}{R}_B^{a_3}>+<R_B^{a_4}{R}_{P_2}^{a_4}>-<R_{P_1}^{a_1}{R}_{P_2}^{a_1}>-2=4-2=2>0.$$

In particular the CHSH formula

$$<R_{P_1}^{a_2}{R}_{D_2}^{a_2}>+<R_{D_2}^{a_3}{R}_B^{a_3}>+<R_B^{a_4}{R}_{P_2}^{a_4}>-<R_{P_1}^{a_1}{R}_{P_2}^{a_1}>=4$$

taking its maximal value and violating the Tsirel’son bound of $2\sqrt{2}$.

5. Discussion

At the present moment, the consensus interpretation of the Bell inequality agrees with that of Bell himself - that the violation of the Bell inequality (or here, the CHSH type inequality) by certain quanum mechanical systems demonstrates conclusively that the existence of a set of local hidden variables underyling the quantum phenomena is impossible. Reality, at its most fundamental level possesses a form of nonlocality which permits the existence of spooky actions at a distance, influences which may pass instantaneously between certain quantum systems, in particular entangled systems, somethng that Abner Shimony described as "passion at a distance" [19]. These nonlocal influences are said not to transmit signals, which would violate the Special Theory of Relativity, and yet are capable of informing a space-like separated particle of the form of observation being experienced by its entangled counterpart. Surely though the propagation of information constitutes a signal. It is unreasonable to believe that the bounds of Special Relativity apply only to the actions of human observers (or physicists). As noted above, a number of researchers [13,14,15,16] have questioned the validity of this interpretation of the violation of the Bell inequality, suggesting instead that the Bell inequality is not about nonlocality, but is instead about contextuality.

The fixation that physicists have upon nonlocality appears to me to be due to an implicit bias towards interpreting classical situations within an Objectivist worldview, in which classical entities are treated as objects which are enduring, with enduring properties (in the absence of interactions). That is, the view is that classical entities may be described in terms of logical propositions. This Objectivist worldview might hold, with some validity, for inanimate entities, but as noted above, it does not hold for animate entities, especially organisms, which posssess agency. A Processist worldview in which entities are generated, are transient, whose properties are conditional, generated, and contextual, provides much greater homology with observed characteristics.

The Objectivist worldview comes with another feature which has long been appealing; that is determinism. The assumption that entities are objects, with enduring propositional properties, makes the idea that everything is already fixed in place as it were, appealing. Determinism, in the sense that complete knowledge of the present fixes the future (and when time reversible the past as well), has long been a feature of classical physical theory, going back to the time of Newton. Stochasticity does occur in classical physics but it is understood to represent a lack of complete knowledge, so predictions cannot be made with complete accuracy but only with a measure of uncertainty. Quantum mechanics suggested the presence of a fundamental non-determinism in reality, although the equations which govern this non-determinism are still deterministic, so determinism persists, just a step removed from the fundamental events. Those who cling to determinism within quantum mechanics are often forced to resort to elaborate mental gymnastics such as the postulation of an unimaginable infinity of alternative universes, to epxlain away the fundamental uncertainty in quantum mechanics. A simpler explanation pertains if one shifts to a Processist worldview. There, non-determinism, or choice, is simply a fundamental feature of reality and of the entities which manifest within it. The probabilities which we observe are considered to be emergent from the underlying dynamics and interactions among these entities. In a Processist world, everything is contingent and change is fundamental, so that the idea that properties too can be dispositional and contingent in no longer so strange. Change need not be without order, however. Examples of this abound within the world of organisms and can be readily demonstrated as characteristic of organisms [4,5,7,8]. The adoption of a Processist worldview brings coherence back to our understanding of reality, and, to my mind at least, remains in keeping with the principle of Ockham’s razor [32,36].

Previous criticisms of the Bell inequality often resort to deep exploration of probability and measure theory, and are not always easily accessible. The analysis of the Bell inequality present here is simple, succinct, cogent, transparent, and I would hope so blatant that there should no longer be any question that what the Bell inequality shows is the following:

Theorem 1.

If any system possesses a set of random variables whose expectation values lead to a violation of the CHSH (or more generally the Dzhafarov inequality), then those random variables must be contextual, in other words, the relevant joint probability distributions required to calculate the expectation values must be local to each expectation value.

Questions concerning how exactly this contextuality occurs for each individual system depends upon the particular dynamics of the system. It does not depend upon whether the system can be understood in classical or in quantum mechanical terms. It does not depend upon whether there are, or are not, nonlocal influences. The examples presented in this paper of the two player co-operative game shows conclusively that a purely local, classical situation can exist, with consistently connected random variables, whch nevertheless violates the CHSH inequality maximally, and thus also violating the Tisrel’son bound. Niether classicality nor locality are involved in the violation of the inequality. What is necessary is that the probabilities associated with the calculations of the various expectation values be local, contextual. In order for that to occur, it is argued that the properties of the entities involved in such a situation not be propositional, in other words, not enduring throughout an interaction free duration, but rather are generated, disposed, transient, conditional; determined by an underlying process as dispositions (or propensities as Gisin suggests [33,34,35]) but whose values are not fixed over time but allowed to vary depending upon local conditions. The models presented in [32,36] using the Process Algebra show that at leats non-relativistic quantum mechanics can be accurately modeled using processes in which the fundamental entities are generated, rather than merely being. The fact that the mariginal probabilities for the two observers in the Bell setting are the same as in the case of free particles does not imply that they are, in fact, free particles exchanging some instananeous influence. Consistent conenctedness implies marginal degeneracy - the marginal probabilities simply do not suffice to determine the joint probabilities which arise from the dynamics of the system. In the Bell situation, the entanlged particles may appear as if they are two free particles when viewed from the vantage point of the two observers, but a third observer correlating the results their observers, or the experimenter overseeing the process which generates them in the first place, understand that the underlying joint state, or generating process, is that of entanglement, not freeness.

One cannot truly understand a system without knowing and understanding its dynamics. Indeed, I argued in [7] that what distingushed classical from quantum systems had nothing to do with size, scale, number of subcomponents, but rather the structure of the interacting processes involved in its generation. Macroscopic systems can exhibit quantum behaviour, microscopic systems can exhibit classical behaviour - it all depends upon the structure of their dynamics. The two player game presented here shows a classical level system which nevertheless exhibits quantum-like features. In this case these arise because the local properties manifested by the game depend upon the initial plays, and more generally upon the entire history of play, becuase they are generated moment to moment over a duration [5].

This the deep understanding to be derived from the violation of the Bell inequality (and its variants) is that reality cannot, in general, take an enduring, propositional form. In simpler terms, reality cannot "shine". Instead the entities of reality which comprise any system capable of violating thse inequalities must manifest properties which vary over time and over situations, even in the absence of any disturbing interactions. They must be contextual. Again in simpler terms, reality must fluctuate over time and context, in other words, reality "twinkles".