Introduction

The Problem with the Dirac field

When Dirac wrote down his equation, he was expecting the two state spin that is observed. His gamma matrices, however, were four dimensional, containing two SU(2) spins with two states, and each being the mirror image twin of the other.

The SU(2) description of Dirac spin shows it is a point particle with three spatial components, $\left({\sigma }_X,{\sigma }_Y,{\sigma }_Z\right)$ of $\mathbf{\sigma }$, with its antimatter twin the opposite, $\left(-{\sigma }_X,-{\sigma }_Y,-{\sigma }_Z\right)$, being the reflection, or mirror image, of its partner. In contrast, under Q${ }_8$, a Q-spin is one particle with four states, and represented by $\left({\gamma }_s^0,{\gamma }_s^1,\pm {\tilde{\gamma }}_s^2,{\gamma }_s^3\right)$, displaying both L and R helicities via the $\pm 1$.

The perplexing point for Dirac was the twin spin also had opposite, or negative, energies from its partner which went to minus infinity. Dirac resolved this by proposing a fermionic definition of a negative energy continuum, and added electrons according to the Pauli principle until he had filled the continuum. Holes of antimatter positrons are formed as electrons jumped to positive energy.

Dirac, seeing his second particle with opposite charge, believed this was a way antimatter can form. Eventually this interpretation took hold, but it has worrying details, such as issues with the negative energy of the antimatter twin, lepton decay in the standard model, [33], and parity violation, the tau-theta puzzle, [34].

Here we do not dispute the existence of antimatter, but question the Dirac interpretation of its production. The four dimensional field describes one, not two, particles. Since the two spins precess oppositely and in phase, Figure 2, the two energies must be equal and opposite. This interpretation resolves the negative energy problem Dirac encountered. Later we discuss the parity violation of beta decay.

From Figure 2, the two coupled axes that give the spin 1 boson, are mirror states. Normally two spins can form four states, the singlet and the triplet. In the triplet, the symmetric in-phase precession of the two spins can be in two directions by simultaneously reversing the precession of each, Figure 2. This generates the two magnetic components of $m=\pm 1$. However, consistent with a photon, Q-spin has no $m=0$ component nor a singlet state because the mirror symmetry between the two axes cannot then be maintained.

We also note that the wave-particle duality of spin is dependent upon its environment. The polarization (particle), $\sigma$, and the coherence (wave), $i\sigma$ are manifest as complementary attributes such that the polarization exists only in an anisotropic environment as a fermion, whereas the coherence exists only in an isotropic environment as a boson,

$$e_F^{-}\left(\mathrm{anisotropy}\right)\leftrightarrow e_B^{-}\left(\mathrm{isotropy}\right)$$

(34)

By finding a 2D planar structure of an electron, it is no longer a point particle, whereas all the elementary particles in the Standard Model are point particles. We can only surmise that in the brief epoch after the Big Bang, electrons and positrons were formed in a two step process, with at first positrons having a charge of $+\frac{1}{2}{e}^{-}$ and electrons with $-\frac{1}{2}{e}^{-}$. Call these pre-spins for now, and indeed are point particle vector quantities. Shortly thereafter, collisions resulted in either: 2D Q-spins (collisions between particles with the same charge), which are stabilized by the resonance of its boson spin, whereas collisions between particles with opposite charge are annihilated.

If such a two-step process occurred, then it can also reverse in collisions, which would resemble a matter-antimatter pair production, [35] but does not provide proof of Dirac’s interpretation.

Bell’s Theorem

The notion of non-locality rests upon the violation of BI, [5] which have nothing to do with quantum mechanics but provide a useful quantitative classical definition of the “Invisible Boundary” [36] between the microscopic and macroscopic. For the CHSH [11] form of Bell’s inequalities, the invisible boundary is 2. Here we accept Bell’s inequalities, but reject Bell’s Theorem which states in his own words [37],

“If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

We first point out that in the presentation here, there are no hidden variables and consequently there can be no loopholes. The only variable is the angle $\theta$ which orients a spin on the Bloch sphere (with an azimuthal component of zero).

Second Q-spin theory does not violate Bell’s theorem when viewed as two complementary sources of events. Each attribute obeys BI with a CHSH of two for polarization and one for coherence, from the simulation.

We do not question the mathematical rigor of the derivation of Bell’s theorem, [37]. Rather we assert that the theorem is only applicable to classical system, giving a CHSH bound of 2, which is never violated. Quantum systems display complementarity which Bell ignored. Simply stated, Bell’s theorem considered only polarized states and not coherence. His theorem’s proof shows that using classical systems, the only way to violate his classical bound was to invoke non-local connectivity. This, he proves cannot be due to Local Hidden Variables. The work here proves his conclusion wrong.

Despite the wide acceptance of Bell’s theorem, there exist a number of critics that raise cogent doubts of the applicability of Bell’s Theorem , see [38]–49].

Non-locality vs. local realism

The immediate consequence of the inapplicability of Bell’s Theorem is that non-local connectivity is not viable. Out of the myriad descriptions we feel that Wikipedia sums up non-locality the best, [50]:

“The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle ... and hence assured the “proper" outcome of the measurement of the other part of the entangled pair."

Scholarpedia states [51], “Bell’s theorem asserts that if certain predictions of QM are correct then our world is non-local. “Non-local" here means that there exist interactions between events that are too far apart in space and too close together in time for the events to be connected even by signals moving at the speed of light."

Without Bell’s Theorem, these definitions cannot be supported.

The violation of BI now takes on entirely different meaning. Spin polarization and helicity both simultaneously exist as elements of reality; coincidence experiments are sensitive to both polarization and helicity Eq.(13); the experimental violation of Bell’s inequalities confirms local realism.

The diametrically opposite conclusion is reached in Bell-type experiments of which an enormous literature exists, see $e.g.$ “The Big Bell test" [52]. The paper’s conclusion is incorrect claiming local realism is challenged. Rather their data shows just the opposite: evidence for local realism by the existence of elements of reality that account for the apparent violation.

Quantum teleportation

Quantum teleportation, [21], depends solely upon long range correlation between EPR pairs which cannot be justified in the absence of Bell’s Theorem. The opening paragraph in that paper confirms this, “The existence of long range correlations between Einstein-Podolsky-Rosen (EPR) pairs of particles raises the question of their use for information transfer."

Teleportation claims that non-locality is mediated by “Einstein, Podolsky, Rosen Channels", but nowhere are these “channels" formulated and there is no evidence they exist. Our work shows they do not exist. In particular the step from their Eq.(4) to Eq.(5), which is critical to quantum teleportation, cannot be justified. This requires swapping entanglement over spacetime separations. When Alice’s spin is far from Bob’s entangled pair, 23, then the following process is applied which is physically unfeasible:

$$\left|{\varphi }_1\right.〉\left|{\Psi }_{23}^{\left(-\right)}\right.〉\leftarrow \times \to \left|{\Psi }_{12}^{\left(-\right)}\right.〉\left|{\phi }_3\right.〉$$

(35)

Therefore their Eq.(5) is impossible and the notion of quantum teleportation collapses. Recall Longuet-Higgins [53] made the distinction between mathematical operations and feasible operations, like ammonia, NH${ }_3$, and methyl fluoride, FCH${ }_3$, both have inversion symmetry, but only for ammonia is it feasible. The concept of quantum teleportation is unfeasible, along with emerging technologies [20,22] that use it.

Quantum computing

We do not suggest that quantum computing is unfeasible, but it too partly rests upon Bell’s theorem and entirely rests on the persistence and stability of qubits. We suggest this needs to be re-examined since the polarized states of up and down that make a qubit, cannot persist in free-flight when the fermions change to bosons. Rather, the spacing between the gates must be carefully controlled to allow, in principle, the phase of the helicity to determine the polarized state when entering the next gate.

Entanglement decomposition

Entanglement, as Schrödinger famously said [54] was $\underline{\mathrm{the}}$ difference from classical systems. Without Bell’s theorem, persistence of entanglement between separated pairs is not possible. Therefore we accept entanglement as a vital property of QM but reject that it persists after separation, [8], to non-local situations. Whereas QM gives Tsirel’son’s bound, [12], of $2\sqrt{2}=2.828$, the simulation gives a maximum violation with a CHSH of 3. The reason for this is that the singlet state, Eq.[5] is an approximation. We show this later in this paper.

The theory presented here agrees with the usual quantum result of $-cos({\theta }_a-{\theta }_b)$ without non-local “EPR channels", (see the appendix). The violation is due to correlation between the helicities.

The simulated value of CHSH = 3 is due to the modeling of Q-spin giving a CHSH value 1 per axis. Here we suggest the simulation gives a more accurate accounting for the correlation than from QM. Essentially the Q-spin treatment here supports the notion that polarization carries two axes while the coherence initially carries only one, being the coupled boson of spin 1, see Figure 6. The left and right parts of this figure show the two possible polarization axes after the boson has decoupled. The center part shows the coupled boson. These axes are attracted to the filter via two mechanisms. For polarized states, only the sign of the aligning vector determines the correlation and this is easily modeled by finding if the polarization vector is positive or negative. This generates the triangle correlation in Figure 3 with a CHSH value of 2.

Coherent states, in contrast, initially precess in a filter about the intact boson axis, see Figure 6. It must, however, eventually decompose into the fermionic axes the closer the boson gets to the filter. Since each axis carries opposite spin, it is important to know which axis aligns. To this end, first we determine which axis lies closer to the applied filter direction. We assume this closer axis is favoured to align. The click of $\pm 1$ is found again by determining if that closer axis is positive, plus click, or negative, minus click. This process generates the mustache function in Figure 3, which is closely fit by the following function which is plotted in Figure 7

$$E_q=\left\{\begin{array}{c}-\frac{1}{4}sin\left(2{\theta }_{ab}\right)0\le {\theta }_{ab}\le \pi \\ +\frac{1}{4}sin\left(2{\theta }_{ab}\right)\pi <{\theta }_{ab}\le 2\pi \end{array}\right.$$

(36)

We propose that the true correlation between the two spins is more accurately determined by this classical simulation than from QM.

The alignment and polarization of these axes is determined by their relative orientation with respect to the filter direction. Additionally the strength of the applied field relative to the spin-spin coupling between the two axes is also relevant. In Figure 8, the left panel shows the aligned coherent spin. At this orientation, and say up to 22.5 degrees on either side of the filter, the boson precesses and nutates intact with magnetic moment of $2\mu$, twice that of the two fermionic axes with a magnetic moment of $\mu$ each. Notice in Eq.(36) that the filter angle between Alice and Bob is $2{\theta }_{ab}$ which is consistent with a magnetic moment of $2\mu$ and a spinning boson.

If the boson is offset from the filter to one direction by 45 degrees, second panel, then one of the fermionic axes aligns leading to the precession of the fermion with a magnetic moment of $\mu$. The third panel shows the boson perpendicular to the field. In this case there is a competition between the two axes. One axis aligns with the positive direction of the filter, or the second axis is pulled to the negative direction along the filter. The panel on the right of Figure 8 shows that in a strong field, one quarter of the time Q-spin acts as a boson, and three quarters it acts as a fermion. We conclude that in a strong polarizing field, in some cases the boson is encountered intact and in others it decouples to the fermion.

Figure 9. Displaying the maximum correlation from coherence. Alice sets her filter to ${\mathbf{a}}_3$ and the boson spin of 1 aligns. Bob has two settings that will give the maximum violation by applying his filter either along $\mathbf{d}$ or along $-\mathbf{c}$ at 45 degrees from ${\mathbf{a}}_3$. Alternately, Alice can set her filter angle to ${\mathbf{a}}_1$ and Bob has two filter settings, $\mathbf{d}$ and $\mathbf{c}$ that lead to the maximum correlation. Note that a spin of magnitude 1 makes an angle of 45 degrees with filter directions.

Figure 9. Displaying the maximum correlation from coherence. Alice sets her filter to ${\mathbf{a}}_3$ and the boson spin of 1 aligns. Bob has two settings that will give the maximum violation by applying his filter either along $\mathbf{d}$ or along $-\mathbf{c}$ at 45 degrees from ${\mathbf{a}}_3$. Alternately, Alice can set her filter angle to ${\mathbf{a}}_1$ and Bob has two filter settings, $\mathbf{d}$ and $\mathbf{c}$ that lead to the maximum correlation. Note that a spin of magnitude 1 makes an angle of 45 degrees with filter directions.

Interpreting the maximum CHSH correlation

In the last section we showed how different orientations of Q-spins in a polarizing field favour fermions over bosons, or vice versa. In EPR coincidence experiments, we seek the correlation between two Q-spins for Alice and Bob at different settings.

Figure 10 top left shows the BFF of Q-spin whence each quadrant is bisected by a boson spin. At separation Alice and Bob share the same BFF so before filtering, both spins of the EPR pair are anti-parallel bosons. When Alice and Bob set their polarizers to be parallel, the anti-correlation is $-1$ whereas if the polarizers are anti-parallel, the correlation is $+1$. Turning the applied filters to 90 degrees apart, gives no correlation. These four points are seen in Figure 3 at ${\theta }_{ab}=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$.

If we choose the filter setting of Alice and Bob to both lie at 45 degrees in the LFF, then both Alice and Bob simultaneously encounter their boson together in the same way as if the filters were applied along the Z axis. That is, since the two bosons are anti-parallel along any direction, the situation becomes identical to the anti-correlation of $-1$, already discussed for the Z direction.

If, however, the two filters are separated by increasing ${\theta }_{ab}$ to $\frac{\pi }{4}$, by moving Alice’s filter to 22.5 degrees from the bisector, to ${\mathbf{a}}_3$, and Bob’s down by 22.5 degrees to $\mathbf{d}$, then both still remain coupled bosons, Figure 10. Alice’s boson then aligns along ${\mathbf{a}}_3$ and Bob’s aligns along $\mathbf{d}$, Figure 10. Both bosons are forced to align with their applied field along these two directions, ${\mathbf{a}}_3$ for Alice and $\mathbf{d}$ for Bob.

The BFF precesses with the helicity left or right in the LFF, so with fixed polarizer settings of $\frac{\pi }{4}$, it can be shown that one quarter of the time, both bosons lie within that $\frac{\pi }{4}$ wedge without decoupling, which can be deduced by placing two panels, one for Bob and one for Alice, on top of each other in Figure 8. It is shown in the Appendix and [6] that for coherence correlation, both Alice and Bob spins must be bosons. This is the origin of $\frac{1}{4}$ factor in Eq.11. Three quarters of the time, the correlation is due to fermion polarization and one quarter of the time, the correlation is due to the boson coherence.

A similar situation occurs in the other quadrants of the BFF, as seen from the second, lower, $\frac{\pi }{4}$ wedge in the same figure. Alice’s boson then precesses about ${\mathbf{a}}_1$ and Bob’s around $\mathbf{c}$. This is shown in Figure 9. Note that the angle a spin makes with its polarized direction differs from fermion to boson, given by $cos\chi =\frac{m}{\sqrt{S\left(S+1\right)}}$. A spin $\frac{1}{2}$ precesses at the magic angle of 54.74 degrees while a spin 1 makes an angle of 45 degrees. In Figure 9 the 45 degree precession of the bosons is depicted, and this shows considerable correlation as the axes coincide as they precess. The 45 degrees precession angle supports the assumption that the bosons have not decoupled, and remain intact within the wedge.

Note also from Figure 10, that similar wedges occur so that $-\mathbf{c}$ is also strongly correlated to ${\mathbf{a}}_3$, and $\mathbf{d}$ is strongly coupled to ${\mathbf{a}}_1$, and so forth around the 2D plane.

The maximum correlation from coherence for the S=CHSH inequality is determined by the filter setting that differ by $\frac{\pi }{4}$, as shown in Figure 10, and giving,

$$\begin{array}{cc}\hfill & S=a_1·\left(d+c\right)+a_3·\left(d-c\right)\hfill \\ \hfill & =\frac{1}{\sqrt{2}}\left(d+c\right)·\left(d+c\right)+\frac{1}{\sqrt{2}}\left(d-c\right)·\left(d-c\right)=2\sqrt{2}\hfill \end{array}$$

(37)

We do not treat the dynamics of the 2D spin approaching the filter, but if the magnetic moment for each spin axis is $\mu$, then the resonance spin of 1 has a Larmor precession proportional to $2\mu$ before it decouples. The Larmor frequency of the resonance spin is twice that of the aligned or polarized spin $\frac{1}{2}$.

The singlet state

Above we stated that the singlet state is an approximation. Here we summarize the treatment in [6]. A singlet state is usually expressed by opposing polarized states which are anti-symmetrized, $\uparrow \downarrow -\downarrow \uparrow$, see Eq.(5). Here a spin has structure and we suggest a singlet state of an electron is more realistically expressed as shown in Figure 11 where the two magnetic axes are attracted which is balanced by the repulsion of the charges. The helicity now spins the axis as shown.

First the singlet state can be separated into the sum of eight product states if non-Hermitian states are used [56],

$$\begin{array}{cc}\hfill & {\rho }_{{\Psi }_{12}^{-}}=\hfill \\ \hfill & \frac{1}{8}\sum _{\begin{array}{c}{n}_1,n_2,\\ n_3=\pm 1\end{array}}{\left(I^1+n_3{\sigma }_3^1+n_1{\sigma }_1^1+n_{2}i{\sigma }_2^1\right)\otimes \left(I^2-n_3{\sigma }_3^2-n_1{\sigma }_1^2-n_{2}i{\sigma }_2^2\right)}^{†}\hfill \end{array}$$

(38)

Each time an EPR pair separates, it does so randomly into only one of the eight non-hermitian terms. Each term represents a polarization direction and its helicity. That is why the states are non-Hermitian.

There are two product states in Eq.(38) for each BFF quadrant that can emerge, see Figure 10. One term describe left helicity and the other right helicity for each quadrant.

We suggest that after initial separation, each spin settles into a trajectory with the disc of polarization spinning along the axis of linear momentum, see Figure 1, governed by the Intermediate Axis theorem, [57].

Inserting the Pauli spin matrices and summing the eight terms reproduces the entangled singlet matrix shown in Eq.(7). In the sum, the outer coherent states cancel, and those, we suggest, are responsible for the difference between QM and the classical simulation.

To show this, each quadrant has two terms that are Hermitian conjugates of each other, describing L and R helicity. Upon separation, either the left or right handed helicity occurs. For the first quadrant, $n_1,n_3=1$, adding the left ${\rho }^1{{\rho }^2}^{†}$ and right ${{\rho }^1}^{†}{\rho }^2$, $(n_2=\pm 1)$, Eq.(8), matrices leads to a Hermitian matrix from the two terms. That is, for each quadrant, at separation, the EPR pair is one of the eight products, for example for the first quadrant,

$${\rho }^1{\rho }^{2†}=\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right)\left(\begin{array}{cc}0& 0\\ -1& 1\end{array}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -1& -1& 1& 1\\ 0& 0& 0& 0\end{array}\right)$$

(39)

This shows Alice and Bob’s states with opposite polarization (diagonal elements), and opposite helicity, (off-diagonal elements) in the two by two matrices. Adding the Hermitian conjugate gives a Hermitian matrix for the state in the first quadrant,

$$\begin{array}{cc}\hfill & \rho \left(1^{\mathrm{st}}\mathrm{quadrant},\mathrm{L}\mathrm{and}\mathrm{R}\right)={\rho }_1{\rho }_2^{†}+{\rho }_1^{†}{\rho }_2\hfill \\ \hfill & =\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -1& -1& 1& 1\\ 0& 0& 0& 0\end{array}\right)+\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& -1& 0\\ 0& 0& 1& 0\\ 0& 0& 1& 0\end{array}\right)=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& -1& 0\\ -1& -1& 2& 1\\ 0& 0& 1& 0\end{array}\right)\hfill \end{array}$$

(40)

Permutations of $n_{i}s$ gives similar matrices for the other three quadrants. Finally summing over the quadrants gives the entangled singlet, Eq.(7). Notice the outer coherent terms cancel between quadrants, which is responsible for the difference in correlation between quantum, CHSH$=2\sqrt{2}$ and the simulation, CHSH$=3$.

We conclude that use of the singlet state in QM, Eq.(5) is an approximation which misses these canceled terms. That is entanglement is a property of QM, but not of Nature. Entanglement simplifies calculation but missing correlation is the price.

Conclusions

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