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PACS-Code: 03.65.Bz
e-mail : guillaume.adenier@ulp.u-strasbg.fr
This paper is also available at http://fr.arxiv.org/abs/quant-ph?0006014
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Bell's theorem, Bell inequalities
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% ----------------------------------------------------------------
% Refutation of Bell's theorem ************************************************
% by Guillaume ADENIER
% **** -----------------------------------------------------------
\documentclass[pra,secnumarabic,groupedaddress,showpacs]{revtex4}
\usepackage{amsmath}
\usepackage[latin1]{inputenc}
\newcommand{\vect}[1]{\mathbf{#1}}
\newcommand{\gvect}[1]{\boldsymbol{#1}}
% ----------------------------------------------------------------
\begin{document}
\title{Refutation of Bell's Theorem}%
\author{Guillaume ADENIER}
\email{guillaume.adenier@ulp.u-strasbg.fr} \affiliation{Université
de Strasbourg } \pacs{03.65.Bz}
\date{June 3, 2000}
% ----------------------------------------------------------------
\begin{abstract}
Bell's theorem is based on a linear combination of spin
correlation functions, each of these functions being characterized
by a different couple of arguments. The meaning of the
simultaneous presence of these different couples of arguments in
the same equation can be understood in two radically different
ways: either as a strongly objective meaning, that is, all
correlation functions are counterfactual properties of the same
set of particle pairs, or as a weakly objective meaning, that is,
each correlation function is measured on a different (and
contextual) set of particle pairs. It is demonstrated that once
this meaning is explicated, no discrepancy can appear between
local realistic theories and quantum mechanics, and that the
discrepancy exhibited by Bell's theorem is due to a meaningless
comparison between the local realistic inequality written within
strongly objective interpretation (thus relevant to a single set
of particle pairs) and the quantum mechanical prediction written
within weakly objective interpretation (thus relevant to several
different sets of particle pairs).
\end{abstract} \maketitle
% ----------------------------------------------------------------
\section{Introduction}
Bell's theorem\cite{JSB2} is exhibiting a peculiar discrepancy
between any local realistic theory and quantum mechanics, the
choice between alternatives to be settled by experimental means.
The trouble is that neither local realistic conceptions nor
quantum mechanics are easy to abandon. Indeed, classical physics
and common sense are usually based upon the former, when the
latter is often presented as the most successful theory of all
times.
Many tests have been performed, all but few\cite{FS1} showing
violations of Bell inequalities. Yet, the ideas brought forth by
Bell's theorem are so disconcerting that there is still
incredulity, not to mention a certain reluctance, before the
verdict. Some physicists, though fewer every day, are still
striving to find bias or loopholes capable of explaining the
apparent violation of Bell's inequalities\cite{CT1}, but there are
very few attempts to refute the theorem itself. The purpose of
this article is to provide such a refutation, within a strictly
quantum theoretical framework and without the need of any
additional assumptions.
Although the mere idea of a refutation might seem very unlikely,
as experiments showing violation of Bell's theorem are getting
increasingly accurate and loophole-free\cite{ASP1}, it must be
stressed that experimental tests, however accurate and close to
the ideal scheme, cannot prove the validity of the discrepancy
exhibited by Bell's theorem but only the validity of quantum
mechanics. Actually, it will be assumed in this article that all
tests conducted so far are proving with quite a good accuracy the
validity of quantum mechanics. In other words, the purpose of this
article is not to criticize the numerous experiments, nor quantum
mechanics for that matter, but Bell's theorem itself.
\section{The EPRB gedanken experiment}\label{EPRB}
\subsection{Spin observables and singlet state}\label{observsing}
Bell's theorem is usually based on an heuristic formulation of the
EPR (Einstein, Podolski and Rosen\cite{EPR1}) gedanken experiment,
due to David Bohm\cite{DB1}: in this EPRB gedanken experiment, a
pair of spin-½ particles with total spin zero is produced at a
source, each particle moving in opposite directions along the
y-axis. Two Stern-Gerlach devices are placed at opposite ends
(left and right) of the y-axis, and are oriented respectively
along the directions $\vect{u}$ and $\vect{v}$.
The spin observable associated to a measurement with a
Stern-Gerlach device oriented along the unit vector $\gvect{u}$ is
$\gvect{\sigma}.\vect{u}$, the components of $\gvect{\sigma}$
being the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$.
Let $\mathcal{H}_\mathrm{L}$ and $\mathcal{H}_\mathrm{R}$ be the
hilbert space respectively associated to each Stern-Gerlach
devices. The hilbert space $\mathcal{H}$ associated to the entire
EPRB system is the direct product of the hilbert spaces associated
to each Stern-Gerlach devices:
\begin{equation}\label{ep}
\mathcal{H}\equiv\mathcal{H}_\mathrm{L}\otimes\mathcal{H}_\mathrm{R}
\end{equation}
The spin observables of spaces $\mathcal{H}_\mathrm{L}$ and
$\mathcal{H}_\mathrm{R}$ have their respective counterpart in this
new product space $\mathcal{H}$ as
\begin{subequations}
\label{prols}
\begin{eqnarray}
\gvect{\sigma}_\mathrm{L}.\vect{u}\equiv\gvect{\sigma}.\vect{u}\otimes
1\negmedspace \mathrm{l}_\mathrm{R}\label{s1}\\
\gvect{\sigma}_\mathrm{R}.\vect{v}\equiv
1\negmedspace
\mathrm{l}_\mathrm{L}\otimes\gvect{\sigma}.\vect{v}\label{s2}
\end{eqnarray}
\end{subequations}
where $1\negmedspace \mathrm{l}_\mathrm{L}$ and $1\negmedspace
\mathrm{l}_\mathrm{R}$ are respectively the identity operators of
$\mathcal{H}_\mathrm{L}$ and $\mathcal{H}_\mathrm{R}$.
Contrary to observables $\gvect{\sigma}.\vect{u}$ and
$\gvect{\sigma}.\vect{v}$ which are not commuting with each others
if $\vect{u}\neq\vect{v}$, these new observables
$\gvect{\sigma}_\mathrm{L}.\vect{u}$ and
$\gvect{\sigma}_\mathrm{R}.\vect{v}$ are commuting observables,
thus reflecting the fact that each Stern-Gerlach devices are
arbitrary far from each others.
The product of these two observables
\begin{equation}\label{spincor}
(\gvect{\sigma}_\mathrm{L}.\vect{u}).(\gvect{\sigma}_\mathrm{R}.\vect{v})=
\gvect{\sigma}.\vect{u}\otimes\gvect{\sigma}.\vect{v}
\end{equation}
is therefore also an observable and can be understood as a
\emph{spin correlation observable} corresponding to the
\emph{joint spin measurement} of both Stern-Gerlach devices.
The product space $\mathcal{H}$ is spanned by the product basis
formed by the four eigenvectors of the spin correlation observable
$(\gvect{\sigma}_\mathrm{L}.\vect{n})(\gvect{\sigma}_\mathrm{R}.\vect{n})$
where $\vect{n}$ is a unitary vector. These eigenvectors are
written in the compact notation: $\{ |++\rangle, |+-\rangle,
|-+\rangle, |--\rangle \}$.
A system constituted by two spin one half particles has a total
spin of either zero or one. In an EPRB gedanken experiment, the
source produces particle pairs with zero total spin represented by
the singlet state
\begin{equation}\label{enf}
|\psi\rangle=\frac{1}{\sqrt{2}}\Big[|+-\rangle-|-+\rangle\Big]
\end{equation}
This singlet state has the important property of being invariant
under rotation, allowing not to mention explicitly the chosen
vector $\vect{n}$ in the writings of the $\mathcal{H}$ basis (see
for instance reference \cite{GHSZ1}).
\subsection{Statistical properties of the singlet state}
Yet, nothing certain can be said of a single spin measurement, nor
of a single spin correlation measurement, performed on a system
represented by this singlet state. Only probabilistic predictions,
such as expectation values can be made over numerous measurements
in the same context, according to Born interpretation of the state
vector.
It can be shown (see for instance reference\cite{ctdl1}, chapter
IV), that the expectation value is
\begin{equation}
{\langle \hat{A} \rangle}_{\phi}=\langle\phi | \hat{A} | \phi\rangle
\end{equation}
and therefore with equations (\ref{prols}) and (\ref{enf}), that
the \emph{expectation value of a spin observable} for the singlet
state $|\psi\rangle$ is zero:
\begin{subequations}
\label{expspin}
\begin{eqnarray}
\langle\gvect{\sigma}_\mathrm{L}.\vect{u}\rangle_\psi &=&
\langle\psi|\gvect{\sigma}.\vect{u}\otimes1\negmedspace\mathrm{l}_\mathrm{R}|\psi\rangle=0
\\
\langle\gvect{\sigma}_\mathrm{R}.\vect{v}\rangle_\psi &=&
\langle\psi|1\negmedspace\mathrm{l}_\mathrm{L}\otimes\gvect{\sigma}.\vect{v}|\psi\rangle=0
\end{eqnarray}
\end{subequations}
whatever $\vect{u}$ and $\vect{v}$, according to the rotational
invariance of the singlet state, and likewise that the
\emph{expectation value of spin correlation observable} is:
\begin{subequations}
\label{qcor}
\begin{eqnarray}
\label{qcor1}
E^\psi(\vect{u},\vect{v})=&
\langle\psi|
(\gvect{\sigma}_\mathrm{L}.\vect{u})
(\gvect{\sigma}_\mathrm{R}.\vect{v})
|\psi\rangle
\\ \label{qcor2}
=&-\vect{u}.\vect{v}
\end{eqnarray}
\end{subequations}
(see for instance references \cite{FS1}, \cite{GHSZ1}, or
\cite{AB1}).
\subsection{Perfect correlation and
hidden-variables}\label{perfectc}
When $\vect{u}=\vect{v}$, the expectation value of spin
correlation observable (\ref{qcor}) is equal to $-1$, meaning that
if both Stern-Gerlach devices are oriented along the same
direction, then with certainty the outcomes will be found to be
opposite.
Since the Stern-Gerlach devices are arbitrary far from each other,
thus locals, this perfect correlation can be understood from a
realistic point of view as relevant to the idea that somehow the
measurement result is \emph{predetermined}.
Yet, this predetermination seems contrary to the stochastic
description given by quantum mechanics, which is against all
better judgement the most complete description of a physical
system. Hence, if measurements on this system are predetermined,
then it should be possible to give a more complete description of
this system by means of additional hidden-variables.
Bell's idea is thence to write down mathematical requirements of
such a local hidden-variables theory, and to confront it both to
quantum mechanics and experiments. A single particle pair is thus
supposed to be entirely characterized by means of a set of
hidden-variables, which are represented altogether by a parameter
$\lambda$, so that the measurement result $A$ on the left along
$\vect{u}$ is $A(\lambda,\vect{u})$, and the result $B$ on the
right along $\vect{v}$ is $B(\lambda,\vect{v})$. Hence, the
perfect correlation condition is written
$B(\lambda,\vect{u})=-A(\lambda,\vect{u})$ whatever the particle
pair and its corresponding parameter .
Although the hidden-variable theory is supposed to be
deterministic, it must be capable of reproducing the stochastic
nature of the EPRB gedanken experiment expressed in equations
(\ref{expspin}) and (\ref{qcor}). For that purpose, the complete
state $\lambda_i$ of any particle pair $i$ is a random variable:
it is supposed to be drawn randomly according to a probability
distribution $\rho$ (see references \cite{JSB2} and \cite{JSB4}):
the probability of having $\lambda_i$ equal to a particular
$\lambda$ is equal to $\rho (\lambda)$.
Consider a set of $N$ particle pairs $\{i=1,\ldots,N\}$, the
\emph{mean value of joint measurements} over this set of $N$
particle pairs is
\begin{equation}\label{objcor2}
E^\rho(\vect{u},\vect{v})=
\frac{1}{N}\sum_{i=1}^{N}
-A(\vect{u},\lambda_i).A(\vect{v},\lambda_i)
\end{equation}
The probability distribution $\rho$ is supposed to grant the
equality between this equation and its quantum mechanical
counterpart (\ref{qcor1}) when $N$ is arbitrary large.
\section{The CHSH
function}\label{CHSH}
Thence, Bell's idea is to study linear combinations of correlation
functions given by quantum mechanics (equation (\ref{qcor})) and
by local hidden-variables theories (equation (\ref{objcor2})) with
\emph{different arguments}\cite{JSB8}, and to compare the results.
A well known choice of such a linear combination is the CHSH
\cite{CHSH1} combination, written with four different arguments
$\vect{a}$, $\vect{a'}$, $\vect{b}$ and $\vect{b'}$ :
\begin{equation}\label{CHSHeq}
S\equiv|
E(\vect{a},\vect{b})
-E(\vect{a},\vect{b'})
+E(\vect{a'},\vect{b})
+E(\vect{a'},\vect{b'})|
\end{equation}
The meaning of the simultaneous presence of these \emph{different
arguments} in this CHSH function must be explicated.
Basically, there are two possible interpretation, the
\emph{strongly objective} interpretation and the \emph{weakly
objective} interpretation \cite{BDE2,BDE1}:
\begin{description}
\item[strongly objective interpretation]
These four correlation functions have a counterfactual meaning. They are not
relevant to real experiments but rather with what result
\emph{would have been} obtained if measured on the same $N$
particle pairs along different directions.
\item[weakly objective interpretation]
These four correlation functions have a contextual meaning.
Each correlation function is actually to be measured
on a distinct set of $N$ particle pairs, that is,
4 correlation functions with \emph{alternative settings
of the instruments}\cite{JSB4} measured on 4 respective sets of
$N$ particle pairs.
\end{description}
Actually, the CHSH function was designed for testing
purpose\cite{CHSH1}, and many experiments where thus conducted
(the most famous being undoubtedly Aspects's test\cite{ADR1}), so
that the most likely interpretation of the linear combination of
expectation values is the weakly objective one. Nevertheless, the
strongly objective interpretation must be surveyed as well, since
it remains a possible interpretation and since this choice between
strong and weak objectivity is not at all explicit in many papers,
including John Bell's.
It must be stressed that these interpretations are radically
different, not only epistemologically, but also physically.
Indeed, the strongly objective interpretation requires a single
set of $N$ particle pairs characterized by the corresponding set
of parameters $\{\lambda_i\}$, whereas the weakly objective
interpretation requires no less than 4 sets of $N$ particle pairs
(in this article, it is assumed that the experimenter can control
the number $N$ of particle pairs constituting a distinct set).
The trouble is that a set of $N$ particle pairs characterized by
$\{\lambda_i\}$ cannot be reproduced to identical, neither
theoretically (for each complete state $\lambda_i$ of any particle
pair $i$ is a random variable, as defined in section
\ref{perfectc}), nor empirically (for the experimenter has no
control over the complete state of a particle pair).
Of course, if $N$ is arbitrary large, these four sets of $N$
particle pairs have necessarily the same statistical properties
(described by the probability distribution $\rho$), but they are
nonetheless \emph{four different sets of particle pairs} (see
reference \cite{AB1}, page 348) respectively characterized by four
different sets of hidden-variables parameters $\{\lambda_{1,i}\}$,
$\{\lambda_{2,i}\}$, $\{\lambda_{3,i}\}$ and $\{\lambda_{4,i}\}$.
The difference between each interpretation can therefore be
embodied in the degrees of freedom of the whole system. Let $f$ be
the degrees of freedom of a single particle pair. In the strongly
objective interpretation the degrees of freedom of the whole
system is $Nf$, whereas in the weakly objective interpretation the
degrees of freedom is 4 times as large, that is equal to $4Nf$,
which is something completely different.
That is why before establishing Bell's theorem, one has to choose
explicitly one interpretation and stick to it. Unfortunately, this
is not what has been done. It will be shown here that the
discrepancy exhibited by Bell's theorem is due to a meaningless
comparison between strongly objective and weakly objective
results, which means comparing the numerical value of the CHSH
function for two systems, one with $Nf$ degrees of freedom, the
other with $4Nf$ degrees of freedom, hence the discrepancy.
\section{Strongly objective interpretation: counterfactual properties of $N$
particle pairs}\label{obj}
\subsection{Local realistic inequality within strongly objective
interpretation}
Counterfactuality being a corollary to realism \cite{BDE1}, the
local realistic formulation of the CHSH function within strongly
objectivity is perfectly meaningful:
\begin{alignat}{2}
S^\rho_{\text{strong}}=\Big|
E^\rho(\vect{a},\vect{b})
-E^\rho(\vect{a},\vect{b'})
+E^\rho(\vect{a'},\vect{b})
+E^\rho(\vect{a'},\vect{b'})
\Big|
\end{alignat}
which is explicitly (using equation (\ref{objcor2}))
\begin{equation}
\begin{split}
S^\rho_{\text{strong}}=\bigg|\frac{1}{N}\sum_{i=1}^{N}
&A(\vect{a},\lambda_i)A(\vect{b},\lambda_i)
-A(\vect{a},\lambda_i)A(\vect{b',\lambda_i)}
\\
+&A(\vect{a'},\lambda_i)A(\vect{b},\lambda_i)
+A(\vect{a'},\lambda_i)A(\vect{b'},\lambda_i)\bigg|
\end{split}
\end{equation}
and after factorisation
\begin{equation}\label{Srhoobj1}
\begin{split}
S^\rho_{\text{strong}}=
\bigg|
\frac{1}{N}\sum_{i=1}^{N}
&A(\vect{a},\lambda_i)\Big[A(\vect{b},\lambda_i)-A(\vect{b',\lambda_i)}\Big]
\\
-&A(\vect{a'},\lambda_i)\Big[A(\vect{b},\lambda_i)+A(\vect{b'},\lambda_i)\Big]
\bigg|
\end{split}
\end{equation}
with two possible values for each term of the
summation\cite{FS1,AB1}:
\begin{equation}\label{rhoobj1}
\begin{split}
A(\vect{a},\lambda_i)&\Big[A(\vect{b},\lambda_i)-A(\vect{b',\lambda_i)}\Big]
\\
-&A(\vect{a'},\lambda_i)\Big[A(\vect{b},\lambda_i)+A(\vect{b'},\lambda_i)\Big]
=\pm2
\end{split}
\end{equation}
so that the narrowest local realistic local inequality within
strongly objective interpretation is :
\begin{equation}\label{locinobj}
S^{\rho}_{\text{strong}}\leq2
\end{equation}
which is the generalized formulation of Bell inequality due to
CHSH \cite{CHSH1}. It must be stressed however once more that this
inequality has been established within strongly objective
interpretation, so that each expectation value is relevant to the
same set of $N$ particle pairs. Hence, this result cannot be
applied directly to real experimental tests, where mean values are
measured upon four distinct sets of $N$ particle pairs. The
question whether the same inequality can be derived from the idea
that the different arguments have only a weakly objective meaning
will be discussed in section \ref{qmobj2}.
\subsection{Quantum mechanical prediction within strongly objective
interpretation}\label{qmobj}
The quantum prediction for the \mbox{CHSH} function within
strongly objective interpretation is written
\begin{equation}\label{bellq}
S^{\psi}_{\text{strong}}=|
E^\psi(\vect{a},\vect{b})
-E^\psi(\vect{a},\vect{b'})
+E^\psi(\vect{a'},\vect{b})
+E^\psi(\vect{a'},\vect{b'})|
\end{equation}
This equation is usually directly evaluated by replacing each
expectation value by the result of equation (\ref{qcor2}). This is
unfortunately all too hasty.
Indeed, in order to understand the quantum mechanical meaning of
equation (\ref{bellq}), it is better to take a step backward using
equation (\ref{qcor1}):
\begin{equation}\label{bof}
\begin{split}
S^\psi_{\text{strong}}=
\Big|
\langle\psi|
(\gvect{\sigma}_\mathrm{L}.\vect{a})&(\gvect{\sigma}_\mathrm{R}.\vect{b})
|\psi\rangle
-\langle\psi|
(\gvect{\sigma}_\mathrm{L}.\vect{a})(\gvect{\sigma}_\mathrm{R}.\vect{b'})
|\psi\rangle
\\
+&\langle\psi|
(\gvect{\sigma}_\mathrm{L}.\vect{a'})(\gvect{\sigma}_\mathrm{R}.\vect{b})
|\psi\rangle
+\langle\psi|
(\gvect{\sigma}_\mathrm{L}.\vect{a'})(\gvect{\sigma}_\mathrm{R}.\vect{b'})
|\psi\rangle
\Big|
\end{split}
\end{equation}
Now, it can be shown (by calculating the commutator of
$(\gvect{\sigma}_\mathrm{L}.\vect{u})(\gvect{\sigma}_\mathrm{R}.\vect{v})$
and
$(\gvect{\sigma}_\mathrm{L}.\vect{u})(\gvect{\sigma}_\mathrm{R}.\vect{v'})$
with $\vect{v}\neq\vect{v'}$) that the four spin correlation
observables in this equation are \emph{non commuting observables}.
According to Von Neumann \cite{JVN1}, the linear combination of
expectation values of different observables $\hat{R}$,
$\hat{S},\ldots$ is always meaningful in quantum mechanics:
\begin{equation}\label{vn1}
\langle \hat{R}+\hat{S}+\ldots\rangle_\phi=
\langle \hat{R} \rangle_\phi
+\langle \hat{S}\rangle_\phi
+\ldots
\end{equation}
even if $\hat{R}$, $\hat{S},\ldots$ are non commuting observables,
Yet, it is important restate that quantum mechanics is a weakly
objective theory, and that expectation values given by quantum
mechanics are as well weakly objective statements
\cite{BDE2,BDE3}, that is to say statements relevant to
observations.
Hence, when $\hat{R}$, $\hat{S},\ldots$ are non commuting
observables, the expectation values cannot be simultaneously
relevant to the same set of $N$ systems. The only possible meaning
of equation (\ref{vn1}) for non commuting observables is that each
expectation value is relevant to a distinct set of $N$ systems
(all systems being represented by the quantum state
$|\phi\rangle$). Likewise, the meaning of the linear combination
of expectation values of equation (\ref{bof}) is therefore but
contextual, that is to say weakly objective.
A remark might be necessary here, since these expectation values
are known with certainty, one could be tempted to claim the right
to consider them as counterfactuals. This would be a mistake, for
conterfactuality requires measurement compatibility, that is
commuting observables. The certainty of a contextual prediction is
not sufficient to make it a counterfactual prediction, or in other
words, \emph{weakly objective results known with certainty are not
strongly objective results} (incidentally, this is also true in
case of perfect correlation).
Therefore, $S^\psi_{\text{strong}}$ is meaningless and deceitful,
for its only possible meaning is weakly objective when it was
intended to be strongly objective. Hence,
$S^{\rho}_{\text{strong}}$ cannot be compared with any strongly
objective prediction given by quantum mechanics. Bell's theorem
cannot be established within strongly objective interpretation of
the CHSH function.
This denial is the first part of the refutation of Bell's theorem,
though maybe not the most conclusive (at least for the CHSH
formulation of the theorem), since the strength of this theorem is
mainly its ability to be confronted to experimental tests with
apparent success. Yet, this step had to be overcome, for now that
strongly objective interpretation is right out, there is no choice
but to rely on the weakly objective interpretation in order to
compare hidden-variables theories and quantum mechanics.
\section{Weakly objective interpretation : contextual measurements on $4$
distinct sets of $N$ particle pairs}
\subsection{Quantum mechanical prediction within weakly objective
interpretation}\label{qmobj2}
In last section, it was shown that quantum mechanical formalism
could be deceitful regarding the meaning of linear combination of
expectation values of non commuting observables. In this section,
a simple method will be provided in order to avoid such misleading
writings.
It was stressed in section \ref{CHSH} that strong objectivity and
weak objectivity are bound to physically different systems. This
difference should therefore appear in our equations. Indeed, the
correlation expressed in equation (\ref{qcor2}) is relevant to
spin measurements performed on particles that were once
constituting a single particle pair. Yet, two particles issued
from two distinct particle pairs never interact with each others,
so that spin measurements performed on these particles should not
be correlated. Hence, if left and right spin measurements are
performed on two distinct sets of $N$ particle pairs, instead of
the same set, there should not be any correlation either, and this
property should appear in a generalized spin correlation function
(i.e. generalized to the case of spin measurements performed on
different sets of particle pairs).
This can be easily done by means of a distinct EPRB space for each
set of $N$ particle pairs. Let $\mathcal{H}_j$ be the EPRB hilbert
space associated with the $j$th set of particle pairs. In this
hilbert space, the EPRB gedanken experiment is represented by the
singlet state $|\psi_j\rangle$ (see section \ref{EPRB}),
\begin{equation}\label{enfj}
|\psi_j\rangle=\frac{1}{\sqrt{2}}\Big[|+-\rangle_j-|-+\rangle_j\Big]
\end{equation}
The spin observables and spin correlation observables are
respectively similar to equations (\ref{prols}) and
(\ref{spincor}).
The whole experiment with the four sets of particle pairs can be
expressed in a new direct product space $\mathcal{H}_{1234}$
\begin{equation}\label{espacepos}
\mathcal{H}_{1234}\equiv
\mathcal{H}_1\otimes\mathcal{H}_2\otimes\mathcal{H}_3\otimes\mathcal{H}_4
\end{equation}
by the state vector
\begin{equation}\label{vectorpos}
|\psi_{1234}\rangle=
|\psi_1\rangle\otimes|\psi_2\rangle\otimes|\psi_3\rangle\otimes|\psi_4\rangle\
\end{equation}
The counterparts of observables in $\mathcal{H}_{1234}$ are
obtained akin to what was done in section \ref{observsing}. For
instance, the observables pertaining to the right Stern-Gerlach
device for the 1st, 2nd, 3rd and 4th set of particle pairs are
respectively
\begin{subequations}
\label{counterp}
\begin{eqnarray}
\gvect{\sigma}_{1,\mathrm{R}}.\vect{u}\equiv
(\gvect{\sigma}_\mathrm{R}.\vect{u})
\otimes 1\negmedspace\mathrm{l}_2
\otimes 1\negmedspace\mathrm{l}_3
\otimes 1\negmedspace\mathrm{l}_4
\\
\gvect{\sigma}_{2,\mathrm{R}}.\vect{u}\equiv
1\negmedspace\mathrm{l}_1
\otimes(\gvect{\sigma}_\mathrm{R}.\vect{u})
\otimes 1\negmedspace\mathrm{l}_3
\otimes 1\negmedspace\mathrm{l}_4
\\
\gvect{\sigma}_{3,\mathrm{R}}.\vect{u}\equiv
1\negmedspace\mathrm{l}_1
\otimes 1\negmedspace\mathrm{l}_2
\otimes(\gvect{\sigma}_\mathrm{R}.\vect{u})
\otimes 1\negmedspace\mathrm{l}_4
\\
\gvect{\sigma}_{4,\mathrm{R}}.\vect{u}\equiv
1\negmedspace\mathrm{l}_1
\otimes 1\negmedspace\mathrm{l}_2
\otimes 1\negmedspace\mathrm{l}_3
\otimes(\gvect{\sigma}_\mathrm{R}.\vect{u})
\end{eqnarray}
\end{subequations}
where $1\negmedspace\mathrm{l}_j$ is the identity operator of the
EPRB space $\mathcal{H}_j$, the counterparts for the left
Stern-Gerlach device being obtained by simply replacing the letter
$\mathrm{R}$ by the letter $\mathrm{L}$ in these equations.
Hence, the expectation value of the product of two spin
observables, the first belonging to the $k$th set and the second
to the $l$th set, is
\begin{equation}\label{expectation2a}
E^\psi_{kl}(\vect{u},\vect{v})\equiv
\langle\psi_{1234}|
(\gvect{\sigma}_{k,L}.\vect{u})(\gvect{\sigma}_{l,R}.\vect{v})
|\psi_{1234}\rangle
\end{equation}
and this is the \emph{generalized expectation value of spin
correlation observables} that was looked for.
The expectation value for measurements performed on the same set
($k=l$) of particle pairs is already known (see equations
(\ref{qcor})), and $E^\psi_{kk}(\vect{u},\vect{v})$ should provide
the same result. Indeed, using equations (\ref{vectorpos}) and
(\ref{counterp}) leads to
\begin{align}\label{expectation2b}
E^\psi_{kk}(\vect{u},\vect{v})&=
\langle\psi_k|(\gvect{\sigma}_{L}.\vect{u}).(\gvect{\sigma}_{R}.\vect{v})
|\psi_k\rangle
\\\nonumber
&=-\vect{u}.\vect{v}
\end{align}
but, if $k\neq l$ the result is quite different:
\begin{align}\label{expectation2c}
E^\psi_{\begin{subarray}{l} {kl}\\ \nonumber
k\neq l
\end{subarray}}
(\vect{u},\vect{v})&=
\langle\psi_k|(\gvect{\sigma}_{L}.\vect{u})|\psi_k\rangle
.\langle\psi_l|(\gvect{\sigma}_{R}.\vect{v})|\psi_l\rangle
\\
&=
\langle\psi_k|\gvect{\sigma}.\vect{u}\otimes1\negmedspace\mathrm{l}_\mathrm{R}
|\psi_k\rangle
.\langle\psi_l|1\negmedspace\mathrm{l}_\mathrm{L}\otimes\gvect{\sigma}.\vect{v}
|\psi_l\rangle
\\\nonumber
&=0
\end{align}
this according to equation (\ref{expspin}). There are indeed no
correlations between two sets of particle pairs, as requested in
the beginning of this section.
Now, contrary to what was done in section \ref{qmobj}, it is
possible to proceed much more according to quantum mechanical
postulates, for the spin correlation observables of equations
(\ref{counterp}) are commuting with each others, so that it is
possible to define a new observable describing the whole
experiment:
\begin{equation}\label{Sobserv}
\begin{split}
\hat{S}_{\text{weak}}\equiv
(\gvect{\sigma}_{1,L}.\vect{a}).&(\gvect{\sigma}_{1,R}.\vect{b})
-(\gvect{\sigma}_{2,L}.\vect{a}).(\gvect{\sigma}_{2,R}.\vect{b'})
\\
+&(\gvect{\sigma}_{3,L}.\vect{a'}).(\gvect{\sigma}_{3,R}.\vect{b})
+(\gvect{\sigma}_{4,L}.\vect{a'}).(\gvect{\sigma}_{4,R}.\vect{b'})
\end{split}
\end{equation}
and the quantum prediction for the CHSH function within weakly
objective interpretation is therefore
\begin{equation}\label{Sq4a}
S^{\psi}_{\text{weak}}=
\Big|
\langle\psi_{1234}|
\hat{S}_{\text{weak}}
|\psi_{1234}\rangle
\Big|
\end{equation}
which using equations (\ref{vectorpos}) and(\ref{Sobserv}) is
\begin{equation}\label{Sq4c}
\begin{split}
S^{\psi}_{\text{weak}}=
\Big|
\langle\psi_1|
(\gvect{\sigma}_L.\vect{a})&(\gvect{\sigma}_R.\vect{b})
|\psi_1\rangle
-\langle\psi_2|
(\gvect{\sigma}_L.\vect{a'})(\gvect{\sigma}_R.\vect{b})
|\psi_2\rangle
\\
+&\langle\psi_3|
(\gvect{\sigma}_L.\vect{a})(\gvect{\sigma}_R.\vect{b'})
|\psi_3\rangle
+\langle\psi_4|
(\gvect{\sigma}_L.\vect{a'})(\gvect{\sigma}_R.\vect{b'})
|\psi_4\rangle
\Big|
\end{split}
\end{equation}
that is, using equation (\ref{expectation2b}),
\begin{equation}\label{Sq4e}
S^{\psi}_{\text{weak}}=\Big|
E^\psi_{11}(\vect{a},\vect{b})
-E^\psi_{22}(\vect{a},\vect{b'})
+E^\psi_{33}(\vect{a'},\vect{b})
+E^\psi_{44}(\vect{a'},\vect{b'})
\Big|
\end{equation}
This equation is not ambiguous (as was equation (\ref{bof})): it
is a linear combination of expectation values, each of these being
relevant to a distinct and contextual set of $N$ particle pairs.
This equation is therefore weakly objective, as requested.
Finally, using equation (\ref{expectation2b}),
\begin{equation}\label{Sq4d}
S^{\psi}_{\text{weak}}=
\Big|
\vect{a}.\vect{b}
-\vect{a}.\vect{b'}
+\vect{a'}.\vect{b}
+\vect{a'}.\vect{b'}
\Big|
\end{equation}
with a maximum equal to
\begin{equation}\label{Sq4f}
\max(S^{\psi}_{\text{weak}})=2\sqrt{2}
\end{equation}
Hence, not surprisingly, quantum mechanics, which is a weakly
objective theory \cite{BDE2}, provides a clear answer to the CHSH
function understood as a weakly objective question.
\subsection{Local realistic inequality within weakly objective
interpretation}\label{lrweak}
The last step consists in comparing $S^{\psi}_{\text{weak}}$ with
$S^{\rho}_{\text{weak}}$. It was stressed in section \ref{CHSH}
that the $j$th set of particle pairs must be characterized by a
distinct set of hidden-variables parameters
$\{\lambda_{j,i}\,;\,i=1,\ldots,N \}$. Hence, to the generalized
expectation value of spin correlation observable (equation
(\ref{expectation2a})) is responding the \emph{generalized mean
value of joint spin measurements}:
\begin{equation}
E^\rho_{kl}(\vect{u},\vect{v})\equiv
\frac{1}{N}\sum_{i=1}^{N}
-A(\vect{u},\lambda_{k,i}).A(\vect{v},\lambda_{l,i})
\end{equation}
so that the local realistic CHSH function within weakly objective
interpretation is
\begin{equation}\label{Srhopos1}
S^{\rho}_{\text{weak}}=
\Big|
E^\rho_{11}(\vect{a},\vect{b})
-E^\rho_{22}(\vect{a},\vect{b'})
+E^\rho_{33}(\vect{a'},\vect{b})
+E^\rho_{44}(\vect{a'},\vect{b'})
\Big|
\end{equation}
and that is explicitly
\begin{equation}\label{Srhopos2}
\begin{split}
S^{\rho}_{\text{weak}}=
\Big|
\frac{1}{N}\sum_{i=1}^{N}
\big[
A(\vect{a},\lambda_{1,i}).&A(\vect{b},\lambda_{1,i})
-A(\vect{a},\lambda_{2,i}).A(\vect{b'},\lambda_{2,i})
\\
+&A(\vect{a'},\lambda_{3,i}).A(\vect{b},\lambda_{3,i})
+A(\vect{a'},\lambda_{4,i}).A(\vect{b'},\lambda_{4,i})
\Big]
\Big|
\end{split}
\end{equation}
This expression should be compare with the one pertaining to the
strongly objective interpretation (equation (\ref{objcor2})) which
contained terms that could be factorized. Here, since each terms
are different from the others, no factorisation is possible.
\emph{There is no way to derive Bell's inequality}. This, of
course, is of extreme importance. Surprisingly, this fact is not
pointed out for the first time, as Arno BOHM already has (page
351,352 of his book\cite{AB1}). Unfortunately, he remarked this in
a matter-of-fact way.
Yet, this impossibility cannot be ignored, for it has been shown
(in section \ref{qmobj}) that Bell's theorem could not be
established within strongly objective interpretation.
The only local realistic inequality that can be drawn is obtained
by considering (as was done with equation (\ref{rhoobj1})) the
possible numerical values of each term of the summation in
equation (\ref{Srhopos2}), that is
\begin{equation}\label{rhopos1}
\begin{split}
A(\vect{a},\lambda_{1,i}).A(\vect{b},\lambda_{1,i})
-A(\vect{a},\lambda_{2,i}).&A(\vect{b'},\lambda_{2,i})
+A(\vect{a'},\lambda_{3,i}).A(\vect{b},\lambda_{3,i})
\\
+&A(\vect{a'},\lambda_{4,i}).A(\vect{b'},\lambda_{4,i})
=+4,+2,0,-2,-4
\end{split}
\end{equation}
which extrema are +4 and -4, so that the narrowest local realistic
inequality that can be derived from equation (\ref{Srhopos2}) is
nothing but
\begin{equation}\label{Srhopos3}
S^{\rho}_{\text{weak}}\leq 4
\end{equation}
This narrowest local realistic inequality cannot be violated by
quantum mechanics, as the maximum of $S^{\psi}_{\text{weak}}$ is
$2\sqrt{2}$. Therefore, Bell's theorem cannot be established
within weakly objective interpretation of the CHSH function.
It must be stressed that the local realistic inequality
(\ref{Srhopos3}) for $S^{\rho}_{\text{weak}}$ would have been
impossible to reckon with the integral formulation of the
expectation value
\begin{equation}\label{intexp}
E(\vect{a},\vect{b})\equiv
\int\rho(\lambda)A(\vect{a},\lambda)B(\vect{b},\lambda)\text{d}\lambda
\end{equation}
instead of the discrete formulation used in this article. With
this integral writing (which can be found in most demonstrations
of Bell's theorem), no differences can be made between expectation
values of two distinct sets of $N$ particles, for $\lambda$ is but
a mute variable. A linear combination of such integral expectation
value can therefore only be strongly objective, not weakly
objective.
\section{Conclusion : Quantum mechanics should be approached with
prudence}
The apparent discrepancy exhibited by John Bell is therefore only
due to a meaningless comparison between $S^{\rho}_{\text{strong}}$
and $S^{\psi}_{\text{weak}}$, the former being relevant to a
system with $Nf$ degrees of freedom, the latter to a system with
$4Nf$ degrees of freedom (see section \ref{CHSH}).
Bell's theorem could not be established, neither within strongly
objective interpretation of the CHSH function, for quantum
mechanics could not provide a strongly objective result about non
commuting observables (see section \ref{qmobj}), nor within a
weakly interpretation, for the only local realistic inequality
that could be written was wide enough to avoid any violation by
quantum mechanics (see section \ref{lrweak}). Bell's theorem is
therefore refuted.
% ----------------------------------------------------------------
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\end{document}
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