# Bell’s Theorem Begs the Question

Joy Christian<EMAIL_ADDRESS>Einstein Centre for Local-Realistic Physics, Oxford
OX2 6LB, United Kingdom

###### Abstract

I demonstrate that Bell’s theorem is based on circular reasoning and thus a
fundamentally flawed argument. It unjustifiably assumes the additivity of
expectation values for dispersion-free states of contextual hidden variable
theories for non-commuting observables involved in Bell-test experiments,
which is tautologous to assuming the bounds of $\pm 2$ on the Bell-CHSH sum of
expectation values. Its premises thus assume in a different guise the bounds
of $\pm 2\,$ it sets out to prove. Once this oversight is ameliorated from
Bell’s argument by identifying the impediment that leads to it and local
realism is implemented correctly, the bounds on the Bell-CHSH sum of
expectation values work out to be ${\pm 2\sqrt{2}}$ instead of ${\pm 2}$,
thereby mitigating the conclusion of Bell’s theorem. Consequently, what is
ruled out by any of the Bell-test experiments is not local realism but the
linear additivity of expectation values,which does not hold for non-commuting
observables in any hidden variable theories to begin with.

KEYWORDS: Bell’s theorem, local realism, Bell-CHSH inequalities, quantum
correlations, Bell-test experiments

##  I Introduction

Bell’s theorem Bell-1964  is an impossibility argument (or “proof”) that
claims that no locally causal and realistic hidden variable theory envisaged
by Einstein Einstein  that could “complete” quantum theory can reproduce all
of the predictions of quantum theory. But some such claims of impossibility in
physics are known to harbor unjustified assumptions. In this paper, I show
that Bell’s theorem against locally causal hidden variable theories is no
exception. It is no different, in this respect, from von Neumann’s theorem
against all hidden variable theories vonNeumann , or the Coleman-Mandula
theorem overlooking the possibilities of supersymmetry Coleman . The implicit
and unjustified assumptions underlying the latter two theorems seemed so
innocuous to many that they escaped notice for decades. By contrast, Bell’s
theorem has faced skepticism and challenges by many from its very inception
(cf. footnote 1 in IEEE-1 ), including by me IEEE-1 ; Disproof ; Christian ;
IJTP ; Oversight ; RSOS ; IEEE-2 ; IEEE-3 ; IEEE-4 ; Local ; Symmetric ; RSOS-
Reply , because it depends on a number of questionable implicit and explicit
physical assumptions that are not difficult to recognize RSOS ; RSOS-Reply .
In what follows, I bring out one such assumption and demonstrate that Bell’s
theorem is based on a circular argument Oversight . It unjustifiably assumes
the additivity of expectation values for dispersion-free states of hidden
variable theories for non-commuting observables involved in the Bell-test
experiments Clauser , which is tautologous to assuming the bounds of $\pm 2$
on the Bell-CHSH sum of expectation values. Its premises thus assume in a
different guise what it sets out to prove. Once this oversight is ameliorated
from Bell’s argument, the local-realistic bounds on the Bell-CHSH sum of
expectation values work out to be ${\pm 2\sqrt{2}}$ instead of ${\pm 2}$,
thereby mitigating the conclusion of Bell’s theorem. As a result, what is
ruled out by the Bell-test experiments is not local realism but the additivity
of expectation values, which does not hold for non-commuting observables in
dispersion-free states of hidden variable theories to begin with.

##  II Heuristics for completing quantum mechanics

The goal of any hidden variable theory vonNeumann ; Bell-1966 ; Gudder  is to
reproduce the statistical predictions encoded in the quantum states
$|\psi\rangle\in{\mathscr{H}}$ of physical systems using hypothetical
dispersion-free states
$|\psi,\,\lambda):=\\{|\psi\rangle,\,\lambda\\}\in{\mathscr{H}}\otimes{\mathscr{L}}$
that have no inherent statistical character, where the Hilbert space
${\mathscr{H}}$ is extended by the space ${\mathscr{L}}$ of hidden variables
$\lambda$, which are hypothesized to “complete” the states of the physical
systems as envisaged by Einstein Einstein . If the values of
$\lambda\in{\mathscr{L}}$ can be specified in advance, then the results of any
measurements on a given physical system are uniquely determined.

To appreciate this, recall that expectation value of the square of any self-
adjoint operator ${\Omega}\in{\mathscr{H}}$ in a normalized quantum mechanical
state $|\psi\rangle$ and the square of the expectation value of ${\Omega}$
will not be equal to each other in general:

${\langle\psi|\,{\Omega}^{2}\,|\psi\rangle}\not={\left\langle\psi\left|\,{\Omega}\,\right|\psi\right\rangle}^{2}.$
(1)

This gives rise to inherent statistical uncertainty in the value of $\Omega$,
indicating that the state $|\psi\rangle$ is not dispersion-free:

$\Delta\Omega=\sqrt{\langle\psi|\\{\,\Omega-\langle\psi|\,\Omega\,|\psi\rangle\hbox{${1\hskip-2.4pt{\rm
l}}$}\\}^{2}\,|\psi\rangle}\not=0.$ (2)

By contrast, in a normalized dispersion-free state $|\psi,\,\lambda)$ of
hidden variable theories formalized by von Neumann vonNeumann , the
expectation value of ${\Omega}$, by hypothesis, is equal to one of its
eigenvalues ${\omega}({\lambda})$, determined by the hidden
variables$\;\lambda$,

$\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)={\omega}({\lambda})\;\Longleftrightarrow\;\Omega\,|\,\psi,\,\lambda)={\omega}({\lambda})\,|\,\psi,\,\lambda),$
(3)

so that a measurement of $\Omega$ in the state
$\left|\,\psi,\,\lambda\,\right)$ would yield the result ${\omega({\lambda})}$
with certainty. How this can be accomplished in a dynamical theory of
measurement process remains an open question Bell-1966 . But accepting the
hypothesis (3) implies

$(\psi,\,\lambda\,|\,\Omega^{2}\,|\,\psi,\,\lambda)=(\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda)^{2}.$
(4)

Consequently, unlike in a quantum sate $|\psi\rangle$, in a dispersion-free
state $|\psi,\,\lambda)$ observables $\Omega$ have no inherent uncertainty:

$\Delta\Omega=\sqrt{(\,\psi,\,\lambda\,|\,\\{\,\Omega-\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)\hbox{${1\hskip-2.4pt{\rm
l}}$}\\}^{2}\,|\,\psi,\,\lambda)}=0.$ (5)

The expectation value of $\Omega$ in the quantum state $|\psi\rangle$ can then
be recovered by integrating over the hidden variables$\;\lambda$:

$\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle\,=\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}{\omega}({\lambda})\;p(\lambda)\,d\lambda\,,$
(6)

where ${p(\lambda)}$ denotes the normalized probability distribution over the
space ${\mathscr{L}}$ of thus hypothesized hidden variables. The quantum
mechanical dispersion (2) in the measured value of the observer $\Omega$ can
thus be interpreted as due to thedistribution $p(\lambda)$ in the values of
the hidden variables $\lambda$ over the statistical ensemble of the physical
systems measured.

As it stands, this prescription amounts to assignment of unique eigenvalues
${\omega}({\lambda})$ to all observables $\Omega$ simultaneously, regardless
of whether they are actually measured. In other words, according to (6) every
physical quantity of a given system represented by $\Omega$ would possess a
unique preexisting value, irrespective of any measurements being performed.
The prescription (6) thus mathematically encodes Einstein’s conception of
realism. In Einstein , Einstein explained his point of view in terms of the
position and momentum of a free particle as follows: “The (free) particle
really has a definite position and a definite momentum, even if they cannot
both be ascertained by measurement in the same individual case [as in quantum
mechanics]. According to this point of view, the $\psi$-function represents an
incomplete description of the real state of affairs.” The acceptance of this
point of view — Einstein continues — “would lead to an attempt toobtain a
complete description of the real state of affairs as well as the incomplete
one (my emphasis), and to discover physical laws for such a description.”
Accordingly, the left-hand side of the first equality in (6) provides the
incomplete description of the system and its right-hand side provides the
complete one, with all possible statements one can make about the system
encoded in the expectation values of the observables being measured in the
state of the system vonNeumann . If $\Omega_{1}$ and $\Omega_{2}$ are two non-
commuting observables, then the uncertainty relation between them can also be
interpreted as

$\Delta\Omega_{1}\,\Delta\Omega_{2}\,\geqslant\,\frac{1}{2}\left|\left\langle\,\psi\,|\,\left[\Omega_{1},\,\Omega_{2}\right]\,|\,\psi\,\right\rangle\right|=\frac{1}{2}\left|\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\left[\Omega_{1},\,\Omega_{2}\right]\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,\right|=\frac{1}{2}\left|\int_{\mathscr{L}}{\omega_{1,2}}({\lambda})\;p(\lambda)\,d\lambda\,\right|$
(7)

in terms of the probability distribution ${p(\lambda)}$ in the values of the
hidden variables $\lambda$, where the first inequality is the one established
by Robertson Robertson , and $i\,\omega_{1,2}(\lambda)$ is a purely imaginary
eigenvalue of the skew-Hermitian operator
$\left[\Omega_{1},\,\Omega_{2}\right]$. Similarly, using the kinematical
equivalence seen in (6), the dynamical equivalence between the quantum
mechanical description and Einstein’s “complete” description can also be
established, as demonstrated in the Appendix A below:

$\left[\frac{d\,}{dt}\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle=\frac{1}{i\hbar}\left\langle\,\psi\,|\,\left[\,\Omega,\,H\,\right]\,|\,\psi\,\right\rangle+\langle\,\psi\,|\,\frac{\partial\Omega}{\partial
t}\,|\,\psi\,\rangle\right]=\int_{\mathscr{L}}\left[\frac{d\;}{dt}\,\omega(\lambda)=\\{\omega(\lambda),\,\mathscr{H}(\lambda)\\}+\frac{\partial\,\omega(\lambda)}{\partial
t}\right]\,p(\lambda)\,d\lambda\,.$ (8)

If particular values of $\lambda$ are precisely known with $p(\lambda)=1$,
then the right-hand side of (8) would reduce to the classical Hamiltonian
equations of motion. Otherwise, Ehrenfest’s equation in quantum mechanics on
the left-hand side of (8) can be understood as an ensemble average of
classical dynamics over the probability distribution $p(\lambda)$ of
$\lambda$. Thus, eachprobabilistic statement about the system can be traced
back to the incompleteness of our knowledge about the system.

In Section 2 of Bell-1966 , Bell works out an instructive example to
illustrate how the prescription (6) works for a system of two-dimensional
Hilbert space. It fails, however, for Hilbert spaces of dimensions greater
than two, because in higher dimensions degeneracies prevent simultaneous
assignments of unique eigenvalues to all observables in dispersion-free states
$\left|\,\psi,\,\lambda\,\right)$ dictated by the ansatz (3), giving
contradictory values for the same physical quantities. This was proved
independently by Bell Bell-1966 , Kochen and Specker Kochen , and Belinfante
Belinfante , as a corollary to Gleason’s theorem Gleason ; Shimony .

These proofs – known as the Kochen-Specker theorem – do not exclude contextual
hidden variable theories in which the complete state $|\,\psi,\,\lambda)$ of a
system assigns unique values to physical quantities only relative to
experimental contexts Gudder ; Shimony . If we denote the observables as
$\Omega(c)$ with $c$ being the environmental contexts of their measurements,
then thenon-contextual prescription (6) can be easily modified to accommodate
contextual hidden variable theories as follows:

$\left\langle\,\psi\,|\,\Omega(c)\,|\,\psi\,\right\rangle\,=\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}{\omega}(c,\,\lambda)\;p(\lambda)\,d\lambda\,.$
(9)

Each observable $\Omega(c)$ is still assigned a unique eigenvalue
${\omega}(c,\,\lambda)$, but now determined cooperatively by the complete
state $|\,\psi,\,\lambda)$ of the system and the state $c$ of its
environmental contexts. Consequently, even though some of its features are no
longer intrinsic to the system, contextual hidden variable theories do not
have the inherent statistical character of quantum mechanics, because outcome
of an experiment is a cooperative effect just as it is in classical physics
Shimony .Therefore, such theories interpret quantum entanglement at the level
of the complete state $|\,\psi,\,\lambda)$ only epistemically.

For our purposes here, it is also important to recall that in the Hilbert
space formulation of quantum mechanics vonNeumann  the correspondence between
observables and Hermitian operators is one-to-one. Moreover, a sum
$\widetilde{\Omega}(\tilde{c})=\sum_{i=1}^{n}\Omega_{i}(c_{i})$ of several
observables such as
$\Omega_{1}(c_{1}),\,\Omega_{2}(c_{2}),\,\Omega_{3}(c_{3}),\dots,\,\Omega_{n}(c_{n})$
is also an observable representing a physical quantity, and consequently the
sum of the expectation values of $\Omega_{i}(c_{i})$ is the expectation value
of the summed operator $\widetilde{\Omega}(\tilde{c})$,

$\sum_{i=1}^{n}\left\langle\,\psi\,|\,\Omega_{i}(c_{i})\,|\,\psi\,\right\rangle=\langle\,\psi\,|\left[\sum_{i=1}^{n}\Omega_{i}(c_{i})\right]|\,\psi\,\rangle,$
(10)

regardless of whether the observables are simultaneously measurable or
mutually commutative Bell-1966 . The question then is, since within any
contextual hidden variable theory characterized by (9) all of the observables
$\Omega_{i}(c_{i})$ and their sum $\widetilde{\Omega}(\tilde{c})$ are assigned
unique eigenvalues $\omega_{i}(c_{i},\,\lambda)$ and
$\widetilde{\omega}(\tilde{c},\,\lambda)$, respectively, would these
eigenvalues satisfy the equality

$\sum_{i=1}^{n}\left[\int_{\mathscr{L}}{\omega_{i}}(c_{i},\,\lambda)\;p(\lambda)\,d\lambda\right]\,\stackrel{{\scriptstyle?}}{{=}}\int_{\mathscr{L}}\left[\sum_{i=1}^{n}{\omega_{i}}(c_{i},\,\lambda)\right]p(\lambda)\,d\lambda$
(11)

in dispersion-free states $|\,\psi,\,\lambda)$ of physical systems in analogy
with the linear quantum mechanical relation (10) above? The answer is: Not in
general, because the eigenvalue ${\widetilde{\omega}(\tilde{c},\,\lambda)}$ of
the summed operator $\widetilde{\Omega}(\tilde{c})$ is not equal to the sum
$\sum_{i=1}^{n}\omega_{i}(c_{i},\,\lambda)$ of eigenvalues
$\omega_{i}(c_{i},\,\lambda)$ for given $\lambda$, unless the constituent
observables $\Omega_{i}(c_{i})$ are mutually commutative. In other words,
${\widetilde{\omega}(\tilde{c},\,\lambda)}\not=\sum_{i=1}^{n}\omega_{i}(c_{i},\,\lambda)$
in general, and therefore the correct counterpart of relation (10) is not (11)
but

$\sum_{i=1}^{n}\left[\int_{\mathscr{L}}{\omega_{i}}(c_{i},\,\lambda)\;p(\lambda)\,d\lambda\right]\,=\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\,\lambda)\;p(\lambda)\,d\lambda\,.$
(12)

As Bell points out in Section 3 of Bell-1966 , the linear relation (10) is an
unusual property of quantum mechanical states $|\psi\rangle$. There is no
reason to demand it individually of the dispersion-free states
$|\,\psi,\,\lambda)$, whose function is to reproduce themeasurable features of
quantum systems only when averaged over, as in (9). There is no reason why the
value of $\Omega(c)$should not be determined by some nonlinear function
$\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)$. I will
come back to this issue in Section V.

In Bell-1966 , Bell explains this non-linearity using spin components of a
spin-$\frac{1}{2}$ particle. Suppose we make a measurement of the component
$\sigma_{x}$ of the spin with a Stern-Gerlach magnet suitably oriented in
$\mathrm{I\\!R}^{3}$. That would yield an eigenvalue $s_{x}$ of $\sigma_{x}$
as a result. However, if we wish to measure the component $\sigma_{y}$ of the
spin, then that would require a different orientation of the magnet in
$\mathrm{I\\!R}^{3}$, and would give a different eigenvalue, $s_{y}$ of
$\sigma_{y}$, as a result. Moreover, a measurement of the sum of the $x$\- and
$y$-components of the spin, $\sigma_{x}+\sigma_{y}$, would again require a
very different orientation of the magnet in $\mathrm{I\\!R}^{3}$. Therefore,
the result obtained as an eigenvalue of the summed operators
$\sigma_{x}+\sigma_{y}$ will not be the sum $s_{x}+s_{y}$ of an eigenvalue of
the operator $\sigma_{x}$ added linearly to an eigenvalue of the operator
$\sigma_{y}$. Indeed, the eigenvalues of $\sigma_{x}$ and $\sigma_{y}$ are
both $\pm 1$, while the eigenvalues of $\sigma_{x}+\sigma_{y}$ are
$\pm\sqrt{2}$, so a linear relation cannot hold. As Bell points outin
Bell-1966 , the additivity of expectation values
$\langle\,\psi\,|\,\sigma_{x}\,|\,\psi\,\rangle+\langle\,\psi\,|\,\sigma_{y}\,|\,\psi\,\rangle=\langle\,\psi\,|\,\sigma_{x}+\,\sigma_{y}\,|\,\psi\,\rangle$
is a rather unusual property of the quantum states $|\psi\rangle$. The
linearity of it is effectuated in quantum mechanics by promoting observable
quantities to self-adjoint operators Grete . It does not hold for the
dispersion-free states $|\psi,\,\lambda)$ of hidden variable theories in
general because the eigenvalues of non-commuting observables such as
$\sigma_{x}$ and $\sigma_{y}$ do not add linearly, as we noted above.
Consequently, the additivity relation (10) that holds for quantum states would
not hold for the dispersion-free states.

##  III Special case of the singlet state and EPR-Bohm observables

${{\bf s}_{2}}$${{\bf s}_{1}}$${\bf 1}$${\bf 2}$${\bf a}$${\bf
a^{\prime}}$${\bf b}$${\bf
b^{\prime}}$source${\pi^{0}\longrightarrow\,e^{-}+\,e^{+}\,}$total spin = 0

Figure 1: In an EPR-Bohm-type experiment, a spin-less fermion – such as a
neutral pion – is assumed to decay from a source into an electron-positron
pair, as depicted. Then, measurements of the spin components of each separated
fermion are performed at space-like separated observation stations
${\mathbf{1}}$ and ${\mathbf{2}}$, obtaining binary results $\mathscr{A}=\pm
1$ and $\mathscr{B}=\pm 1$ along directions ${\mathbf{a}}$ and ${\mathbf{b}}$.
The conservation of spin momentum dictates that the total spin of the system
remains zero during its free evolution. After Ref. IEEE-1 .

Now, the proof of Bell’s famous theorem Bell-1964  is based on Bohm’s spin
version of the EPR’s thought experiment Bohm-1951 , which involves an
entangled pair of spin-$\frac{1}{2}$ particles emerging from a source and
moving freely in opposite directions, with particles ${1}$ and ${2}$ subject,
respectively, to spin measurements along independently chosen unit directions
${\bf a}$ and ${\bf b}$ by Alice and Bob, who are stationed at a spacelike
separated distance from each other (see Fig. 1). If initially the pair has
vanishing total spin, then the quantum mechanical state of the system is
described by the entangled singlet state

$|\Psi\rangle=\frac{1}{\sqrt{2}}\Bigl{\\{}|{\bf k},\,+\rangle_{1}\otimes|{\bf
k},\,-\rangle_{2}\,-\,|{\bf k},\,-\rangle_{1}\otimes|{\bf
k},\,+\rangle_{2}\Bigr{\\}},$ (13)

where ${\bf k}$ is an arbitrary unit vector in ${\mathrm{I\\!R}^{3}}$ and

${\boldsymbol{\sigma}}\cdot{\bf k}\,|{\bf k},\,\pm\rangle\,=\,\pm\,|{\bf
k},\,\pm\rangle$ (14)

defines quantum mechanical eigenstates in which the two fermions have spins
“up” or “down” in the units of ${\hbar=2}$, with ${\boldsymbol{\sigma}}$ being
the Pauli spin “vector” ${({\sigma_{x}},\,{\sigma_{y}},\,{\sigma_{z}})}$. Once
the state (13) is prepared, the observable $\Omega(c)$ of interest is

$\Omega(c)={\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,,$ (15)

whose possible eigenvalues are

$\omega(c,\,\lambda)={\mathscr{A}}{\mathscr{B}}({\bf a},\,{\bf b},\lambda)=\pm
1,$ (16)

where ${\mathscr{A}}=\pm 1$ and ${\mathscr{B}}=\pm 1$ are the results of spin
measurements made jointly by Alice and Bob along their randomly chosen
detector directions ${\bf a}$ and ${\bf b}$. In the singlet state (13), the
joint observable (15) predicts sinusoidal correlations
$\langle\Psi|{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}|\Psi\rangle=-{\bf
a}\cdot{\bf b}$ between the values of the spins observed about the freely
chosen contexts ${\bf a}$ and ${\bf b}$ Christian .

For locally contextual hidden variable theories there is a further requirement
that the results of local measurements must be describable by functions that
respect local causality, as first envisaged by Einstein Einstein  and later
formulated mathematically by Bell Bell-1964 . It can be satisfied by requiring
that the eigenvalue $\omega(c,\lambda)$ of the observable $\Omega(c)$ in (15)
representing the joint result ${\mathscr{A}}{\mathscr{B}}({\bf a},\,{\bf
b},\lambda)=\pm 1$ is factorizable as
$\omega(c,\lambda)=\omega_{1}(c_{1},\lambda)\,\omega_{2}(c_{2},\lambda)$, or
in Bell’s notation as

${\mathscr{A}}{\mathscr{B}}({\bf a},\,{\bf b},\lambda)={\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda),$ (17)

with the factorized functions ${\mathscr{A}}({\bf a},\lambda)=\pm 1$ and
${\mathscr{B}}({\bf b},\lambda)=\pm 1$ satisfying the following condition of
local causality:

> Apart from the hidden variables ${\lambda}$, the result ${{\mathscr{A}}=\pm
> 1}$ of Alice depends only on the measurement context ${\bf a}$, chosen
> freely by Alice, regardless of Bob’s actions Bell-1964 . And, likewise,
> apart from the hidden variables ${\lambda}$, the result ${{\mathscr{B}}=\pm
> 1}$ of Bob depends only on the measurement context ${\bf b}$, chosen freely
> by Bob, regardless of Alice’s actions. In particular, the function
> ${{\mathscr{A}}({\bf a},\,\lambda)}$ does not depend on ${\bf b}$ or
> ${\mathscr{B}}$ and the function ${{\mathscr{B}}({\bf b},\,\lambda)}$ does
> not depend on ${\bf a}$ or ${\mathscr{A}}$. Moreover, the hidden variables
> ${\lambda}$ do not depend on either ${\bf a}$, ${\bf b}$, ${\mathscr{A}}$,
> or ${\mathscr{B}}$ IEEE-2 .

The expectation value ${\cal E}({\mathbf{a}},{\mathbf{b}})$ of the joint
results in the dispersion-free state $|\,\psi,\,\lambda)$ should then satisfy
the condition

$\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,|\Psi\rangle=\,{\cal
E}({\mathbf{a}},{\mathbf{b}}):=\\!\int_{\mathscr{L}}{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;p(\lambda)\,d\lambda\,,$
(18)

where the hidden variables ${\lambda}$ originate from a source located in the
overlap of the backward light cones of Alice and Bob, and the normalized
probability distribution ${p(\lambda)}$ is assumed to remain statistically
independent of the contexts ${\bf a}$ and ${\bf b}$ so that
$p(\lambda\,|\,{\mathbf{a}},{\mathbf{b}})=p(\lambda)$, which is a reasonable
assumption. In fact, relaxing this assumption to allow $p(\lambda)$ to depend
on ${\mathbf{a}}$ and ${\mathbf{b}}$ introduces a form of non-locality, as
explained by Clauser and Horne in footnote 13 of Horne . Then, since
${\mathscr{A}}({\bf a},\lambda)=\pm 1$ and ${\mathscr{B}}({\bf b},\lambda)=\pm
1$, their product
${\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)=\pm
1$, setting the following bounds on ${\cal E}({\mathbf{a}},{\mathbf{b}})$:

$-1\leqslant\,{\cal E}({\mathbf{a}},{\mathbf{b}})\leqslant+1.$ (19)

These bounds are respected not only by local hidden variable theories but also
by quantum mechanics and experiments.

##  IV Mathematical core of Bell’s theorem

By contrast, at the heart of Bell’s theorem is a derivation of the bounds of
$\pm 2$ on an ad hoc sum of the expectation values ${\cal
E}({\mathbf{a}},{\mathbf{b}})$ of local results ${\mathscr{A}}({\bf
a},\lambda)$ and ${\mathscr{B}}({\bf b},\lambda)$, recorded at remote
observation stations by Alice and Bob, from four different sub-experiments
involving measurements of non-commuting observables such as
${\boldsymbol{\sigma}}_{1}\cdot{\bf a}$ and
${\boldsymbol{\sigma}}_{1}\cdot{\bf a^{\prime}}$ Bell-1964 ; Clauser :

${\cal E}({\bf a},\,{\bf b})+{\cal E}({\bf a},\,{\bf b^{\prime}})+{\cal
E}({\bf a^{\prime}},\,{\bf b})-{\cal E}({\bf a^{\prime}},\,{\bf
b^{\prime}})\,.$ (20)

Alice can freely choose a detector direction ${\bf a}$ or ${\bf a^{\prime}}$,
and likewise Bob can freely choose a detector direction ${\bf b}$ or ${\bf
b^{\prime}}$, to detect, at a space-like distance from each other, the spins
of fermions they receive from the common source. Then, from (19), we can
immediately read off the upper and lower bounds on the combination (20) of
expectation values:

$-4\,\leqslant\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf
b^{\prime}})\,+\,{\cal E}({\bf a^{\prime}},\,{\bf b})\,-\,{\cal E}({\bf
a^{\prime}},\,{\bf b^{\prime}})\,\leqslant\,+4\,.$ (21)

###  IV.1 Standard derivation of the Bell-CHSH inequalities (25)

The next step in Bell’s derivation of the bounds $\pm 2$ instead of $\pm 4$ is
the assumption of the additivity of expectation values:

$\displaystyle{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf
b^{\prime}})\,+\,{\cal E}({\bf a^{\prime}},\,{\bf b})\,-\,{\cal E}({\bf
a^{\prime}},\,{\bf b^{\prime}})$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{A}}({\bf
a},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\int_{\mathscr{L}}{\mathscr{A}}({\bf
a},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\\!\int_{\mathscr{L}}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda-\\!\\!\int_{\mathscr{L}}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda$
$\displaystyle=\\!\int_{\mathscr{L}}\\!\big{\\{}\,{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)+{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf b},\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)\big{\\}}\;p(\lambda)\,d\lambda\,.$ (22)

We will have much to discuss about this seemingly harmless mathematical step
employing the built-in linear additivity of integrals. I will demonstrate in
Section V that, far from being harmless, it is, in fact, an unjustified
assumption that harbors a profound mistake of assuming the very thesis of the
theorem to be proven, just as von Neumann’s theorem did Oversight ; Bell-1966
; Grete ; Mermin ; BohmBub . It assumes, without proof, that linear additivity
of integrals leading to (22) can be meaningfully applied, not only to the
eigenvalues of commuting observables but also to the eigenvalues of non-
commuting observables that cannot be measured simultaneously. But, as we saw
in the paragraph following equation (12), this assumption is quite mistaken.
However, if we overlook this mistake and accept equality (22), then the bounds
of $\pm 2$ on the Bell-CHSHcombination (20) of expectation values are not
difficult to work out by rewriting the integrand on its right-hand side as

${\mathscr{A}}({\bf a},\lambda)\,\big{\\{}\,{\mathscr{B}}({\bf
b},\lambda)+{\mathscr{B}}({\bf
b^{\prime}},\lambda)\,\big{\\}}\,+\,{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,\big{\\{}\,{\mathscr{B}}({\bf
b},\lambda)-{\mathscr{B}}({\bf b^{\prime}},\lambda)\,\big{\\}}.$ (23)

Since ${{\mathscr{B}}({\bf b},\lambda)=\pm 1}$, if ${|{\mathscr{B}}({\bf
b},\lambda)+{\mathscr{B}}({\bf b^{\prime}},\lambda)|=2}$, then
${|{\mathscr{B}}({\bf b},\lambda)-{\mathscr{B}}({\bf
b^{\prime}},\lambda)|=0}$, and vice versa. Consequently, since
${{\mathscr{A}}({\bf a},\lambda)=\pm 1}$, the integrand (23) is bounded by
$\pm 2$ and the absolute value of the last integral in (22) does not exceed 2:

$-2\,\leqslant\int_{\mathscr{L}}\\!\big{\\{}\,{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)+{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf b},\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)\big{\\}}\;p(\lambda)\,d\lambda\;\leqslant\,+2\,.$ (24)

Therefore, the equality (22) implies that the absolute value of the
combination of expectation values is bounded by 2:

$-2\,\leqslant\,{\cal E}({\bf a},\,{\bf b})\,+\,{\cal E}({\bf a},\,{\bf
b^{\prime}})\,+\,{\cal E}({\bf a^{\prime}},\,{\bf b})\,-\,{\cal E}({\bf
a^{\prime}},\,{\bf b^{\prime}})\,\leqslant\,+2\,.$ (25)

But since the bounds on (20) predicted by quantum mechanics and observed in
experiments are $\pm 2\sqrt{2}$, Bell concludes that no local and realistic
theory envisaged by Einstein can reproduce the statistical predictions of
quantum mechanics. In particular, contextual hidden variable theories
specified by (9) that respect the factorizability (17) are not viable.

In many derivations of (25) in the literature, factorized probabilities of
observing binary measurement results are employed rather than measurement
results themselves I have used in (17) in my derivation following Bell
Bell-1964 ; Clauser . But employing probabilities would only manage to
obfuscate the logical flaw in Bell’s argument I intend to bring out here.

###  IV.2 Converse derivation of the additivity (22) by assuming (25)

Now, it is not difficult to demonstrate the converse of the above derivation
in which the additivity of expectation values (22) is derived by assuming the
stringent bounds of $\pm 2$ on the sum (20). Employing (18), (20) can be
written as

$\int_{\mathscr{L}}{\mathscr{A}}({\bf a},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\int_{\mathscr{L}}{\mathscr{A}}({\bf
a},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\\!\int_{\mathscr{L}}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda-\\!\\!\int_{\mathscr{L}}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda\,.$ (26)

Since each product ${\mathscr{A}}({\bf a},\lambda){\mathscr{B}}({\bf
b},\lambda)$ in the above integrals is equal to $\pm 1$, each of the four
integrals is bounded by $\pm 1$:

$-1\,\leqslant\int_{\mathscr{L}}{\mathscr{A}}({\bf
a},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda\;\leqslant\,+1.$ (27)

Thus the sum of four integrals in (26) is bounded by $\pm 4$, not $\pm 2$.
However, we started with (25), which contends that the sum of integrals in
(26) is bounded by $\pm 2$. But the only way to reduce the bounds on (26) from
$\pm 4$ to $\pm 2$ without compromising the independence of the results
${\mathscr{A}}({\bf a},\lambda){\mathscr{B}}({\bf b},\lambda)$, etc., or of
their averages appearing in the sum (26), orviolating the rules of anti-
derivatives, is by equating the sum of integrals in (26) to the following
integral of the sum:

$\int_{\mathscr{L}}\\!\big{\\{}\,{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)+{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf b},\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)\big{\\}}\;p(\lambda)\,d\lambda\,.$ (28)

To be sure, we can easily reduce the bounds on the sum in (26) from $\pm 4$ to
$\pm 2$, for example, by setting the last two integrals in it equal to each
other, with the minus sign between them. That would reduce (26) to a sum of
only the first two integrals, which is then easily seen to be bounded by $\pm
2$ using (27). But to achieve this, we have compromised the independence of at
least the last two sub-ensemble averages, which is not permitted by the
conditions adhered to in Bell’s argument. Similarly, again as an example, we
can set the results themselves in the last two integrals in (26) equal to each
other: ${\mathscr{A}}({\bf a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b},\lambda)={\mathscr{A}}({\bf a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)$. That would again reduce (26) to the sum of only the
first two integrals, which is then easily seen to be bounded by $\pm 2$ using
(27). But we have again compromised the independence of the last two results
to achieve this, which is not permitted. One can also easily reduce the bounds
on (26) from $\pm 4$ to $\pm 2$ by violating some of the rules of anti-
derivatives, or of statistical averages, etc., as an extreme measure, but that
again amounts to forfeiting the game from the start. Thus, the only admissible
means of reducing the bounds on (26) from $\pm 4$ to $\pm 2$ is by using the
standard rules of anti-derivatives that allow us to equate the sum ofintegrals
appearing in (26) to the integral (28) of the sum of eigenvalues, which, as we
saw in (24), is bounded by $\pm 2$.

We have thus derived the additivity of expectation values (22) by imposing the
bounds of $\pm 2$ on the Bell-CHSH sum (20) as our starting assumption.
Therefore, given the previous derivation that led us to (25) by assuming (22)
and the current derivation that led us to (22) by assuming (25), we have
proved that the assumption (22) of the additivity of expectation values is
tautologous to assuming the bounds of $\pm 2$ on the Bell-CHSH sum (20) of
expectation values.

##  V Additivity of expectation values (22) is an unjustified assumption,
equivalent to the thesis to be proven

The key step that led us to the bounds of $\pm 2$ on (20) that are more
restrictive than $\pm 2\sqrt{2}$ is the step (22) of the linear additivity of
expectation values. In what follows, I will demonstrate that this step is, in
fact, an unjustified assumption, equivalent to the main thesis of the theorem
to be proven, just as it is in von Neumann’s now discredited theorem Oversight
; Bell-1966 ; Grete ; Mermin ; BohmBub . But this fact is obscured by the
seemingly innocuous built-in linear additivity of integrals used in step (22).
However, as we noted around (12) and will be further demonstrated in Section
VII, the built-in linear additivity of integrals is physically meaningful only
for simultaneously measurable or commuting observables Grete ; Mermin . It is,
therefore, not legitimate to invoke it at step (22) without proof. Step (22)
would be valid also in classical physics in which the value of a sum of
observable quantities would be the same as the sum of the values each quantity
would take separately, because, unlike in quantum mechanics, they would all be
simultaneously measurable, yielding only sharp values. Perhaps for this reason
it is usually not viewed as an assumption but mistaken for a benign
mathematical step. It is also sometimes claimed to be necessitated by
Einstein’s requirement of realism Einstein . But I will soon explain why it is
a much overlooked unjustified assumption, and demonstrate in Section VII that,
far from being required by realism, the right-hand side of step (22), in fact,
contradicts realism, which requires that every observable of a physical
system,including any sums of observables, must be assigned a correct
eigenvalue, quantifying one of its preexisting properties.

Moreover, realism has already been adequately accommodated by the very
definition of the local functions ${\mathscr{A}}({\bf a},\lambda)$ and
${\mathscr{B}}({\bf b},\lambda)$ and their counterfactual juxtaposition on the
left-hand side of (22), as contextually existing properties of the system.
Evidently, while a result in only one of the four expectation values
corresponding to a sub-experiment that appears on the left-hand side of (22)
can be realized in a given run of a Bell-test experiment, the remaining three
results appearing on that side are realizable at least counterfactually, thus
fulfilling the requirement of realism Oversight . Therefore, the requirement
of realism does not necessitate the left-hand side of (22) to be equated with
its right-hand side in the derivation of (25). Realism requires definite
results ${\mathscr{A}}({\bf a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)$ to
exist as eigenvalues only counterfactually, not allfour at once, as they are
written on the right-hand side of (22). What is more, as we will soon see,
realism implicit in the prescription (9) requires the quantity (23) to be a
correct eigenvalue of the summed operator (39), but it is not.

On the other hand, given the assumption
$p(\lambda\,|\,{\mathbf{a}},{\mathbf{b}})=p(\lambda)$ of statistical
independence and the additivity property of anti-derivatives, mathematically
the equality (22) follows at once because of the linearity built into the
integrals, provided we adopt a double standard for additivity: we reject (22)
for von Neumann’s theorem as Bell did in Bell-1966 , but accept it
unreservedly for Bell’s theorem Oversight ; Grete . The binary properties of
the functions ${\mathscr{A}}({\bf a},\lambda)$ and ${\mathscr{B}}({\bf
b},\lambda)$ then immediately lead us to the bounds of $\pm 2$ on (20). But,
as we saw above, assuming the bounds of $\pm 2$ on (20) leads, conversely, to
the assumption (22) of additivity of expectation values. Thus, assuming the
additivity of expectation values (22) is mathematically equivalent to assuming
the bounds of $\pm 2$ on the Bell-CHSH sum (20). In other words, Bell’s
argument presented in Section IV assumes its conclusion (25) in the guise of
assumption (22), by implicitly assuming that the expectation functions
$(\Psi,\,\lambda\,|\,{\Omega}({c})\,|\,\Psi,\,\lambda)$ determining the
eigenvalues ${\omega}({c},\lambda)$ of ${\Omega}({c})$ are linear BohmBub :

$\widetilde{\omega}(\widetilde{c},\,\lambda)=(\,\Psi,\,\lambda\,|\left[\sum_{i=1}^{4}\Omega_{i}(c_{i})\right]|\,\Psi,\,\lambda\,)=\sum_{i=1}^{4}\,(\,\Psi,\,\lambda\,|\,\Omega_{i}(c_{i})\,|\,\Psi,\,\lambda\,)=\sum_{i=1}^{4}\,\omega_{i}(c_{i},\,\lambda).$
(29)

But, as explained by Bohm and Bub in BohmBub  (see Appendix B below), this
assumed linearity of $(\Psi,\,\lambda\,|\,{\Omega}({c})\,|\,\Psi,\,\lambda)$
is unreasonably restrictive for dispersion-free states $|\,\Psi,\,\lambda)$,
because the observables defined in (15) are not simultaneously measurable.
However, it allows us to reduce the following correct relation within quantum
mechanics as well as hidden variable theories,

$\sum_{i=1}^{4}\left[\int_{\mathscr{L}}\,(\,\Psi,\,\lambda\,|\,\Omega_{i}(c_{i})\,|\,\Psi,\,\lambda\,)\;p(\lambda)\,d\lambda\,\right]=\int_{\mathscr{L}}\,(\,\Psi,\,\lambda\,|\left[\sum_{i=1}^{4}\Omega_{i}(c_{i})\right]|\,\Psi,\,\lambda\,)\;p(\lambda)\,d\lambda\,,$
(30)

to the relation

$\sum_{i=1}^{4}\left[\int_{\mathscr{L}}\,(\,\Psi,\,\lambda\,|\,\Omega_{i}(c_{i})\,|\,\Psi,\,\lambda\,)\;p(\lambda)\,d\lambda\,\right]=\int_{\mathscr{L}}\left[\sum_{i=1}^{4}\,(\,\Psi,\,\lambda\,|\,\Omega_{i}(c_{i})\,|\,\Psi,\,\lambda\,)\right]p(\lambda)\,d\lambda\,,$
(31)

which is the same as assumption (22), albeit written in a more general
notation. The equality (30), on the other hand, is equivalent to the quantum
mechanical relation (32) discussed below, which can be verified using the
prescription (9).The same equality (30) is also valid for hidden variable
theories, because it does not make the mistake of relying on the linearity
assumption (29). This can be verified also using (9) and the ansatz (3). Thus,
the innocuous-looking linear additivity of integrals in assumption (22), while
mathematically correct, is neither innocent nor physically reasonable.

It is not difficult to understand why appealing to the built-in linear
additivity of anti-derivatives is not as innocent or physically reasonable as
it may seem. In fact, for non-commuting observables that are not
simultaneously measurable, justification of (22) or (31) by appealing to the
built-in linear additivity of integrals leads to incorrect equality between
unequal physical quantities. The reasons for this were recognized by Grete
Hermann Grete  some three decades before the formulation of Bell’s theorem
Bell-1964 , as part of her insightful criticism of von Neumann’s alleged
theorem vonNeumann ; Mermin . As she explained in Grete , we are not concerned
here with classical physics in which all observable quantities are
simultaneously measurable yielding only sharp values, and therefore the value
of a sum of observable quantities is nothing other than the sum of the values
each of those quantities would separately take. Consequently, in classical
physics, the averages of such values over individual initial states $\lambda$
of the system can also be meaningfully added linearly, just as assumed in step
(22) or (31), because there is no scope for any contradiction between the
averages obtained by evaluating the left-hand side and the right-hand side of
these equations. Therefore, in classical physics linear additivity of
expectation values remains consistent with the built-in linear additivity of
anti-derivatives. However, the same cannot be assumed without proof for the
dispersion-free states $|\psi,\,\lambda)$ of hidden variable theories,
because, in that case, the values of the observable quantities are eigenvalues
of the corresponding quantum mechanical operators dictated by the ansatz (3),
and, as we noted above and toward the end of Section II, the eigenvalue
${\widetilde{\omega}(\tilde{c},\,\lambda)}$ of the summed observable
$\widetilde{\Omega}(\tilde{c})$ is not equal to the sum
$\sum_{i=1}^{n}\omega_{i}(c_{i},\,\lambda)$ of the eigenvalues
$\omega_{i}(c_{i},\,\lambda)$ of $\Omega_{i}(c_{i})$, unless the observables
$\Omega_{i}(c_{i})$ constituting the sum $\widetilde{\Omega}(\tilde{c})$ are
simultaneously measurable. Thus, an important step in the proof of (25) is
missing. A necessary step that would prove the consistency of the built-in
linear additivity of anti-derivatives with the non-additivity of expectation
values for the non-commuting observables. In equation (42) of Section VII
below we will see the difference between the eigenvalue of the summed operator
and the sum of individual eigenvalues explicitly. It will demonstrate how, in
hidden variable theories equation (22) or (31) involving averages of
eigenvalues ends up equating unequal averages of physical quantities in
general. It will thereby prove that, while valid in classical physics and for
simultaneously measurable observables, equation (22) or (31) is not valid for
hidden variable theories in general. Insisting otherwise thus amounts to
assuming the validity of this equation without proof, despite the contrary
evidence just presented Grete . That, in turn, amounts to assuming the very
thesis to be proven — namely, the bounds of $\pm 2$ on the Bell-CHSH sum (20).
Consequently, the onlycorrect meaning assignable to (22) or (25) is that it is
valid only in classical physics and/or for commuting observables.

Sometimes assumption (22) is justified on statistical grounds. It is argued
that the four sub-experiments appearing on the left-hand side of (22) with
different experimental settings $\\{{\bf a},\,{\bf b}\\}$, $\\{{\bf a},\,{\bf
b^{\prime}}\\}$, etc. can be performed independently of each other, on
possibly different occasions, and then the resulting averages are added
together at a later time for statistical analysis. If the number of
experimental runs for each pair of settings is sufficiently large, then,
theoretically, the sum of the four averages appearing on the left-hand side of
(22) are found not to exceed the bounds of $\pm 2$, thus justifying the
equality (22). This can be easily verified in numerical simulations (see Ref.
[27] cited in IEEE-4 ). However,this heuristic argument is not an analytical
proof of the bounds. What it implicitly neglects to take into account by
explicitly assuming that the four sub-experiments can be performed
independently, is that the sub-experiments involve mutually exclusive pairs of
settings such as $\\{{\bf a},\,{\bf b}\\}$ and $\\{{\bf a},\,{\bf
b^{\prime}}\\}$ in physical space, and thus involve non-commuting observables
that cannot be measured simultaneously Oversight . Unless the statistical
analysis takes this physical fact into account, it cannot be claimed to have
any relevance for the Bell-test experiments RSOS-Reply . For ignoring this
physical fact amounts to incorrectly assuming that the spin observables
${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. are mutually
commuting, and thus simultaneously measurable, for which assumption (22) is
indeed valid, as demonstrated below in Section VII (see the discussion around
(45)). On the other hand, when the non-commutativity of the observables
involved in the sub-experiments is taken into account in numerical
simulations, the bounds on (20) turn out to be $\pm 2\sqrt{2}$, as shown in
RSOS ; IEEE-2  and Ref. [27] cited in IEEE-4 . In other words, such a
statistical argument is simply assumption (22) in disguise.

Another important point to recognize here is that the above derivation of the
stringent bounds of $\pm 2$ on (20) for a locally causal dispersion-free
counterpart $|\,\Psi,\,\lambda)$ of the quantum mechanical singlet state (13)
must comply with the heuristics of the contextual hidden variable theories we
discussed in Section II. If it does not, then the bounds of $\pm 2$ cannot be
claimed to have any relevance for the viability of local hidden variable
theories Shimony . Therefore, as discussed in Section II, in a contextual
hidden variable theory all of the observables $\Omega_{i}(c_{i})$ of any
physical system, including their sum
$\widetilde{\Omega}(\tilde{c})=\sum_{i=1}^{n}\Omega_{i}(c_{i})$, which also
represents a physical quantity in the Hilbert space formulation of quantum
mechanics vonNeumann  whether or not it is observed, must be assigned unique
eigenvalues $\omega_{i}(c_{i},\,\lambda)$ and
$\widetilde{\omega}(\tilde{c},\,\lambda)$, respectively, in the dispersion-
free states $|\,\psi,\,\lambda)$ of the system, regardless of whether these
observables are simultaneously measurable. In particular, while the summed
observable (39) discussed below is never observed in the Bell-test
experiments, realism nevertheless requires it to be assigned a unique
eigenvalue in accordance with the ansatz (3) and the prescription (9).

##  VI Additivity of expectation values is respected by quantum states

Now, within quantum mechanics, expectation values do add in analogy with the
equality (22) assumed by Bell for local hidden variable theories vonNeumann ;
Bell-1966 . In quantum mechanics, the statistical predictions of which any
hidden variable theory is obliged to reproduce, the joint results
${\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)$ observed
by Alice and Bob would be eigenvalues of the operators
${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, and the linearity in the
rules of Hilbert space quantum mechanics ensures that these operators satisfy
the additivity of expectation values. Thus, for any quantum state
$|\psi\rangle$, the following equality holds:

$\displaystyle\langle\psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf a}\,$
$\displaystyle\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}\,|\psi\rangle+\langle\psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\psi\rangle+\langle\psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}\,|\psi\rangle-\langle\psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\psi\rangle$
$\displaystyle=\,\langle\psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}-{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\psi\rangle.$ (32)

The linearity of this equation has been achieved by promoting observable
quantities to Hermitian operators vonNeumann ; Grete . Any local hidden
variable theory is then obliged to reproduce the predictions of both sides of
this equation. Comparing equations (22) and (32), the equality between the two
sides of (22) seems reasonable, even physically. Furthermore, since the
condition (18) for any hidden variable theory obliges us to set the four terms
on the left-hand side of (32) as

$\displaystyle\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,|\Psi\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;p(\lambda)\,d\lambda\,,$
(33) $\displaystyle\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}}\,|\Psi\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}^{\prime}},\,\lambda)\;p(\lambda)\,d\lambda\,,$
(34) $\displaystyle\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,|\Psi\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{A}}({\mathbf{a}^{\prime}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;p(\lambda)\,d\lambda\,,$
(35)
$\displaystyle\text{and}\;\;\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\Psi\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{A}}({\mathbf{a}^{\prime}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}^{\prime}},\,\lambda)\;p(\lambda)\,d\lambda\,,$
(36)

it may seem reasonable that, given the quantum mechanical equality (32), any
hidden variable theory should satisfy

$\displaystyle\langle\Psi|\,\widetilde{\Omega}(\tilde{c})\,|\Psi\rangle$
$\displaystyle=\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}-{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\Psi\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}\\!\\!\big{\\{}\,{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)+{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf b},\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)\big{\\}}\;p(\lambda)\,d\lambda\,,$ (37)

adhering to the prescription (9), which would then justify equality (22).
Since hidden variable theories are required to satisfy the prescription (9),
should not they also reproduce equation (37)? The answer to this is not
straightforward.

##  VII Additivity of expectation values does not hold for dispersion-free
states

The problem with equation (37) is that, while the joint results
${\mathscr{A}}(\mathbf{a},\lambda){\mathscr{B}}(\mathbf{b},\lambda)$, etc.
appearing on the left-hand side of equation (22) are possible eigenvalues of
the products of spin operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc., their summation

${\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b},\,\lambda)+{\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\,\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\,\lambda)$ (38)

appearing as the integrand on the right-hand side of equation (37) or (22) is
not an eigenvalue of the summed operator

$\widetilde{\Omega}(\tilde{c})={\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}-{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}},$ (39)

because the spin operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf a}$ and
${\boldsymbol{\sigma}}_{1}\cdot{\bf a^{\prime}}$, etc., and therefore
${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc., do not commute with
each other:

$\displaystyle\left[\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b},\;{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}}\,\right]$
$\displaystyle=2\,{\boldsymbol{\sigma}}\cdot\left\\{\left({\bf a}\times{\bf
b^{\prime}}\right)\times\left({\bf a}\times{\bf b}\right)\right\\}$
$\displaystyle\not=0\;\,\text{if}\;\,{\bf b^{\prime}}\not={\bf b}\not={\bf
a}.$ (40)

Consequently, equation (37) would hold within any hidden variable theory only
if the operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. were commuting
operators. As we discussed, this is well known from the famous criticisms of
von Neumann’s theorem against hidden variable theories Oversight ; Bell-1966 ;
Grete ; Mermin . While the equality (22) of the sum of expectation values with
the expectation value of the sum is respected in quantum mechanics, it does
not hold for hidden variable theories Bell-1966 . Nor does local realism
necessitate the linear additivity (29) of eigenvalues for individual
dispersion-free states $|\,\Psi,\,\lambda)$.

This problem, however, suggests its own resolution. We can work out the
correct eigenvalue $\widetilde{\omega}(\tilde{c},\,\lambda)$ of the summed
operator (39), at least formally, as I have worked out in Appendix C below.
The correct version of equation (37) is then

$\langle\Psi|\,{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}-{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}\,|\Psi\rangle=\\!\int_{\mathscr{L}}{\widetilde{\omega}}({\bf
a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}},\lambda)\;p(\lambda)\,d\lambda\,,$ (41)

where

${\widetilde{\omega}}\\!=\pm\sqrt{\big{\\{}{\mathscr{A}}({\bf
a},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)+{\mathscr{A}}({\bf
a},\,\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\,\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\,\lambda)\big{\\}}^{2}+(\Psi,\,\lambda\,|\,{\widetilde{\Theta}}\,|\,\Psi,\,\lambda)\,}$
(42)

is the correct eigenvalue of the summed operator (39), with its non-commuting
part separated out as the operator

${\widetilde{\Theta}}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}})=2\,{\boldsymbol{\sigma}}\cdot{\bf n}({\bf a},{\bf
a^{\prime}},{\bf b},{\bf b^{\prime}})\,.$ (43)

Here $(\Psi,\,\lambda\,|\,{\widetilde{\Theta}}\,|\,\Psi,\,\lambda)\not=0$ in
general, because the vector ${\bf n}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}})$ does not vanish in general. It works out to be

$\displaystyle{\bf n}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}})=\big{\\{}\left({\bf a}\times{\bf
b^{\prime}}\right)\times\left({\bf a}\times{\bf b}\right)$
$\displaystyle+\left({\bf a^{\prime}}\times{\bf b}\right)\times\left({\bf
a}\times{\bf b}\right)+\left({\bf a^{\prime}}\times{\bf
b}\right)\times\left({\bf a}\times{\bf b^{\prime}}\right)$
$\displaystyle-\left({\bf a^{\prime}}\times{\bf
b^{\prime}}\right)\times\left({\bf a}\times{\bf b}\right)-\left({\bf
a^{\prime}}\times{\bf b^{\prime}}\right)\times\left({\bf a^{\prime}}\times{\bf
b}\right)-\left({\bf a^{\prime}}\times{\bf b^{\prime}}\right)\times\left({\bf
a}\times{\bf b^{\prime}}\right)\big{\\}}.$ (44)

The details of how this separation is accomplished using (40) can be found in
Appendix C below. From (42), it is now easy to appreciate that the additivity
of expectation values (22) assumed by Bell can hold only if the expectation
value $(\Psi,\lambda\,|\,{\widetilde{\Theta}}\,|\,\Psi,\lambda)=\pm 2\,||{\bf
n}||$ of the non-commuting part within the eigenvalue
${\widetilde{\omega}}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}},\lambda)$ of the summed operator (39) is zero. But that is
possible only if the operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. constituting the sum
(39) commute with eachother. In general, if the operators
${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. in (39) do not
commute with each other, then we would have

${\widetilde{\omega}}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}},\lambda)\not={\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b},\,\lambda)+{\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\,\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\,\lambda).$ (45)

But the operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. indeed do not
commute with each other, because the pairs of directions $\\{{\bf a},\,{\bf
a^{\prime}}\\}$, etc. in (39) are mutually exclusive directions in
${\mathrm{I\\!R}^{3}}$. Therefore, the additivity of expectation values
assumed at step (22) in the derivation of (25) is unjustifiable. Far from
being necessitated by realism, it actually contradicts realism.

Since three of the four results appearing in the expression (38) can be
realized only counterfactually, their summation in (38) cannot be realized
even counterfactually Oversight . Thus, in addition to not being a correct
eigenvalue of the summed operator (39) as required by the prescription (9) for
hidden variable theories, the quantity appearing in (38) is, in fact, an
entirely fictitious quantity, with no counterpart in any possible world, apart
from in the trivial case when all observables are commutative. By contrast,
the correct eigenvalue (42) of the summed operator (39) can be realized at
least counterfactually because it is a genuine eigenvalue of that operator,
thereby satisfying the requirement of realism correctly, in accordance with
the prescription (9) for hidden variable theories. Using (42), all five of the
observables appearing on both sides of the quantum mechanical equation (32)
can be assigned unique and correct eigenvalues Oversight .

Once this oversight is ameliorated, it is not difficult to show that the
conclusion of Bell’s theorem no longer follows. For then, using the correct
eigenvalue (42) of (39) instead of (38) on the right-hand side of (22), we
have the equation

${\cal E}({\bf a},\,{\bf b})+{\cal E}({\bf a},\,{\bf b^{\prime}})+{\cal
E}({\bf a^{\prime}},\,{\bf b})-{\cal E}({\bf a^{\prime}},\,{\bf
b^{\prime}})=\\!\int_{\mathscr{L}}{\widetilde{\omega}}({\bf a},{\bf
a^{\prime}},{\bf b},{\bf b^{\prime}},\lambda)\;p(\lambda)\,d\lambda\,$ (46)

instead of (22), which implements local realism correctly on both of its
sides, as required by the prescription (9) we discussed in Section II. This
equation (46) is thus the correct dispersion-free counterpart of the
equivalence (32) for the quantum mechanical expectation values Oversight . It
can reduce to Bell’s assumption (22) only when the expectation value
$(\Psi,\lambda\,|\,{\widetilde{\Theta}}\,|\,\Psi,\lambda)$ of the non-
commuting part within the eigenvalue ${\widetilde{\omega}}({\bf a},{\bf
a^{\prime}},{\bf b},{\bf b^{\prime}},\lambda)$ of the summed operator (39)
happens to be vanishing. It thus expresses the correct relationship (30) among
the expectation values for the singlet state (13) in the local hidden variable
framework considered by Bell Bell-1964 . Recall again from the end of Section
II that the quantum mechanical relation (32) is an unusual property of the
quantum states $|\psi\rangle$. As Bell stressed in Bell-1966 , “[t]here is no
reason to demand it individually of the hypothetical dispersion free states,
whose function it is to reproduce the measurable peculiarities of quantum
mechanics when averaged over.” Moreover, in Section V of Oversight  I have
demonstrated that the bounds on the right-hand side of (46) are $\pm
2\sqrt{2}$ instead of $\pm 2$. An alternative derivation of these bounds
follows from the magnitude $||{\bf n}||$ of the vector defined in (44), which,
as proved in Appendix D below, is bounded by $2$, and thereforethe eigenvalue
$\pm 2\,||{\bf n}||$ of the operator (43) obtained as its expectation value
$(\Psi,\lambda\,|\,{\widetilde{\Theta}}\,|\,\Psi,\lambda)$ is bounded by $\pm
4$, giving

$-4\,\leqslant(\Psi,\lambda\,|\,{\widetilde{\Theta}}({\bf a},{\bf
a^{\prime}},{\bf b},{\bf b^{\prime}})\,|\,\Psi,\lambda)\leqslant+4\,.$ (47)

Substituting these into (42), together with the bounds of $\pm 2$ we worked
out before on the commuting part (38), gives

$-2\sqrt{2}\,\leqslant\,{\widetilde{\omega}}({\bf a},{\bf a^{\prime}},{\bf
b},{\bf b^{\prime}},\lambda)\leqslant+2\sqrt{2}\,,$ (48)

which is constrained to be real despite the square root in the expression (42)
because the operator (39) is Hermitian. Consequently, we obtain the following
Tsirel’son’s bounds in the dispersion-free state, on the right-hand side of
(46):

$-2\sqrt{2}\,\leqslant\int_{\mathscr{L}}{\widetilde{\omega}}({\bf a},{\bf
a^{\prime}},{\bf b},{\bf
b^{\prime}},\lambda)\;p(\lambda)\,d\lambda\,\leqslant+2\sqrt{2}\,.$ (49)

Given the correct relation (46) between expectation values instead of the
flawed assumption (22), we thus arrive at

$-2\sqrt{2}\,\leqslant\,{\cal E}({\bf a},\,{\bf b})+{\cal E}({\bf a},\,{\bf
b^{\prime}})+{\cal E}({\bf a^{\prime}},\,{\bf b})-{\cal E}({\bf
a^{\prime}},\,{\bf b^{\prime}})\leqslant+2\sqrt{2}\,.$ (50)

Since the bounds of $\pm 2\sqrt{2}$ we have derived on the Bell-CHSH sum of
expectation values are the same as those predicted by quantum mechanics and
observed in the Bell-test experiments, the conclusion of Bell’s theorem is
mitigated. What is ruled out by these experiments is not local realism but the
assumption of the additivity of expectation values, which does not hold for
non-commuting observables in dispersion-free states of any hidden variable
theories to begin with.

It is also instructive to note that the intermediate bounds $\pm 2\sqrt{2}$ on
the Bell-CHSH sum (20), instead of the extreme bounds $\pm 2$ or $\pm 4$,
follow in the above derivation of (50) as a consequence of the geometry of
physical space IEEE-1 ; Christian . Thus, what is brought out in it is the
oversight of the non-commutative or Clifford-algebraic attributes of the
physical space in Bell’s derivation of the bounds $\pm 2$ in (25). Indeed, it
is evident from Appendix D below that the geometry of physical space imposes
the bounds $0\leqslant||{\bf n}||\leqslant 2$ on the magnitude of the vector
(44), which, in turn, lead us to the bounds $\pm 2\sqrt{2}$ in (50). This is
in sharp contrast with the traditional view of these bounds as due to non-
local influences,stemming from a failure of the locality condition (17). But
in the derivation of (50) above, the condition (17) is strictly respected.
Therefore, the strength of the bounds $\pm 2\sqrt{2}$ in (50) is a consequence
— not of non-locality or non-reality, but of the geometry of physical space
IEEE-1 ; Christian . Non-locality or non-reality is necessitated only if one
erroneously insists on linear additivity (22) of eigenvalues of non-commuting
observables for each individual dispersion-free state $|\,\Psi,\,\lambda)$.

##  VIII Conclusion: Bell’s theorem assumes its conclusion (petitio principii)

Let me reiterate the main points discussed above. Together, they demonstrate
that Bell’s theorem begs the question.

(1) The first point is that the derivation in Section IV of the bounds of $\pm
2$ on (20) for the dispersion-free counterpart $|\,\Psi,\,\lambda)$ of the
singlet state (13) must comply with the heuristics of the contextual hidden
variable theories discussed in Section II. Otherwise, the stringent bounds of
$\pm 2$ cannot be claimed to have any relevance for hidden variable theories.
This requires compliance with the prescription (9) that equates the quantum
mechanical expectation values with theirhidden variable counterparts for all
observables, including any sums of observables, pertaining to the singlet
system.

(2) The most charitable view of the equality (22) is that it is an assumption,
over and above those of locality, realism, and all other auxiliary assumptions
required for deriving the inequalities (25), because it is valid only for
commuting observables. Far from being required by realism, it contradicts
realism, because it fails to assign the correct eigenvalue (42) to the summed
observable (39) as its realistic counterpart, as required by the prescription
(9). Realism requiresthat all observables, including their sums, must be
assigned unique eigenvalues, regardless of whether they are observed.

(3) Expectation values in dispersion-free states of hidden variable theories
do not add linearly for observables that are not simultaneously measurable.
And yet, Bell assumed linear additivity (22) within a local hidden variable
model. Conversely, in the light of the heuristics of contextual hidden
variable theories we discussed in Section II, assuming (22)is equivalent to
assuming that the spin observables ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc. commute with each
other, but they do not.

(4) When the correct eigenvalue (42) is assigned to the summed operator (39)
replacing the incorrect step (22), the bounds on Bell-CHSH sum (20) work out
to be $\pm 2\sqrt{2}$ instead of $\pm 2$, thus mitigating the conclusion of
Bell’s theorem.

(5) As we proved in Section IV, the assumption (22) of the additivity of
expectation values is equivalent to assuming the strong bounds of $\pm 2$ on
Bell-CHSH sum (20) of expectation values. In other words, (22) and (25) are
tautologous.

(6) For observables that are not simultaneously measurable or commuting, the
built-in linear additivity of integrals in step (22) or (31) leads to
incorrect equality between averages of unequal physical quantities. Therefore,
the view that this step is a harmless mathematical step is mistaken. It is, in
fact, an unjustified assumption that is equivalent to the very thesis of the
theorem to be proven, and is valid only in classical physics and/or for
commuting observables.

The first four points above invalidate assumption (22), and thus inequalities
(25) on physical grounds, and the lasttwo demonstrate that Bell’s theorem
assumes its conclusion in a different guise, and is thus invalid on logical
grounds.

Bell’s theorem is useful, however, for ruling out classical local theories. By
relying on the assumption (22), which is valid for classical theories Grete ,
it proves that no classical local theory can reproduce all of the predictions
of quantummechanics. But no serious hidden variable theories I am aware of
have ever advocated returning to classical physics Gudder .

In this paper, I have focused on a formal or logical critique of Bell’s
theorem. Elsewhere RSOS ; Local ; RSOS-Reply , I have developed a
comprehensive local-realistic framework for understanding quantum correlations
in terms of the geometry of the spatial part of one of the well-known
solutions of Einstein’s field equations of general relativity — namely, that
of a quaternionic 3-sphere — taken as a physical space within which we are
confined to perform Bell-test experiments. Thisframework is based on Clifford
algebra and thus explicitly takes the non-commutativity of observables into
account. It thus shows, constructively, that contextually local hidden
variable theories are not ruled out by Bell-test experiments. Since, as we
discussed in Section III, the formal proof of Bell’s theorem is based on the
entangled singlet state (13), in Christian ; IJTP ; IEEE-1 ; IEEE-2 ; IEEE-3 ;
IEEE-4 ; Symmetric  I have reproduced the correlations predicted by (13) as a
special case within the local-realistic framework proposed in RSOS ; Local ;
RSOS-Reply . I especially recommend the calculations presented in IJTP  and
Symmetric , which also discuss a macroscopic experiment that would be able to
falsify the 3-sphere hypothesis I have proposed in these publications.

## Appendix

##  Appendix A Dynamical equivalence of quantum mechanical description and
Einstein’s description

To establish the equivalence between two dynamics exhibited in Eq. (8), let us
begin with the Ehrenfest equation,

$\frac{d\,}{dt}\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle\,=\,\frac{1}{i\hbar}\left\langle\,\psi\,|\,\left[\,\Omega,\,H\,\right]\,|\,\psi\,\right\rangle\,+\,\langle\,\psi\,|\,\frac{\partial\Omega}{\partial
t}\,|\,\psi\,\rangle,$ (51)

where $H$ is a Hamiltonian operator. Using prescription (6) for the complete
description, this equation can be written as

$\frac{d\,}{dt}\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,=\,\frac{1}{i\hbar}\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\left[\,\Omega,\,H\,\right]\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,+\int_{\mathscr{L}}(\,\psi,\,\lambda\,|\,\frac{\partial\Omega}{\partial
t}\,|\,\psi,\,\lambda\,)\;p(\lambda)\,d\lambda\,.$ (52)

Since in the ansatz (3) we have implicitly assumed that the hidden variables
$\lambda$ do not depend on time explicitly, the Leibniz integral rule for
differentiation allows us to reduce the first term on the left-hand side of
the above equation to

$\frac{d\,}{dt}\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,=\frac{d\,}{dt}\int_{\mathscr{L}}\omega(\lambda)\;p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}\left[\frac{d\,}{dt}\,\omega(\lambda)\right]p(\lambda)\,d\lambda\,.$
(53)

Next, we recall the Dirac rule for canonical quantization, which promotes the
Poisson brackets to operators and sets

$\left[\,\Omega,\,H\,\right]=i\hbar\,\widehat{\\{\omega,\,\mathscr{H}\\}},$
(54)

where $\omega$ is an eigenvalue of the observable $\Omega$ and the Hamiltonian
function $\mathscr{H}$ is an eigenvalue of the corresponding operator $H$.
Then the Poisson bracket $\\{\omega,\,\mathscr{H}\\}$ is an eigenvalue of the
operator $\widehat{\\{\omega,\,\mathscr{H}\\}}$. Now it is well known that the
Dirac rule is not unique in general when promoting classical variables to
quantum operators. It is unique, however, for the reverse process of reducing
the quantum operators to classical variables. Given any two operators
$\Omega_{1}$ and $\Omega_{2}$ with eigenvalues $\omega_{1}$ and $\omega_{2}$,
their commutator $[\Omega_{1},\,\Omega_{2}]$ uniquely reduces to the Poisson
bracket $\\{\omega_{1},\,\omega_{2}\\}$ in correspondence limit. Consequently,
using (54) and the ansatz (3), we can transform the first term on the right-
hand side of (52) to

$\frac{1}{i\hbar}\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\left[\,\Omega,\,H\,\right]\,|\,\psi,\,\lambda\,\right)\,p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}(\,\psi,\,\lambda\,|\,\widehat{\\{\omega,\,\mathscr{H}\\}}\,|\,\psi,\,\lambda\,)\,p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}\\{\omega(\lambda),\,\mathscr{H}(\lambda)\\}\;p(\lambda)\,d\lambda\,,$
(55)

where a consistent application of ansatz (3) to all observables of the system
requires that
$\\{\omega,\,\mathscr{H}\\}(\lambda)=\\{\omega(\lambda),\,\mathscr{H}(\lambda)\\}$.
Finally, using the ansatz (3) for evaluating the second term on the right-hand
side of (52), it is easy to prove that

$\int_{\mathscr{L}}(\,\psi,\,\lambda\,|\,\frac{\partial\Omega}{\partial
t}\,|\,\psi,\,\lambda\,)\,p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}\frac{\partial\omega(\lambda)}{\partial
t}\;p(\lambda)\,d\lambda\,.$ (56)

Putting all of the results we have obtained in (52), (53),(55), and (56), we
can finally conclude using (51) that

$\left[\frac{d\,}{dt}\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle=\frac{1}{i\hbar}\left\langle\,\psi\,|\,\left[\,\Omega,\,H\,\right]\,|\,\psi\,\right\rangle+\langle\,\psi\,|\,\frac{\partial\Omega}{\partial
t}\,|\,\psi\,\rangle\right]=\int_{\mathscr{L}}\left[\frac{d\;}{dt}\,\omega(\lambda)=\\{\omega(\lambda),\,\mathscr{H}(\lambda)\\}+\frac{\partial\,\omega(\lambda)}{\partial
t}\right]p(\lambda)\,d\lambda\,.$ (57)

For specific values of $\lambda$ with $p(\lambda)=1$, the right-hand side of
(57) reduces to classical equations of motion. If, however, only a probability
distribution $p(\lambda)$ of $\lambda$ is known, then Ehrenfest’s equation in
quantum mechanics can be understood as an average of an ensemble of classical
dynamics, with each of its members specifying a trajectory in phase space. All
probabilistic statements about the system would then reduce to the
incompleteness of our knowledge about the system.

##  Appendix B Demonstration of linearity of the expectation function
$\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)$ assumed in
(22)

For the demonstration of linearity of expectation function
$\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)$ assumed in
(22), let us follow similar demonstration by Bohm and Bub in BohmBub  within
the context of von Neumann’s theorem. To that end, recall that in the Hilbert
space model of quantum mechanics vonNeumann , every ensemble can be
characterized by a statistical operator, say ${\cal W(\psi)}$, of unit trace,
representing the quantum state $|\,\psi\,\rangle$ of the system. In other
words, there exists a linear Hermitian matrix ${\cal W}_{pq}(\psi)$ such that
the expectation value in the state $|\,\psi\,\rangle$ of observable $\Omega$
can be expressed in the linear form

$\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle=\text{Tr}\left\\{{\cal
W}(\psi)\,\Omega\right\\}=\sum_{pq}\,{\cal W}_{qp}(\psi)\,\Omega_{pq}\,.$ (58)

However, if dispersion in the measured values of $\Omega$ is assumed to be due
to a distribution $p(\lambda)$ in the values of hidden variables $\lambda$
over the ensemble of the system as in (6), then the expectation value of
$\Omega$ should be the average over $\lambda$,

$\left\langle\,\psi\,|\,\Omega\,|\,\psi\,\right\rangle\,=\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)\;p(\lambda)\,d\lambda=\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,\Omega_{pq}\;p(\lambda)\,d\lambda\,,$ (59)

which amounts to replacing the matrix ${\cal W}_{qp}(\psi)$ with its average
$\overline{\cal W}_{qp}(\psi)$ over the distribution of hidden variables
$\lambda$:

${\cal W}_{qp}(\psi)\longrightarrow\overline{\cal
W}_{qp}(\psi)\,=\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\;p(\lambda)\,d\lambda\,.$ (60)

For each particular value of $\lambda$, (59) thus assumes expectation function
to be of a linear form, similar to that in (58):

$\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)=\text{Tr}\left\\{{\cal
W}(\psi,\lambda)\,\Omega\right\\}=\sum_{pq}\,{\cal
W}_{qp}(\psi,\lambda)\,\Omega_{pq}\,.$ (61)

By now it is well-recognized that demanding this linear form of expectation
function $\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)$ is
not justifiable for hidden variable theories Bell-1966 ; Grete ; Mermin ;
BohmBub . Ironically, however, after criticizing von Neumann for retaining the
linear form (61), Bell also ended up making the same mistake for contextual
hidden variable theories, by implicitly assuming

$\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)\;p(\lambda)\,d\lambda\,=\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,\Omega_{pq}(c)\;p(\lambda)\,d\lambda$ (62)

in the equality (22). It is not difficult to see that (22) follows from the
linearity (61) of the function
$\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)$:

$\displaystyle\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega_{1}(c_{1})+\Omega_{2}(c_{2})\,|\,\psi,\,\lambda\,\right)\;p(\lambda)\,d\lambda\,$
$\displaystyle=\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,\left\\{\,\Omega_{1}(c_{1})+\Omega_{2}(c_{2})\right\\}_{pq}\;p(\lambda)\,d\lambda$
(63)
$\displaystyle\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\,$
$\displaystyle=\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,\left\\{(\Omega_{1})_{pq}(c_{1})+(\Omega_{2})_{pq}(c_{2})\right\\}\;p(\lambda)\,d\lambda$
(64)
$\displaystyle\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\,$
$\displaystyle=\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,(\Omega_{1})_{pq}(c_{1})\;p(\lambda)\,d\lambda$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\sum_{pq}\int_{\mathscr{L}}{\cal
W}_{qp}(\psi,\lambda)\,(\Omega_{2})_{pq}(c_{2})\;p(\lambda)\,d\lambda$ (65)
$\displaystyle\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\,$
$\displaystyle=\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega_{1}(c_{1})\,|\,\psi,\,\lambda\,\right)\;p(\lambda)\,d\lambda\;+\int_{\mathscr{L}}\left(\,\psi,\,\lambda\,|\,\Omega_{2}(c_{2})\,|\,\psi,\,\lambda\,\right)\;p(\lambda)\,d\lambda$
(66)
$\displaystyle\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\,$
$\displaystyle=\int_{\mathscr{L}}\omega_{1}(c_{1},\lambda)\;p(\lambda)\,d\lambda\;+\int_{\mathscr{L}}\omega_{2}(c_{2},\lambda)\;p(\lambda)\,d\lambda\,,$
(67)

where only two terms of (22) is used for simplicity, and the ansatz (3) is
used to give
$\left(\,\psi,\,\lambda\,|\,\Omega_{1}(c_{1})\,|\,\psi,\,\lambda\,\right)=\omega_{1}(c_{1},\lambda)$,
$\left(\,\psi,\,\lambda\,|\,\Omega_{2}(c_{2})\,|\,\psi,\,\lambda\,\right)=\omega_{2}(c_{2},\lambda)$,
and
$\left(\,\psi,\,\lambda\,|\,\Omega_{1}(c_{1})+\Omega_{2}(c_{2})\,|\,\psi,\,\lambda\,\right)=\widetilde{\omega}(\tilde{c},\lambda)$.
Then linearity of anti-derivatives gives

$\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\,=\int_{\mathscr{L}}\left\\{\,\omega_{1}(c_{1},\lambda)+\omega_{2}(c_{2},\lambda)\right\\}\;p(\lambda)\,d\lambda\,,$
(68)

implying that the eigenvalue $\widetilde{\omega}(\tilde{c},\lambda)$ of the
sum $\Omega_{1}(c_{1})+\Omega_{2}(c_{2})$ of operators is equal to the sum of
their eigenvalues:

$\widetilde{\omega}(\tilde{c},\lambda)=\omega_{1}(c_{1},\lambda)+\,\omega_{2}(c_{2},\lambda)\,,$
(69)

for each particular value of the hidden variables $\lambda$. But, as we noted
in the last paragraph of Section II, this equality among the eigenvalues is
mistaken in general (recall the example $\sqrt{2}\not=1+1$). It is valid only
in classical physics, and for commuting observables. We have arrived at it by
assuming the linear form of the expectation function
$\left(\,\psi,\,\lambda\,|\,\Omega\,|\,\psi,\,\lambda\,\right)$defined in
(61). Recognizing this difficulty, Bohm and Bub considered a more general,
non-linear expectation function:

$\left(\,\psi,\,\lambda\,|\,\Omega(c)\,|\,\psi,\,\lambda\,\right)=F(\psi,\lambda,\Omega_{pq}(c))\not=\sum_{pq}{\cal
W}_{qp}(\psi,\lambda)\,\Omega_{pq}(c)\,,$ (70)

where (at least for non-commuting observables)
$F(\psi,\lambda,\Omega_{pq}(c))$ is a non-linear function of $\psi$,
$\lambda$, and $\Omega_{pq}(c)$ such that

$\displaystyle\widetilde{\omega}(\tilde{c},\lambda)\not=\omega_{1}(c_{1},\lambda)+\omega_{2}(c_{2},\lambda)\,.$
(71)

Indeed, there is no reason why these eigenvalues should not be determined by
some non-linear $F(\psi,\lambda,\Omega_{pq}(c))$, giving

$\int_{\mathscr{L}}\omega_{1}(c_{1},\lambda)\;p(\lambda)\,d\lambda\;+\int_{\mathscr{L}}\omega_{2}(c_{2},\lambda)\;p(\lambda)\,d\lambda\;=\int_{\mathscr{L}}\widetilde{\omega}(\tilde{c},\lambda)\;p(\lambda)\,d\lambda\;\not=\int_{\mathscr{L}}\left\\{\,\omega_{1}(c_{1},\lambda)+\omega_{2}(c_{2},\lambda)\right\\}\;p(\lambda)\,d\lambda\,.$
(72)

As Bell stressed in Bell-1966 , there is no reason for demanding (69) of the
dispersion-free state $|\,\psi,\,\lambda\,)$ for each value of ${\lambda}$.

##  Appendix C Separating the commuting and non-commuting parts of the summed
operator (39)

Before considering the specific operator (39), in this appendix let us prove
that, in general, the eigenvalue of a sum
${r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\;{\mathcal{U}}}$ of
operators is not equal to the sum
$r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}$ of the
individual eigenvalues of the operators ${\mathcal{R}}$, ${\mathcal{S}}$,
${\mathcal{T}}$, and ${\mathcal{U}}$, unless these operators commute with each
other. Here $r$, $s$, $t$, and $u$ are real numbers. It is not difficult to
prove this known fact by evaluating the square of the operator
${\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}}$
as follows:

$\displaystyle\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}$
$\displaystyle=r^{2}{\mathcal{R}}^{2}+rs\,{\mathcal{R}}{\mathcal{S}}+rt\,{\mathcal{R}}{\mathcal{T}}+ru\,{\mathcal{R}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+sr\,{\mathcal{S}}{\mathcal{R}}+s^{2}{\mathcal{S}}^{2}+st\,{\mathcal{S}}{\mathcal{T}}+su\,{\mathcal{S}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+tr\,{\mathcal{T}}{\mathcal{R}}+ts\,{\mathcal{T}}{\mathcal{S}}+t^{2}{\mathcal{T}}^{2}+tu\,{\mathcal{T}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+ur\,{\mathcal{U}}{\mathcal{R}}+us\,{\mathcal{U}}{\mathcal{S}}+ut\,{\mathcal{U}}{\mathcal{T}}+u^{2}{\mathcal{U}}^{2}.$
(73)

Now, assuming that the operators ${\mathcal{R}}$, ${\mathcal{S}}$,
${\mathcal{T}}$, and ${\mathcal{U}}$ do not commute in general, let us define
the following operators:

$\displaystyle{\mathcal{L}}$
$\displaystyle:={\mathcal{S}}{\mathcal{R}}-{\mathcal{R}}{\mathcal{S}}\;\Longleftrightarrow\;{\mathcal{S}}{\mathcal{R}}={\mathcal{R}}{\mathcal{S}}+{\mathcal{L}},$
(74) $\displaystyle{\mathcal{M}}$
$\displaystyle:={\mathcal{T}}{\mathcal{R}}-{\mathcal{R}}{\mathcal{T}}\;\Longleftrightarrow\;{\mathcal{T}}{\mathcal{R}}={\mathcal{R}}{\mathcal{T}}+{\mathcal{M}},$
(75) $\displaystyle{\mathcal{N}}$
$\displaystyle:={\mathcal{T}}{\mathcal{S}}-{\mathcal{S}}{\mathcal{T}}\;\Longleftrightarrow\;{\mathcal{T}}{\mathcal{S}}={\mathcal{S}}{\mathcal{T}}+{\mathcal{N}},$
(76) $\displaystyle{\mathcal{O}}$
$\displaystyle:={\mathcal{U}}{\mathcal{R}}-{\mathcal{R}}{\mathcal{U}}\;\Longleftrightarrow\;{\mathcal{U}}{\mathcal{R}}={\mathcal{R}}{\mathcal{U}}+{\mathcal{O}},$
(77) $\displaystyle{\mathcal{P}}$
$\displaystyle:={\mathcal{U}}{\mathcal{T}}-{\mathcal{T}}{\mathcal{U}}\;\Longleftrightarrow\;{\mathcal{U}}{\mathcal{T}}={\mathcal{T}}{\mathcal{U}}+{\mathcal{P}},$
(78) $\displaystyle\text{and}\;\;\;{\mathcal{Q}}$
$\displaystyle:={\mathcal{U}}{\mathcal{S}}-{\mathcal{S}}{\mathcal{U}}\;\Longleftrightarrow\;{\mathcal{U}}{\mathcal{S}}={\mathcal{S}}{\mathcal{U}}+{\mathcal{Q}}.$
(79)

These operators would be null operators with vanishing eigenvalues if the
operators ${\mathcal{R}}$, ${\mathcal{S}}$, ${\mathcal{T}}$, and
${\mathcal{U}}$ did commute with each other. Using these relations for the
operators ${\mathcal{S}}{\mathcal{R}}$, ${\mathcal{T}}{\mathcal{R}}$,
${\mathcal{T}}{\mathcal{S}}$, ${\mathcal{U}}{\mathcal{R}}$,
${\mathcal{U}}{\mathcal{T}}$ and ${\mathcal{U}}{\mathcal{S}}$, equation (73)
can be simplified to

$\displaystyle\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}$
$\displaystyle=r^{2}{\mathcal{R}}^{2}+2rs\,{\mathcal{R}}{\mathcal{S}}+2rt\,{\mathcal{R}}{\mathcal{T}}+2ru\,{\mathcal{R}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+rs\,{\mathcal{L}}+s^{2}{\mathcal{S}}^{2}+2st\,{\mathcal{S}}{\mathcal{T}}+2su\,{\mathcal{S}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+rt\,{\mathcal{M}}+st\,{\mathcal{N}}+t^{2}{\mathcal{T}}^{2}+2tu\,{\mathcal{T}}{\mathcal{U}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+ru\,{\mathcal{O}}+su\,{\mathcal{Q}}+tu\,{\mathcal{P}}+u^{2}{\mathcal{U}}^{2}$
(80)
$\displaystyle=\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}_{\bf
c}^{2}\,+\,{\mathcal{Y}},$ (81)

where

${\mathcal{Y}}:=rs\,{\mathcal{L}}+rt\,{\mathcal{M}}+st\,{\mathcal{N}}+ru\,{\mathcal{O}}+tu\,{\mathcal{P}}+su\,{\mathcal{Q}}\,.$
(82)

We have thus separated out the commuting part
${\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}_{\bf
c}}$ and the non-commuting part ${\mathcal{Y}}$ of the summed operator
${{\mathcal{X}}:=\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}}$.
Note that the operators ${\mathcal{L}}$, ${\mathcal{M}}$, ${\mathcal{N}}$,
${\mathcal{O}}$, ${\mathcal{P}}$, and ${\mathcal{Q}}$ defined in (74) to (79)
will not commute with each other in general unless their constituents
${\mathcal{R}}$, ${\mathcal{S}}$, ${\mathcal{T}}$, and ${\mathcal{U}}$
themselves are commuting. Next, we work out the eigenvalue ${\mathscr{X}}$ of
the operator ${\mathcal{X}}$ in a normalized eigenstate $|\,\xi\,\rangle$
using the eigenvalue equations

${\mathcal{X}}\,|\,\xi\,\rangle={\mathscr{X}}\,|\,\xi\,\rangle$ (83)

and

${\mathcal{X}}\,{\mathcal{X}}\,|\,\xi\,\rangle={\mathcal{X}}\big{\\{}{\mathcal{X}}\,|\,\xi\,\rangle\big{\\}}={\mathcal{X}}\,\big{\\{}{\mathscr{X}}\,|\,\xi\,\rangle\big{\\}}={\mathscr{X}}\,\big{\\{}{\mathcal{X}}\,|\,\xi\,\rangle\big{\\}}={\mathscr{X}}^{2}\,|\,\xi\,\rangle,$
(84)

in terms of the eigenvalues ${\mathscr{R}}$, ${\mathscr{S}}$, ${\mathscr{T}}$,
and ${\mathscr{U}}$ of the operators ${\mathcal{R}}$, ${\mathcal{S}}$,
${\mathcal{T}}$, and ${\mathcal{U}}$ and the expectation value
$\langle\,\xi\,|\,{\mathcal{Y}}\,|\,\xi\,\rangle$:

${\mathscr{X}}=\pm\sqrt{\langle\,\xi\,|\,{\mathcal{X}}\,{\mathcal{X}}\,|\,\xi\,\rangle}=\pm\sqrt{\langle\,\xi\,|\big{\\{}r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\big{\\}}_{\bf
c}^{2}\,|\,\xi\,\rangle+\langle\,\xi\,|\,{\mathcal{Y}}\,|\,\xi\,\rangle\,},$
(85)

where we have used (81). But the eigenvalue of the commuting part
${\\{r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\\}_{\bf
c}}$ of ${\mathcal{X}}$ is simply the linear sum
${r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}}$ of the
eigenvalues of the operators ${\mathcal{R}}$, ${\mathcal{S}}$,
${\mathcal{T}}$, and ${\mathcal{U}}$. Consequently, using the equation
analogous to (84) for the square of the operator
${\big{\\{}r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}\big{\\}}_{\bf
c}}$ we can express the eigenvalue ${\mathscr{X}}$ of ${\mathcal{X}}$ as

${\mathscr{X}}=\pm\sqrt{\big{\\{}r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}\big{\\}}^{2}+\,\langle\,\xi\,|\,{\mathcal{Y}}\,|\,\xi\,\rangle\,}.$
(86)

Now, because the operators ${\mathcal{L}}$, ${\mathcal{M}}$, ${\mathcal{N}}$,
${\mathcal{O}}$, ${\mathcal{P}}$, and ${\mathcal{Q}}$ defined in (74) to (79)
will not commute with each other in general if their constituent operators
${\mathcal{R}}$, ${\mathcal{S}}$, ${\mathcal{T}}$, and ${\mathcal{U}}$ are
non-commuting, the state ${|\,\xi\,\rangle}$ will not be an eigenstate of the
operator ${\mathcal{Y}}$ defined in (82). Moreover, while a dispersion-free
state $|\,\psi,\,\lambda)$ would pick out one of the eigenvalues
${\mathscr{Y}}$ of ${\mathcal{Y}}$, it will not be equal to the linear sum of
the corresponding eigenvalues ${\mathscr{L}}$, ${\mathscr{M}}$,
${\mathscr{N}}$, ${\mathscr{O}}$, ${\mathscr{P}}$, and ${\mathscr{Q}}$ in
general,

${\mathscr{Y}}\not=rs\,{\mathscr{L}}+rt\,{\mathscr{M}}+st\,{\mathscr{N}}+ru\,{\mathscr{O}}+tu\,{\mathscr{P}}+su\,{\mathscr{Q}}\,,$
(87)

even if we assume that the operators ${\mathcal{X}}$ and ${\mathcal{Y}}$
commute with each other so that
$(\,\psi,\,\lambda\,|\,{\mathcal{Y}}\,|\,\psi,\,\lambda\,)={\mathscr{Y}}$ is
an eigenvalue of ${\mathcal{Y}}$. That is to say, just like the eigenvalue
${\mathscr{X}}$ of ${\mathcal{X}}$, the eigenvalue ${\mathscr{Y}}$ of
${\mathcal{Y}}$ is also a nonlinear function in general. On the other hand,
because we wish to prove that the eigenvalue of the sum
${r\,{\mathcal{R}}+s\,{\mathcal{S}}+t\,{\mathcal{T}}+u\,{\mathcal{U}}}$ of the
operators ${\mathcal{R}}$, ${\mathcal{S}}$, ${\mathcal{T}}$, and
${\mathcal{U}}$ is not equal to the sum
$r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}$ of the
individual eigenvalues of the operators ${\mathcal{R}}$, ${\mathcal{S}}$,
${\mathcal{T}}$, and ${\mathcal{U}}$ unless they commute with each other, we
must make sure that the eigenvalue ${\mathscr{Y}}$ does not vanish for the
unlikely case inwhich the operators ${\mathcal{L}}$, ${\mathcal{M}}$,
${\mathcal{N}}$, ${\mathcal{O}}$, ${\mathcal{P}}$, and ${\mathcal{Q}}$ commute
with each other. But even in that unlikely case, we would have

$\overline{\mathscr{Y}}=rs\,{\mathscr{L}}+rt\,{\mathscr{M}}+st\,{\mathscr{N}}+ru\,{\mathscr{O}}+tu\,{\mathscr{P}}+su\,{\mathscr{Q}}\,$
(88)

as eigenvalue of the operator ${\mathcal{Y}}$ defined in (82), and
consequently the eigenvalue ${\mathscr{X}}$ in (86) will at best reduce to

${\mathscr{X}}=\pm\sqrt{\big{\\{}r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}\big{\\}}^{2}+rs\,{\mathscr{L}}+rt\,{\mathscr{M}}+st\,{\mathscr{N}}+ru\,{\mathscr{O}}+tu\,{\mathscr{P}}+su\,{\mathscr{Q}}\,}.$
(89)

In other words, even in such an unlikely case ${\mathscr{Y}}$ will not vanish,
and consequently the eigenvalue ${\mathscr{X}}$ will not reduce to

$\overline{\mathscr{X}}=r\,{\mathscr{R}}+s\,{\mathscr{S}}+t\,{\mathscr{T}}+u\,{\mathscr{U}}.$
(90)

Consequently, unless
$\left(\,\psi,\,\lambda\,|\,{\mathcal{Y}}(c)\,|\,\psi,\,\lambda\,\right)\equiv
0$, the expectation value of ${\cal X}(c)$ equating the average of
${\mathscr{X}}(c,\,\lambda)$ will be

$\displaystyle\langle\,\psi\,|\,{\mathcal{X}}(c)\,|\,\psi\,\rangle$
$\displaystyle=\\!\int_{\mathscr{L}}{\mathscr{X}}(c,\,{\lambda})\;p(\lambda)\,d\lambda$
(91)
$\displaystyle=\\!\int_{\mathscr{L}}\\!\left[\pm\sqrt{\big{\\{}r\,{\mathscr{R}}(c,\,{\lambda})+s\,{\mathscr{S}}(c,\,{\lambda})+t\,{\mathscr{T}}(c,\,{\lambda})+u\,{\mathscr{U}}(c,\,{\lambda})\big{\\}}^{2}+\,\left(\,\psi,\,\lambda\,|\,{\mathcal{Y}}(c)\,|\,\psi,\,\lambda\,\right)\,}\,\right]p(\lambda)\,d\lambda$
(92)
$\displaystyle\not=\\!\int_{\mathscr{L}}\\!\left[\pm\sqrt{\big{\\{}r\,{\mathscr{R}}(c,\,{\lambda})+s\,{\mathscr{S}}(c,\,{\lambda})+t\,{\mathscr{T}}(c,\,{\lambda})+u\,{\mathscr{U}}(c,\,{\lambda})\big{\\}}^{2}+\,\overline{\mathscr{Y}}(c,\,\lambda)}\,\right]p(\lambda)\,d\lambda\;\;\text{if}\;[\,{\mathcal{X}},\,{\mathcal{Y}}\,]\not=0$
(93)
$\displaystyle\not=\\!\int_{\mathscr{L}}\pm\big{\\{}r\,{\mathscr{R}}(c,\,{\lambda})+s\,{\mathscr{S}}(c,\,{\lambda})+t\,{\mathscr{T}}(c,\,{\lambda})+u\,{\mathscr{U}}(c,\,{\lambda})\big{\\}}\;p(\lambda)\,d\lambda\;\;\text{if}\;{\mathscr{L}},\,{\mathscr{M}},\,{\mathscr{N}},\,{\mathscr{O}},\,{\mathscr{P}},\,{\mathscr{Q}}\not=0,$
(94)

where $c$ indicates the contexts of experiments as discussed in Section II.
The above result confirms the inequality (45) we discussed in Section VII.
Note that, because ${\mathscr{X}}(c,\,\lambda)$ and
${\mathscr{Y}}(c,\,\lambda)$ are highly nonlinear functions in general
(recall, e.g., that $\sqrt{x^{2}\pm
y^{2}\,}\not=\sqrt{x^{2}}\,\pm\sqrt{y^{2}}\,$), the inequality in (94) can
reduce to equality if and only if the operators ${\mathcal{R}}$,
${\mathcal{S}}$, ${\mathcal{T}}$, and ${\mathcal{U}}$ commute with each other.
In that case, the operators ${\mathcal{L}}$, ${\mathcal{M}}$, ${\mathcal{N}}$,
${\mathcal{O}}$, ${\mathcal{P}}$, and ${\mathcal{Q}}$ defined in (74) to (79)
will also commute with each other, as well as being null operators, with each
of the eigenvalues ${\mathscr{L}}$, ${\mathscr{M}}$, ${\mathscr{N}}$,
${\mathscr{O}}$, ${\mathscr{P}}$, and ${\mathscr{Q}}$reducing to zero.
Consequently, in that case
$\left(\,\psi,\,\lambda\,|\,{\mathcal{Y}}(c)\,|\,\psi,\,\lambda\,\right)$ will
vanish identically and (86) will reduce to (90).

It is now straightforward to deduce the operator ${\widetilde{\Theta}}({\bf
a},{\bf a^{\prime}},{\bf b},{\bf b^{\prime}})$ specified in (43) using (40).
For this purpose, we first note that for the Bell-CHSH sum (20) the real
numbers ${r=s=t=+1}$ and ${u=-1}$, and therefore (90) simplifies to

$\overline{\mathscr{X}}(c,\,\lambda)={\mathscr{R}}(c,\,\lambda)+{\mathscr{S}}(c,\,\lambda)+{\mathscr{T}}(c,\,\lambda)-{\mathscr{U}}(c,\,\lambda).$
(95)

This quantity is tacitly assumed in the derivation of Bell’s theorem to be the
eigenvalue of the summed operator (39), implying the following
identifications:

$\displaystyle{\mathscr{A}}({\bf a},\,{\lambda})\,{\mathscr{B}}({\bf
b},\,{\lambda})$ $\displaystyle\equiv{\mathscr{R}}({\bf a},\,{\bf
b},\,{\lambda})$ $\displaystyle=\pm 1\;\;\text{is an eigenvalue of the
observable}\;\;{\mathcal{R}}({\bf a},\,{\bf
b})\equiv{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,,$ (96)
$\displaystyle{\mathscr{A}}({\bf a},\,{\lambda})\,{\mathscr{B}}({\bf
b^{\prime}},\,{\lambda})$ $\displaystyle\equiv{\mathscr{S}}({\bf a},\,{\bf
b^{\prime}},\,{\lambda})$ $\displaystyle=\pm 1\;\;\text{is an eigenvalue of
the observable}\;\;{\mathcal{S}}({\bf a},\,{\bf
b^{\prime}})\equiv{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}}\,,$ (97)
$\displaystyle{\mathscr{A}}({\bf a^{\prime}},\,{\lambda})\,{\mathscr{B}}({\bf
b},\,{\lambda})$ $\displaystyle\equiv{\mathscr{T}}({\bf a^{\prime}},\,{\bf
b},\,{\lambda})$ $\displaystyle=\pm 1\;\;\text{is an eigenvalue of the
observable}\;\;{\mathcal{T}}({\bf a^{\prime}},\,{\bf
b})\equiv{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}\,,$ (98)
$\displaystyle\text{and}\;\;\;{\mathscr{A}}({\bf
a^{\prime}},\,{\lambda})\,{\mathscr{B}}({\bf b^{\prime}},\,{\lambda})$
$\displaystyle\equiv{\mathscr{U}}({\bf a^{\prime}},\,{\bf
b^{\prime}},\,{\lambda})$ $\displaystyle=\pm 1\;\;\text{is an eigenvalue of
the observable}\;\;{\mathcal{U}}({\bf a^{\prime}},\,{\bf
b^{\prime}})\equiv{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}}.$ (99)

The non-commuting part of the operator (39) can therefore be identified using
(82) and the above identifications as

${\widetilde{\Theta}}({\bf a},{\bf a^{\prime}},{\bf b},{\bf
b^{\prime}})=\big{\\{}{\mathcal{L}}+{\mathcal{M}}+{\mathcal{N}}-{\mathcal{O}}-{\mathcal{P}}-{\mathcal{Q}}\big{\\}}({\bf
a},{\bf a^{\prime}},{\bf b},{\bf b^{\prime}})\,,$ (100)

where the operators ${\mathcal{L}}$, ${\mathcal{M}}$, ${\mathcal{N}}$,
${\mathcal{O}}$, ${\mathcal{P}}$, and ${\mathcal{Q}}$ are defined in (74) to
(79). The result is the operator specified in (43).

##  Appendix D Establishing bounds on the magnitude of the vector
${\mathbf{n}}$ defined in (44)

The vector ${\bf n}$ defined in (44) is a function of four unit vectors, ${\bf
a}$, ${\bf a^{\prime}}$, ${\bf b}$, and ${\bf b^{\prime}}$, in
${\mathrm{I\\!R}^{3}}$, and involves various cross products among these
vectors. Consequently, as the vectors ${\bf a}$, ${\bf a^{\prime}}$, ${\bf
b}$, and ${\bf b^{\prime}}$ vary in their directions within
${\mathrm{I\\!R}^{3}}$ due to various choices made by Alice and Bob, the
extremum values of the magnitude $||{\bf n}||$ is obtained by setting the
vectorsorthogonal to each other, with angles between them set to 90 or 270
degrees. However, in three dimensions that is possible only for three of the
four vectors, so one of the four would have to be set either parallel or anti-
parallel to one of the remaining three. Therefore, let us first choose to set
${\bf b^{\prime}}=-{\bf b}$. Substituting this into (44) then gives ${\bf
n}={\bf 0}$, andthus $||{\bf n}||=0$. We have thus found the lower bound on
the magnitude $||{\bf n}||$. To determine the upper bound on $||{\bf n}||$, we
set ${\bf a^{\prime}}=-{\bf a}$ instead. Substituting this into (44) reduces
the vector ${\bf n}$ to the following function of ${\bf a}$, ${\bf
a^{\prime}}$, ${\bf b}$ and ${\bf b^{\prime}}$:

${\bf n}=2\,\big{\\{}\left({\bf a}\times{\bf
b^{\prime}}\right)\times\left({\bf a}\times{\bf b}\right)\big{\\}}.$ (101)

Consequently, in this case, the magnitude of the vector ${\bf n}$ works out to
be

$\displaystyle||{\bf n}||$ $\displaystyle=2\,||({\bf a}\times{\bf
b^{\prime}})||\,||({\bf a}\times{\bf b})||\,\sin\beta_{({\bf a}\times{\bf
b^{\prime}}),({\bf a}\times{\bf b})}$ (102) $\displaystyle=2\,\big{\\{}||{\bf
a}||\,||{\bf b^{\prime}}||\,\sin\beta_{{\bf a},{\bf
b^{\prime}}}\big{\\}}\,\big{\\{}||{\bf a}||\,||{\bf b}||\,\sin\beta_{{\bf
a},{\bf b}}\big{\\}}\,\big{\\{}\sin\beta_{({\bf a}\times{\bf
b^{\prime}}),({\bf a}\times{\bf b})}\big{\\}},$ (103)

where $\beta_{{\bf a},{\bf b}}$ is the angle between ${\bf a}$ and ${\bf b}$,
etc. But since the vectors ${\bf a}$, ${\bf a^{\prime}}$, ${\bf b}$, and ${\bf
b^{\prime}}$ are all unit vectors and we have set them orthogonal to each
other (apart from ${\bf a^{\prime}}=-{\bf a}$), we obtain $||{\bf n}||=2$ as
the maximum possible value for the magnitude of ${\bf n}$. We have thus
established the following bounds on the magnitude of the vector ${\bf n}$ as
specified in (44):

$0\,\leqslant\,||{\bf n}||\,\leqslant\,2.$ (104)

## References

  *

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