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- Summary
- Did Bell have make three errors proving in his famous theorem?

I have some questions about J. S. Bell’s famous theorem as presented in his1964 paper.

As in Bell's paper, Alice and Bob are distant observers of spin.

1. My first question is practical. Bell considers “a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions.” While Bell’s description seems innocuous, is it even possible to prepare such a state? Suppose, for example, that we allow a spin-0 boson to decay into two spin-1/2 fermions (say

Of course, cloud, bubble and spark chamber observations show well-defined tracks, but such observations involve von Neumann’s process 1 (the physics of observation) and not process 2 (the physics of unobserved time development). Only process 2 physics is relevant until the quanta strike Alice’s and Bob’s detectors -- and it entails spherical symmetry.

So, Bell has one particle arriving at Alice’s detector, and another at Bob’s detector. Yet if, as in my example, the quanta have equal mass, both will arrive simultaneously at both detectors. This is not a mere conceptual difference, but has physical implications. If the electron and positron waves are co-extensive, their four-currents will cancel, and no EM field will be produced. If they are separate, each will produce an EM field and, collectively, a dipole field. This physical difference requires a different mathematical model of detection. (My example also seems to imply charge-exchange (particle-antiparticle) entanglement.)

2. My central question deals with detector independence. Bell states: “Measurements can be made, say by Stern-Gerlach magnets, on selected components of the Spins

This is crucial in questioning local realism, yet it seems to contradict accepted physics. While not proven from more fundamental considerations, the antisymmetry of multi-fermion wavefunctions under interchange of coordinates is an accepted principle of quantum physics. To be fully covariant, the interchange must include time as well as space coordinates. In Dirac's many-time formulation

If we take values of

Alice and Bob can select the orientation of their detectors, but they can’t violate the antisymmetry constraining their electrons. As those electrons interact with incident quanta to produce detection events, observing a detection result

3. Finally, Bell further writes: “Since we can predict in advance the result of measuring any chosen component of

This conclusion ignores the role of the detector state in process 1. Let Alice’s and Bob’s Stern-Gerlach magnets be orthogonally aligned. Then, no matter what the direction of the measured spins, the magnitude of their vector sum will be 2

Since the detection system can contribute to the measured result, predetermination need not imply the possibility of a more complete specification

1. J. S. Bell, "On the Einstein Podolsky Rosen Paradox,"

2. Alain Aspect, Philippe Grangier, Gérard Roger, "Experimental Tests of Realistic Local Theories via Bell's Theorem,"

3. Paul A. M. Dirac, “Relativistic Quantum Mechanics,”

4. Werner Heisenberg, “Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik,”

^{1}These are about his theoretical assumptions and reasoning, not about experimental observations such as Aspect-type experiments. While some questions relate to the experiments, others do not because Aspect’s work^{2}used photons from a calcium radiative cascade, not the spin-1/2 particles discussed by Bell.As in Bell's paper, Alice and Bob are distant observers of spin.

1. My first question is practical. Bell considers “a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions.” While Bell’s description seems innocuous, is it even possible to prepare such a state? Suppose, for example, that we allow a spin-0 boson to decay into two spin-1/2 fermions (say

*π*→^{o}*e*^{-}+*e*^{+}). Since the initial state is spherically symmetric, its decay products should radiate in spherical waves, like ripples on a pond, not as particles moving in opposition directions. As singlet states are spherically symmetric, this will be true in general. (We can separate electrons from positrons electromagnetically, but is it possible to do so without transferring angular momentum?)Of course, cloud, bubble and spark chamber observations show well-defined tracks, but such observations involve von Neumann’s process 1 (the physics of observation) and not process 2 (the physics of unobserved time development). Only process 2 physics is relevant until the quanta strike Alice’s and Bob’s detectors -- and it entails spherical symmetry.

So, Bell has one particle arriving at Alice’s detector, and another at Bob’s detector. Yet if, as in my example, the quanta have equal mass, both will arrive simultaneously at both detectors. This is not a mere conceptual difference, but has physical implications. If the electron and positron waves are co-extensive, their four-currents will cancel, and no EM field will be produced. If they are separate, each will produce an EM field and, collectively, a dipole field. This physical difference requires a different mathematical model of detection. (My example also seems to imply charge-exchange (particle-antiparticle) entanglement.)

2. My central question deals with detector independence. Bell states: “Measurements can be made, say by Stern-Gerlach magnets, on selected components of the Spins

**σ**_{l}and**σ**_{2}, … we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.”This is crucial in questioning local realism, yet it seems to contradict accepted physics. While not proven from more fundamental considerations, the antisymmetry of multi-fermion wavefunctions under interchange of coordinates is an accepted principle of quantum physics. To be fully covariant, the interchange must include time as well as space coordinates. In Dirac's many-time formulation

^{3}, this constrains the multi-electron wave function,*ψ*(**x**_{1},*t*_{1},**x**_{2},*t*_{2}, ...), by transtemporal equations of the form*ψ*(*…,***x**_{i}*, t*…,_{i},**x**_{j},*t*_{j}, …) = -*ψ*(*…,***x**_{j}*, t*_{j}*, .*…,*,***x**_{i}*t*_{i}, …).If we take values of

**x**_{i}*, t*_{i}in Alice’s detector at the time of her observations and of**x**_{j}*, t*_{j}in Bob’s detector at the time of his observations, we can see that the wavefunctions of detector electrons are not independent as Bell assumes, but highly cross-constrained.Alice and Bob can select the orientation of their detectors, but they can’t violate the antisymmetry constraining their electrons. As those electrons interact with incident quanta to produce detection events, observing a detection result

*could*inform us about correlative remote detection results, at least statistically. This doesn’t predict entanglement, but it seems to refute Bell’s independence assumption.3. Finally, Bell further writes: “Since we can predict in advance the result of measuring any chosen component of

**σ**2, by previously measuring the same component of**σ**l, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.”This conclusion ignores the role of the detector state in process 1. Let Alice’s and Bob’s Stern-Gerlach magnets be orthogonally aligned. Then, no matter what the direction of the measured spins, the magnitude of their vector sum will be 2

^{1/2}/2. Since the initial angular momentum was zero, the measured valued cannot represent the state prior to measurement. Conservation requires another source of angular momentum, which can only be the detection system.Since the detection system can contribute to the measured result, predetermination need not imply the possibility of a more complete specification

*of the state*. It may only imply that a full description of process 1 must include the state of the detection system. This is the point made by Heisenberg in his 1927 uncertainty paper^{4},*i.e*. that measurements invariably combine detector state information with system state information.**Notes:**1. J. S. Bell, "On the Einstein Podolsky Rosen Paradox,"

*Physics*1:3 (1964), 195-290. https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf2. Alain Aspect, Philippe Grangier, Gérard Roger, "Experimental Tests of Realistic Local Theories via Bell's Theorem,"

*Phys. Rev. Lett.*47 :7, (1981), 460–3. Alain Aspect, Jean Dalibard, Gérard Roger. "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers,"*Phys. Rev. Lett.*49:25 (1982), 1804-7.3. Paul A. M. Dirac, “Relativistic Quantum Mechanics,”

*Proc. R. Soc. Lond.*A 136 (1932), 453-64. doi:10.1098/rspa.1932.00944. Werner Heisenberg, “Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik,”

*Z. für Physik,*43 (1927), 172-98.