**Nature of an antiparticle**

A positron can be interpreted as an electron travelling backwards in time.

Feynman and earlier Stueckelberg, proposed an interpretation of the positron as an electron moving backward in time, reinterpreting the negative-energy solutions of the Dirac equation. Electrons moving backward in time would have a positive electric charge. (https://en.wikipedia.org/wiki/Positron)

If the positron is an electron moving backwards in time, its positive charge is positive only in relation to normal matter moving forwards in time. But if the positron is an electron travelling backwards in time, its reverse direction in time should not be disregarded and its reversed attributes viewed only as legitimate from an observer moving forwards in time. A positron will of necessity have its spin measured according to how the positron approaches the measuring apparatus from our future and its new spin will carry on into our past. This one concept completely invalidates the application of Bell’s Inequalities to the physical interpretation of a traditional Bell experiment. Bell’s Theorem is wholly valid and the inequalities are unbreakable but they are irrelevant to the physics actually occurring when an antiparticle moving backwards in time is involved. The Bell Inequalities are only valid for particle/antiparticle pairs both moving forwards in time.

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**Malus’s Law: the true determiner of the Bell correlation**

** **The mathematical equivalence of Malus’s Law results and a Bell experiment result is not usually relevant as Malus’s Law acts on polarised inputted beams whereas Bell acts on inputted unpolarised beams. But this lack of equivalence is only true for particle/antiparticle pairs travelling forwards in time. When the antiparticle travels backwards in time, Malus’s Law becomes the sole determiner of the Bell correlation of – cos θ.

My paper: *Malus’s Law and Bell’s Theorem with Local Hidden Variables* at https://vixra.org/pdf/2006.0160v1.pdf explains in detail the importance of Malus’s Law in deriving the Bell correlation in a Bell’s Theorem experiment.

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**Realism, locality, hidden variables and counterfactual determinism**

** **Realism and locality are features relevant to Bell’s Theorem but are superfluous in an explanation of the Bell correlation when antiparticles are travelling backwards in time. It may be difficult, nevertheless, to accept the reality of this interpretation of antiparticles. The particles and antiparticles can have hidden variables which could be their spin vector directions. It is not possible to keep track of a particle’s hidden variable even in a simulation of the particle passing through a measurement (in a Stern Gerlach apparatus). There is a statistical element in the outcomes and this chance element means that counterfactual determinism cannot apply. So a table of simulated results cannot be constructed using counterfactual determinism to fill in blank spaces in a 2×2 table of results of paired measurements. Say a pair of particles has identical hidden variables to another pair of particles. The two pairs can have different outcomes to one another because of an element of chance in the measurement. This chance is straightforwardly derived from Malus’s Law which is a probabilistic law and has been known for two hundred years. So it is true that the Quantum Randi challenge cannot be met. However, the challenge is irrelevant to issues of locality and realism as the physics of a Bell experiment with backwards-in-time antiparticles is classically solved using a very old statistical law.

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**The physics of a Bell Experiment**

** **In the conventional or forwards-in-time explanation of the physics of a Bell experiment a particle / antiparticle pair are emitted together from a source and travel to Alice and Bob who measure their spins with Stern Gerlach apparatuses. The two spins are entangled which means, at the least, that they have coordinated spins, with net angular momentum of zero. Entanglement of spins is the focus of the interpretation of the results. Although there are Bell deniers, it is completely clear to me that Bell’s Theorem is correct and that in a classical sense the Bell correlation cannot be as great as – cos θ. To see this, one could simulate a Bell correlation just using one setting of Alice and Bob devices set at an angle of 45^{o} and using a large number of pairs in the calculation. It is impossible to exceed, in absolute magnitude, a simulated correlation of -0.5. But the Bell correlation is -0.707 which is greater than can possibly be simulated using local hidden variables such as particle spin vector directions.

In a backwards-in-time antiparticle solution, the antiparticles arrive from outside the experiment and head towards the source. Alice and Bob make measurements on them on their way to the source. The antiparticles are unpolarised before measurement (This is ‘before’ from the perspective of the antiparticles whereas Alice and Bob of course see this as ‘afterwards’!) and polarised after measurement, either in Alice’s or in Bob’s setting direction. One cannot pre-determine, even in a simulation, the track of a particle through a measurement because of the statistical chance involved in the measurement outcome. But using Malus’s Law one can derive a distribution of spin vector directions immediately after measurement and follow the beam through the rest of the experiment. After reaching the source, the beam of positrons is replaced by a beam of electrons travelling forwards through time. Their spins are entangled with the spins of their partner positrons, so entanglement does play a role, but not the dominant role, which in fact belongs to Malus’s Law.

The electrons travelling to be measured by Alice are pre-polarised in the direction of Bob’s setting and those to be measured by Bob are pre-polarised in the direction of Alice’s setting. So Alice and Bob are both measuring beams pre-polarised at 45^{o} to the new measurement. Even if a range of settings is used with random and late choices and even if randomised using distant astronomical data, the usual settings in a Bell experiment all involve angles related to 45^{o}, and that is the only key value in a Malus’s Law outcome.

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**Duality of Malus’s Law results and Bell’s Experiment table of results **

Table 1 shows the generalised table of results of a Bell experiment where A and B are Alice’s and Bob’s measurements which can take the values of +1 or -1. Alice sets her apparatus at a vector setting of **a** while Bob uses a vector setting of **b**. The difference in angles between **a **and **b** is θ.

** ****Table 1 Bell results for electrons with a – b = θ **

Results as proportions | B = 1 | B = -1 | Total |

A = 1 | {P++ = } 0.5*sin^{2} θ/2 |
0.5*cos^{2} θ/2 |
0.5 |

A = -1 | 0.5*cos^{2} θ/2 |
0.5*sin^{2} θ/2 |
0.5 |

Total | 0.5 | 0.5 | 1.0 |

It is easy to show that p++ = (1 + correlation coefficient)/4 and, conversely,

the correlation coefficient = 4 * p++ – 1

So, the correlation coefficient between A and B measurements = 4 * 0.5* sin^{2} θ/2 – 1

= 2* sin^{2} θ/2 – 1 = (sin^{2} θ/2 – 1) + sin^{2} θ/2 = sin^{2} θ/2 – cos^{2} θ/2 = – cos θ.

Malus’s Law shows that the intensity of the beam measured by Bob is cos^{2} (θ/2) for electrons where θ = angle a – b. This value is needed for the cell of Table 1 where A = +1 and B = -1. Alice’s measurements of A = +1 on her antiparticles form one quarter of the total measurements so she contributes 0.25*cos^{2} θ/2 to this cell of Table 2. Similarly Bob’s measurement of B = -1 on his antiparticles contributes another 0.25*cos^{2} θ/2 to this cell making a total of 0.5*cos^{2} θ/2 for the final cell value. Table 1 only has one degree of freedom as the marginal values are fixed, so the other three cells of results are filled in automatically once one cell value is known.

**Photons versus electrons in Malus’s Law**

Malus’s Law for photons is:

Intensity of a beam passing through a filter at an angle θ to an incoming polarised beam = cos^{2} θ * original intensity of the beam.

Malus’s Law for electrons is:

Intensity of a beam passing through a filter at an angle θ to an incoming polarised beam = cos^{2} (θ/2) * original intensity of the beam.

3 September 2020