Contextual Hidden Fields Preclude the Derivation of Bell-Type Inequalities

Abstract

We show that loophole-free Bell-type no-go theorems cannot be derived in theories involving local hidden fields. At the time of measurement, a contextuality loophole appears because each particle’s electromagnetic field interacts with the field of its respective apparatus, preventing the expression of the probability density as a function independent of the orientation of the measuring devices. Then, we use the dynamical evolution of the probability distribution to show that the spin-correlation integral cannot be expressed in terms of initial Cauchy data restricted to the particles. A measurement independence loophole ensues, which prevents the usage of the non-contextual correlation integrals required to demonstrate the CHSH-Bell inequality. We propose that correlated fields are the missing hidden variable triggering the coupled nonlinear oscillations of the particles, which bring about the synchronicities observed in the Einstein–Podolsky–Rosen–Bohm (EPRB) experiment.

1. Introduction

In the seminal work of Einstein, Podolsky, and Rosen, the following dilemma is proposed [1]: either complementary variables associated with non-commuting observables cannot be computed from quantum mechanics but have a simultaneous physical reality or, on the contrary, it is just that quantum mechanics does not fail in computing all aspects of reality for the simple reason that some of these observables do not exist when specific experimental settings are considered. To this end, they adopt a sufficient empirical definition of reality, which states that a property is real if it can be measured without disturbing the system. Then, these authors demonstrate how quantum mechanics encounters a contradiction by using two entangled particles [1]. More specifically, they show that, if we assume that quantum mechanics is complete, we can overcome the lack of simultaneous reality imposed via the principle of uncertainty by making measurements using only one of the particles, which, presumably, cannot have any effect on the other particle (locality). Thus, beginning with the assumption of completeness, the assignation of realistic properties to non-commuting observables is achieved. It follows, consequently, that completeness and realism can be logically implied in entangled quantum mechanical systems. This contradicts the original dilemma, which ruled out any option apart from the mutual exclusion between completeness and realism. Thus, completeness must fail.

From an analytical perspective, there are two solid criticisms of their reasoning regarding the two fundamental philosophical assumptions made in their work. Indeed, we can avoid their dilemma by renouncing either their definition of realism or locality (or both). Bohr criticized their definition of realism by appealing to the physical interaction between the measuring apparatus and the particles [2]. According to Bohr, two different experimental contexts were used to measure non-commuting observables, and therefore, the two settings could not be combined to attribute properties to the same particle. On the other front, Bell investigated the assumption of the separability between the two electrodynamic bodies whose properties are to be measured [3]. Following an example advocated by Bohm and Ahronov [4], Bell studied the statistical correlation of different components of the spin for a pair of entangled particles (see Figure 1). He proved that statistical theories based on hidden classical magnitudes must obey certain inequalities, which are violated by quantum mechanical correlations of entangled pairs.

Many works have been published since the discovery of Bell’s theorem [5], highlighting the numerous loopholes that experiments can encounter. Most of these loopholes have been experimentally closed [6], some of them in conjunction, and it is not expected that they can produce violations of the magnitude appearing in Bell tests [6]. However, here, we adopt Bohr’s contextual perspective and show that a loophole cannot be avoided from the very definition of measurement when classical field theories are at stake. For this purpose, we reconsider Bell-type theorems using dynamical hidden fields. In these theories, the apparatus has an unavoidable effect on the particle when a measurement takes place as a consequence of their electrodynamic interaction [7,8,9]. A contextuality loophole is thus enforced because we cannot impose a priori that the probability distribution of a hidden variable is independent of the experimental arrangement.

Contrary to the assumption that the probability distributions of stationary random hidden variables are independent, the probability density of a hidden field defines a stochastic process [10] that evolves and differs across various experimental settings. Using this fact, we show that Bell’s correlation integral, when expressed in terms of the initial hidden data, is different from the integral used in the derivation of the CHSH-Bell inequality [11] unless a correlation loophole is closed. Furthermore, we demonstrate a novel feature in that this correlation loophole can also be understood as a measurement independence loophole [12]. Finally, we extend our arguments to situations where last-instant and free choices can be made regarding the orientation of the Stern–Gerlach apparatuses.

It is not the purpose of the present work to provide a quantitative mechanism that explains EPRB correlations but, rather, to demonstrate that Bell-type theorems cannot yet discard classical field theories as a foundation of quantum mechanics. Nevertheless, we suggest a qualitative explanation for the mysterious correlations between the measurements of the two electrodynamic particles by invoking the electromagnetic nature of entanglement and the concept of the synchronization of self-oscillations [7,8,13].

2. Hypotheses in Bell-Type Theorems

Bell’s program is based on computing correlations of the spin of two entangled particles [3]. In particular, a singlet state of zero total spin $S=S_1+S_2=0$ is considered, which can be written as

$$|0,0\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|\uparrow \rangle \otimes |\downarrow \rangle -|\downarrow \rangle \otimes |\uparrow \rangle \right),$$

(1)

where $|\uparrow \rangle$ and $|\downarrow \rangle$ are eigenstates of the operator $S_z$ representing the spin angular momentum component along the z-axis. As depicted in Figure 1c, from a physical point of view, we can think of this entangled state as two electrodynamic particles, a positron and an electron, flying away from each other from the location where the disintegration of a neutral meson took place, for example. These two particles are heading toward two identical Stern–Gerlach (S-G) apparatuses.

In principle, each of these two measuring devices can be oriented along any possible direction described by the unit vector a in the Euclidean space so that one measures the spin component $S_{1a}=S_1·a$, while the other can measure some other component of the spin $S_{2b}=S_2·b$, along the direction given by the unit vector b. Then, if we consider these two components of the spin, related to our two respective entangled particles, the correlation $C_{ab}$ of the product of spin projections can be computed as

$$C_{ab}=\langle 0,0|S_{1a}\otimes S_{2b}|0,0\rangle =-{\displaystyle \frac{{\hslash }^2}{4}}a·b,$$

(2)

where Equation (1) has been used, together with the expression $S_n=\hslash n^i{\sigma }_i/2$, with ${\sigma }_i$ being the Pauli matrices and n being some unit vector in the three-dimensional Euclidean space. Note that we are just computing a second-order moment using the random variables $S_{1a}$ and $S_{2b}$. For convenience, we neglect the ${\hslash }^2/4$ term in the computation of correlations by considering units of $\hslash /2$ and assigning values of plus one or minus one to the spin.

Bell’s point is that, for a deterministic hidden variable theory to exist, we must be able to compute the averages, the correlations, and the moments of any order appearing in quantum mechanics from the evolution of the system by using a probability density, $\rho$, depending on the hidden variables, $\lambda$. Any quantum operator must be defined as a function of these hidden variables in such a way that their knowledge would allow them to precisely determine the value of the quantum operator and the average of any function of this and other operators, no matter how complicated. In particular, the correlation integral can be written as

$$C_{ab}=\int A_a\left(\lambda \right)B_b\left(\lambda \right)\rho \left(\lambda \right)d\lambda ,$$

(3)

where $A_a\left(\lambda \right)$ denotes the result of measuring the spin of the first particle along the direction given by a. Given a value of $\lambda$, we must have a specific value of $A_a\left(\lambda \right)$, whether this value is one (spin up) or minus one (spin down). Correspondingly, we denote as $B_b\left(\lambda \right)$ the value of the spin of the second particle when measured along the direction given by b. If we now consider four different orientations of the apparatuses $a,a^{\prime },b,b^{\prime }$, it can be shown after some algebraic manipulations [11] that

$$|C_{ab}+C_{a{b}^{\prime }}|+|C_{a^{\prime }b}-C_{a^{\prime }{b}^{\prime }}|\le 2.$$

(4)

This inequality sets a restriction among the values that the correlations between two spin variables measured along different directions can have. By considering the quantum mechanical result $C_{ab}=-a·b$, it is very easy to see that many values do not obey this inequality. A common counterexample is to let the four directions be $\pi /4$ radians apart in a plane, in the order $b^{\prime },a,b,a^{\prime }$, so that the violation is maximized [6], yielding a value of $2\sqrt{2}$ in Equation (4).

Four hypotheses can be identified in Bell’s work [3] regarding the expression of the correlation integral appearing in Equation (3):

1.

Hidden variables $\lambda$ can be anything, from simple constant vectors to dynamical vector fields evolving in spacetime $\lambda (x,t)$.

2.

There exist two functions, $A_a\left(\lambda \right)$ and $B_b\left(\lambda \right)$, with outputs $\pm 1$, representing the two possible results of measuring the spin of any of the two particles, which depend on the hidden variables $\lambda$ related to them.

3.

There exists a probability measure, $\rho \left(\lambda \right)d\lambda$, that does not depend on the orientation of the apparatuses.

4.

The principle of locality, which states that $A_a\left(\lambda \right)$ does not depend on b and $B_b\left(\lambda \right)$ does not depend on a.

We now thoroughly discuss these four postulates about the EPRB experiment. The first assumption is of great importance since the primary purpose of Bell’s no-go theorem is to reject classical field theories as the foundation of quantum mechanics. Electrodynamic fields are primarily concerned in this regard since they do not explicitly appear in the Schödinger equation when studying the EPRB experiment, which involves two electrically charged bodies. This also concerns experiments involving correlated photon pairs, which are represented in quantum mechanics without any allusion to the electromagnetic field [9].

The second assumption is undisputed as long as the correlations are computed at the time of measurement since, in a classical field theory, the magnetic field inside the particle specifies its internal angular momentum. Together with the orientation of the measuring device, which also determines the direction of the non-uniform magnetic field created by the SG apparatus, this would suffice to determine the measurement result. However, as we shall see in Sec. IV, the expression of the functions $A_a\left(\lambda \right)$ and $B_b\left(\lambda \right)$ in terms of initial hidden field variables $\lambda$ related exclusively to the particles [11] is prevented due to the contextuality of classical field theories.

Concerning the third assumption, it is evident that, when Bell defines the probability density function $\rho \left(\lambda \right)$, he does not rigorously address the question of whether these variables depend on any time-dependent parameter. If we assume that any initial $\rho \left(\lambda \right)$ does not evolve in time, it is evident that we are also assuming that the hidden variables have constant values all over their trip from the place where the disintegration took place until they arrive at the apparatus. This is frequently considered, assuming that measured observables correspond to properties possessed by the particles before and after measurement, which is tantamount to realism, according to the definition provided by Einstein–Podolsky–Rosen [1]. Quite the opposite, in contextual field theories, this definition of reality becomes over-restrictive because the apparatus plays an active role in the production of the outcomes of an observable.

It is crucial to recall that, if hidden variables are dynamic, the probability density must inherit such a dependence, as it occurs in the theory of stochastic processes [10]. For example, if the hidden variables depend on time, $\lambda \left(t\right)$, we would have a probability density, $\rho (\lambda ,t)$, even if this time is made the same by resetting clocks at each row of the experiment. If hidden fields are under investigation, in principle, we would have to consider a probability density, $\rho (\lambda ,x_a^{\prime },x_b^{\prime },t)$, defined over the entire space, where $x_a^{\prime }$ and $x_b^{\prime }$ would be the position at which the two point particles would be found at the time of measurement.

Alternatively, if particles are considered extended and composed of fields, the probability density would need to be integrated throughout the region of space where these particles are located at time t of measurement. In this case, when we compute the correlation $C_{ab}$ using the distribution at the precise moment of arrival at the apparatus, we must face the question of whether this probability is the same for all experimental settings, a and b. In the next section, we demonstrate that a contextuality loophole gives rise to a dependence of the probability density on the measurement direction at the time of measurement, even in Newtonian mechanical systems.

We can try to avoid the effect of the apparatus on the hidden variables by assuming that uncertainty arises from a lack of control of initial conditions. The knowledge of these hidden variables would entail a determination of the spin of the particles in the very moment in which they are created. This is an example that Bell has in mind as he manifests [3]. To this end, we must use a probability distribution that describes the frequency with which these hidden variable values are uncontrollably chosen as initial conditions. Then, the hidden variables would determine, at the moment of arrival at the apparatus, the values of the experiment’s result. In this case, the probability can be expressed as independent of the measurement settings only if hidden fields in the initial Cauchy hypersurface are uncorrelated. Quite the opposite, we suggest that correlated electrodynamic field fluctuations prevent the expression of Bell-type correlation integrals.

Here, we assume the fourth hypothesis as evident, i.e., that $A_a$ does not depend on the direction of the second apparatus (say b) at the time of measurement, and vice versa. Otherwise, the correlation could be perfectly explained by assuming that the well-distanced experiments are communicating during the collapse of the simultaneous (in the laboratory frame) measurements of the two spins. This locality loophole was mainly closed in experiments with entangled photons [14,15].

3. Bell-Type Correlation Integrals

Given the evidence presented in Bell’s works [16], it is clear that he was fully aware of whether the values of the hidden variables were specified before the measurement process occurred or not. Indeed, in such works, Bell demonstrates a vast knowledge of Bohr’s ideas, where he also recognizes the misuse of the word “measurement” by acknowledging that it is the interaction between the system and the measuring apparatus that determines the result of an experiment. In other words, hidden variables are contextual. Thus, in the first place, we show that a stationary probability density is unattainable, even in ordinary classical systems. This is important since many no-hidden variables theorems rely on this assumption [17], ascribing it to realism, as previously explained.

If we make such an assumption, the main conclusion that can be drawn from the violation of such relations is that there exists no local hidden variables theory compatible with the predictions of quantum mechanics as long as these hidden variables are assigned values before the measurement of the quantum physical magnitudes being studied. The assumption of no interference between the apparatus and the hidden variables is sometimes made because, otherwise, we would still have to explain why the collapse of the wave function does not destroy correlations. However, it is challenging to determine whether this assumption is correct in the absence of a clear physical understanding of entanglement and the mechanism through which such a collapse of the wave function occurs.

Consequently, in what follows, we present a possible scenario for how a process of collapse can occur in a physical system. In this respect, we propose that the collapse of the wave function is a real dissipative process, as it has been recently demonstrated that the wave function is the wave function is the average of a real force field related to internal and external electromagnetic forces, rather than just a probabilistic entity [7,9,18,19]. The present example is used for illustration purposes only. It is not intended to replace the real case of fundamental particles under electrodynamic retarded potentials, even though it shares common features due to the dissipative character of the interaction between the apparatus and the physical system. In any case, this example aims to demonstrate that avoiding the effect of the measuring apparatus on the hidden variables cannot be easily and generally accomplished without detailed experimental justification.

The experiment that we propose is ostensibly simple. Consider that we let a coin fall freely along the vertical line with some initial angular velocity, ${\omega }_0$, from some initial height, $z_0$. The dynamics of the coin, when it is falling, are governed by Newton’s second law and Euler’s rotation equations and can be described using the variables $\theta \left(t\right)$ and $z\left(t\right)$ in the configuration space. We suppose that the value of the angle $\theta \left(t\right)$ is taken concerning an axis of rotation that, for simplicity, we assume to be orthogonal with respect to the vertical axis. Then, to measure the angle (modulo $2\pi$) of the coin, we let our approximately rigid body evolve until it experiences an inelastic collision with a table underneath, which sets it to one of its two sides (the edge of the coin is thin enough). There can be only two possible outcomes of this measurement: heads ($\theta =0$) or tails ($\theta =\pi$). We can safely affirm that, before the measurement, this system was in a combined state, oscillating between the two possible results of the experiment. Its interaction with the table, here playing the role of the apparatus, forces it to acquire one of the two possible outcomes that, following the simile, play the role of the eigenstates.

Of course, we could do better by using another apparatus with much smaller action than the table so as not to disturb the coin. We must also recall that both the final coordinates and the momenta of the two degrees of freedom, z and $\theta$, are known. Thus, no uncertainty principle is taking place, as in quantum mechanics. Moreover, we do not find here any interfering waves whose modes can be quantized, as they occur with self-oscillating electrodynamic moving bodies [7,8] or in experiments with walking droplets [20,21]. Nevertheless, this example allows us to illustrate one important point, which is that the interaction with the table produced the experiment’s outcome. Therefore, the result cannot be considered a property possessed by the system before the collision taking place. Interestingly, we note that this interaction is mediated by a dissipative phenomenon, a consequence of the electrodynamic nature of both the coin and the table.

In the modern jargon of dynamical systems theory, we would say that the coin and the table comprise a nonlinear dynamical system that possesses two attractors in the phase space [22]: $(\theta ,\omega ,z,v_z)=(0,0,0,0)$ and $(\theta ,\omega ,z,v_z)=(\pi ,0,0,0)$. It is also worth mentioning that, as a consequence of the dissipative dynamics, all the information concerning the angle and the angular momentum at the moment immediately previous to the contact between the two bodies is erased with the interaction. In other words, once an attractive limit set is attained, we have no empirical procedure to find out how this collapse toward the attractor occurred. This is just another manifestation of the irreversible nature of classical electrodynamics of moving sources, which is governed by the non-conservative functional differential equations arising from the field solutions of Jefimenko [8,23].

Moreover, we can study how the initial conditions affect the outcome of the experiment by computing the basins of attraction. Given a dynamical system with two or more attractors, a basin of attraction is a color plot defined over a region of the phase space, where every initial condition within a range of values is assigned a color according to its asymptotic fate [24]. For example, in our case, we can use two colors, white and blue, corresponding to the outcomes of heads and tails, respectively. It has been shown that, if we do not let the coin rebound on the table one or more times, the basins of attraction are manifestly separated by a smooth and predictable boundary at our visual scale [22]. However, as shown in Figure 30 of Ref. [22], when we allow the coin to rebound twice on the table, the basins become increasingly random. If we allow more rebounds, they completely mess up at the original scale [22]. Further inspection reveals that a fractal structure does not appear as we reduce the scale by zooming into the picture, as it commonly occurs with chaotic dynamical systems [24]. Therefore, from a classical point of view, the system is not unpredictable if sufficiently small scales are accessible to the experimenter [22]. On the contrary, for atomic systems, a sufficient reduction in the scale cannot be achieved. This occurs because fundamental particles are limiting structures, and, as a consequence of this fact as well, it is impossible to reduce arbitrarily the action of the measuring apparatus on the system.

These last considerations lead us to the second important moral of the present analysis. The key point that we would like to stress is that, if our lack of knowledge of the final result of the coin orientation arises from our inability to control the initial conditions with which the coin is released in the beginning, a modification of the materials or the geometry of the table, here playing the role of the measuring apparatus, certainly can affect the nature of the potential results. These effects have two consequences. On the one hand, they modify the function $B_b\left(\lambda \right)$, for example, to some other $B_c\left(\lambda \right)$, as Bell wisely devised when defining these variables.

On the other hand, since the dynamical evolution of the system is now guided by new differential equations, because of the different arrangements of the apparatus, the dynamically evolving solutions might not be the same. This introduces a dependence of the probability density on the measuring apparatus, which can be represented in the form ${\rho }_{ab}(\lambda ,t)$, as it frequently occurs in contextual theories [25,26]. In principle, $\lambda$ can also incorporate hidden variables of the apparatus. However, it must be highlighted that independence between the hidden variables related to the apparatus and those of its corresponding particle at the time of measurement is unattainable, contrary to expressions appearing in some contextual models [26].

If the reader hesitates to ascertain that this last argument is correct, perhaps they can consider the more familiar case of conservative physical systems. There, the evolution of a probability distribution of an ensemble of initial conditions $\rho \left(\lambda \right)$ is governed by Liouville’s equation [27]. This hive of initial points distributed over the phase space, each representing a repetition of the same experimental arrangement, evolves under the symplectic flow defined by Hamilton’s equations. But then, we should not deny that, if the Hamiltonian changes throughout the evolution of the system, so does the nature of the probability density $\rho (\lambda ,t)$, which is advanced by the Liouvillian operator. In our more complicated case, which is neither conservative nor Hamiltonian [7,8], this change in the probability distribution is also reflected in the basins of attraction, which in turn affect the functions $A_a\left(\lambda \right)$ and $B_b\left(\lambda \right)$.

We now proceed to demonstrate Bell’s correlation integral by using the insights gained from the previous example. In the case of traditional mechanical systems (e.g., Hamiltonian systems described by ordinary differential equations), where the feedback between the environmental electrodynamic fields and the dynamics of particles is neglected, particles can be represented by some finite set of generalized coordinates that are affected by some potentials defined in the region where the apparatuses rest. Since the dynamics of the fields are being disregarded, we can neglect the explicit spatial dependence and write down the correlation as

$$C_{ab}\left(t\right)=\int A_a\left(\lambda \right)B_b\left(\lambda \right){\rho }_{ab}(\lambda ,t)d\lambda .$$

(5)

We can return to a probability density in terms of the hidden initial conditions ${\lambda }_0$ and obtain a relation similar to the one used by Bell and his advocates. This can be done through a change in random variables in the form

$${\rho }_{ab}(\lambda ,t)=\int \rho \left({\lambda }_0\right)\delta ({\lambda }_A-{\mathsf{\Lambda }}_{At}^a\left({\lambda }_0\right))\delta ({\lambda }_B-{\mathsf{\Lambda }}_{Bt}^b\left({\lambda }_0\right))d{\lambda }_0,$$

(6)

where the flow ${\mathsf{\Lambda }}_t$ relates the hidden variables at the time of measurement and at the initial time $\lambda ={\mathsf{\Lambda }}_t^{a,b}\left({\lambda }_0\right)$ so that the vector $\lambda$ separates as a direct sum of two other vectors ${\lambda }_A$ and ${\lambda }_B$, which are uncorrelated. In other words, we assume that the flow can also be written as ${\lambda }_A={\mathsf{\Lambda }}_{At}^a\left({\lambda }_0\right)$ and ${\lambda }_B={\mathsf{\Lambda }}_{Bt}^b\left({\lambda }_0\right)$. Finally, we must consider that each observable depends on its respective and disconnected hidden variables as well (i.e., $A_a\left({\lambda }_A\right)$ and $B_b\left({\lambda }_B\right)$ must hold), as suggested by the principle of locality. In this manner, we obtain the integral appearing in Equation (3) by simply redefining the functions $A_a$ and $B_b$, and Bell’s agenda [3,11] can be perfectly carried out.

4. Hidden Field Theories

Now, we assume that the hidden variables represent a vector field, $\lambda (x,t)$, where t is the present time in the laboratory frame, while x represents the position in the space. Here, we accept that electrodynamic bodies are made of extended electromagnetic fields. This entails thinking of fundamental particles as some electromagnetic solitons [28], which can be dynamically stabilized as a consequence of spacetime curvature [9]. Then, any other properties relying on their internal structure, for example, the spin, can be connected to the different physical properties of such gauge fields [29,30]. For example, the spin could be related to the magnetic field configuration, while the charge would correspond to the topological charge of the vortex knot [31]. However, we recall that no particular assumptions are made here about our argument concerning what particles are. Our results are perfectly extensible to point particles, where their properties (charge, mass, and spin) are all embedded in the point.

It is evident that the different orientations of the non-uniform magnetic field of a Stern–Gerlach apparatus will affect our dynamical hidden fields. Then, we have to rewrite the correlation function as

$$C_{ab}\left(t\right)=\int \int A_a\left[{\lambda }_A(x_a^{\prime },t)\right]B_b\left[{\lambda }_B(x_b^{\prime },t)\right]{\rho }_{ab}[\lambda ,t]\mathcal{D}{\lambda }_A\mathcal{D}{\lambda }_B,$$

(7)

where t represents the time at which the particle’s spin is first aligned with the S-G apparatus, while $x_a^{\prime }$ and $x_b^{\prime }$ are the spatial coordinates of the region where the two particles at such time instant are found. This integral is a functional integral over a set of hidden fields at the time of measurement, integrated over the region where most of the particle’s energy is stored. We also notice that the integral runs through all the possible values of the hidden fields $\lambda =({\lambda }_A(x_a^{\prime },t),{\lambda }_B(x_b^{\prime },t))$ of the two particles, which have been gathered in one hidden variable for simplicity. Again, recall that the effect of the apparatus on the probability distribution appears explicitly in ${\rho }_{ab}[\lambda ,t]$.

We now demonstrate that, when fields defined throughout the entire space are considered, the derivation of the correlation integral becomes more complicated, yielding a different result. The hurdle arises because the hidden fields at the particle’s position when its magnetic moment has completely aligned with the external field of the Stern-Gerlach apparatus (i.e., when the collapse has completed) are determined from its causal past lying in some initial Cauchy hypersurface. As an example, the reader can consider the well-known theorem of Cauchy–Kovalévskaya [32]. In the study of Bell-type theorems, the Cauchy surface must be chosen at the time when the entangled pair is created at the source.

In Figure 2, we have represented the light cones of two entangled bodies that are measured at some instant of time to illustrate this effect. We observe that, when attempting to express the correlation integral in terms of the initial hidden fields, which are defined on the aforementioned Cauchy surface, the probability density must inherit a dependence on the orientation of the apparatus. Indeed, the fields at the time of measurement are related to the initial fields through Green’s function, $G(x^{\prime },x,t)$, in field theory and its time derivatives. For simplicity, we write this relation in the form

$${\lambda }_a(x^{\prime },t)=\int G(x^{\prime },x,t){\lambda }_0\left(x\right)dx.$$

(8)

Indeed, the propagator $G(x^{\prime },x,t)$ will be zero between two regions that are not causally connected, and consequently, only the region inside the initial domain ${\mathsf{\Omega }}_a$ affects the values of the hidden variables ${\lambda }_a(x^{\prime },t)$ and similarly for ${\lambda }_b(x^{\prime },t)$ and ${\mathsf{\Omega }}_b$ (see Figure 2 again). If we now consider that the change in variables between the initial probability density and the probability density at the time of measurement is given via the mathematical relation ${\rho }_{ab}[\lambda ,t]=\int \delta [{\lambda }_A(x_a^{\prime },t)-{\lambda }_a(x_a^{\prime },t)]\delta [{\lambda }_B(x_b^{\prime },t)-{\lambda }_b(x_b^{\prime },t)]{\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]\mathcal{D}{\lambda }_0$, the correlation integral in terms of the initial fields can be written in the form

$$C_{ab}=\int {\widehat{A}}_a\left[{\lambda }_0\left(x\right)\right]{\widehat{B}}_b\left[{\lambda }_0\left(x\right)\right]{\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]\mathcal{D}{\lambda }_0,$$

(9)

where this integral is again a functional integral over a set of uncontrollable initial hidden fields, defined on the initial Cauchy surface. We have introduced the functional ${\widehat{A}}_a\left[{\lambda }_0\left(x\right)\right]=A_a[\int G(x^{\prime },x,t)\lambda (x,0)dx]$, and the same has been done for $B_b$.

Importantly, we highlight that, when these functionals were defined, the spatiotemporal coordinates t and $x^{\prime }$ have been omitted. The temporal coordinate can be neglected by assuming that clocks are reset at the beginning of each experimental row and that measurements are always performed at the same instant of time. The spatial coordinate can be neglected if we assume that the measured particle remains at the exact location and maintains the same orientation during the measurement. These two restrictions are experimentally impossible to achieve [7,8] and would hinder the derivation of Bell’s inequalities, as has already been proposed [33]. Nevertheless, here, we take these impossible conditions for granted to focus on a more severe loophole due to a correlation of hidden field fluctuations [34].

If we assume that the initial fields are only correlated in the domain ${\mathsf{\Sigma }}_c={\mathsf{\Omega }}_a\cap {\mathsf{\Omega }}_b$ depicted in Figure 2, the CHSH-Bell inequality can be derived. We have to average the initial hidden fields over the regions ${\mathsf{\Sigma }}_a={\mathsf{\Omega }}_a/{\mathsf{\Sigma }}_c$ and ${\mathsf{\Sigma }}_b={\mathsf{\Omega }}_b/{\mathsf{\Sigma }}_c$. Indeed, under such a hypothesis, the probability density ${\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]$ can be expressed as a product of densities ${\rho }_a\left[{\lambda }_a\left(x\right)\right]{\rho }_b\left[{\lambda }_b\left(x\right)\right]\rho \left[{\lambda }_c\left(x\right)\right]$, where ${\lambda }_a\left(x\right)=\{{\lambda }_0\left(x\right):x\in {\mathsf{\Sigma }}_a\}$, ${\lambda }_b\left(x\right)=\{{\lambda }_0\left(x\right):x\in {\mathsf{\Sigma }}_b\}$ and ${\lambda }_c\left(x\right)=\{{\lambda }_0\left(x\right):x\in {\mathsf{\Sigma }}_c\}$. This is possible under the assumption that the fields are uncorrelated in space-like separated sets. Similarly, we can express the functions ${\widehat{A}}_a[{\lambda }_a\left(x\right),{\lambda }_c\left(x\right)]$ and ${\widehat{B}}_b[{\lambda }_b\left(x\right),{\lambda }_c\left(x\right)]$. This yields the functional integral

$$C_{ab}=\int \int \int {\widehat{A}}_a[{\lambda }_a\left(x\right),{\lambda }_c\left(x\right)]{\widehat{B}}_b[{\lambda }_b\left(x\right),{\lambda }_c\left(x\right)]{\rho }_a\left[{\lambda }_a\left(x\right)\right]{\rho }_b\left[{\lambda }_b\left(x\right)\right]\rho \left[{\lambda }_c\left(x\right)\right]\mathcal{D}{\lambda }_a\mathcal{D}{\lambda }_b\mathcal{D}{\lambda }_c,$$

(10)

which, afterfluctuations ${\overline{A}}_a\left[{\lambda }_c\left(x\right)\right]=\int {\widehat{A}}_a[{\lambda }_a\left(x\right),{\lambda }_c\left(x\right)]{\rho }_a\left[{\lambda }_a\left(x\right)\right]\mathcal{D}{\lambda }_a$ are averaged out, and the same is conducted for ${\overline{B}}_b\left[{\lambda }_c\left(x\right)\right]$, yields a Bell-type integral. As has been pointed out in previous works [26,35], these experiments are impossible to accomplish since we would have to repeat each experimental row with the same two entangled particles a significant number of times, letting the electrodynamic fields in ${\mathsf{\Sigma }}_a$ and ${\mathsf{\Sigma }}_b$ change to average out their fluctuations. However, Equation (10) allows us to derive the CHSH-Bell inequality and must describe the same type of correlations.

The reader might wonder why there should be a dependence of ${\widehat{A}}_a[{\lambda }_a\left(x\right),{\lambda }_c\left(x\right)]$ and ${\widehat{B}}_b[{\lambda }_b\left(x\right),{\lambda }_c\left(x\right)]$ on the fields outside the particles ${\lambda }_a\left(x\right)$ and ${\lambda }_b\left(x\right)$ at all. The importance of the environmental fluctuating hidden fields in all three sets, ${\mathsf{\Sigma }}_a$, ${\mathsf{\Sigma }}_b$, and ${\mathsf{\Sigma }}_c$, is explained as follows. On the one hand, these fluctuations must not be strong in the former regions. Otherwise, decoherence phenomena would appear. In both the former and latter areas, as has been recently suggested, these field fluctuations, which travel at the speed of light and reach the particle along their journey to the SG apparatus, can be indispensable to promote their coherent synchronized oscillatory motion [36]. In turn, these oscillations can be crucial in explaining the entanglement of electrodynamic particles. We develop these ideas more thoroughly in Section 6.

A simple correlation between ${\mathsf{\Sigma }}_a$ (or ${\mathsf{\Sigma }}_b$) and ${\mathsf{\Sigma }}_c$ still allows us to factorize Equation (10) and derive the CHSH-Bell inequality. Therefore, our main conclusion is that, to prove this inequality for classical field theories, we must first show that hidden electrodynamic field fluctuations are uncorrelated for the space-like separated sets in ${\mathsf{\Sigma }}_a$ and ${\mathsf{\Sigma }}_b$ in the Cauchy hypersurface [37]. Only this condition allows us to write the correlation integral as in Equation (3) and ensure the mathematical condition of measurement independence between hidden fields in ${\mathsf{\Sigma }}_a$ and ${\mathsf{\Sigma }}_b$. This assumption deserves more attention since correlations of field fluctuations far from equilibrium can extend through vast regions of spacetime in the Cauchy hypersurface [34]. The distance between ${\mathsf{\Sigma }}_a$ and ${\mathsf{\Sigma }}_b$ depends considerably on the nature of the experiment, being very small for experiments involving photons receding from each other and critically depending on the speed at which charges separate in experiments with entangled fermions. Finally, we can wonder whether the size of ${\mathsf{\Sigma }}_a$ (and ${\mathsf{\Sigma }}_b$) can be reduced to a tiny region in the initial Cauchy hypersurface comprising the mechanism that sets the apparatus’s orientation. However, this is contrary to the problem of initial conditions in field theory, where all the Cauchy data is required to compute the field at $x^{\prime }$, which includes both the field created by the apparatus and its orientation.

As has already been pointed out [34], this correlation loophole was overlooked by Shimony, Horne, and Clauser in reply to a work by Bell [38], where they discuss another possible source of conspiracy [39]. Unless the correlation loophole is closed, we cannot restrict the parameters a and b to the functions ${\overline{A}}_a\left[{\lambda }_c\left(x\right)\right]$ and ${\overline{B}}_b\left[{\lambda }_c\left(x\right)\right]$ or remove them from the probability density. Equation (9) prevents the derivation of the CHSH-Bell inequality. Unless the loophole is experimentally closed, we cannot safely affirm that classical field theories can be rejected as a foundation of quantum mechanics. First, it must be shown that all the fields and their fluctuations are uncorrelated or irrelevant to the production and maintenance of entangled pairs, which is at odds with recent findings explaining the origin of the wave-particle duality in terms of self-oscillations [7,40]. Importantly, our derivation of Equation (9) involves hidden fields that are local in the sense of classical field theories [9].

5. Last-Instant and Random Choices

We might be tempted to avoid any possible effect on each particle by the environmental field fluctuations, which depend on its related apparatus, by making a last-instant choice, i.e., by dynamically setting its orientation through some physical mechanism, before the particle enters the S-G device [15]. The main purpose is to avoid the dependence of ${\rho }_{ab}[{\lambda }_a\left(x\right),{\lambda }_b\left(x\right),{\lambda }_c\left(x\right)]$ on a and b, both in the hidden fields and the probability density. This would entail that the hidden fields appearing in ${\widehat{A}}_a[{\lambda }_a\left(x\right),{\lambda }_c\left(x\right)]$, and ${\widehat{B}}_b[{\lambda }_b\left(x\right),{\lambda }_c\left(x\right)]$ are also independent of a and b.

Now, suppose the orientation of the apparatuses is set dynamically. In that case, the system of partial differential equations that describe the evolution of the fields is required to compute the evolution of such orientations, which can be expressed as their functions $f_a$ and $f_b$ in the form $a({\widehat{x}}_a,t)=f_a\left({\lambda }_A({\widehat{x}}_a,t)\right)$ and $b({\widehat{x}}_b,t)=f_b\left({\lambda }_B({\widehat{x}}_b,t)\right)$. By composition, these functions become dynamical fields as well, where ${\widehat{x}}_a$ and ${\widehat{x}}_b$ represent the positions of the center of the top part of each respective S-G (see Figure 1), measured from the center of mass of the apparatus, for example.

In this case, the resulting change in variables relating the probability density at the time of measurement, and the probability density defined on the initial Cauchy hypersurface is given by the relation ${\rho }_{ab}[\lambda ,t]=\int \delta [{\lambda }_A(x_a^{\prime },t)-{\lambda }_a(x_a^{\prime },t)]\delta [{\lambda }_B(x_b^{\prime },t)-{\lambda }_b(x_b^{\prime },t)]\delta [a({\widehat{x}}_a,t)-f_a\left({\lambda }_A({\widehat{x}}_a,t)\right)]\delta [b({\widehat{x}}_b,t)-f_b\left({\lambda }_B({\widehat{x}}_b,t)\right)]{\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]\mathcal{D}{\lambda }_0$, where we have now gathered the four variables in the vector $\lambda =({\lambda }_A(x_a^{\prime },t),{\lambda }_B(x_b^{\prime },t),{\lambda }_A({\widehat{x}}_a,t),{\lambda }_B({\widehat{x}}_b,t))$. However, we must not neglect the fact that, just as it occurred before, the initial probability density ${\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]$ changes when a different orientation of the S-G apparatus results in measuring the spin. The dependence of the probability density of the initial hidden fields on the ${\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]$ can be considered, from a mathematical point of view, a superdeterministic loophole [41], which is not covered in experiments closing empirical loopholes up to the date [42].

The reason for this dependence is that, to compute the correlations for a definite orientation of the apparatus, all of those initial field configurations that lead to a different orientation are disregarded. Indeed, in a deterministic field theory, the uniqueness of solutions of the partial differential equations enforces that two different orientations of the apparatuses always have different hidden configurations in the initial Cauchy surface. Consequently, a restriction is imposed on the events in the sample space of initial fields. Thus, a dependence of the probability density ${\rho }_{ab}\left[{\lambda }_0\left(x\right)\right]$ on the orientations a and b cannot be circumvented. As readers can verify for themselves, the replacement of ${\rho }_{ab}[\lambda ,t]$ in a corresponding correlation integral, similar to Equation (7), leads to Equation (9) anew.

Moreover, we can also assume that the orientations of the S-G apparatuses are set via an inherently random mechanism (freely, if desired) at some brief instant of time before the measurement takes place. The consideration of these stochastic hidden variable models is relevant because, even though one is inclined to think that randomness is just a byproduct of deterministic chaos [43], there exist complexity measures that distinguish between low-dimensional chaotic dynamical systems and computer-generated noise [44].

If stochastic fields are considered fundamental [45], we can mathematically represent the selection of the orientation of the apparatus, and any other randomness in the experiment as well, using a Langevin current $j(x,t)$ acting on the dynamical fields, which ultimately decides what orientation of the S-G is used. This turns the system of equations describing the evolution of the fields $\lambda (x,t)$ into a system of stochastic partial differential equations. The relation between the hidden field configuration of the apparatus and the initial data ${\lambda }_0\left(x\right)$ is now given by the convolution ${\lambda }_a(\widehat{x},t)=\int G(\widehat{x},x,t){\lambda }_0\left(x\right)dx+\int {\int }_0^{t}G(\widehat{x},x,t^{\prime })j_a(x,t^{\prime })dxd{t}^{\prime }$, where, in this example, we have considered a situation in which the final orientation of the S-G is given by the vector a. A similar equation holds for any other orientation concerning the other measuring device by appropriately changing the Langevin current.

Nevertheless, the previous convolution prevents the expression of the probability density as being independent of the apparatus setting. As long as there is any deterministic component in Green’s function of the field’s dynamical equations (no matter how small), in addition to the stochastic Langevin force, we cannot guarantee the ergodic hypothesis. It is required that any orientation of the apparatuses is accessible from a particular initial field configuration in the entire initial Cauchy surface. This argument also concerns very complicated situations where human decision-making is used to generate random choices [46] or when cosmic photons are used to ensure that the space-like separation between the events that trigger the orientation of the apparatuses are independent [47]. Simply put, the values of a and b can be assumed to be statistically independent and causally disconnected. Irrespective of how this choice is made, a particular setting of the apparatus affects the particle’s hidden variables when its internal angular momentum is measured. Because different probability densities computed from the Cauchy data in its causal past correspond to different orientations of the SG devices, the correlation loophole induced via contextual hidden fields shall persist in stochastic hidden field variable models [25,45].

6. Entanglement as Chaotic Synchronization

We now provide a qualitative explanation of why quantum collapse preserves the correlations of entangled pairs. Assuming that no communication between apparatuses is allowed (locality loophole), as recent experiments guarantee [48], the authors can only envisage one possible solution: that such correlations are dynamically preserved throughout the entire physical process. In other words, it must be the property of entanglement that guarantees the observed correlation at all times. From this point of view, the internal angular momentum of the two particles is evolving, but both particles are electromagnetically synchronized to render the total conservation of the spin. This locking of the phases of their evolving internal angular momentum must not be a product of chance but can only be maintained using some nonlinear interaction [49].

It has recently been demonstrated, by deriving the quantum potential from classical electrodynamics, that the wave-particle duality originates in the electrodynamic Liénard–Wiechert potentials [7,8,40]. These retarded potentials lead to state-dependent, time-delayed differential equations, reflecting the memory effects arising from self-interactions of electrodynamic accelerated extended bodies. The feedback interaction of radiative and Coulombian fields can unleash self-oscillatory dynamics [13], which manifest the excitable character of fundamental particles and enables them to operate far from equilibrium, especially in the presence of fluctuating external fields [8]. The resulting motion is a violent oscillation with a frequency similar to the zitterbewegung oscillation appearing in Dirac’s equation.

When several particles are considered in interaction, the delays of their self-interactions, which lead to internal oscillation, can couple with the delays affecting their mutual interactions. Since these delays depend on the kinematic variables of the bodies at different times, the dynamics of these fundamental particles become subsequently entangled. In other words, the internal oscillations of the particles can become synchronized, as this phenomenon is technically named in the theory of nonlinear oscillations. In the case of spin-entangled pairs, a chaotic synchronization between the evolving internal magnetic moment of the electromagnetic particles is expected [36].

We can further assume that the process of collapse is of such a nature that it destroys the entanglement in accordance with deterministic laws, even if there is a sensitivity to initial conditions. This means that the basin of attraction, which determines which eigenmode of vibration (limit cycle) is selected, does not occur by chance. It depends on a dynamic relation between both the system and the apparatus. The only mandatory condition is that the collapse occurs more or less simultaneously in the laboratory frame. In this regard, the two particles must be kept entangled throughout their respective journeys, which is a rather challenging task to accomplish, given the tendency of these particles to decohere as they strongly couple to their surrounding electromagnetic fluctuating environment and approach other external bodies.

7. Discussion

We have shown that Bell-type inequations cannot be derived from classical theories involving local fields. This occurs because the hidden fields circumscribed to the particle, from which the internal angular momentum is defined, are dynamically evolving. Their values are determined by the interaction of the particle’s fields and the hidden fields of the contextual measuring devices [35,50]. These hidden fields are related to initial field data from its causal past, which also involves the apparatuses. In turn, this fact implies a measurement independence loophole that can only be circumvented by closing the correlation loophole. Our argument is reminiscent of arguments presented in previous works [34]. However, here, we have not assumed that the fields need to be random; we have explicitly represented hidden fields all over the physical process and connected superdeterminism to the contextual paradigm.

We also recall that the present arguments are extensible to other inequalities [11], which rely on correlation integrals as defined by Bell, and also to stochastic hidden variable models that are tantamount to them [51]. For the same reason, the present work demonstrates the non-Kolmogorov nature of joint probability distributions [52] comprising more than one orientation of the same measuring apparatus due to field correlations. Our work also rules out other formulations of no-hidden-variables theorems, for example, the Bell–Kochen–Specker theorems [17], since we have shown that stationary probability densities do not comply with classical electrodynamics. If the contextuality loophole is not circumvented, traditional experimental works carried out so far to test Bell’s inequalities [14,48] do not prove the impossibility of a foundation of quantum mechanics in terms of classical field theories [9].

The point of view of the present work aligns well with a description of reality in terms of two layers, as explained in previous essays [53]. The first layer consists of fields, which are hidden insofar as they are not directly measurable. Following Bell, we could refer to these hidden fields as the beables [38]. On the other hand, there is what we get to know through the affection of different parts of the field, one related to the system and another to the apparatus, which manifests as some perceivable change. Such interacting forces manifest as a dynamical effect of the fields (e.g., the appearance of a spot on a fluorescent screen), which is the only thing that is accessible to us directly, as opposed to the fields themselves. This would constitute the phenomenal part of reality, which is frequently called the observables.

But even if we decide not to perform the measurement, the fields must be there as the precondition of any experimental knowledge, at least if we are willing to accept that this knowledge is nothing other than the result of a dynamical effect of the fields [9]. Thus, whatever the fundamental equations that constitute the dynamics of elementary fields, they must be used simultaneously for the representation of the system and the measuring apparatus, which have the same physical basis and, in this respect, form a unified, inseparable physical reality.

The frequent avoidance of the representation of the apparatus and its fields in quantum mechanical formalism, which reveals the non-completeness of quantum theory itself, is justified because of its tremendous size and complexity. This theoretical convenience, which led Bohr to formulate the principle of correspondence, might be at the core of the measurement problem in quantum mechanics. Hopefully, our increasing ability to perform numerical simulations will help us address these multi-scale problems, thereby avoiding the need for the concept of measurement as a fundamental concept in physical theories [54].