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European Journal of Applied Sciences – Vol. 12, No. 3
Publication Date: June 25, 2024
DOI:10.14738/aivp.123.17174.
Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
Services for Science and Education – United Kingdom
Mystery’s End: Analysis of Bell’s Theorem
Darrell Bender
New Mexico Institute of Mining and Technology
ABSTRACT
Entanglement requires that spin measurements satisfy those just given for a⃗ ≠ b⃗⃗.
Bell’s hidden variable quantum mechanical local particle with a⃗ = b⃗⃗ cannot
provide the values for a⃗ ≠ b⃗⃗ and, hence, the values for an entangled particle
system. The perception that any hidden variable quantum mechanical description
of an entangled system of particles is non-local depends on Bell’s inappropriate
definition of locality, which excludes entanglement for a distant particle and
ignores that entanglement is a local phenomenon, and the condition that a⃗ = b⃗⃗,
which does not give the values for a⃗ ≠ b⃗⃗ as required to give the quantum
mechanical expectation values whether entanglement occurs for systems of
particles that are close, not distant, or not. Expecting the condition a⃗ = b⃗⃗ and a
local particle analysis with no distant entangled particle to give the quantum
mechanical result is absurd. Any disentanglement of once entangled particles
moving apart sets the spin measurements at those given for a⃗ ≠ b⃗⃗ if a⃗ ≠ b⃗⃗ and
those given for a⃗ = b⃗⃗ if a⃗ = b⃗⃗. Any subsequent measurement of the set values
agrees with those for a⃗ ≠ b⃗⃗ if a⃗ ≠ b⃗⃗ and those given for a⃗ = b⃗⃗ if a⃗ = b⃗⃗ just as if the
particles are entangled still. In examples, including Bell’s, where Bell’s inequality
is supposedly violated, the conditions for which the inequality is true are not met.
There is no contradiction.
In Quantum Gravity, Energy Wave Spheres, and the Proton Radius, we considered electrons and
protons in the hydrogen atom as energy wave spheres that shed mass corresponding to, in the
case of the electron wave sphere, the Bohr radius electron orbit and, in the case of the proton
wave sphere, the outer proton radius so that the corresponding radius of the shed mass is
given by
r =
ħ
mc
.
The reflection that this mass, in the case of Bohr radius electron wave sphere orbit, was the
mass remaining changed to the shed mass after considering that
m = αme
,
with me
the initial electron mass so that the mass of the remaining electron wave sphere is
(1 − α)me
and the radius of the remaining electron wave sphere is
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1
(1 − α)
re
.
Similarly, the mass shed by the proton energy wave sphere is
m =
1
4
mp,
with mp the initial proton mass so that the mass of the remaining proton wave sphere is
3
4
mp,
with radius
4
3
rp.
Thus, the radius of the shed proton energy wave sphere in the hydrogen atom is
rshed = 4
ħ
mpc
.
For
mp = 1.67272 × 10−27kg,
we have for the radius of the shed proton energy wave sphere inside of an electron energy
wave sphere
rpeshed
=
ħ
mpeshed
c
= 4
ħ
mpc
= 4
1
2π
6.62607015
(1.67272)(2.99792458)
10−15 meters
= 0.8411863173145236 × 10−15 meters.
The concept of energy remaining arose from the consideration of clock rates in a gravitational
field and the realization that energy wave spheres are clocks such that the frequency of the
energy wave is the clock rate if the wavelength is the circumference of the energy wave
sphere.
In Quantum Gravity, Energy Wave Spheres, and the Proton Radius, pages 9-10, our copy, we
wrote,
“We stated in the abstract to On the Nature of Being: Gravitation,
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Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
URL: http://dx.doi.org/10.14738/aivp.123.17174
“The experimental result for the rate of a clock in a gravitational field is given in the paper,
“Optical Clocks and Relativity” by C. W. Chou et al., Science 329, 1630 (2010). The clock rate f
satisfies
δf
f0
=
g∆h
c
2
. [i]
By elementary functional analysis, Einstein’s clock rate,
f = (g44)
−
1
2 = (1 −
α
r
)
−
1
2
, [ii]
from The Foundation of the General Theory of Relativity is greater than 1, the clock rate in flat
space-time, and does not satisfy, as we show in Part 2, the equation that is experimentally
verified. On the other hand, the multiplicative inverse of Einstein’s clock rate,
f = (1 −
α
r
)
1
2
, [iii]
gives a clock rate that is less than 1, thus a slower rate than that in flat space-time, and
satisfies the equation that is experimentally verified.”
The clock rate in the Chou paper is not the clock rate from The Foundation of the General
Theory of Relativity, but rather, the clock rate from Einstein’s 1911 paper, On the Influence of
Gravitation on the Propagation of Light. The true clock rate is not the one implied by the
General Theory of Relativity; but, if General Relativity is true, then the clock rate arrived at by
assuming General Relativity to be true must be true for General Relativity to imply it. More
generally, no true statement implies one that is false. On the other hand, a false statement
implies one that is true; moreover, assuming statements to be true, for example, the statement
that the clock rate in a gravitational field is Einstein’s clock rate, does not prove the premises
or theory from which the statements follow; yet Einstein, referring to the facts given above,
claims proof: “these facts must be taken as convincing proof of the correctness of the theory.”
In the situation in which no theory of science can be proven, Einstein claims proof based on
his fallacious logic. When no proof can be given, Einstein steps forward with a proof of his
theory. No one is surprised and no one cares enough to challenge it; certainly, no one who is
convinced is going to consider obvious evidence that contradicts the theory. This is a logical
crime that cries out hysterically into the darkness of thought. No one, laziness and ignorance
come to mind as applicable traits, knows the theory and no one cares, just that the theory is
proven. The obvious contradictory evidence must be ignored. The experiments that find that q
is true are repeatedly done, with increasing precision perhaps, as if they constitute proof of
the theory.”
In on the Nature of Being: Gravitation, pages 36-37, our copy, we showed that Einstein’s clock
rate from The Foundation of the General Theory of Relativity does not satisfy the equation that
is experimentally verified in the Chou paper:
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“More generally, for ds = dt and x4 = t′, we have
dt2 = (1 −
α
r
) dt′
2
, [2.25]
or
dt′
dt
= (1 −
α
r
)
−
1
2
. [2.26]
Thus, according to Foundation, this last equation gives, to at least a first approximation, the
instantaneous clock rate for a clock in a gravitational field. The derivative dt
′
dt
is just f, the clock
frequency in the gravitational field.
Simply by inspection, we already know that this clock rate is greater than 1, the clock rate in
the “local” system; so, this gives the clock rate of a faster clock. Once again, for dt = ds = 1 in
the local system, Einstein obtains this:
Image 11: Scanned image from The Foundation of the General Theory of Relativity from The
Principle of Relativity, Dover Publications, Inc., 1952, page 161.
For average undergraduate students, every one of whom is better than Bender at
mathematics, it is the last line:
dx4 =
1
√g44
, [2.27]
with
g44 = 1 −
α
r
. [2.28]
By the chain rule and since the unit of time is chosen so that c = 1, we thus have
1
f
df
dr
=
1
(1−
α
r
)
−
1
2
−
1
2
(1 −
α
r
)
−
3
2 α
r
2 = −
1
(1−
α
r
)
(
1
2
α
r
2
) = −
g
c
2
1
(1−
α
r
)
. [2.29]
This is obviously not, the problem being the – sign,
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Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
URL: http://dx.doi.org/10.14738/aivp.123.17174
g
c
2
, [2.30]
the value of
1
f
df
dr
[2.31]
according to the Chou, Wineland paper.”
For
f′ = (1 −
α
r
)
1
2
,
the multiplicative inverse of Einstein’s clock rate, we have
1
f
′
df
′
dr = f
df
−1
dr = f
−1
f
2
df
dr = −
1
f
df
dr =
g
c
2
1
(1 −
α
r
)
,
so that it satisfies the equation that is experimentally verified in the Chou, Wineland paper.
For Einstein’s clock rate f, the negative value of
1
f
df
dr,
which arises from the negative value of
df
dr,
implies that the clock rate f increases as r decreases, contradicting the slower clock rate in
the gravitational field as r decreases.
Thus, the frequency of an energy wave sphere clock in a gravitational field decreases by a
factor of
(1 −
α
r
)
1
2
,
so that its energy at radius r becomes
(1 −
α
r
)
1
2
hν,
with
hν = mc
2
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for m the mass of the energy wave sphere in the absence of gravitation. The energy that the
energy wave sphere has at radius r is the remaining energy of the energy wave sphere; thus,
we have the origin of the concept of energy remaining and, from it, the concept of energy lost.
We had these ideas beforehand when we considered energy wave spheres in the hydrogen
atom.
If the radius of an energy wave sphere decreases by the factor β(r) at the radius r in a
gravitational field, then the frequency and, hence, velocity of the energy wave must decrease
by both factors β(r) and (1 −
α
r
)
1
2
in order that the wave sphere clock frequency decrease by
the factor (1 −
α
r
)
1
2
at radius r. In on the Nature of Being: Gravitation, we assumed that the
radius r does not decrease so that the frequency and, hence, velocity of the energy wave
decreases by the factor (1 −
α
r
)
1
2
, resulting in the frequency of the energy wave clock
decreasing by the same factor.
For any energy wave sphere we have
(1 −
α
r
)
1
2
hν = (1 −
α
r
)
1
2
mc
2
,
so that
(1 −
α
r
)
1
2
ν
h
c
= (1 −
α
r
)
1
2
cm.
Thus
1
λ
h
c
= m,
which is the same value for m that it had in the absence of gravitation. Any change in m by the
multiplicative factor k occurs if and only if the circumference λ changes by the multiplicative
factor 1
k
. By the law of conservation of mass, as well as Newton’s law of gravitation, neither
changes. Moreover, we have
(1 −
α
r
)
1
2
ν
(1 −
α
r
)
1
2
c
=
1
λ
,
so that the velocity of the energy wave in the energy wave sphere at the distance r is
(1 −
α
r
)
1
2
c.
If we consider the velocity of the energy wave sphere along a radius to be
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sphere in a gravitational field. We found the rule by thinking about it. We want and encourage
the possibility that some other explanation exists when the Newtonian energy of motion of an
energy wave sphere is the first order term of the energy lost. Specifically, the energy lost,
hν − (1 −
α
r
)
1
2
hν = (1 − (1 −
α
r
)
1
2
) hν = hν (
1
2
α
r
+
1
8
(
α
r
)
2
+
1
16 (
α
r
)
3
+
5
128 (
α
r
)
4
+ ⋯ ),
is such that that, after multiplication by hν, the first term of the last expression is the
Newtonian value for the gravitational energy of motion of the energy wave sphere. The first
term of this last series,
1
2
α
r
hυ,
is just the kinetic energy,
1
2
mv
2
,
that an energy wave sphere gains in moving from r = ∞ to some finite r. Since v is not
constant, the energy wave sphere does not move in a geodetic line as Einstein would have it.
Somehow, the collection of falsehoods that comprise Einstein’s relativity theories is an
ignorable fault in the game of science.
In the article, “Optical Clocks and Relativity” by C. W. Chou et al., Science 329, 1630 (2010), in
the magazine, Science, the first sentence states, “Albert Einstein’s theory of relativity forced us
to alter our concepts of reality,” so the relativity referenced in the work should be that of
Einstein, but the reference for the difference in clock rates, given in equation (2), is R. F. C.
Vessot, M. W. Levine, E. M. Mattison, E. L. Blomberg, T. E. Hoffman, G. U. Nystrom, B. F. Farrel, R.
Decher, P. B. Eby, C. R. Baugher, J. W. Watts, D. L. Teuber, and F. D. Wills, Phys. Rev. Lett. 45, 2081
– Published 29 December 1980. Since the reference, Test of Relativistic Gravitation with a
Space-Borne Hydrogen Maser, for the relativistic difference in clock rates in a gravitational
field is not a work of Einstein and the given difference in clock rates is not the one presented
in Einstein’s Foundation of the General Theory of Relativity, the dishonesty here is obvious. The
journals Science and Physical Review Letters are part of the fraud, which has gone on for well
over a hundred years. When we can show that Einstein’s clock rate from the Foundation of the
General Theory of Relativity does not satisfy the equation that is experimentally verified, this is
fraud. The fox is guarding the henhouse.
At the beginning of The Material Point Universe Revisited, applicable here as well, we wrote,
“This work is dedicated to every child, boy or girl, who ever dreamed of being a physicist. The
dedication is thus put as part of the work.
The intent here is to revisit the material point universe as put forth by Albert Einstein in the
Foundation of the General Theory of Relativity. The approach is, by necessity, a logical one,
giving arguments based on facts. The value of this work depends only on these arguments; it
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Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
URL: http://dx.doi.org/10.14738/aivp.123.17174
“Bell’s Theorem is the collective name for a family of results, all of which involve the
derivation, from a condition on probability distributions inspired by considerations of local
causality, together with auxiliary assumptions usually thought of as mild side-assumptions, of
probabilistic predictions about the results of spatially separated experiments that conflict, for
appropriate choices of quantum states and experiments, with quantum mechanical
predictions. These probabilistic predictions take the form of inequalities that must be satisfied
by correlations derived from any theory satisfying the conditions of the proof, but which are
violated, under certain circumstances, by correlations calculated from quantum mechanics.
Inequalities of this type are known as Bell inequalities, or sometimes, Bell-type inequalities.
Bell’s theorem shows that no theory that satisfies the conditions imposed can reproduce the
probabilistic predictions of quantum mechanics under all circumstances.
The principal condition used to derive Bell inequalities is a condition that may be called Bell
locality, or factorizability. It is, roughly, the condition that any correlations between distant
events be explicable in local terms, as due to states of affairs at the common source of the
particles upon which the experiments are performed. See section 3.1 for a more careful
statement.
The incompatibility of theories satisfying the conditions that entail Bell inequalities with the
predictions of quantum mechanics permits an experimental adjudication between the class of
theories satisfying those conditions and the class, which includes quantum mechanics, of
theories that violate those conditions. At the time that Bell formulated his theorem, it was an
open question whether, under the circumstances considered, the Bell inequality-violating
correlations predicted by quantum mechanics were realized in nature. Beginning in the
1970s, there has been a series of experiments of increasing sophistication to test whether the
Bell inequalities are satisfied. With few exceptions, the results of these experiments have
confirmed the quantum mechanical predictions, violating the relevant Bell Inequalities. Prior
to 2015, however, each of these experiments was vulnerable to at least one of two loopholes,
referred to as the communication, or locality loophole, and the detection loophole (see section
5). In 2015, experiments were performed that demonstrated violation of Bell inequalities with
these loopholes blocked. Experimental demonstration of violation of the Bell inequalities has
consequences for our physical worldview, as the conditions that entail Bell inequalities are,
arguably, an integral part of the physical worldview that was accepted prior to the advent of
quantum mechanics. If one accepts the lessons of the experimental results, then someone or
other of these conditions must be rejected.”
Any contradiction of Bell’s Theorem, since it is a theorem, must be the result of a logical
argument, not an experiment. The condition of Bell locality is stated roughly rather than
precisely as if there were a need for clarification later by a rough statement now. There is no
guarantee that the definitions of the conditions, the premises, necessary for proof are
mathematically acceptable. Multiple theorems with different settings, concepts of local
causality, auxiliary assumptions usually thought of as mild side-assumptions, a series of
experiments with increasing sophistication but vulnerable to loopholes prior to 2015 all of
which give us reason to be dubious when everything is set up before us like it has to be true.
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Just as, metaphorically speaking, Einstein, when no proof is possible, came forward with a
proof of his relativity theories, John Stewart Bell emerged with a theorem, which the evidence
disproved. When a proof was needed, where no proof was possible, that quantum theory met
the Einstein-Rosen-Podolsky paradox and won, John Stewart Bell, with the middle name
Stewart and a host of cicadas, came forth with a false theorem, contradicted by evidence
implied by quantum theory.
Yet this is all so hyped that one could get convinced that “Bell’s theorem shows that no theory
that satisfies the conditions imposed can reproduce the probabilistic predictions of quantum
mechanics under all circumstances” via the assumption that the theoretical world is as the
theorem states or that the theorem implies the theory so that the contradiction of the theory
by the evidence, which is not a logical argument, is in turn a contradiction of Bell’s Theorem.
Perhaps even more convincing illogic is that experiments that “with few exceptions, the results
of” which “have confirmed the quantum mechanical predictions, violating the relevant Bell
Inequalities” show, as in prove, that the quantum mechanical predictions and, hence, quantum
mechanics, itself, are true.
Even if, as is the case here, the contradiction to Bell’s inequality, from which the theorem
results, is a logical one, following from the predictions of quantum mechanics, the truth of
quantum mechanics is not established.
The current hype of Bell’s Theorem is reflected in the same Stanford Encyclopedia of
Philosophy article, continuing as follows:
“For much of the interval between the original publication of Bell’s theorem and the
experiments of Aspect and his collaborators, interest in Bell’s theorem was confined to a
handful of physicists and philosophers. During that period, much of the discussions on the
foundations of physics occurred in a mimeographed publication entitled Epistemological
Letters. In the wake of the Aspect experiments (Aspect, Grangier, and Roger, 1982; Aspect,
Dalibard, and Roger 1982), there was considerable philosophical discussion of the
implications of Bell’s theorem; see Cushing and McMullin, eds. (1989), for a snapshot of the
philosophical discussions of the time. Interest was also stimulated by the publication of a
collection of Bell’s papers on the foundations of quantum mechanics (Bell 1987b). The rise of
quantum information theory, which, among other things, explores the ways in which quantum
entanglement can be used to perform tasks that would not be feasible classically, also
contributed to raising awareness of the significance of Bell’s theorem, which throws into
sharp relief the difference between quantum entanglement-based correlations and classical
correlations. The year 2014 was the 50th anniversary of the original publication of Bell’s
theorem, and was marked by a special issue of Journal of Physics A (47, number 42, 24
October 2014), a collection of essays (Bell and Gao, eds., 2016), and a large conference
comprising over 400 attendees (see Bertlmann and Zeilinger, eds., 2017). The interested
reader is urged to consult these collections for an overview of current discussions on topics
surrounding Bell’s theorem.
The shift in attitude of the physics community towards the importance of Bell’s theorem was
dramatically illustrated by the awarding of the Nobel Prize in Physics for 2022 to Alain
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Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
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Aspect, John Clauser, and Anton Zeilinger “for experiments with entangled photons,
establishing the violation of Bell inequalities and pioneering quantum information science.””
Considering our list of significant results that Quantum Theory has yet to consider, let alone
give, we consider Bell’s Theorem to be the shill in an effort to further the perception that the
theory gives a complete description of natural phenomena. The occurrence of spooky action
at a distance is not, to the exclusion of other explanations, necessarily part of this complete
description since the authors, Wayne Myrvold, Marco Genovese, and Abner Shimony, of Bell’s
Theorem, in Section 2. Proof of a Theorem of Bell’s Type, note,
“Often, in popular writings, the case of aligned devices is the only one mentioned,
and the perfect anticorrelation (for spin- 1
2
particles in the singlet state) or
correlation (for photons in state ||Φ⟩) of results in this case is offered as evidence
of “spooky action at a distance.” In fact, as Bell (1964, 1966) demonstrated by
means of simple toy models, this behaviour can be reproduced by entirely local
means.”
In “Bells Theorem,” February 1999, by David M. Harrison, Department of Physics, University
of Toronto, Harrison, at the beginning of the introduction, states,
“In 1975 Stapp called Bell's Theorem "the most profound discovery of science."
Note that he says science, not physics. I agree with him.”
So, there is something to be said of the profundity of a theorem that assumes that locality,
“or more precisely that the result of a measurement on one system be unaffected
by operations on a distant system with which it has interacted in the past”
According to Bell, holds and concludes
“that no theory that satisfies the conditions imposed can reproduce the
probabilistic predictions of quantum mechanics under all circumstances,” the
condition of locality being the culprit. According to Bell in the Introduction to “On
the Einstein Rosen Podolsky Paradox,”
“Moreover, a hidden variable interpretation of elementary quantum theory [5]
has been explicitly constructed. That particular interpretation has indeed a
grossly nonlocal structure. This is characteristic, according to the result to be
proved here, of any such theory which reproduces exactly the quantum
mechanical predictions.”
Contrary to the perception that Bell’s results, Bell’s Theorem, and his concept of locality hold
profoundly and inescapably, entanglement, if it occurs, is a local phenomenon if the two
systems are not distant from each other, but close. What quantum theory fails to explain gets
explained anyway. Quantum theory is stuck with providing a complete description of reality
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even if reality, which is not that difficult to see, eludes it. The experiments that are claimed to,
in the case of Bell’s Theorem, confirm the predictions of quantum mechanics simply measure
spin as if it were the quantum mechanical expectation of spin when quantum theory is not
required to explain the measurement. The calculation of spin requires a projection by cosθ of
the spin.
Paradigmatically and ironically, reflecting on how dirty this is, the very situation, the same one
as in the Einstein Rosen Podolsky Paradox, which quantum theory cannot describe, is used as
propaganda for its success.
Bell’s definition of locality “that the result of a measurement on one system be unaffected by
operations on a distant system with which it has interacted in the past” compares to the
version of Bell locality, or factorizability of Wayne Myrvold, Marco Genovese, and Abner
Shimony: “It is, roughly, the condition that any correlations between distant events be
explicable in local terms, as due to states of affairs at the common source of the particles upon
which the experiments are performed.” What is not stated in this condition of Bell’s locality is
Bell’s definition of locality.
To Bell, locality implies non-entanglement for distant systems that interacted in the past.
Assuming non-entanglement for distant systems that interacted in the past, he derives an
inequality, Bell’s inequality, and concludes that an entangled system does not satisfy this
inequality. Assuming that some hidden variable quantum theory and non-entanglement, or
with non-entanglement as part of the theory, imply the results of quantum theory in this case,
Bell shows that the prediction of quantum theory does not satisfy Bell’s inequality, which in
turn implies that no hidden variable theory with non-entanglement, or locality, holds. The
condition of non-entanglement cannot describe an entangled system. Let locality, or non- entanglement, take the blame. No non-entangled system, but call it local, can be entangled. Let
experiments be run that measure the spin, for which there is no great mystery if one gives a
proper analysis, which quantum theory lacks, but call what is measured the quantum
mechanical expectation of spin.
If, somehow, Bell gets credit, via Bell’s Theorem, for the most profound discovery of science,
then the theorem deserves our attention until we take it apart. Supposed variants, some of
which may be true, of Bell’s Inequality have the property that they all have three legs. Variants
divert attention from the inequality, which is false. Bell’s Theorem, omitting for now, section
III. Illustration, begins as follows:
“
INTRODUCTION
THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum
mechanics could not be a complete theory but should be supplemented by additional
variables. These additional variables were to restore to the theory causality and locality [2]. In
this note that idea will be formulated mathematically and shown to be incompatible with the
statistical predictions of quantum mechanics. It is the requirement of locality, or more
precisely that the result of a measurement on one system be unaffected by operations on a
distant system with which it has interacted in the past, that creates the essential difficulty.
There have been attempts [3] to show that even without such a separability or locality
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Bender, D. (2024). Mystery’s End: Analysis of Bell’s Theorem. European Journal of Applied Sciences, Vol - 12(3). 521-542.
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requirement no "hidden variable" interpretation of quantum mechanics is possible. These
attempts have been examined elsewhere [ 4] and found wanting. Moreover, a hidden variable
interpretation of elementary quantum theory [5] has been explicitly constructed. That
particular interpretation has indeed a grossly nonlocal structure. This is characteristic,
according to the result to be proved here, of any such theory which reproduces exactly the
quantum mechanical predictions.
FORMULATION
With the example advocated by Bohm and Aharonov [6], the EPR argument is the following.
Consider a pair of spin one-half particles formed somehow in the singlet spin state and
moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach
magnets, on selected components of the spins σ⃗⃗⃗1⃗ and σ⃗ 2⃗ . If measurement of the component
σ⃗⃗⃗1⃗ ∙ a⃗, where a⃗ is some unit vector, yields the value + 1 then, according to quantum mechanics,
measurement of σ⃗ 2⃗ ∙ a⃗ must yield the value -1 and vice versa. Now we make the hypothesis
[2], 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. Since we can predict in advance the result of measuring any chosen
component of σ⃗ 2⃗, by previously measuring the same component of σ⃗⃗⃗1⃗, 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.
Let this more complete specification be affected by means of parameters λ. It is a matter of
indifference in the following whether λ denotes a single variable or a set, or even a set of
functions, and whether the variables are discrete or continuous. However, we write as if λ
were a single continuous parameter. The result A of measuring a σ⃗⃗⃗1⃗ ∙ a⃗ is then determined by
a⃗ and λ, and the result B of measuring σ⃗ 2⃗ ∙ b⃗⃗ in the same instance is determined by b⃗⃗ and λ,
and
A(a⃗, λ) = ± 1, B(b⃗⃗, λ) = ± 1. (1)
The vital assumption [2] is that the result B for particle 2 does not depend on the setting a⃗ of
the magnet for particle 1, nor A on b⃗⃗. If ρ(λ) is the probability distribution of λ then the
expectation value of the product of the two components σ⃗⃗⃗1⃗ ∙ a⃗ and σ⃗ 2⃗ ∙ b⃗⃗ is
Ρ(a⃗, b⃗⃗) = ∫ dλ ρ(λ) A(a⃗, λ) B(b⃗⃗, λ) (2)
This should equal the quantum mechanical expectation value, which for the singlet state is
〈σ⃗⃗⃗1⃗ ∙ a⃗ σ⃗ 2⃗ ∙ b⃗⃗〉 = −a⃗ ∙ b⃗⃗. (3)
But it will be shown that this is not possible. Some might prefer a formulation in which the
hidden variables fall into two sets, with A dependent on one and B on the other; this
possibility is contained in the above, since λ stands for any number of variables and the
dependences thereon of A and B are unrestricted. In a complete physical theory of the type
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envisaged by Einstein, the hidden variables would have dynamical significance and laws of
motion; our λ can then be thought of as initial values of these variables at some suitable
instant.
CONTRADICTION
The main result will now be proved. Because ρ is a normalized probability distribution,
∫ dλρ(λ) = 1 (12)
and because of the properties (1), Ρ in (2) cannot be less than - 1. It can reach - 1 at a⃗ = b⃗⃗ only
if
A(a⃗, λ) = −B(a⃗, λ) (13)
except at a set of points A of zero probability. Assuming this, (2) can be rewritten
Ρ(a⃗, b⃗⃗) = − ∫ dλ ρ(λ) A(a⃗, λ)A(b⃗⃗, λ). (14)
It follows that c⃗ is another unit vector
Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗) = − ∫ dλρ(λ) [A(a⃗, λ)A(b⃗⃗, λ) − A(a⃗, λ)A(c⃗, λ)]
= ∫ dλ ρ(λ) A(a⃗, λ) A(b⃗⃗, λ)[A(b⃗⃗, λ)A(c⃗, λ) − 1]
using (1), whence
|Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)| ≤ ∫ dλ ρ(λ)[1 − A(b⃗⃗, λ)A(c⃗, λ)]
The second term on the right is Ρ(b⃗⃗, c⃗), whence
1 + Ρ(b⃗⃗, c⃗) ≥ |Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)| (15)
Unless Ρ is constant, the right-hand side is in general of order |b⃗⃗ − c⃗| for small |b⃗⃗ − c⃗|. Thus
Ρ(b⃗⃗, c⃗) cannot be stationary at the minimum value (−1 at b⃗⃗ = c⃗) and cannot equal the
quantum mechanical value (3).”
At the beginning of section III. Illustration, Bell states, “The proof of the main result is quite
simple. Before giving it, however, a number of illustrations may serve to put it in perspective.”
Evidently, as opposed to simple, the more complicated the better it is for Bell in the sense that
intricate deception is more difficult to see through than plain deception.
Pre-contradiction, Bell dangles a different description of locality, “The vital assumption [2] is
that the result B for particle 2 does not depend on the setting a⃗ of the magnet for particle 1,
nor A on b⃗⃗,” before pulling it back and apparently using equation (3) for the quantum
mechanical expectation value, “because of the properties (1), Ρ in (2) cannot be less than - 1. It
can reach - 1 at a⃗ = b⃗⃗ only if
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A(a⃗, λ) = −B(a⃗, λ) (13)
except at a set of points A of zero probability. Assuming this, (2) can be rewritten
Ρ(a⃗, b⃗⃗) = − ∫ dλ ρ(λ) A(a⃗, λ)A(b⃗⃗, λ) ," (14)
which is Bell’s local version of entanglement.
Other than obvious problems, requiring thought perhaps, with the result B of measuring σ⃗ 2⃗ ∙ b⃗⃗
in the same instance, which should mean after the act of measuring σ⃗⃗⃗1⃗ ∙ a⃗ and obtaining
A(a⃗, λ), being −A(b⃗⃗, λ), if this result, −A(b⃗⃗, λ), is equal to A(a⃗, λ) for a⃗ = b⃗⃗, then
A(a⃗, λ) = B(b⃗⃗, λ)
for a⃗ ≠ b⃗⃗. If the result B for particle 2 does not depend on the setting a⃗ of the magnet for
particle 1, nor A on b⃗⃗, it is not because Bell assumed it, but rather because the quantum
mechanical expectation value 〈σ⃗⃗⃗1⃗ ∙ a⃗ σ⃗ 2⃗ ∙ b⃗⃗〉 for the entangled system is −a⃗ ∙ b⃗⃗.
If we were Bell and could get away with little or no analysis, when the time is right for proof of
a result referred to as “the most profound discovery of science,” then all we could give would
be another version of Bell’s inequality in the midst of mutterings of locality. If Bell’s inequality
is false, then it is because of Bell’s errors in logic. In just a few lines, from equation (14) to
inequality (15), the error must lie if the inequality is false. If Bell had any missing argument
for the steps, then that argument is false if the inequality is false. Bell’s inequality, if false, is
false for the functions of the quantum mechanical expectation of the product of spins as set up
in the theorem.
The equation following Bell’s equation (14) has two steps, both of which may hold or not. The
same type of statement, that the next two steps resulting in inequality (15) hold or not,
applies. Bell finds that inequality (15) does not hold after giving an apparently unjustifiable
argument for it. If every step that Bell makes were justifiable, then inequality (15) holds.
The problem with concluding that Bell’s inequality does not hold because of a logical error on
the part of Bell in the indicated steps is that when Bell states, “Ρ(b⃗⃗, c⃗) cannot be stationary at
the minimum value (−1 at b⃗⃗ = c⃗),” he, in a way, reveals that a⃗ = b⃗⃗ = c⃗, by his assumption at
least although this is not clearly stated. The near triviality of the result for a⃗ = b⃗⃗ = c⃗ allows
one to obtain the inequality immediately with only the condition that Ρ(a⃗, b⃗⃗) = Ρ(a⃗, c⃗) =
Ρ(b⃗⃗, c⃗) = −1, which follows from a⃗ = b⃗⃗ = c⃗. Even if a⃗ ≠ b⃗⃗, as long as Bell’s equation (1) holds,
we have the equality,
1 + Ρ(b⃗⃗, c⃗) = |Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)|.
From Bell’s equation (2)
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Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗) = ∫ dλρ(λ) [A(a⃗, λ)B(b⃗⃗, λ) − A(a⃗, λ)B(c⃗, λ)],
we obtain
Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)
= ∫ dλρ(λ)A(a⃗, λ)B(b⃗⃗, λ)[1 + A(b⃗⃗, λ)B(c⃗, λ)] − ∫ dλρ(λ) A(a⃗, λ)B(c⃗, λ)[1 + A(b⃗⃗, λ)B(b⃗⃗, λ)].
Thus, using Bell’s equation (1)
|Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)| ≤ ∫ dλρ(λ) [1 + A(b⃗⃗, λ)B(c⃗, λ)] + ∫ dλρ(λ) [1 + A(b⃗⃗, λ)B(b⃗⃗, λ)] (13),
or
|Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗)| ≤ 1 + Ρ(b⃗⃗, c⃗).
From Bell’s equation (3) we have
|a⃗ ∙ c⃗ − a⃗ ∙ b⃗⃗| ≤ 1 − b⃗⃗ ∙ c⃗.
For a⃗ ∙ c⃗ = 0, a⃗ ∙ b⃗⃗ = b⃗⃗ ∙ c⃗ = 1⁄√2 we have
1
√2
≤ 1 −
1
√2
,
or
√2 ≤ 1,
violating Bell’s inequality. Considering that a⃗ = b⃗⃗ = c⃗ does not hold, recalling Bell’s equation
(13) for which the assumption a⃗ = b⃗⃗ was made and that Bell uses equation (13) here in the
right side of the inequality (13), and that Bell’s equation (1) does not hold, there is no reason
to expect Bell’s inequality to hold in the first place. The second term on the right side of
inequality (13),
∫ dλρ(λ) [1 + A(b⃗⃗, λ)B(b⃗⃗, λ)],
vanishes if a⃗ = b⃗⃗ since
A(a⃗, λ) = −B(a⃗, λ) (13)
except at a set of points A of zero probability. If the conditions for which the inequality holds
are not met, there is no contradiction. Any use of Bell’s equation (14),
Ρ(a⃗, b⃗⃗) = − ∫ dλ ρ(λ) A(a⃗, λ)A(b⃗⃗, λ), (14)
twice in the expression for
Ρ(a⃗, b⃗⃗) − Ρ(a⃗, c⃗),
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assumes equation (13), and, thus, that a⃗ = b⃗⃗ = c⃗ for the above expression.
If
A(a⃗, λ) = σ⃗⃗⃗1⃗ ∙ a⃗ = +1,
then
B(a⃗, λ) = σ⃗ 2⃗ ∙ a⃗ = −1
and
B(b⃗⃗, λ) = σ⃗ 2⃗ ∙ b⃗⃗ = −1.
If
A(a⃗, λ) = σ⃗⃗⃗1⃗ ∙ a⃗ = −1,
Then
B(a⃗, λ) = σ⃗ 2⃗ ∙ a⃗ = +1
And
B(b⃗⃗, λ) = σ⃗ 2⃗ ∙ b⃗⃗ = +1.
Thus, if a⃗ = b⃗⃗, the result for B depends on whether the result for A was +1 or −1, the result
for B being the opposite or negative of the value for A in each case.
Similarly, for a⃗ ≠ b⃗⃗, if
A(a⃗, λ) = σ⃗⃗⃗1⃗ ∙ a⃗ = +1,
Then
B(a⃗, λ) = σ⃗ 2⃗ ∙ a⃗ = −1
And
B(b⃗⃗, λ) = σ⃗ 2⃗ ∙ b⃗⃗ = +1.
For a⃗ ≠ b⃗⃗, if
A(a⃗, λ) = σ⃗⃗⃗1⃗ ∙ a⃗ = −1,
Then
B(a⃗, λ) = σ⃗ 2⃗ ∙ a⃗ = +1
And
B(b⃗⃗, λ) = σ⃗ 2⃗ ∙ b⃗⃗ = −1.
Thus, if a⃗ ≠ b⃗⃗, the result for B(b⃗⃗, λ) is the same as the result for A(a⃗, λ) and the result for
B(a⃗, λ) is the opposite or negative of the value for A(a⃗, λ) in each case.
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In Bell’s theory and unstated conditions under which he derives Bell’s inequality, there is no b
for which a⃗ ≠ b⃗⃗ and there is no distant entangled particle with which the particle under
consideration interacted in the past. Bell’s hidden variable local system with only one-unit
vector a⃗ cannot possibly give the quantum mechanical expectation value, which for the singlet
state is
〈σ⃗⃗⃗1⃗ ∙ a⃗ σ⃗ 2⃗ ∙ b⃗⃗〉 = −a⃗ ∙ b⃗⃗. (3)
Entanglement requires that spin measurements satisfy those just given for a⃗ ≠ b⃗⃗. Bell’s
hidden variable quantum mechanical local particle with a⃗ = b⃗⃗ cannot provide the values for
a⃗ ≠ b⃗⃗ and, hence, the values for an entangled particle system. The perception that any hidden
variable quantum mechanical description of an entangled system of particles is non-local
depends on Bell’s inappropriate definition of locality, which excludes entanglement for a
distant particle and ignores that entanglement is a local phenomenon, and the condition that
a⃗ = b⃗⃗, which does not give the values for a⃗ ≠ b⃗⃗ as required to give the quantum mechanical
expectation values whether entanglement occurs for systems of particles that are close, not
distant, or not. Expecting the condition a⃗ = b⃗⃗ and a local particle analysis with no distant
entangled particle to give the quantum mechanical result is absurd. Any disentanglement of
once entangled particles moving apart sets the spin measurements at those given for a⃗ ≠ b⃗⃗ if
a⃗ ≠ b⃗⃗ and those given for a⃗ = b⃗⃗ if a⃗ = b⃗⃗. Any subsequent measurement of the set values
agrees with those for a⃗ ≠ b⃗⃗ if a⃗ ≠ b⃗⃗ and those given for a⃗ = b⃗⃗ if a⃗ = b⃗⃗ just as if the particles
are entangled still. In examples, including Bell’s, where Bell’s inequality is supposedly
violated, the conditions for which the inequality is true are not met. There is no contradiction.
We have other ideas, but going through everything takes time and tends to overwhelm when
little is known of previous works, which we have considered in part here. Errors in logic in
science last forever if one never considers the science. Deceptive, false theories, no matter
how inept the argument, have propaganda, ignorance, and nothing else on their side.
Einstein’s relativity theories and quantum theories come to mind. Bell’s locally entangled
particle has only one-unit vector with which to measure spin. In the introduction to David M.
Harrison’s “Bell’s Theorem,” Harrison states,
“In the early 1950's David Bohm (not "Bohr") was a young Physics professor at Princeton
University. He was assigned to teach Quantum Mechanics and, as is common, decided to write
a textbook on the topic; the book is still a classic. Einstein was at Princeton at this time, and as
Bohm finished each chapter of the book Einstein would critique it. By the time Bohm had
finished the book Einstein had convinced him that Quantum Mechanics was at least
incomplete. Bohm then spent many years in search of hidden variables, unobserved factors
inside, say, a radioactive atom that determines when it is going to decay. In a hidden variable
theory, the time for the decay to occur is not random, although the variable controlling the
process is hidden from us. We will discuss Bohm's work extensively later in this document.”
The of irony of Einstein, with the fraudulent relativity theories but with the criticism of the
completeness of quantum theory via arguments regarding entangled particles, critiquing
chapters in Bohm’s book and convincing Bohm that quantum theory was incomplete