# Symmetric derivation of singlet correlations in a quaternionic 3-sphere
model

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

###### Abstract

Using the powerful language of geometric algebra, we present an
observationally symmetric derivation of the strong correlations predicted by
the entangled singlet state in a deterministic and locally causal model,
usually also referred to as a local-realistic model, in which the physical
space is assumed to be a quaternionic 3-sphere, or $S^{3}$, available as the
spatial part of a solution of Einstein’s field equations of general
relativity, and compare it in quantitative detail with Bell’s local-realistic
model for the singlet correlations set within a flat Euclidean space
${\mathrm{I\\!R}^{3}}$. Since the quantitatively detailed expressions of
relative-angle-dependent probabilities of observing measurement outcomes for
Bell’s local model do not seem to have been fully articulated before, our
novel analysis exploiting the non-commutative properties of quaternions, in
addition to allowing the comparison with the quaternionic 3-sphere model, may
also provide useful comparisons for other less compelling local-realistic
models, such as those relying on retrocausality or superdeterminism. Apart
from the conservation of zero spin angular momentum, the key attribute
underlying the strong singlet correlations within $S^{3}$ in comparison with
Bell’s local model turns out to be the spinorial sign changes intrinsic to
quaternions that constitute the 3-sphere. In addition, we also discuss anew a
macroscopic experiment that can, in principle, test our 3-sphere hypothesis.

##  I Introduction

It is well known that a cosmological solution of Einstein’s field equations of
general relativity known as the Friedmann-Lemaître-Robertson-Walker spacetime
admits three possible geometries for the three-dimensional physical space,
isomorphic to ${\mathrm{I\\!R}^{3}}$, $H^{3}$, or $S^{3}$, where
${\mathrm{I\\!R}^{3}}$ represents an open Euclidean space, $H^{3}$ represents
an open hyperboloid of negative curvature, and $S^{3}$ represents a closed
3-sphere of constant positive curvature d'Inverno . Among these possible
geometries, only $S^{3}$ represents a closed universe with compact geometry.
Moreover, observationally, the cosmic microwave background spectra mapped by
the space observatory Planck now prefers a positive curvature, or $S^{3}$, at
more than 99% confidence level closed ; Handley .

On the other hand, Bell’s argument (or Bell’s theorem Bell-1964 ) against
possible local-realistic models for quantum correlations is set within a flat
and immutable spacetime. Indeed, in Chapter 7 of his book Speakable , while
exploring possible strategies to overcome his argument, Bell wonders: “The
space time structure has been taken as given here. How then about
gravitation?” Considering this shortcoming of Bell’s argument and other
physical considerations in the foundations of quantum mechanics, together with
the universal validity of Einstein’s field equations of general relativity at
all macroscopic scales including terrestrial and other non-cosmological
scales, in a series of works since 2007 Disproof ; IJTP ; RSOS ; IEEE-1 ;
IEEE-2 ; local ; RSOS-Reply  we have proposed the following experimentally
testable hypothesis:

###### Hypothesis 1

The strong quantum correlations observed in Nature can be understood as
manifestly local, realistic, and deterministic correlations if we model the
three-dimensional physical space as a closed and compact quaternionic 3-sphere
$S^{3}$ using geometric (or Clifford) algebra Clifford  rather than as a flat
and open space ${\mathrm{I\\!R}^{3}}$ using “vector algebra.”

Here, the correlations are said to be strong if the absolute value of the
Bell-CHSH combination Bell-1964 ; Speakable  of expectation values

${\cal E}({\bf a},\,{\bf b})+{\cal E}({\bf a},\,{\bf b^{\prime}})+{\cal
E}({\bf a^{\prime}},\,{\bf b})-{\cal E}({\bf a^{\prime}},\,{\bf b^{\prime}})$
(1)

calculated for them exceeds the bound of $2$, where the functions ${\cal
E}({\bf a},\,{\bf b})$ relevant for us here are defined in equation (34)
below. Moreover, the geometry of physical space we are concerned with here is
its intrinsic geometry at the terrestrial, solar, or galactic scales IEEE-1 .
Extrinsic geometries, cosmological scales, and local gravitational
interactions will not be relevant to our analysis. In other words, it is not
the strength or weakness of gravitational interactions that will be relevant
for us here but the qualitative algebraic differences between the geometries
of $S^{3}$ and $\mathrm{I\\!R}^{3}$, which are admissible spatial parts of the
solution of Einstein’s field equations of general relativity we alluded to
above. On the other hand, geometrically the tangent space ${T_{p}S^{3}}$ at
each point $p$ of $S^{3}$ is isomorphic to ${{\mathrm{I\\!R}}^{3}}$, analogous
to how the tangent space at each point of the surface $S^{2}$ of a ball is
isomorphic to the plane ${{\mathrm{I\\!R}}^{2}}$. Moreover, the tangent bundle
$TS^{3}$ of ${S^{3}}$ is trivial:

$TS^{3}:=\\!\bigcup_{p\in S^{3}}\\{p\\}\times
T_{p}S^{3}=S^{3}\times{\mathrm{I\\!R}}^{3}.$ (2)

Therefore, locally the space $S^{3}$ is indistinguishable from
${{\mathrm{I\\!R}}^{3}}$. As a result, the local experiences of experimenters
within $S^{3}$ will be no different from those of their counterparts within
${{\mathrm{I\\!R}}^{3}}$, the flat geometry of which is usually taken for
granted in all investigations and analyses concerning Bell’s theorem. Our
hypothesis is thus similar to ancient suspicions that Earth is spherical
despite the local appearance suggesting that it is flat.

The above Hypothesis 1 has far-reaching consequences, some of which we have
explored previously in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local ;
RSOS-Reply . Our goal here is to explicitly derive one of these consequences
using the powerful language of geometric algebra, namely the strong
correlations predicted by the entangled singlet state within the local-
realistic framework of Bell Bell-1964 , and compare it, in quantitative
details, with Bell’s local model111In this paper we are only concerned with
the probabilities of observing measurement outcomes for the explicit local-
realistic model presented by Bell in Section 3 of Bell-1964  and not directly
with the so-called theorem he has presented in the same paper. The latter we
have criticized elsewhere, such as in Refs. RSOS ; Begs ; Oversight . In
Sections II and III of Begs , a succinct review of what is meant by local
realism and hidden variable theories can also be found. for the singlet
correlations set within a flat Euclidean space ${\mathrm{I\\!R}^{3}}$. This
requires a different set of hidden variables from the one employed in Disproof
; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local ; RSOS-Reply  (cf. footnote 3 below).
Since, as far as we are able to ascertain, the probabilities of observing
measurement outcomes for Bell’s local model have never been fully worked out
before, our novel analysis exploiting the non-commutative properties of
quaternions may also provide useful comparisons for other less compelling
local-realistic models, such as those relying on retrocausality and
superdeterminism. In addition, we will also discuss anew a macroscopic
experiment proposed in IJTP  that can, in principle, test the above
hypothesis.

To that end, let us begin by defining a quaternionic 3-sphere in the language
of Geometric Algebra Clifford  whose tangent spaces are locally
${\mathrm{I\\!R}^{3}}$:

$S^{3}:=\left\\{\,{\bf
q}(\beta,\,{\mathbf{r}})=\varrho_{r}\exp\left[{\mathbf{J}}({\mathbf{r}})\,\frac{\beta}{2}\right]\Bigg{|}\;\left|\left|\,{\bf
q}\left(\beta,\,{\mathbf{r}}\right)\,\right|\right|=\varrho_{r}\right\\},$ (3)

where

$\exp\left[{\mathbf{J}}({\mathbf{r}})\,\frac{\beta}{2}\right]=\cos\left(\frac{\beta}{2}\right)+{\mathbf{J}}({\mathbf{r}})\,\sin\left(\frac{\beta}{2}\right),$
(4)

${{\mathbf{J}}({\mathbf{r}})=I_{3}{\mathbf{r}}}$ is a unit bivector (or pure
quaternion) rotating about a unit axis vector ${{\bf
r}\in{\mathrm{I\\!R}}^{3}}$ with rotation angle ${0\leq\beta<4\pi}$,
$\varrho_{r}$ is the radius of the 3-sphere, and
$I_{3}={\mathbf{e}}_{x}\wedge{\mathbf{e}}_{y}\wedge{\mathbf{e}}_{z}={\mathbf{e}}_{x}{\mathbf{e}}_{y}{\mathbf{e}}_{z}$
is the standard trivector in the Clifford algebra $\text{Cl}_{3,0}$ of
orthogonal directions in ${\mathrm{I\\!R}^{3}}$ Clifford . The basis bivectors
of ${{\mathbf{J}}({\mathbf{r}})}$ generate the even subalgebra of
$\text{Cl}_{3,0}$:

$\mathbf{J}_{j}\,\mathbf{J}_{k}=-\,\delta_{jk}\,-\sum_{l}\epsilon_{jkl}\,\mathbf{J}_{l}\,,$
(5)

where ${\delta_{jk}}$ is the Kronecker delta and ${\epsilon_{jkl}}$ is the
Levi-Civita symbol. Note that this bivector subalgebra is isomorphic to the
familiar algebra of Pauli matrices ${\boldsymbol{\sigma}_{j}}$ (${j=x,y,z}$)
specified by

$\boldsymbol{\sigma}_{j}\boldsymbol{\sigma}_{k}=\,\delta_{jk}\,{\hbox{${1\hskip-2.4pt{\rm
l}}$}}\,+\,i\,\sum_{l}\epsilon_{jkl}\,\boldsymbol{\sigma}_{l}\,,$ (6)

where ${1\hskip-2.4pt{\rm l}}$ is $2\times 2$ identity matrix and
${i\equiv\sqrt{-1}}$ is a unit imaginary. The isomorphism between the algebras
(5) and (6) can be easily established by the transformations
${\mathbf{J}_{k}\Longleftrightarrow{i\boldsymbol{\sigma}_{k}}}$ and
$\hbox{${1\hskip-2.4pt{\rm l}}$}\Longleftrightarrow 1$.

It is easy to verify that quaternions ${\bf q}(\beta,\,{\mathbf{r}})$
appearing in (3) respect the rotational symmetries exhibited by spinors:

${\bf q}(\beta+2\kappa\pi,\,{\bf r})=(-1)^{\kappa}\,{\bf q}(\beta,\,{\bf
r})\;\;\,\text{for}\;\,\kappa=0,1,2,3,\dots$ (7)

Thus, we can use ${{\bf q}(\beta,\,{\bf r})}$ to represent states of a
physical system that returns to itself only after even multiples of ${2\pi}$
rotations. Given two unit vectors ${\bf x}$ and ${\bf y}$ and a rotation axis
${\bf r}$, each of the quaternions ${\bf q}(\eta_{{\bf x}{\bf y}},\,{\bf r})$
in a unit ${S^{3}}$ can be factorized into a product of corresponding
bivectors ${{\bf J}({\bf x})}$ and ${{\bf J}({\bf y})}$ [which can be expanded
using the basis bivectors in (5)] as follows:

$\displaystyle{\bf q}(\eta_{{\bf x}{\bf y}},\,{\bf r})$ $\displaystyle=-\,{\bf
J}({\bf x})\,{\bf J}({\bf y})=-\,(I_{3}{\bf x})\,(I_{3}{\bf y})$ (8)
$\displaystyle=-(I_{3})^{2}{\bf x}\,{\bf y}={\bf x}\,{\bf y}={\bf x}\cdot{\bf
y}\,+\,{\bf x}\wedge{\bf y}$ (9) $\displaystyle=\cos(\eta_{{\bf x}{\bf
y}})+{\bf J}({\bf r})\,\sin(\eta_{{\bf x}{\bf y}}),$ (10)

which are now evidently unit quaternions, $||{\bf q}(\eta_{{\bf x}{\bf
y}},\,{\bf r})||=1$, if ${\eta_{{\bf x}{\bf y}}}$ is the angle between ${\bf
x}$ and ${\bf y}$, ${{\bf x}\,{\bf y}}$ is the geometric product between ${\bf
x}$ and ${\bf y}$, ${{\bf x}\wedge{\bf y}}$ is the wedge product between ${\bf
x}$ and ${\bf y}$, and ${{\bf J}({\bf r})}$ is identified with ${\frac{{\bf
x}\wedge{\bf y}}{||{\bf x}\wedge{\bf y}||}}$. Comparing ${\bf q}(\eta_{{\bf
x}{\bf y}},\,{\bf r})$ in (8) with ${{\bf q}(\beta,\,{\bf r})}$ defined in (3)
for $\varrho_{r}=1$, we see that the rotation angle ${\beta}$ of ${{\bf
q}(\beta,\,{\bf r})}$ is twice the angle ${\eta_{{\bf x}{\bf y}}}$ between the
vectors ${\bf x}$ and ${\bf y}$ in any factorization such as that in (8):

$\beta=2\,\eta_{{\bf x}{\bf y}}.$ (11)

As a result, the spinorial sign changes exhibited by quaternions as shown in
(7) can be expressed also using $\eta_{{\bf x}{\bf y}}$ as

${\bf q}(\eta_{{\bf x}{\bf y}}+\kappa\pi,\,{\bf r})=(-1)^{\kappa}\,{\bf
q}(\eta_{{\bf x}{\bf y}},\,{\bf r})\,\;\;\text{for}\;\,\kappa=0,1,2,3,\dots$
(12)

As we shall demonstrate below, relation (7) exhibits the key property that
induces the singlet correlations within $S^{3}$.

##  II Locally causal, realistic, and deterministic measurement functions

${{\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: A spin-less neutral pion decays into an electron-positron pair.
Measurements of spin components of each separated fermion are performed by
experimenters Alice and Bob at remote 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 angular momentum dictates that the total spin of the
system remains zero during its free evolution after the fermions have
separated and ceased to interact. Their initial rotation senses and directions
are thus unaltered before they interact with the remote detectors to produce
results. After IEEE-1 .

Now, for our purposes here, it is convenient to express the unit quaternions
defined in (3) together with $\varrho_{r}=1$ as pairs of products of two
bivectors such as ${\bf D}({\bf a})\,{\bf L}({\bf s}_{1})$ and ${\bf D}({\bf
b})\,{\bf L}({\bf s}_{2})$, with unit bivectors ${\bf D}({\bf a})=I_{3}{\bf
a}$ and ${\bf D}({\bf b})=I_{3}{\bf b}$ defined within $S^{3}$ about freely
chosen unit detector directions ${\bf a}$ and ${\bf b}$ representing the
detector orientations used by Alice and Bob, and unit bivectors ${\bf L}({\bf
s}_{1})=I_{3}{\bf s}_{1}$ and ${\bf L}({\bf s}_{2})=I_{3}{\bf s}_{2}$ defined
similarly about the unit spin directions ${\bf s}_{1}$ and ${\bf s}_{2}$
representing the spins detected along the detector orientations ${\bf a}$ and
${\bf b}$, as in a typical setup for Bell-test experiments Aspect  shown in
Fig. 1.

Recall that, geometrically IEEE-1 ; Clifford , a bivector such as $I_{3}{\bf
a}$ is an abstraction of a directed plane segment, with only a magnitude and a
sense of rotation [counterclockwise ($+$) or clockwise ($-$)] as its
properties. Neither the shape of the plane segment nor its rotation axis
represented by the vector ${\bf a}$ is an intrinsic part of the bivector
$I_{3}{\bf a}$. Thus, just as a unit vector ${\bf a}$ mathematically
represents an orientation in $\mathrm{I\\!R}^{3}$ or a point on $S^{2}$, a
unit bivector $I_{3}{\bf a}$ represents an orientation in $\mathrm{I\\!R}^{4}$
or a point on $S^{3}$. Therefore, a bivector such as ${\bf D}({\bf a})$
provides the correct representation of a detector orientation within $S^{3}$.
On the other hand, as we noted above, the tangent space at each point of
$S^{3}$ is isomorphic to $\mathrm{I\\!R}^{3}$. Therefore, the local
experiences of Alice and Bob in $S^{3}$ would be no different from those of
their counterparts in $\mathrm{I\\!R}^{3}$. In other words, Alice and Bob will
not be aware of the fact that when they are choosing detector orientations
${\bf a}$ and ${\bf b}$ within $\mathrm{I\\!R}^{3}$, they are in fact choosing
bivectors ${\bf D}({\bf a})$ and ${\bf D}({\bf b})$ within $S^{3}$, just as
our local experiences do not inform us that Earth is a sphere (for further
discussion, see Appendix B).

Now, during the free evolution222 Here ‘free evolution’ refers to the time
evolution of a physical system in the absence of any external influences or
interactions. For further explanation, see Appendix B. of the two spins from
the central source to detectors as shown in Fig. 1, the conservation of the
net zero angular momentum requires our representation of the total spin to
respect the condition

$\displaystyle-\,{\bf L}({\bf s}_{1})\,+\,{\bf L}({\bf s}_{2})=0\;$
$\displaystyle\Longleftrightarrow\;{\bf L}({\bf s}_{1})={\bf L}({\bf s}_{2})$
$\displaystyle\Longleftrightarrow\;{\bf s}_{1}=\,{\bf s}_{2}\,\equiv\,{\bf
s}.$ (13)

Consequently, during free evolution, the spins do not change in either their
senses or directions. Therefore we are justified in representing them by time-
independent bivectors ${\bf L}({\bf s}_{1})$ and ${\bf L}({\bf s}_{2})$.
Evidently, in the light of the product rule (5) for the unit bivectors, the
above condition is equivalent to the condition

${\bf L}({\bf s}_{1})\,{\bf L}({\bf s}_{2})=\,I_{3}{\bf s}\,I_{3}{\bf
s}=(I_{3})^{2}\,{\bf s}\,{\bf s}=-1.$ (14)

Next, we introduce the following two sign functions analogous to the ones
introduced by Bell in his local model Bell-1964 :

$\mu_{1}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})=\pm
1\;\;\;\text{and}\;\;\;\mu_{2}=\text{sign}(\mathbf{s}^{i}_{2}\cdot\mathbf{b})=\pm
1,$ (15)

where the subscripts 1 and 2 on $\mu$ and ${\mathbf{s}}$ refer to the
observation stations of Alice and Bob (cf. Fig. 1), and the superscript $i$ on
${\mathbf{s}}$ indicates its initial direction at the source with respect to
their chosen detector directions. The spin direction
${\mathbf{s}}={\mathbf{s}}_{1}={\mathbf{s}}_{2}$ acts as a hidden variable,
just as in Bell’s local model Bell-1964 ; Peres . The function
${\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})}$ can be
understood as follows. If, initially (i.e., before the detection process
defined by the measurement functions to be specified below), the two unit
vectors ${\mathbf{a}}$ and ${\mathbf{s}^{i}_{1}}$ happen to be pointing
through the same hemisphere of $S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$
centered at the origin of ${\mathbf{s}^{i}_{1}}$, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be equal
to $+1$, and if the two unit vectors ${\mathbf{a}}$ and ${\mathbf{s}^{i}_{1}}$
happen to be pointing through the opposing hemispheres of $S^{2}$ centered at
the origin of ${\mathbf{s}^{i}_{1}}$, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be equal
to $-1$, provided that ${{\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1}\not=0}$. If
${{\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1}}$ happens to be zero, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be assumed
to be equal to the sign of the first nonzero component of ${\bf a}$ from the
set ${\\{a_{x},\,a_{y},\,a_{z}\\}}$. And likewise for the function
${\mu_{2}=\mathrm{sign}({\mathbf{s}}^{i}_{2}\cdot{\mathbf{b}})}$.

With the above preliminaries in mind333It is worth noting that the hidden
variable used in the 3-sphere model presented in RSOS ; IEEE-1 ; IEEE-2  is
quite different from the one used in the present model. In the model presented
in RSOS ; IEEE-1 ; IEEE-2  the orientation $\lambda=\pm 1$ of $S^{3}$ is
assumed to be a hidden variable. This implies that the bivector subalgebra (5)
respected by the spin bivectors ${\mathbf{L}}({\mathbf{s}})$ and that
respected by the detector bivectors ${\mathbf{D}}({\mathbf{a}})$ are
different, but related by the orientation $\lambda$ so that
${\mathbf{L}}({\mathbf{r}})\,=\,\lambda\,{\mathbf{D}}({\mathbf{r}})\,\,\Longleftrightarrow\,\,{\mathbf{D}}({\mathbf{r}})\,=\,\lambda\,{\mathbf{L}}({\mathbf{r}})\,.$
On the other hand, in the model presented here, the initial spin direction
${\mathbf{s}}^{i}$ is taken to be a hidden variable, just as it is in Bell’s
local model published in Section 3 of Bell-1964 . This choice of hidden
variable is made here in order to facilitate a closer comparison of the
$S^{3}$ model with Bell’s local model in Bell-1964  based on
$S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$. The difference between the two
choices of hidden variables is at least algebraically quite significant. In
the present model, the spin bivectors ${\mathbf{L}}({\mathbf{s}})$ and the
detector bivectors ${\mathbf{D}}({\mathbf{a}})$ respect the same algebra
defined by (5), so that, algebraically,
${\mathbf{L}}({\mathbf{r}})\equiv{\mathbf{D}}({\mathbf{r}})$., we can now
state the central theorem, proved in several different ways in Disproof ; IJTP
; RSOS ; IEEE-1 ; IEEE-2 ; local .

###### Theorem 1

The strong quantum mechanical correlations predicted by the entangled singlet
state can be understood as classical, local, realistic, and deterministic
correlations among the pairs of limiting scalar points ${{\mathscr{A}}({\bf
a},\,{\bf s}^{i}_{1})=\pm 1}$ and ${{\mathscr{B}}({\bf b},\,{\bf
s}^{i}_{2})=\pm 1}$ of a quaternionic 3-sphere, or $S^{3}$, assumed to be a
model of the three-dimensional physical space.

Here the precise definition of what is meant by “limiting scalar points
${{\mathscr{A}}({\bf a},\,{\bf s}^{i}_{1})=\pm 1}$” is given in equations (17)
to (22) below. The proof of this theorem requires us to compute the
correlations while preserving the geometrical properties of $S^{3}$:

${\cal
E}({\mathbf{a}},\,{\mathbf{b}})\,=\int_{S^{2}}{\mathscr{A}}({\mathbf{a}},\,{\bf
s}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},\,{\bf s}^{i}_{2})\;p({\bf
s}^{i})\,d{\bf s}^{i}\,=-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}}),$ (16)

where $S^{2}$ is the base manifold of $S^{3}$ IEEE-1 ; IEEE-2 , the function
$p({\bf s}^{i})$ specifies the normalized initial probability distribution of
the spin direction ${\bf s}^{i}$ over $S^{2}$, and the functions
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$ encode local physical
interactions taking place during the detection processes at the two ends of
the experiment, producing results observed by Alice and Bob. Since the non-
uniform magnetic fields within detectors such as Stern–Gerlach devices are
known to align spin directions of incoming particles to their orientations, we
may define these functions as follows:

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
(17)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-(I_{3}{\mathbf{a}})(I_{3}{\mathbf{s}}_{1})\right\\}$
(18)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{{\mathbf{a}}\cdot{\mathbf{s}}_{1}+I_{3}({\mathbf{a}}\times{\mathbf{s}}_{1})\right\\}$
(19)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\right\\}$
(20)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\},$
(21)

where $\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}$ is the angle between the spin
direction ${\mathbf{s}}_{1}$ and the detector direction ${\mathbf{a}}$ chosen
by Alice as depicted in Fig. 1, and
${\mathbf{r}}_{1}=\frac{{\mathbf{a}}\times{\mathbf{s}}_{1}}{||{\mathbf{a}}\times{\mathbf{s}}_{1}||}$
is the normalized rotation axis of the quaternion
${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})$ in
$S^{3}$. Thus, given the definition (15) of $\mu_{1}$ as the function
$\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})$, the spin direction
${\mathbf{s}}_{1}$, which mathematically acts as a dummy variable in the above
limits while approaching the detector ${\mathbf{D}}({\mathbf{a}})$, will tend
to $+{\mathbf{a}}$ if initially the two unit vectors ${\mathbf{a}}$ and
${\mathbf{s}^{i}_{1}}$ happen to be pointing through the same hemisphere of
$S^{2}$ centered at the origin of ${\mathbf{s}^{i}_{1}}$, and otherwise the
spin direction ${\mathbf{s}}_{1}$ will tend to $-{\mathbf{a}}$. Consequently,
as ${\mathbf{s}}_{1}\rightarrow\mu_{1}{\mathbf{a}}=\pm{\mathbf{a}}$ during the
detection process by Alice so that the angle
$\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!\rightarrow 0$ or $\pi$, the binary
value of the result observed by her is obtained because
$\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\rightarrow\pm 1$ and
$\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\rightarrow 0$, giving

${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\longrightarrow+\mu_{1}=\pm
1.$ (22)

Thus, using a limit function to model the detection process of the spin values
allows us to retain their directional or contextual information passively,
without having to explicitly use directed numbers such as vectors or bivectors
to represent scalar-valued measurement results, $\pm 1$, observed in the Bell-
test experiments.

Similarly, the measurement function for Bob is defined as

$\displaystyle S^{3}\ni{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}$
(23)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+(I_{3}{\mathbf{s}}_{2})(I_{3}{\mathbf{b}})\right\\}$
(24)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{s}}_{2}\cdot{\mathbf{b}}-I_{3}({\mathbf{s}}_{2}\times{\mathbf{b}})\right\\}$
(25)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})-(I_{3}{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\right\\}$
(26)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\},$
(27)

where $\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}$ is the angle between the spin
direction ${\mathbf{s}}_{2}$ and the detector direction ${\mathbf{b}}$ chosen
by Bob as depicted in Fig. 1, and
${\mathbf{r}}_{2}=\frac{{\mathbf{s}}_{2}\times{\mathbf{b}}}{||{\mathbf{s}}_{2}\times{\mathbf{b}}||}$
is the normalized rotation axis of the quaternion
${\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})$ in
$S^{3}$. Consequently, analogous to Alice’s detection process, as
${\mathbf{s}}_{2}\rightarrow\mu_{2}{\mathbf{b}}=\pm{\mathbf{b}}$ during the
detection process by Bob so that the angle
$\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}\\!\rightarrow 0$ or $\pi$, the binary
value of the result observed by him is obtained because
$\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\rightarrow\pm 1$ and
$\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\rightarrow 0$, giving

${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\longrightarrow-\mu_{2}=\mp
1.$ (28)

Recall that the spin variables in the EPR-Bohm experiments do not change until
detection. Their initial spin directions and rotation senses acquired from the
source remain the same until the spins reach the detectors, because of the
conservation of the initial zero spin angular momentum. The limiting process
appearing in the definition (17) of the measurement function
${{\mathscr{A}}({\bf a},\,{\bf s}^{i}_{1})}$ thus physically describes the
process of detection of the spin ${\bf L}({\bf s}_{1})$ by the detector ${\bf
D}({\bf a})$, analogous to what occurs in a Stern–Gerlach device, whereas
mathematically the limiting process is a Hopf map $h:S^{3}\rightarrow{S^{2}}$
that projects the Hopf circle $S^{1}$ seen in (20) onto a point $\mu_{1}=\pm
1$ on the $S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$ assumed by Bell Bell-1964
(for further details see Appendix B).

Now, it follows from (22) and (28) that, in general, for the choices
${\mathbf{a}}\not={\mathbf{b}}$, the product of the results observed by Alice
and Bob will fluctuate between $-1$ and $+1$:

$\displaystyle{\mathscr{A}}$
$\displaystyle({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
$\displaystyle=-\mu_{1}\mu_{2}=-\,(\pm 1)(\pm 1)=\mp 1.$ (29)

On the other hand, for the choice ${\mathbf{b}}={\mathbf{a}}$, we have
${\mu}_{2}={\mu}_{1}$ because ${\mathbf{s}}^{i}_{2}=\,{\mathbf{s}}^{i}_{1}$,
and perfect anti-correlations are predicted:

$\displaystyle{\mathscr{A}}$
$\displaystyle({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{a}},{\mathbf{s}}^{i}_{2})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{a}})\right\\}\right]$
$\displaystyle=-(\mu_{1})^{2}=-1.$ (30)

It is important to note that the functions
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$ defined in (17) and (23)
are manifestly local-realistic in the sense espoused by Einstein and
formalized by Bell in Bell-1964 . Apart from the hidden variable ${\bf
s}^{i}_{1}$, the result ${{\mathscr{A}}=\pm 1}$ depends only on the
measurement direction ${\bf a}$, chosen freely by Alice, regardless of Bob’s
actions. And, analogously, apart from the hidden variable ${\bf s}^{i}_{2}$,
the result ${{\mathscr{B}}=\pm 1}$ depends only on the measurement direction
${\bf b}$, chosen freely by Bob, regardless of Alice’s actions. In particular,
the function ${{\mathscr{A}}({\bf a},\,{\bf s}^{i}_{1})}$ does not depend on
${\bf b}$ or ${\mathscr{B}}$ and the function ${{\mathscr{B}}({\bf b},\,{\bf
s}^{i}_{2})}$ does not depend on ${\bf a}$ or ${\mathscr{A}}$. Moreover, the
hidden variables ${\bf s}^{i}_{1}$ and ${\bf s}^{i}_{2}$ do not depend on
${\bf a}$, ${\bf b}$, ${\mathscr{A}}$, or ${\mathscr{B}}$ (see also Appendix
B).

##  III Computing the singlet correlations within 3-sphere

It is also important to note that the conservation of the initial zero spin
angular momentum (13) requires the equality
${\mathbf{s}}_{1}=\,{\mathbf{s}}_{2}$ to hold throughout the free evolution of
the spins, but not necessarily during their detection processes IEEE-2 .
Consequently, while the physical interactions are taking place during the
independent detection processes at the spacelike separated observation
stations, ${\mathbf{s}}_{2}$ can tend to $\mu_{2}{\mathbf{b}}$ jointly as
${\mathbf{s}}_{1}$ tends to $\mu_{1}{\mathbf{a}}$, despite the fact that
${\mathbf{s}}_{1}=\,{\mathbf{s}}_{2}$ must hold during the free evolution of
the two spins. If now the initial direction
${\mathbf{s}}^{i}={\mathbf{s}}^{i}_{1}={\mathbf{s}}^{i}_{2}$ of the spins
originating at the source is assumed to be uniformly distributed over $S^{2}$
as in Bell’s local model Bell-1964 ; Peres , specified by a normalized
probability measure $p({\mathbf{s}}^{i})$ so that

$\int_{S^{2}}p({\mathbf{s}}^{i})\;d{\mathbf{s}}^{i}=1,$ (31)

then the results (22) and (28) independently observed by Alice and Bob will be
equal to $+1$ or $-1$ with 50/50 chance. Consequently, in analogy with the
quantum mechanical predictions, the expectation values of these results will
vanish:

${\cal
E}(\mathbf{a})=\int_{S^{2}}\mathscr{A}({\mathbf{a}},\mathbf{s}^{i}_{1})\,p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}=0$
(32)

and

${\cal
E}(\mathbf{b})=\int_{S^{2}}\mathscr{B}({\mathbf{b}},\mathbf{s}^{i}_{2})\,p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}=0,$
(33)

where the subscripts 1 and 2 on the spin directions are retained for clarity
even though they are the same direction.

The question now is: What will be the correlations within $S^{3}$ between the
results ${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$ observed jointly but
independently by Alice and Bob, in coincident counts, at a spacelike distance
from each other? To answer this question, recall that the standard function
for computing correlations as expectation values employed by Bell in Bell-1964
is

${\cal
E}({\mathbf{a}},{\mathbf{b}})=\int_{S^{2}}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\;p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}.$
(34)

This expectation function respects two important properties. First, it is a
scalar-valued function: ${\cal
E}({\mathscr{A}},{\mathscr{B}})\\!\\!:\mathrm{I\\!R}\times\mathrm{I\\!R}\longrightarrow\mathrm{I\\!R}$.
Second, it is invariant under the change of order of the product of the
results ${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$. In other words, it is a
symmetric function: ${\cal E}({\mathscr{A}},{\mathscr{B}})={\cal
E}({\mathscr{B}},{\mathscr{A}})$. These properties are trivially satisfied in
Bell’s local model set within $\mathrm{I\\!R}^{3}$, because in it the results
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$ are scalar-valued functions
by construction, and therefore commute with each other. With some care, these
two properties of the expectation function can be satisfied also within our
$S^{3}$ model despite the fact that the measurement functions in our model are
composed of products of bivectors, or quaternions, that do not commute in
general:

$\displaystyle{\mathscr{A}}$
$\displaystyle({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}},$
(35)

but

$\displaystyle{\mathscr{B}}$
$\displaystyle({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\Big{\\}}.$
(36)

Here we have used the “product of limits equal to limits of product” rule.
Now, a very important property of the 3-sphere defined in (3) is that it
remains closed under multiplications of its constituent quaternions.
Therefore, the product of the unit quaternions
${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\in
S^{3}$ and
${\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\in
S^{3}$ appearing in the definitions (21) and (27) of the measurement outcomes,

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})={\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0}),$
(37)

is also a unit quaternion in $S^{3}$ (while we will not need explicit
expression of
${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})$ here, its
derivation can be found in appendix A.5 of local ). As a result, the following
equalities hold:

$\displaystyle\Big{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\big{\\{}+{\mathbf{q}}$
$\displaystyle(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\big{\\}}\Big{]}\Big{[}\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\big{\\{}-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\big{\\}}\Big{]}$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\big{\\}}$
(38)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})\big{\\}}$
(39)
$\displaystyle\equiv\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}.$
(40)

Thus, the products such as
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
of measurement outcomes are also limiting scalar points of some quaternions
within $S^{3}$. However, it is easy to recognize from equations (35) and (36)
that, since bivectors do not commute in general, the products
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
are limiting scalar points of two entirely different quaternions within
$S^{3}$. Moreover, the symmetry of the experimental setup shown in Fig. 1
necessitates that the product of the results
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$ would be equally likely to
be a limiting scalar value of the quaternion
$-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})$
as it would be a limiting scalar value of the quaternion
$-{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})$.
Therefore, since the expectation value is essentially the average value of the
product of the results ${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$, we must average over both
orders of the product with equal weight to respect the defining properties of
the expectation function we discussed above, by writing (34) in the symmetric
form444Incidentally, in the model presented in RSOS ; IEEE-1 ; IEEE-2  the
symmetric form (41) of the expectation function is automatically achieved by
the choice of an orientation $\lambda=\pm 1$ of $S^{3}$ as a hidden variable
(cf. footnote 3). Recall that, unlike $S^{2}$ whose tangent bundle is non-
trivial, $TS^{2}\not=S^{2}\times{\mathrm{I\\!R}}^{2}$, the quaternionic
3-sphere, or $S^{3}$, is an orientable manifold whose tangent bundle is
trivial, $TS^{3}=S^{3}\times{\mathrm{I\\!R}}^{3}$, as we noted in (2). That is
to say, unlike on $S^{2}$, a consistent sense of orientation or handedness can
be assigned to each point on $S^{3}$, so that “hair” on $S^{3}$ can be
“combed” without creating a “cowlick.” Thus, one can consistently assign
either a right-handed sense of rotation or a left-handed sense of rotation at
each point on $S^{3}$. This allows us to identify the orientation $\lambda=\pm
1$ of $S^{3}$ as a hidden variable with a 50/50 chance of either orientation
being initially realized, which restores the symmetry in the expectation
function (34), because a flip in the orientation induces a flip in the order
of the product of any two quaternions, such as those seen in the products (35)
and (36). as

$\displaystyle{\cal
E}({\mathbf{a}},{\mathbf{b}})=\int_{S^{2}}\Big{[}\frac{1}{2}$
$\displaystyle\Big{\\{}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
$\displaystyle+{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\Big{\\}}\Big{]}p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}.$
(41)

The statistical properties of the expectation functions (34) and (41) are
identical, however, because of the arithmetical identity

$\displaystyle{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
$\displaystyle\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$
$\displaystyle\equiv\frac{1}{2}\Big{\\{}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})+{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\Big{\\}}.$
(42)

Indeed, in analogy with (29) it is easy to verify that

$\displaystyle{\mathscr{B}}$
$\displaystyle({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]\\!\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]$
$\displaystyle=-\mu_{2}\mu_{1}=-\mu_{1}\mu_{2}=\mp 1,$ (43)

which follows from the definitions (17) and (23). Consequently, the
equivalence of formulae (34) and (41) follows:

$\displaystyle\frac{1}{2}\Big{\\{}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})+{\mathscr{B}}$
$\displaystyle({\mathbf{b}},{\mathbf{s}}^{i}_{2})\;{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\Big{\\}}$
$\displaystyle=\frac{1}{2}\Big{\\{}-\mu_{1}\mu_{2}-\mu_{2}\mu_{1}\,\Big{\\}}$
$\displaystyle=-\mu_{1}\mu_{2}=\mp 1.$ (44)

Now, the equality (36) can be simplified by multiplying its right-hand side
with ${\mathbf{L}}({\mathbf{s}}_{2})$ from the left and from the right in its
numerator and denominator to give

$\displaystyle{\mathscr{B}}$
$\displaystyle({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\left\\{-\frac{{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})}{{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{L}}({\mathbf{s}}_{2})}\right\\}$
(45)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\Big{\\}},$
(46)

where in the last step we have used the fact that bivectors square to $-1$:
${\bf L}({\bf s}_{2})\,{\bf L}({\bf s}_{2})=\,I_{3}{\bf s}_{2}\,I_{3}{\bf
s}_{2}=(I_{3})^{2}\,{\bf s}_{2}\,{\bf s}_{2}=-1.$ Consequently, using the
equality (46), we see that the limit relation (36) reduces to the following
convenient form:

$\displaystyle{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$
$\displaystyle=\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\Big{\\}}.$
(47)

The equalities (35) and (47) are easy to verify by taking limits explicitly on
both sides. Nevertheless, for completeness of our derivation, we have proved
them in Appendix A below. With them, we can now derive the correlations using

$\displaystyle{\cal E}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\\!\\!\\!\\!\int_{S^{2}}\frac{1}{2}\Big{[}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})+{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\;{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\Big{]}\,p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
$\displaystyle=\\!\\!\\!\\!\int_{S^{2}}\frac{1}{2}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}\,$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;+\\!\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\Big{\\}}\Bigg{]}p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}.$
(48)

Now, before proceeding further we have to make a choice. Either we ignore the
fact that the spin angular momentum is conserved during the free evaluation of
the spins (cf. Fig. 1), which amounts to ignoring the conditions (13) and
(14), or set
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ respecting
the conservation law. We first proceed by ignoring this condition, which leads
to

$\displaystyle{\cal E}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\int_{S^{2}}\frac{1}{2}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}\,+\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\Big{\\}}\Bigg{]}p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(49)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\Big{\\{}-\mu_{1}\mu_{2}\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}\,+\Big{\\{}-\mu_{1}\mu_{2}\,{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{b}})\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(50)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\Big{\\{}-\mu_{1}\mu_{2}\,I_{3}{\mathbf{a}}\,I_{3}{\mathbf{a}}\,I_{3}{\mathbf{b}}\,I_{3}{\mathbf{b}}\Big{\\}}\,+\Big{\\{}-\mu_{1}\mu_{2}\,I_{3}{\mathbf{b}}\,I_{3}{\mathbf{a}}\,I_{3}{\mathbf{a}}\,I_{3}{\mathbf{b}}\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(51)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\Big{\\{}-\mu_{1}\mu_{2}\,(I_{3}{\mathbf{a}})^{2}\,(I_{3}{\mathbf{b}})^{2}\Big{\\}}\,+\Big{\\{}-\mu_{1}\mu_{2}\,(I_{3}{\mathbf{b}})\,(I_{3}{\mathbf{a}})^{2}\,(I_{3}{\mathbf{b}})\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(52)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\Big{\\{}-\mu_{1}\mu_{2}\,\Big{\\}}\,+\Big{\\{}-\mu_{1}\mu_{2}\,\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(53)
$\displaystyle=\int_{S^{2}}\Big{\\{}-\mu_{1}\mu_{2}\,\Big{\\}}\;p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(54)
$\displaystyle=\int_{S^{2}}\Big{\\{}\\!-\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\,\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\Big{\\}}\;p({\mathbf{s}}^{i})\;d{\mathbf{s}}^{i},$
(55)

where in the step from (52) to (53) we have used
$(I_{3}{\mathbf{a}})^{2}=(I_{3}{\mathbf{b}})^{2}=-1$, and in the last step the
expressions for $\mu_{1}$ and $\mu_{2}$ are substituted from (15). Needless to
say, we could have inferred the last two steps immediately from (29), but have
preferred to carry out longer calculation to demonstrate that our derivation
is consistent and reproduces the known result. Indeed, the integral in (55) is
identical to the one that appears in Bell’s local model Bell-1964 ; Peres .
However, both Bell and Peres claim that it leads to the following weak
correlations that are incapable of exceeding the bounds set by Bell
inequalities:

$\displaystyle{\cal E}({\mathbf{a}},{\mathbf{b}})\,$
$\displaystyle=\int_{S^{2}}\\!\left\\{-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\right\\}\,p({\mathbf{s}}^{i})\;d{\mathbf{s}}^{i}$
$\displaystyle=\,\begin{cases}-\,1\,+\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi,\\\
+\,3\,-\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi.\end{cases}$ (56)

In Fig. 2 these weak correlations are depicted by saw-tooth shaped dashed
straight lines. By contrast, the solid cosine curve in Fig. 2 depicts the
strong singlet correlations predicted by our 3-sphere model as well as quantum
mechanics. They follow if the spin angular momentum is assumed to be conserved
during the free evolution of the constituent spins by setting
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ in the
above derivation of the correlations, as we now proceed to demonstrate:

$\displaystyle{\cal E}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\int_{S^{2}}\frac{1}{2}\Big{[}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\,+\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})\;{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\Big{]}\,p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(57)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}+\\!\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\Big{\\}}\Bigg{]}p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(58)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{D}}({\mathbf{b}})\Big{\\}}+\\!\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\Big{\\{}\,{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(59)
$\displaystyle=\int_{S^{2}}\frac{1}{2}\left[\Big{\\{}\,I_{3}{\mathbf{a}}\,I_{3}{\mathbf{b}}\Big{\\}}\,+\Big{\\{}\,I_{3}{\mathbf{b}}\,I_{3}{\mathbf{a}}\Big{\\}}\right]p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(60)
$\displaystyle=\frac{1}{2}\left[\Big{\\{}\,(I_{3})^{2}\,{\mathbf{a}}\,{\mathbf{b}}\Big{\\}}\,+\Big{\\{}\,(I_{3})^{2}\,{\mathbf{b}}\,{\mathbf{a}}\Big{\\}}\right]\int_{S^{2}}p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
(61)
$\displaystyle=-\frac{1}{2}\big{[}\,{\mathbf{a}}\,{\mathbf{b}}+{\mathbf{b}}\,{\mathbf{a}}\,\big{]}$
(62)
$\displaystyle=-{\mathbf{a}}\cdot{\mathbf{b}}\equiv\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}}),$
(63)

where the step (59) follows from the step (58) by setting
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ in (58),
as required by the conservation of initial zero spin angular momentum, and in
the step from (62) to (63) we have used the definition of the symmetric
product or inner product from Geometric Algebra Clifford . Comparing the
integral in (57) with the result (63) in light of the equality (44), for the
case when ${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$
is respected, then leads to

$\displaystyle{\cal E}({\mathbf{a}},{\mathbf{b}})\,$
$\displaystyle=\int_{S^{2}}\\!\left\\{-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\right\\}\,p({\mathbf{s}}^{i})\;d{\mathbf{s}}^{i}$
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}}).$ (64)

Unlike for the weak correlations (56), for the above correlations the bounds
on the Bell-CHSH sum of expectation values are known to exceed the values of
$\pm 2$ set by Bell’s theorem Bell-1964 ; Begs :

$-2\sqrt{2}\leqslant{\cal E}({\bf a},\,{\bf b})+{\cal E}({\bf a},\,{\bf
b^{\prime}})+{\cal E}({\bf a^{\prime}},\,{\bf b})-{\cal E}({\bf
a^{\prime}},\,{\bf b^{\prime}})\leqslant+2\sqrt{2}\,.$ (65)

It is important to bear in mind that this derivation of the singlet
correlations is a theoretical prediction of the 3-sphere model proposed in
Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local . It should not be confused
with after-the-event data analysis followed by experimenters. It is also worth
noting that the above is but one of several ways the strong correlations can
be derived within $S^{3}$ Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local .

It is evident from the calculations in equations (49) to (55) for Bell’s local
model and equations (57) to (63) for the 3-sphere model that in both cases the
correct predictions of correlations are obtained from essentially the same set
of equations, apart from the condition ${\bf L}({\bf s}_{1}){\bf L}({\bf
s}_{2})=-1$ used in the case of the 3-sphere model. However, because of the
use of this condition, equations (59) to (63) may give the impression that,
unlike in the derivation of (55), the spins ${\bf L}({\bf s}_{1})$ and ${\bf
L}({\bf s}_{2})$ do not play any role in the derivation of (63). But by
setting the value of the product
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})$ different
from $-1$ so that the conservation of spin angular momentum is violated, we
can verify that the spins do play important role in both derivations. To that
end, let us set
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=+0.3$ for the
sake of argument. Then it is straightforward to verify by following the steps
(57) to (63) that, in this case, the final result (63) would be
$+0.3\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$ instead of
$-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$. This demonstrates that the spins
${\bf L}({\bf s}_{1})$ and ${\bf L}({\bf s}_{2})$ do play a fundamental role
in both derivations. For further explanation, see also Appendix B.

##  IV Comparison of the 3-sphere model with Bell’s local model

[axesstyle=frame,linewidth=0.01mm,tickstyle=full,ticksize=0pt,dx=90Dx=180,dy=1
Dy=+2,Oy=-1](0,0)(180,1.0) ${90}$${270}$${+}$${\cal E}{({\bf a},{\bf
b})}$${0}$${\eta_{{\bf a}{\bf b}}}$[linewidth=0.35mm,linecolor=red]0.0180x dup
cos exch cos mul 1.0 mul neg 1 add

Figure 2: Graphs of the correlations (56) and (64). The x-axis depicts the
angle in degrees between the detector directions ${\mathbf{a}}$ and
${\mathbf{b}}$, and the y-axis depicts the corresponding expectation value or
correlations. The dotted straight lines depict the correlations predicted by
Bell’s local model and the solid curve represents the predictions of the
3-sphere model for which
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ holds
during the free evolution of the spins. After IJTP ; IEEE-2 .

In the previous section, we derived two contradictory results in equations
(56) and (64). Remarkably, the left-hand sides of these two equations are
mathematically identical. Their right-hand sides, however, differ
significantly, and this difference is manifest from their graphs shown in Fig.
2. We also identified the physical and mathematical reason for this
difference, namely, the conservation of initial zero spin angular momentum
encoded in the condition
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$, which is
the only global feature of $S^{3}$ that plays a direct role in the derivation
of the correlations (64) (see also Appendix B).

In this section we explain how the constraint
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ induces
the spinorial sign changes (7) in quaternions that have been neglected by Bell
Bell-1964  and Peres Peres  in their derivation (56) of the correlations
within $S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$. Note that in their $S^{2}$
model spins are represented inadequately by ordinary vectors such as ${\bf
s}_{1}$ and ${\bf s}_{2}$ and the conservation of zero spin is stated as ${\bf
s}_{1}+{\bf s}_{2}=0$ or as ${\bf s}_{1}=-{\bf s}_{2}$, whereas in our $S^{3}$
model spins are represented by the bivectors
$-{\mathbf{L}}({\mathbf{s}}_{1})=-I_{3}{\bf s}_{1}$ and
${\mathbf{L}}({\mathbf{s}}_{2})=I_{3}{\bf s}_{2}$ satisfying the conditions
(13) and (14). Consequently, in our derivation (64), the spinorial sign
changes (7) have been taken into account automatically as parts of the
geometrical features of $S^{3}$. However, it is instructive to bring out in
which step in their derivation of the correlations (56) Bell Bell-1964  and
Peres Peres  have ended up neglecting the spinorial sign changes. For this
purpose, it is instructive to reflect on figure 6.5 on page 161 of Peres Peres
, which has been used for calculating the probabilities of various measurement
results jointly observed by Alice and Bob. We have reproduced that figure here
for convenience as Fig. 3, with appropriate changes in notation.

[rot=-17,linestyle=dashed,linewidth=0.5pt,linecolor=blue](2.5,2.5)(2,0.6)0180
[rot=-17,linestyle=solid,linecolor=blue](2.5,2.5)(2,0.6)180360
[rot=40,linestyle=dashed,linewidth=0.5pt,linecolor=red](2.5,2.5)(2,0.7)0180
[rot=40,linestyle=solid,linecolor=red](2.5,2.5)(2,0.7)180360
${\eta_{{\mathbf{a}}{\mathbf{b}}}}$${\eta_{{\mathbf{a}}{\mathbf{b}}}}$${\mathbf{b}}$${\mathbf{a}}$${{\rm
I\\!R}^{3}}$${S^{2}}$${{\color[rgb]{1,0,0}+}\,{\color[rgb]{0,0,1}-}}$${{\color[rgb]{1,0,0}+}\,{\color[rgb]{0,0,1}+}}$${{\color[rgb]{1,0,0}-}\,{\color[rgb]{0,0,1}-}}$${{\color[rgb]{1,0,0}-}\,{\color[rgb]{0,0,1}+}}$[rot=40,linewidth=0.3mm,arrowinset=0.3,arrowsize=2pt
3,arrowlength=1,linecolor=yellow]-(1.355,3.7)(0.075,0.05)12060
[rot=-17,linewidth=0.3mm,arrowinset=0.3,arrowsize=2pt
3,arrowlength=1,linecolor=yellow]-(3.035,4.05)(0.075,0.05)11555

Figure 3: The figure 6.5 of Peres Peres , with appropriate changes in
notation. The calculation of various probabilities of obtaining measurement
results in Bell Bell-1964  and Peres Peres  relies on a unit $S^{2}$ embedded
in ${{\rm I\\!R}^{3}}$. In our quaternionic 3-sphere model hypothesized in
Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local , the unit $S^{2}$ of Bell
and Peres plays the role of a base manifold in the Hopf bundle of $S^{3}$. For
a given pair $\\{{\bf a},\,{\bf b}\\}$ of measurement directions, $S^{2}$ is
divided into four sectors, cut by two equatorial planes perpendicular to ${\bf
a}$ and ${\bf b}$, with alternating signs for the product ${\mathscr{A}}({\bf
a},\,{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\bf b},\,{\mathbf{s}}^{i}_{2})$.
After Peres .

Now in Bell’s local model, the measurement functions of Alice and Bob are the
following sign functions, as in (15):

${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})=+\mu_{1}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})=\pm
1$ (66)

and

${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})=-\mu_{2}=-\,\text{sign}(\mathbf{s}^{i}_{2}\cdot\mathbf{b})=\mp
1.$ (67)

Thus, as we saw in Section III, if the initial direction ${\bf s}^{i}$ of the
two spins is uncontrollable but describable by isotropic probability
distribution $p({\bf s}^{i})$, then the probability that the spin of particle
1 observed by Alice will be detected parallel to the vector ${\mathbf{a}}$
(regardless of whether particle 2 is detected) is unambiguously predicted by
Bell’s local model to be

$P_{1}^{+}({\mathbf{a}})=P_{1}^{-}({\mathbf{a}})=\frac{1}{2}.$ (68)

And, likewise, the probability that the spin of particle 2 observed by Bob
will be detected parallel to ${\mathbf{b}}$ is given by

$P_{2}^{+}({\mathbf{b}})=P_{2}^{-}({\mathbf{b}})=\frac{1}{2}.$ (69)

To calculate the joint probabilities of observing the two results defined in
(66) and (67) following Bell Bell-1964  and Peres Peres , consider a unit
$S^{2}$ embedded in ${{\rm I\\!R}^{3}}$, cut through by the equatorial planes
perpendicular to the detector directions ${\bf a}$ and ${\bf b}$, as shown in
Fig. 3. We then have ${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})=+1$ if
${\bf s}^{i}_{1}$ points through one of the hemispheres, and
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})=-1$ if it points through the
other hemisphere. Likewise, a second equatorial plane perpendicular to ${\bf
b}$ determines the regions where
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})=\pm 1$. The unit $S^{2}$ is
thereby divided by the two equatorial planes into four sectors, with
alternating signs for the product
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})$.
The adjacent sectors have their surface areas proportional to
$\eta_{{\mathbf{a}}{\mathbf{b}}}$ and $\pi-\eta_{{\mathbf{a}}{\mathbf{b}}}$
for $0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi$, and proportional
to $2\pi-\eta_{{\mathbf{a}}{\mathbf{b}}}$ and
$\pi-(2\pi-\eta_{{\mathbf{a}}{\mathbf{b}}})=\eta_{{\mathbf{a}}{\mathbf{b}}}-\pi$
for $\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant 2\pi$, with the area
of each hemisphere being $2\pi$. As a result, the probabilities of observing
the results jointly and simultaneously work out to be

$\displaystyle P_{12}^{++}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\displaystyle=\begin{rcases}\begin{dcases}&\\!\\!\\!\\!\\!\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}}{2\pi}\;\;\;\;\;\;\;\;\;\;\;\text{if}\;\;\,0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi\\\
&\\!\\!\\!\\!\frac{\pi-\eta_{{\mathbf{a}}{\mathbf{b}}}}{2\pi}\;\;\;\;\text{if}\;\;\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi\end{dcases}\end{rcases}$ (70)
$\displaystyle=P_{12}^{--}(\eta_{{\mathbf{a}}{\mathbf{b}}})$

with $P_{12}^{++}(\eta_{{\mathbf{a}}{\mathbf{b}}})\equiv
P_{12}\\{{\mathscr{A}}=+1,\;{\mathscr{B}}=+1\;|\;\eta_{{\mathbf{a}}{\mathbf{b}}}\\}$,
and

$\displaystyle P_{12}^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\displaystyle=\begin{rcases}\begin{dcases}&\\!\\!\\!\\!\frac{2\pi-\eta_{{\mathbf{a}}{\mathbf{b}}}}{2\pi}\;\;\;\text{if}\;\;\,0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi\\\
&\\!\\!\\!\\!\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}-\pi}{2\pi}\;\;\;\;\;\text{if}\;\;\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi\end{dcases}\end{rcases}$ (71)
$\displaystyle=P_{12}^{-+}(\eta_{{\mathbf{a}}{\mathbf{b}}}),$

with $P_{12}^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})\equiv
P_{12}\\{{\mathscr{A}}=+1,\;{\mathscr{B}}=-1\;|\;\eta_{{\mathbf{a}}{\mathbf{b}}}\\}$,
where ${P_{12}^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})}$, etc., are
probabilities of observing the result $+1$ by Alice and $-1$ by Bob, etc., and
the subscripts 1 and 2 label the two remote observation stations of Alice and
Bob as in Fig. 1. The remaining probabilities all vanish:

$\displaystyle
P_{12}^{+0}(\eta_{{\mathbf{a}}{\mathbf{b}}})=P_{12}^{-0}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\displaystyle=P_{12}^{0+}(\eta_{{\mathbf{a}}{\mathbf{b}}})=P_{12}^{0-}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\displaystyle=P_{12}^{00}(\eta_{{\mathbf{a}}{\mathbf{b}}})=0,$ (72)

where the superscript 0 stands for no detection. The correlations predicted by
Bell’s local model thus seem to be

$\displaystyle{\cal E}({\mathbf{a}},\,{\mathbf{b}})\\!$
$\displaystyle=\\!\\!\int{\mathscr{A}}({\mathbf{a}},\,{\mathbf{s}}^{i}_{1})\,{\mathscr{B}}({\mathbf{b}},\,{\mathbf{s}}^{i}_{2})\;p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}\,\approx\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}},\,{\mathbf{s}}_{1}^{k})\;{\mathscr{B}}({\mathbf{b}},\,{\mathbf{s}}_{2}^{k})\right]$
(73)
$\displaystyle=\frac{({\mathscr{A}}{\mathscr{B}}=++)\\!\times\\!P^{++}(\eta_{{\mathbf{a}}{\mathbf{b}}})\\!+\\!({\mathscr{A}}{\mathscr{B}}=--)\\!\times\\!P^{--}(\eta_{{\mathbf{a}}{\mathbf{b}}})\\!+\\!({\mathscr{A}}{\mathscr{B}}=+-)\\!\times\\!P^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})\\!+\\!({\mathscr{A}}{\mathscr{B}}=-+)\\!\times\\!P^{-+}(\eta_{{\mathbf{a}}{\mathbf{b}}})}{\left\\{P^{++}(\eta_{{\mathbf{a}}{\mathbf{b}}})\,+\,P^{--}(\eta_{{\mathbf{a}}{\mathbf{b}}})\,+\,P^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})\,+\,P^{-+}(\eta_{{\mathbf{a}}{\mathbf{b}}})\right\\}=1}$
(74)
$\displaystyle=\\!\begin{dcases}\\!\left[\left\\{(++)\\!\times\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2\pi}\right)\\!\right\\}+\left\\{(--)\\!\times\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2\pi}\right)\\!\right\\}+\left\\{(+-)\\!\times\\!\left(\\!\frac{\pi-\eta_{{\bf
a}{\bf
b}}}{2\pi}\\!\right)\\!\right\\}+\left\\{(-+)\\!\times\\!\left(\\!\frac{\pi-\eta_{{\bf
a}{\bf
b}}}{2\pi}\\!\right)\\!\right\\}\right]&\text{if}\;\,0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi\\\
\\!\left[\\!\left\\{\\!(++)\\!\times\\!\\!\left(\\!\frac{2\pi\\!-\\!\eta_{{\mathbf{a}}{\mathbf{b}}}}{2\pi}\\!\right)\\!\right\\}\\!+\\!\left\\{\\!(--)\\!\times\\!\\!\left(\\!\frac{2\pi\\!-\\!\eta_{{\mathbf{a}}{\mathbf{b}}}}{2\pi}\\!\right)\\!\right\\}\\!+\\!\left\\{\\!(+-)\\!\times\\!\\!\left(\\!\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}\\!-\\!\pi}{2\pi}\\!\right)\\!\right\\}\\!+\\!\left\\{\\!(-+)\\!\times\\!\\!\left(\\!\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}\\!-\\!\pi}{2\pi}\\!\right)\\!\right\\}\\!\right]\\!\\!\\!\\!\\!&\text{if}\;\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi\end{dcases}$ (75)
$\displaystyle=\begin{dcases}-\,1\,+\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi\\\
+\,3\,-\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi\,,\end{dcases}$ (76)

where ${n}$ is the total number of experiments performed, $k$ is the trial
number, $p({\bf s}^{i})=\frac{1}{n}$ is the probability density, and
$({\mathscr{A}}{\mathscr{B}}=+-)$, etc., indicate the values of the products
such as ${\mathscr{A}}{\mathscr{B}}=-1$ for the joint results
${\mathscr{A}}=+1$ and ${\mathscr{B}}=-1$, etc.

As noted, these weak correlations are depicted by dashed straight lines in
Fig. 2. In our derivation (56) within $S^{3}$, they follow if the conservation
of spin angular momentum encoded in the condition
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ is
neglected, which amounts to neglecting the spinorial sign changes (7)
intrinsic to the quaternions that constitute the 3-sphere. On the other hand,
if the spinorial sign changes are taken into account in the derivation by
respecting the condition
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$, then the
resulting correlations are ${\cal
E}({\mathbf{a}},{\mathbf{b}})=-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$, as we
derived in (64). They are depicted by the sinusoidal curve in Fig. 2. We can
appreciate this in more detail as follows. Using the mathematical identity

$\displaystyle-\cos\left(\eta_{{\bf a}{\bf b}}\right)$
$\displaystyle\equiv\frac{\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!+\\!\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!-\\!\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!-\\!\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)}{\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!+\\!\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!+\\!\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\\!+\\!\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)},$ (77)

the correlations we derived in (64) can be expressed as

$\displaystyle{\cal E}({\mathbf{a}},\,{\mathbf{b}})\\!$
$\displaystyle=-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$ (78)
$\displaystyle=\,\frac{\left[(++)\\!\times\\!\left\\{\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2}\right)\right\\}\right]+\left[(--)\\!\times\\!\left\\{\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2}\right)\right\\}\right]+\left[(+-)\\!\times\\!\left\\{\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2}\right)\right\\}\right]+\left[(-+)\times\left\\{\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf
a}{\bf
b}}}{2}\right)\right\\}\right]}{\left[\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf
a}{\bf b}}}{2}\right)+\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)+\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)+\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\bf a}{\bf
b}}}{2}\right)\right]}.$ (79)

Comparing this expression with (75) we thus see that, for
$0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi$, the probability of
obtaining the joint result ${\mathscr{A}}{\mathscr{B}}=++$ is not
$\frac{\eta_{{\bf a}{\bf b}}}{2\pi}$ but
$\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf b}}}{2}\right)$ if the
spinorial sign changes are taken into account. In other words, occasionally
what may be deemed to be a result ${\mathscr{A}}{\mathscr{B}}=++$ according to
the reasoning in Bell Bell-1964  and Peres Peres , would in fact be either
${\mathscr{A}}{\mathscr{B}}=+-$ or ${\mathscr{A}}{\mathscr{B}}=-+$, thereby
changing the number of times the result ${\mathscr{A}}{\mathscr{B}}=++$ occurs
from $\frac{\eta_{{\bf a}{\bf b}}}{2\pi}$-many times to
$\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf b}}}{2}\right)$-many
times. We can appreciate this by considering the example of the result
${\mathscr{A}}$ defined in (17):

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}.$
(80)

Here the spin ${\mathbf{L}}({\mathbf{s}}_{1})$ is a bivector or pure
quaternion, and corresponds to a binary rotation — i.e., rotation by $\pi$
Altmann . We can recognize this by considering the corresponding non-pure
quaternion and noticing that ${\bf
q}(\beta=\pi,\,{\mathbf{s}}_{1})={\mathbf{L}}({\mathbf{s}}_{1})$:

${\bf
q}(\beta,\,{\mathbf{s}}_{1})=\cos\left(\frac{\beta}{2}\right)+{\mathbf{L}}({\mathbf{s}}_{1})\,\sin\left(\frac{\beta}{2}\right)\\!.$
(81)

But the product of two rotations by $\pi$ about the same axis ${\bf s}_{1}$ is
a rotation by $2\pi$, and that is not an identity operation but subject to
spinorial sign changes specified in (7). Indeed, any product of two unit
bivectors, such as
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})$ for ${\bf
s}_{1}={\bf s}_{2}$, is equal to $-1$, which is precisely our condition (14)
for the conservation of angular momentum. Thus the condition
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ enforces
sign changes in the results (80) because they depend on
${\mathbf{L}}({\mathbf{s}}_{1})$, thereby inducing changes also in the
products of results such as
${\mathscr{A}}{\mathscr{B}}=++\,\longrightarrow\,-+$, etc. Consequently, the
probability of occurring the joint result ${\mathscr{A}}{\mathscr{B}}=++$
changes from $\frac{\eta_{{\bf a}{\bf b}}}{2\pi}$ to
$\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\bf a}{\bf b}}}{2}\right)$, as we
noted by comparing the expressions (75) and (79).

Now a sign change in a bivector or pure quaternion, such as
${\mathbf{L}}({\mathbf{s}}_{1})$, means a change in its sense of rotation. In
other words, a counterclockwise rotation, say
${+\mathbf{L}}({\mathbf{s}}_{1})$, changes to a clockwise rotation,
${-\mathbf{L}}({\mathbf{s}}_{1})$. This can be expressed as

$+\,{\bf L}({\bf s}_{1})=+I_{3}{\bf s}_{1}\;\longrightarrow\;-\,{\bf L}({\bf
s}_{1})=-\,I_{3}{\bf s}_{1}=I_{3}(-\,{\bf s}_{1}).$ (82)

Thus a spinorial sign change in the bivector $+{\bf L}({\bf s}_{1})$ can be
expressed equivalently as a sign change in the direction of its axis vector
${\bf s}_{1}$. Indeed, a counterclockwise rotation about $+\,{\bf s}_{1}$
represented by $+{\bf L}({\bf s}_{1})$ is the same as the clockwise rotation
about $-\,{\bf s}_{1}$. But that implies that sign changes in the local result
such as (80) will induce changes such as

$+\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\longrightarrow\;-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})$
(83)

in the integrand on the left-hand side of (56), so that some of its values
will change from $++$ to $+-$, etc., under the spinorial sign changes seen in
(7), with precisely which changes would occur dictated by the condition
${\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})=-1$ for the
conservation of spin angular momentum we imposed in the step (59) of our
derivation of the correlations (64).

##  V Recovering Bell’s Local Model from the $S^{3}$ Model in the Flat
Geometry of $\mathrm{I\\!R}^{3}$

In addition to what has been discussed above, there is a further, purely
geometrical means to recover the traditional interpretation of Bell’s theorem
within the flat geometry of $\mathrm{I\\!R}^{3}$ from the quaternionic
3-sphere model, by recognizing that the strong singlet correlations are a
direct consequence of the Möbius-like twists in the Hopf bundle of $S^{3}$
[10]. To appreciate it, recall that, geometrically, $S^{3}$ is an intricately
interlinked bundle of circles, $S^{1}$, called Hopf circles or Clifford
parallels, which are the fibers of the Hopf bundle, with $S^{2}$ being the
base of this bundle, so that locally (in the topological sense) $S^{3}$ is
isomorphic to the product space $S^{2}\times{S^{1}}$ (see Fig. 4). Globally,
however, $S^{3}$ has no cross-section at all. Consequently, each Hopf circle
$S^{1}$ threads through every other circle in the bundle without sharing a
single point with any of them, thereby producing Möbius-like twists in the
bundle. These twists, and therefore the Hopf bundle, can be constructed using
the relation

$e^{i\psi_{-}}=\,e^{i\omega\phi}\,e^{i\psi_{+}}\,,$ (84)

as shown in Fig. 5, where ${\psi_{-}}$ and ${\psi_{+}}$ in the range
$0\leqslant\psi<4\pi$ are the coordinates of the fibers $S^{1}$ attached to
the hemispheres ${{\mathrm{H}}_{-}}$ and ${{\mathrm{H}}_{+}}$ of the base
space ${S^{2}}$, which is parameterized by spherical coordinates
${(0\leqslant\theta<\pi,\;0\leqslant\phi<2\pi)}$; ${\phi}$ is the angle that
parameterizes a thin intersection ${{\mathrm{H}}_{-}\cap{\mathrm{H}}_{+}}$
around the equator of ${S^{2}}$ [${\theta\sim\frac{\pi}{2}}$]; and
${e^{i\omega\phi}}$ is the transition function that glues the two sections
${{\mathrm{H}}_{-}}$ and ${{\mathrm{H}}_{+}}$ together, thus constituting the
3-sphere. The winding number $\omega$ must be an integer for $\psi$ to be
single-valued and the resulting structure to be a manifold. It is evident from
the relation (84) that, for $\omega=1$, the fibers $S^{1}$ match perfectly at
the angle ${\phi=0}$ (modulo ${2\pi}$), but differ from each other at all
intermediate angles ${\phi}$. For example, ${e^{i\psi_{-}}}$ and
${e^{i\psi_{+}}}$ differ by a minus sign at ${\phi=\pi}$. On the other hand,
for $\omega=0$ the $S^{3}$ bundle reduces to a trivial bundle:
$S^{2}\times{S^{1}}$. While geometrically $S^{3}$ is a highly non-trivial
manifold, algebraically the twists within it can be characterized by the
simple condition

${\bf L}({\bf s}_{1})\,{\bf L}({\bf s}_{2})=-1,$ (85)

which is the condition (14) for the conservation of spin angular momentum
discussed in the paper. I have proved this in two different ways. In Section
VIII of Ref. [9], I have derived the conservation condition (85) from the
relation (84) of the Hopf bundle, and in Appendix X of Ref. [10] I have
conversely proved that the relation (84) of Hopf bundle stems from the
condition (85) for the conservation of vanishing spin angular momentum.

${S^{3}}$${h^{-1}(q)}$${h^{-1}(p)}$${q}$${p}$${S^{2}}$base
space${h:S^{3}\rightarrow S^{2}}$Hopf fibration${h^{-1}:S^{2}\rightarrow
S^{3}}$

Figure 4: The tangled web of linked Hopf circles depicting the geometrical and
topological non-trivialities of $S^{3}$. Locally, in the topological sense,
${S^{3}}$ is isomorphic to the product ${S^{2}\times S^{1}}$. Thus ${S^{3}}$
is ${S^{2}}$ worth of circles. Each circle ${S^{1}}$, as fiber $h^{-1}(p)$,
threads through every other circle in the bundle ${S^{3}}$ without sharing a
single point with any other circle, and projects down to the point $p$ on
${S^{2}}$ via Hopf map ${h:S^{3}\rightarrow S^{2}}$. Adapted from Ref. [10].

Now, as we saw in the paper, Bell’s local model is set within a 2-sphere,
$S^{2}$, embedded in $\mathrm{I\\!R}^{3}$. In the $S^{3}$ model, on the other
hand, the points of this $S^{2}$ are projections of the intricately
interlinked fibers of $S^{3}$ by the Hopf map $h$, as shown in Fig. 4:

$\mathrm{I\\!R}^{4}\hookleftarrow{S^{3}}\xrightarrow{\;\;h\;\;}S^{2}\hookrightarrow{\mathrm{I\\!R}}^{3}.$
(86)

If we parameterize $S^{3}$ with the coordinates $q_{1}$, $q_{2}$, $q_{3}$, and
$q_{4}$ of $\mathrm{I\\!R}^{4}$ so that
$q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{4}^{2}=1$, then
$h\\!:S^{3}\\!\rightarrow{S^{2}}$ is defined by

$h(q_{1},\,q_{2},\,q_{3},\,q_{4})=(x,\,y,\,z)\,,$ (87)

where (as in the original formula discovered by Hopf in 1931)

$\displaystyle x$ $\displaystyle=2(q_{1}q_{3}+q_{2}q_{4})\,,$ (88)
$\displaystyle y$ $\displaystyle=2(q_{2}q_{3}-q_{1}q_{4})\,,$ (89)
$\displaystyle\text{and}\;\;\,z$
$\displaystyle=q_{1}^{2}+q_{2}^{2}-q_{3}^{2}-q_{4}^{2}$ (90)

are the corresponding coordinates in $\mathrm{I\\!R}^{3}$ that parameterize
the base space $S^{2}$ of the Hopf bundle as a unit 2-sphere, and satisfy

$x^{2}+y^{2}+z^{2}=\left(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}+q_{4}^{2}\right)^{2}=1.$
(91)

[linewidth=0.5mm,linestyle=dashed](6.0,8.5)(3.0,0.625)0180
[linewidth=0.5mm,linestyle=dashed](6.0,3.5)(3.0,0.625)1800
[showpoints=true,linewidth=0.5mm](6.0,6.0)(3.0,0.625)300330
[showpoints=false,arrowscale=2,linewidth=0.3mm]-¿(6.0,6.0)(3.0,0.625)300324
${\mathbf{H_{+}}}$${\mathbf{H_{-}}}$${\boldsymbol{\phi}}$${\boldsymbol{\psi}}$

Figure 5: The Hopf bundle of ${S^{3}}$ showing two hemispherical
neighborhoods ${{\mathrm{H}}_{\pm}}$ of the base manifold ${S^{2}}$. A fiber
${S^{1}=h^{-1}(p)}$ parametrized by angle ${\psi}$ is attached to each point
$p$ of ${{\mathrm{H}}_{\pm}}$. The intersection of ${{\mathrm{H}}_{\pm}}$ at
${\theta\sim\frac{\pi}{2}}$ is circular a strip parametrized by angle
${\phi\in[0,\,2\pi)}$. Adapted from Ref. [10].

The Hopf map (87) is thus a surjective map that maps all of the points of a
unit $S^{3}$ embedded in $\mathrm{I\\!R}^{4}$ to the points of a unit $S^{2}$
embedded in $\mathrm{I\\!R}^{3}$, thereby reducing $S^{3}$ to $S^{2}$. In
particular, the entire set of points of a Hopf circle $S^{1}$ in $S^{3}$ is
mapped to a point, say $p$, of the base $S^{2}$ of the bundle, making
$S^{1}=h^{-1}(p)$ a preimage of that point $p$, as shown in Fig. 4. As a
result, the $S^{3}$ model can be naturally projected by the Hopf map to Bell’s
local model set within the flat geometry of $\mathrm{I\\!R}^{3}$. In
particular, the measurement results
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})$, defined in the $S^{3}$
model as

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\right\\}$
(92)
$\displaystyle\longrightarrow\mu_{1}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\in
S^{2}\hookrightarrow\mathrm{I\\!R}^{3},$ (93)

are a part of this projection. We can see this by identifying the components
of the quaternion in (92), such as
$\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})=q_{1}$, etc., and expressing them
in terms of the Euler angles $(\psi,\theta,\phi)$ using the standard
representation of arbitrary rigid rotations:

$\displaystyle q_{1}$
$\displaystyle=\cos\left(\frac{\psi+\phi}{2}\right)\cos\frac{\theta}{2}\,,$
$\displaystyle
q_{2}=\sin\left(\frac{\psi+\phi}{2}\right)\cos\frac{\theta}{2}\,,$ (94)
$\displaystyle q_{3}$
$\displaystyle=\cos\left(\frac{\psi-\phi}{2}\right)\sin\frac{\theta}{2}\,,$
$\displaystyle
q_{4}=\sin\left(\frac{\psi-\phi}{2}\right)\sin\frac{\theta}{2}\,.$ (95)

The Hopf map (87) then projects them to the base space $S^{2}$:

$x=\cos\phi\sin\theta,\;\;\;\;\;y=\sin\phi\sin\theta,\;\;\;\;\;z=\cos\theta.$
(96)

These are the Cartesian coordinates in $\mathrm{I\\!R}^{3}$ of the point
$(\theta,\phi)$ on a unit $S^{2}$ embedded in $\mathrm{I\\!R}^{3}$. Thus, the
Euler angle $\psi$ in this representation can be identified with the
coordinate of a point in $S^{1}$, which is projected onto the point
$(\theta,\phi)$ of the base $S^{2}$. In the $S^{3}$ model, this point is then
assigned observable values,
$\mu_{1}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})=\pm 1$, by the limit
${{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}$ defined in (92),
precisely as assumed in Bell’s local model set within
$S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$.

The Hopf map $h:S^{3}\longrightarrow{S^{2}}$ can also be expressed as

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\longmapsto{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,(I_{3}{\bf
e}_{z})\;{\mathbf{q}}^{\dagger}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})$
(97)

(with $\dagger$ indicating conjugation) in terms of the quaternion

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})=\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})$
(98)

appearing in (92), with respect to the fixed reference bivector $I_{3}{\bf
e}_{z}$ on $S^{2}$. The map then projects
${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})$ to the
bivector

$I_{3}{\bf v}=\,v_{x}\;{{\bf e}_{y}}\,\wedge\,{{\bf e}_{z}}\,+\,v_{y}\;{{\bf
e}_{z}}\,\wedge\,{{\bf e}_{x}}\,+\,v_{z}\;{{\bf e}_{x}}\,\wedge\,{{\bf
e}_{y}}$ (99)

representing a point on $S^{2}$, whose coordinates in $\mathrm{I\\!R}^{3}$ are

$\displaystyle v_{x}$
$\displaystyle=2\,r_{1x}\,r_{1z}\sin^{2}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!)-2\,r_{1y}\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!)\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}),$
(100) $\displaystyle v_{y}$
$\displaystyle=2\,r_{1x}\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!)\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!)+2\,r_{1y}\,r_{1z}\sin^{2}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}),$
(101) $\displaystyle v_{z}$
$\displaystyle=\cos^{2}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+\left(r_{1z}^{2}-\,r_{1x}^{2}-\,r_{1y}^{2}\right)\sin^{2}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}).$
(102)

This Hopf projection from $S^{3}$ to $S^{2}$, however, does not quite recover
Bell’s model, because the condition ${\bf L}({\bf s}_{1})\,{\bf L}({\bf
s}_{2})=-1$ is still satisfied, which induces the Möbius-like twists in the
fibers of $S^{3}$ and thus is responsible for the strong singlet correlations,
as we saw in their derivation in equations (57) to (63). But there are no such
twists in the flat geometry of $\mathrm{I\\!R}^{3}$, which is taken for
granted in Bell’s local model. Therefore, we must relinquish the condition
${\bf L}({\bf s}_{1})\,{\bf L}({\bf s}_{2})=-1$, which is equivalent to
setting the winding number ${\omega}=0$ in (84) so that
$e^{i\psi_{-}}\\!=e^{i\psi_{+}}$ for all azimuthal angles $\phi$, to recover
Bell’s model from the $S^{3}$ model, thus reducing $S^{3}$ globally to the
trivial product space $S^{2}\times{S^{1}}$:

$S^{3}\xrightarrow[\equiv\,\omega\,=\,0]{\;\;\;{\bf L}({\bf s}_{1})\,{\bf
L}({\bf s}_{2})\,\not=\,-1\;\;\;}\,S^{2}\times S^{1}.$ (103)

The resulting bipartite correlations are then no longer strong,

$\displaystyle-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\;\longrightarrow\;\begin{cases}-\,1\,+\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi,\\\
+\,3\,-\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi,\end{cases}$ (104)

as we proved via their derivation in equations (49) to (56). This completes
the recovery of Bell’s local model from the $S^{3}$ model:

${\mathrm{I\\!R}}^{4}\hookleftarrow{S^{3}}\xrightarrow[\equiv\,\omega\,=\,0]{\;\;\;{\bf
L}({\bf s}_{1})\,{\bf L}({\bf
s}_{2})\,\not=\,-1\;\;\;}\,S^{2}\times{S^{1}}\\!\xrightarrow{\;\;h\;\;}\,S^{2}\hookrightarrow{\mathrm{I\\!R}}^{3}.$
(105)

We thus see that the $S^{3}$ model not only reproduces Bell’s local model in
the flat geometry of $\mathrm{I\\!R}^{3}$, but also extends and enriches it to
reproduce the strong correlations predicted by quantum mechanics if the
Möbius-like twists in $S^{3}$ are taken into account.

##  VI Proposed Macroscopic Test of the 3-sphere Hypothesis

${{\bf s}_{2}}$${{\bf s}_{1}}$${\bf 1}$${\bf 2}$${\bf a}$${\bf
a^{\prime}}$${\bf b}$${\bf b^{\prime}}$A bomb at restTotal spin = 0

Figure 6: A bomb, initially at rest, explodes into two unequal fragments
carrying opposite spin angular momenta. Measurements of the spin components on
each fragment are performed at remote stations ${\bf 1}$ and ${\bf 2}$, along
two post-selected directions ${\bf a}$ and ${\bf b}$. The two spins
represented by the bivectors $-{\bf L}({\bf s}_{1})$ and ${\bf L}({\bf
s}_{2})$ are shown rotating in opposite senses about the same vector ${\bf
s}_{1}={\bf s}_{2}$, thanks to the conservation of the initial zero angular
momentum. This description using a coordinate system of fixed handedness
specified in the bivector subalgebra (5) is equivalent to Peres’s description
in Peres  with ${\bf s}_{2}=-{\bf s}_{1}$, because the bivector $-{\bf L}({\bf
s}_{1})$ can also be written as ${\bf L}(-{\bf s}_{1})$. See also the
discussion in footnotes 3 and 4, and in Section V and Appendix B.

In IJTP  we proposed a macroscopic experiment that may be able to detect the
signatures of the above spinorial sign changes under ${2\pi}$ rotations in the
form of strong singlet correlations derived in the previous sections. It is
worthwhile to discuss this experiment again in light of the above analysis,
with some improved considerations.

If realized, the experiment would determine whether Bell inequalities are
violated for the manifestly local and realistic 3-sphere model we considered
above. Needless to say, the proposed experiment has the potential to transform
our understanding of the relationship between classical and quantum physics.
It is based on a macroscopic variant of the local model considered by Bell
Bell-1964  and Peres Peres  we discussed above. In our proposal, we will
closely follow the example of an exploding bomb discussed by Peres Peres
(which is essentially a pedagogical illustration of Bell’s local model
Bell-1964 ). However, our proposal differs from Peres’s illustration in one
important respect. It involves measurements of the actual spin angular momenta
of two fragments of an exploding bomb rather than their normalized spin values
${\pm 1}$. The latter are to be computed only after all runs of the experiment
are completed, which can be executed either in outer space or in a terrestrial
laboratory. In the latter case, the effects of the external gravity of Earth
and air resistance would complicate matters, but it may be possible to choose
experimental parameters judiciously enough to compensate for such unwanted
effects.

With this assumption, consider a “bomb” made out of a hollow toy ball of
diameter, say, three centimeters. The thin hemispherical shells of uniform
density that make up the ball are snapped together at their rims in such a
manner that a slight increase in temperature would pop the ball open into its
two constituents with considerable force IJTP . A small lump of density much
greater than the density of the ball is attached on the inner surface of each
shell at a random location, so that, when the ball pops open, not only would
the two shells propagate with equal and opposite linear momenta orthogonal to
their common plane, but would also rotate with equal and opposite spin momenta
about a random axis in space, as shown in Fig. 6. The volume of the attached
lumps can be as small as a cubic millimeter, whereas their mass can be
comparable to the mass of the ball. This will facilitate some ${10^{6}}$
possible spin directions for the two shells, whose outer surfaces can be
decorated with colors to make their rotations easily detectable IJTP .

Now consider a large ensemble of such balls, identical in every respect except
for the relative locations of the two lumps (affixed randomly on the inner
surface of each shell). The balls are then placed over a heater—one at a
time—at the center of the experimental setup Peres , with the common plane of
their shells held perpendicular to the horizontal direction of the setup.
Although initially at rest, a slight increase in temperature of each ball will
eventually eject its two shells towards the observation stations, situated at
a chosen distance in mutually opposite directions. Instead of selecting the
directions ${\bf a}$ and ${\bf b}$ for observing spin components, however, one
or more contact-less rotational motion sensors—capable of determining the
precise direction of rotation—are placed near each of the two stations,
interfaced with a computer (an improved setup and further practical details
are discussed below). These sensors will determine the exact direction of the
spin angular momentum ${{\bf s}^{k}}$ (or ${-\,{\bf s}^{k}}$) for each shell
in a given explosion, without disturbing them otherwise so that their total
angular momentum would remain zero, at a designated distance from the center.
The interfaced computers can then record this data, in the form of a 3D map of
all such directions, at each station.

Once the actual directions of the angular momenta for a large ensemble of
shells on both sides are fully recorded, the two computers are instructed to
choose a pair of reference directions, ${\bf a}$ for one station and ${\bf b}$
for the other station—from the two 3D maps of already existing data—and then
calculate the corresponding pair of numbers ${-\,\text{sign}({\bf a}\cdot{\bf
s}^{k}_{1})=\pm\,1}$ and ${+\,\text{sign}({\bf b}\cdot{\bf
s}^{k}_{2})=\pm\,1}$. The standard correlation function for the two bomb
fragments with $p({\bf s}^{i})=\frac{1}{n}$ can then be calculated as

$\displaystyle{\cal E}({\bf a},\,{\bf b})\,$
$\displaystyle=\int_{S^{2}}\left\\{-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\right\\}\;p({\mathbf{s}}^{i})\,d{\mathbf{s}}^{i}$
$\displaystyle\approx\lim_{\,n\,\gg\,1}\\!\left[\frac{1}{n}\\!\sum_{k\,=\,1}^{n}\left\\{-\,\text{sign}({\bf
a}\cdot{\bf s}^{k}_{1})\;\text{sign}({\bf b}\cdot{\bf
s}^{k}_{2})\right\\}\right]\\!,$ (106)

together with

${\cal E}({\bf
a})=\\!\lim_{\,n\,\gg\,1}\\!\left[\frac{1}{n}\\!\sum_{k\,=\,1}^{n}\\{\text{sign}(\,+\,{\bf
s}^{k}_{1}\cdot{\bf a})\\}\right]\,=\,0$ (107)

and

${\cal E}({\bf
b})=\\!\lim_{\,n\,\gg\,1}\\!\left[\frac{1}{n}\\!\sum_{k\,=\,1}^{n}\\{\text{sign}(\,-\,{\bf
s}^{k}_{2}\cdot{\bf b})\\}\right]\\!=\,0\,,$ (108)

where ${n}$ is the total number of experiments performed and $k$ specifies a
trial number. As we discussed in the previous sections, naïve computation of
(106) by Bell Bell-1964  and Peres Peres  gives the weak correlations (56),
whereas the quaternionic 3-sphere model predicts strong correlations (64) by
taking the spinorial properties of the 3-sphere (3) into account.

Let us now turn to the practical problem of determining the direction of
rotation of a bomb fragment. In order to minimize the contributions of
precession and nutation about the rotation axis of the fragment, the bomb may
be composed of two flexible squashy balls instead of a single ball. The two
balls can then be squeezed together at the start of a run and released as if
they were two parts of the same bomb. This will retain the spherical symmetry
of the two constituent balls after the explosion, reducing their precession
and nutation effects considerably. Consequently, we assume that during the
narrow time window of the detection process the contributions of precession
and nutation are negligible. In other words, during this narrow time window
the individual spins will remain confined to the plane perpendicular to the
horizontal direction of the setup. This is because we will then have ${{\bf
s}={\bf r}\times{\bf p}}$, with ${\bf r}$ specifying the location of the
massive lump in the constituent ball and ${\bf p}$ being the ball’s linear
momentum. We can now exploit these physical constraints to determine the
direction of rotation of a constituent ball unambiguously, as follows.

Since only the directions of rotation are relevant for computing the
correlation function (1), it would be sufficient for our purposes to determine
only the direction of the vector ${\bf s}$ at each end of the setup. This can
be accomplished by arranging three (or more) successive laser screens
perpendicular to the horizontal path of the constituent balls, say about half
a centimeter apart, and a few judiciously situated cameras around them. To
facilitate the detection of the rotation of a ball as it passes through the
screens, the surface of the balls can be decorated with distinctive marks,
such as dots of different sizes and colors. Then, when a ball passes through
the screens, the entry points of a specific mark on the ball can be recorded
by the system of cameras. Since the ball would be spinning while passing
through the screens, the entry points of the same mark on the successive
screens would be located at different relative positions on the screens. The
rotation axis of the ball can therefore be determined unambiguously by
determining the plane spanned by the entry points and the right-hand rule. In
other words, the rotation axis can be determined as the orthogonal direction
to the plane spanned by the entry points, with the sense of rotation
determined by the right-hand rule. This procedure of determining the direction
of rotation can be followed through manually, or it can be automated with the
help of computer software. Finally, the horizontal distance from the center of
the setup to the location of the middle of the screens can be taken as the
distance of the rotation axis from the center of the setup. This distance
would help in establishing the simultaneity of the spin measurements at the
two ends of the setup.

Undoubtedly, there would be many sources of errors in a mechanical experiment
such as this. But if it is performed carefully enough, then our discussion
above strongly suggests that it will refute the prediction (56), which is
based on an incorrect calculation by Bell Bell-1964  and Peres Peres , and
vindicate the prediction (64) derived within the 3-sphere model.

Considering the practical difficulties in performing the above experiment, it
may seem that its simulation using computer software might provide easier
means to verify the hypothesis 1. However, any attempt to simulate the
experiment would require modeling physical space as a quaternionic 3-sphere,
because, according to the hypothesis, the above experiment is supposed to take
place within a non-flat $S^{3}$, not within a flat $\mathrm{I\\!R}^{3}$.
Providing the non-commutative background geometry of $S^{3}$ would be a nearly
impossible task for any software. Moreover, numerical simulations that do take
the non-commutativity

$[{\bf L}({\bf a}),\,{\bf L}({\bf b})]=-2\,{\bf L}({\bf a}\times{\bf b})$
(109)

of the spin variables ${\bf L(s)}$ into account are known to reproduce
sinusoidal correlations violating Bell inequality, as reported in the
references [8, 9, 10]. Consequently, Bell inequality is very likely to be
violated in the experiment proposed above, just as it is violated in the
microscopic quantum experiments [18].

##  VII Concluding Remarks

In this paper, we have compared Bell’s local model for the singlet
correlations set within ${\mathrm{I\\!R}^{3}}$ with our local model set within
$S^{3}$, and explained in detail why our model succeeds in reproducing the
strong quantum correlations while Bell’s model fails. For this purpose, we
have focused on the singlet correlations involving spin angular momenta,
whereas in most Bell-test experiments such as Aspect ; Weihs-666  what is
actually observed are correlations among the polarization states of entangled
photons. It turns out, however, that the 3-sphere model we have discussed in
this paper can easily accommodate such photon polarization states with
appropriate changes in notation, as we have previously demonstrated in Failure
and in Chapter 8 of Disproof .

Now the important feature of the experiments performed in Orsay Aspect  and
Innsbruck Weihs-666  using the polarization states of photons was that the
measurement settings at the remote polarizers were changed during the flight
of the two particles, thus removing any possibility of communication between
the two ends of the experiment. The Innsbruck team precluded the possibility
of communication between polarizers by using ultrafast random switching of the
orientations of the polarizers. On each side of the experiment, a local
computer registered the polarizer orientation and the result of each
measurement, with timing monitored by an atomic clock, and the data was
gathered and compared for correlation only after the end of each run. These
achievements prompted the principal investigator of the Orsay experiment to
make the following comment Nature-666 :

> I suggest we take the point of view of an external observer, who collects
> the data from the two distant stations at the end of the experiment, and
> compares the two series of results. This is what the Innsbruck team has
> done. Looking at the data a posteriori, they found that the correlation
> immediately changed as soon as one of the polarizers was switched, without
> any delay allowing for signal propagation …

But given the explanation we have offered in this paper by assuming the
geometry of physical space to be that of a quaternionic 3-sphere, this
immediate change in correlation should be no more puzzling than the sudden
change in the color of Dr. Bertlmann’s socks discussed by Bell Sock , as we
have explained elsewhere in considerable detail IEEE-2 . Consequently, in our
view classical men like Hamilton, Grassmann, Clifford, and Hopf would not have
been puzzled by the observed change in the correlation at all. They would have
understood the results of Orsay Aspect  and Innsbruck Weihs-666  experiments
in precisely the same local terms as we have explained them here; namely, in
terms of the algebra of rotations in the physical space ${S^{3}}$, otherwise
known as Clifford algebra Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; local ;
Failure .

While in this paper we have focused only on the correlations predicted by the
entangled singlet state IEEE-1 , elsewhere RSOS ; local  we have developed a
comprehensive local-realistic framework for understanding the origins of all
quantum correlations in terms of the algebraic representation space $S^{7}$ of
the quaternionic 3-sphere $S^{3}$. This demonstrates, constructively, that
local hidden variable theories are not ruled out by the Bell-test experiments.
Moreover, in Begs ; Oversight  we have also explained why Bell’s theorem fails
for the non-commuting local spin variables that respect the relation (109).
While a systematic analysis of the reasons for the failure of Bell’s theorem
for non-commuting variables is beyond the scope of the present paper (because
in this paper we are concerned with only a single expectation value (16)
rather than a sum of expectation values such as (1) required to prove Bell’s
theorem), in Begs  we have shown that the traditional view of the strength
$\pm 2\sqrt{2}$ derived in (65) as due to non-local influences is meaningful
only if one erroneously insists on linear additivity of expectation values of
non-commuting observables for individual dispersion-free states. Once this
error is corrected, the origin of the strength $\pm 2\sqrt{2}$ of correlations
can be traced to the Clifford-algebraic and geometric properties of physical
space Begs ; Oversight .

##  Appendix A Proofs of the Equalities (35) and (47)

In this appendix we prove the equalities (35) and (47), which amounts to
proving that the “product of limits equal to limits of product” rule holds for
quaternions and bivectors. To that end, we begin with the left-hand side of
(35):

$\displaystyle\bigg{[}$
$\displaystyle\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-I_{3}{\mathbf{a}}\,I_{3}{\mathbf{s}}_{1}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+I_{3}{\mathbf{s}}_{2}\,I_{3}{\mathbf{b}}\right\\}\right]$
(110)
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-(I_{3})^{2}\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+(I_{3})^{2}\,{\mathbf{s}}_{2}\,{\mathbf{b}}\right\\}\right]$
(111)
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{s}}_{2}\,{\mathbf{b}}\right\\}\right]$
(112)
$\displaystyle=\,\left[\mu_{1}{\mathbf{a}}\,{\mathbf{a}}\right]\left[-\mu_{2}{\mathbf{b}}\,{\mathbf{b}}\right]$
(113) $\displaystyle=-\mu_{1}\mu_{2}\,,$ (114)

where we have used the fact that all vectors involved in the model are unit
vectors and the fact that the pseudoscalar $I_{3}$ commutes with all other
elements of $\mathrm{Cl}_{3,0}$ and squares to $-1$. Similarly, the right-hand
side of (35) simplifies to

$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}$
$\displaystyle\left[\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{-I_{3}{\mathbf{a}}\,I_{3}{\mathbf{s}}_{1}\right\\}\left\\{+I_{3}{\mathbf{s}}_{2}\,I_{3}{\mathbf{b}})\right\\}\right]$
(115)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{-(I_{3})^{2}\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\left\\{+(I_{3})^{2}\,{\mathbf{s}}_{2}\,{\mathbf{b}})\right\\}\right]$
(116)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{+\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\left\\{-\,{\mathbf{s}}_{2}{\mathbf{b}})\right\\}\right]$
(117)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[-\,{\mathbf{a}}\,{\mathbf{s}}_{1}\,{\mathbf{s}}_{2}{\mathbf{b}}\right]$
(118)
$\displaystyle=\left[-\mu_{1}{\mathbf{a}}\,{\mathbf{a}}\,\mu_{2}{\mathbf{b}}\,{\mathbf{b}}\right]$
(119) $\displaystyle=-\mu_{1}\mu_{2}.$ (120)

Since the right-hand sides of (114) and (120) are equal, “the product of
limits equal to limits of product” rule holds.

Analogously, we can also prove the equality (47) by simplifying its left-hand
side and right-hand side, as follows:

$\displaystyle\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]$
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+I_{3}{\mathbf{s}}_{2}\,I_{3}{\mathbf{b}}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-I_{3}{\mathbf{a}}\,I_{3}{\mathbf{s}}_{1}\right\\}\right]$
(121)
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+(I_{3})^{2}\,{\mathbf{s}}_{2}\,{\mathbf{b}}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-(I_{3})^{2}\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\right]$
(122)
$\displaystyle=\bigg{[}\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{s}}_{2}\,{\mathbf{b}}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\right]$
(123)
$\displaystyle=\,\left[-\mu_{2}{\mathbf{b}}\,{\mathbf{b}}\right]\left[+\mu_{1}{\mathbf{a}}\,{\mathbf{a}}\right]$
(124) $\displaystyle=-\mu_{1}\mu_{2}$ (125)

and

$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}$
$\displaystyle\left[\left\\{-{\mathbf{D}}({\mathbf{b}})\,{\mathbf{D}}({\mathbf{a}})\right\\}\left\\{\,{\mathbf{L}}({\mathbf{s}}_{1})\,{\mathbf{L}}({\mathbf{s}}_{2})\right\\}\right]$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\left[\left\\{-I_{3}{\mathbf{b}}\,I_{3}{\mathbf{a}}\right\\}\left\\{I_{3}{\mathbf{s}}_{1}\,I_{3}{\mathbf{s}}_{2})\right\\}\right]$
(126)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\left[\left\\{-(I_{3})^{2}\,{\mathbf{b}}\,{\mathbf{a}}\right\\}\left\\{(I_{3})^{2}\,{\mathbf{s}}_{1}\,{\mathbf{s}}_{2})\right\\}\right]$
(127)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\left[\left\\{+\,{\mathbf{b}}\,{\mathbf{a}}\right\\}\left\\{-\,{\mathbf{s}}_{1}\,{\mathbf{s}}_{2})\right\\}\right]$
(128)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\\\
{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\end{subarray}}\left[-\,{\mathbf{b}}\,{\mathbf{a}}\,{\mathbf{s}}_{1}\,{\mathbf{s}}_{2}\right]$
(129)
$\displaystyle=\left[-\mu_{1}\mu_{2}\,{\mathbf{b}}\,{\mathbf{a}}\,{\mathbf{a}}\,{\mathbf{b}}\right]$
(130) $\displaystyle=\left[-\mu_{1}\mu_{2}\,{\mathbf{b}}\,{\mathbf{b}}\right]$
(131) $\displaystyle=-\mu_{1}\mu_{2}.$ (132)

Since the right-hand sides of (125) and (132) are equal, “the product of
limits equal to limits of product” rule holds.

##  Appendix B Questions and Answers

In no particular order, in this appendix I answer some questions concerning
the $S^{3}$ model for the singlet correlations presented in this paper.

Question 1: While several good plausibility arguments are presented in the
Introduction of the paper, they do not prove that physical space should be
modeled as $S^{3}$ rather than $\mathrm{I\\!R}^{3}$. How, then, is that
assumption justified in the absence of proof?

Answer 1: Apart from the plausibility arguments presented in the Introduction,
there are several good reasons that justify the assumption of $S^{3}$. While
this assumption is usually not made in the context of Bell’s theorem, the fact
that quaternions play a fundamental role in understanding the algebra,
geometry, and topology of physical space is well-known since the works of
Hamilton and Clifford in the 19th century and those of Pauli and Dirac in the
20th century. Dirac’s belt trick and Feynman’s plate trick are often used
pedagogically to illustrate this fact. It has contributed to the formulation
of Hypothesis 1 stated in the paper, which is a formal statement of my view
that the best terrestrial evidence for $S^{3}$ as the geometry of physical
space is the observed strong correlations in the Bell-test experiments.

But perhaps the strongest justification for the assumption of $S^{3}$ as
physical space stems from the fact that the traditional interpretation of
Bell’s theorem is recovered from the $S^{3}$ model within the flat geometry of
${{\mathrm{I\\!R}^{3}}}$ in which the characteristically non-trivial algebra,
geometry, and topology of ${S^{3}}$ are absent. Consequently, the absolute
upper bound of 2 on the Bell-CHSH combination (1) of expectation values is
respected within $\mathrm{I\\!R}^{3}$, as I have demonstrated, for example, in
[9] and [14]. In fact, it is not difficult to demonstrate that the results
presented in the current paper also reproduce the traditional interpretation
of Bell’s theorem in the flat geometry of ${{\mathrm{I\\!R}}^{3}}$, which is
usually taken for granted in the context of Bell’s theorem. There are several
different ways of appreciating this fact, as explained in Section X of [9],
each providing a different insight into how the standard interpretation of
Bell’s theorem is recovered in ${{\mathrm{I\\!R}}^{3}}$.

As demonstrated in Section VI of [9], one way to appreciate it is by analyzing
$S^{3}\rightarrow\mathrm{I\\!R}^{3}$ limit in the event-by-event numerical
simulations of the strong singlet correlations. A second way to appreciate it
is by setting the parallelizing torsion ${{\mathscr{T}}}$ in $S^{3}$ to zero,
as demonstrated in Section IX of [9], which reduces $S^{3}$ to
$\mathrm{I\\!R}^{3}$ as well as the absolute bound of $2\sqrt{2}$ on the Bell-
CHSH sum (1) of expectation values to 2. A third way to appreciate it is by
recognizing that the eigenvalue of the sum of quantum mechanical operators
involved in Bell-CHSH inequalities and Bell-test experiments necessarily
includes a purely geometrical contribution stemming from the $S^{3}$ geometry
of physical space, and when that contribution is set to zero the absolute
bound of $2\sqrt{2}$ on the Bell-CHSH sum (1) reduces to 2, as shown in [14].

But perhaps the best way to appreciate the recovery of the traditional
interpretation of Bell’s theorem within $\mathrm{I\\!R}^{3}$ from the $S^{3}$
model is geometrical, by recognizing that the strong singlet correlations
$-\cos(\eta_{{\mathbf{a}}{\mathbf{b}}})$ are a direct consequence of Möbius-
like twists in the Hopf bundle of $S^{3}$, as explained in Section V.

Question 2: Does the 3-sphere hypothesis proposed in the paper mean that an
observationally flat universe (which is the view preferred by the presently
dominant $\Lambda$-CDM model of cosmology) would essentially disqualify the
proposed $S^{3}$ model?

Answer 2: No. The hypothesis proposed in the Introduction of the paper stands
on its own. It has strong support from the cosmological observations, and from
the fact that the 3-sphere is a permitted spatial part of one of the well-
known solutions of Einstein’s field equations of general relativity, but it is
not dependent on this evidence from the cosmological observations.

Question 3: The topology of the quaternionic 3-sphere is a global property of
$S^{3}$. Does not that make the $S^{3}$ model a non-local realistic model
rather than a local-realistic one as claimed?

Answer 3: The topology of $S^{3}$ is indeed a global property. But that does
not make the $S^{3}$ model non-local realistic, at least for two fundamental
reasons. To begin with, $S^{3}$, when viewed as physical space as in the
model, is a spatial part of a well-known solution of Einstein’s field
equations of general relativity, which is a locally causal theory. Moreover,
within the context of Bell’s theorem, local causality is very specifically
defined in terms of measurement results, such as the functions
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})=\pm 1$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}^{i}_{2})=\pm 1$ defined in equations
(17) and (23). If these functions are local, then the model is said to be
local. And it is easy to verify that the functions are indeed local in the
sense espoused by Einstein and later mathematically formulated by Bell [4]:
Apart from the hidden variable ${\bf s}^{i}_{1}$, the result
${{\mathscr{A}}=\pm 1}$ depends only on the measurement direction ${\bf a}$,
chosen freely by Alice, regardless of Bob’s actions. And, analogously, apart
from the hidden variable ${\bf s}^{i}_{2}$, the result ${{\mathscr{B}}=\pm 1}$
depends only on the measurement direction ${\bf b}$, chosen freely by Bob,
regardless of Alice’s actions. In particular, the function
${{\mathscr{A}}({\bf a},\,{\bf s}^{i}_{1})}$ does not depend on ${\bf b}$ or
${\mathscr{B}}$ and the function ${{\mathscr{B}}({\bf b},\,{\bf s}^{i}_{2})}$
does not depend on ${\bf a}$ or ${\mathscr{A}}$. Moreover, the hidden
variables ${\bf s}^{i}_{1}$ and ${\bf s}^{i}_{2}$ do not depend on ${\bf a}$,
${\bf b}$, ${\mathscr{A}}$, or ${\mathscr{B}}$. Therefore, the model presented
in the paper is manifestly local-realistic.

Question 4: Is it possible that $S^{3}$ geometry is relevant only for the
space of entangled particles rather than physical space?

Answer 4: That would be a different hypothesis, not the one considered in the
paper. It would amount to assuming that $S^{3}$ is an internal space, as in
gauge theories. That hypothesis would not allow us to overcome Bell’s theorem
while respecting both the conditions of local realism and the observational
constraints set by the Bell-test experiments. In other words, under that
hypothesis, it would not be possible to derive the strong correlations (63),
together with vanishing averages (32) and (33) for the separate results
observed by Alice and Bob, while maintaining the strict locality condition
explained above. This is because Bell’s theorem remains valid within the flat
geometry of $\mathrm{I\\!R}^{3}$, as explained below and in Section X of Ref.
[9].

Question 5: What is meant by the statement “during the free evolution, the
spins do not change in either their senses or directions”? If the spins do not
change in their senses of directions, then in what sense is there “evolution”
of the spins?

Answer 5: Within the context of EPR-Bohm experiments, “free evolution” means
the motion of the emerging spins free from interactions before they reach the
detectors. Consequently, between the source and the detectors, the senses and
directions of the spins are preserved. However, they are still propagating
from the source to the detectors during this time. Soon after the constituent
fermions emerge from the source as shown in Fig. 1, they cease to interact
with each other appreciably but continue their journey toward the detectors.
Moreover, since the spin system is assumed to be isolated, there is no
external interaction between the spins and any other physical system until
they interact with the two detectors. Also, their combined spin angular
momentum would remain equal to zero, thanks to the law of the conservation of
angular momentum. Therefore, the two spins do not change in their senses or
directions while evolving between the source and the detectors. In other
words, the EPR-Bohm correlations are purely kinematical effects. This is well
understood since the first analysis of the singlet correlations using
fermionic spins was carried out by Bohm in Section 22.16 of his 1951 book on
quantum theory Bohm .

Question 6: The Stern-Gerlach device has a non-uniform electromagnetic field
that aligns incoming charged particles to its orientation. In the $S^{3}$
model, the physics of this interaction and the nature of the particle itself
are ignored, with the process of alignment being modeled with limit functions.
But while it may be natural to mathematically represent spins of fermionic
particles with bivectors, there is a vast difference between a Stern-Gerlach
detector and a particle spin, and yet in the $S^{3}$ model both are
represented with bivectors. Is that justified?

Answer 6: Yes, that is perfectly justified and natural. The details of a
specific experimental device are not relevant in the $S^{3}$ model, just as
they are not relevant in Bell’s local model set within $\mathrm{I\\!R}^{3}$.
In Bell’s local model, what is represented by a vector, such as “${\bf a}$”
appearing in the argument of the functions ${\mathscr{A}}({\bf a},{\bf
s}_{1}^{i})$, is an orientation of a detector, in three-dimensional physical
space, mathematically modeled as $\mathrm{I\\!R}^{3}$. The detectors such as
the Stern-Gerlach apparatus themselves are, of course, complicated physical
objects, whereas a vector is a geometrical object that models only the
orientation of a detector in physical space. That is fine when physical space
is a non-compact flat space such as $\mathrm{I\\!R}^{3}$. The compact space
$S^{3}$, however, has very different properties. I have defined it as a set of
quaternions:

$S^{3}:=\left\\{\,{\bf
q}(\beta,\,{\mathbf{r}})=\varrho_{r}\exp\left[{\mathbf{J}}({\mathbf{r}})\,\frac{\beta}{2}\right]\Bigg{|}\;\left|\left|\,{\bf
q}\left(\beta,\,{\mathbf{r}}\right)\,\right|\right|=\varrho_{r}\right\\}.$
(133)

As it is constituted by quaternions, an orientation of a detector within
$S^{3}$ is best represented by a pure quaternion or bivector such as ${\bf
D}({\bf a}):=I_{3}{\bf a}$. Recall that a bivector is also a geometrical
object, much like a vector within $\mathrm{I\\!R}^{3}$. It is an abstraction
of an oriented plane segment with only three properties: magnitude, direction,
and sense of rotation (see Fig. 7). A bivector ${\bf D}({\bf a}):=I_{3}{\bf
a}$ within $S^{3}$ is thus a counterpart of a vector “${\bf a}$” within
$\mathrm{I\\!R}^{3}$. As such, it provides the most adequate representation of
an orientation of a detector in $S^{3}$. A vector cannot serve this purpose
within $S^{3}$.

${\bf a}$${I_{{}_{3}}{\bf a}}$

Figure 7: A bivector, which represents a continuous binary rotation about some
axis, is understood as an abstraction of a directed plane segment, with only a
magnitude and a sense of rotation—i.e., clockwise (${-}$) or counterclockwise
(${+}$). Neither the depicted oval shape of its plane, nor its axis of
rotation such as ${\bf a}$, is an intrinsic part of the bivector ${I_{3}{\bf
a}}$. After [9].

Question 7: What is the motivation for using mathematical limits to model the
physical processes of detecting spin values?

Answer 7: Recall that the measurement function ${{\mathscr{A}}({\bf a},\,{\bf
s}^{i}_{1})}$ for Alice is defined in Bell’s local model using the sign
function:

${\mathscr{A}}({\bf a},\,{\bf s}^{i}_{1})=\text{sign}({\bf a}\cdot{\bf
s}^{i}_{1})=\pm 1.$ (134)

As noted before, the details of specific experimental devices, or the physical
workings of the measurement instruments, are not relevant in formulating
Bell’s local model, set within $\mathrm{I\\!R}^{3}$. What is modeled in (134)
is the experimental fact that measurement instruments like any Stern–Gerlach
device detect a spin by aligning its axis of rotation ${\bf s}^{i}_{1}$ with
the vector $\bf{a}$ that specifies the orientation of the device in space.
This produces an observable spot projected onto a screen, indicating that the
spin is “up” or “down” along the direction $\bf{a}$, which is then
mathematically represented by a number $+1$ or $-1$. Any other details of the
physical workings of the device are not relevant to Bell’s definition (134) of
the measurement results. But this definition is for the detection processes
taking place within $\mathrm{I\\!R}^{3}$. Its use of sign function is not
valid for the detection processes taking place within $S^{3}$. Since the
counterparts of the vectors $\bf{a}\in\mathrm{I\\!R}^{3}$ and
$\bf{s}_{1}\in\mathrm{I\\!R}^{3}$ within $S^{3}$ are the bivectors
${\mathbf{D}}({\mathbf{a}})$ and ${\mathbf{L}}({\mathbf{s}}_{1})$ representing
rotations about the axes $\bf{a}$ and $\bf{s}_{1}$ (see Fig. 7), what is
required is a limit of the product quaternion
${\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})$ representing the
physical interaction between ${\mathbf{D}}({\mathbf{a}})$ and
${\mathbf{L}}({\mathbf{s}}_{1})$ to produce the discrete results $\pm 1$
observed by Alice. Fortunately, definition (134) can be amended to accommodate
this for the detection processes taking place within $S^{3}$, so that the
correct definition of the measurement function ${\mathscr{A}}({\bf a},\,{\bf
s}^{i}_{1})$ is

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}^{i}_{1})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
(135)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-(I_{3}{\mathbf{a}})(I_{3}{\mathbf{s}}_{1})\right\\}$
(136) $\displaystyle=\mu_{1}\,,$ (137)

where the quaternion
$\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
is an element of set $S^{3}$,

$\mu_{1}:=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})=\pm 1,$ (138)

and the substitution rule for limits (and the fact that the unit bivector
$I_{3}{\mathbf{a}}$ squares to $-1$) is used to derive the equality (137).
Thus, definition (135) reproduces Bell’s definition (134) in
$\mathrm{I\\!R}^{3}$, with the use of limit function accomplishing the
experimentally required alignment of the vector $\bf{s}^{i}_{1}$ with the
vector $\bf{a}$. In other words, the use of limits to model the physical
processes within $S^{3}$ of spin detections seen in Stern–Gerlach devices is
not only required by the geometrical properties of $S^{3}$, but also natural.
It is also worth noting that the substitution rule for limits used above,
which essentially amounts to writing

$\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\,I_{3}\,{\mathbf{s}}_{1}\,=\,I_{3}\,{\mathbf{a}}$
(139)

because ${\mathbf{L}}({\mathbf{s}}_{1}):=I_{3}\,{\mathbf{s}}_{1}$ and
${\mathbf{L}}({\mathbf{a}}):=I_{3}\,{\mathbf{a}}$, follows at once from the
rules of Geometric Algebra. Moreover, it is easy to verify that the
substitution rule for limits itself can be derived rigorously for any
functions of vector quantities using the $\epsilon$–$\delta$ definition of
limits from basic calculus Calculus .

Question 8: Why are the directions of the constituent spins anti-aligned in
Bell’s model by setting ${\bf s}_{1}=-{\bf s}_{2}$, whereas in the $S^{3}$
model they are aligned to each other? There might be some subtlety in this
regard, because if one of the observers is rotated by 180 degrees in order to
face oncoming particles, then that will flip the horizontal directions of
their coordinate axes.

Answer 8: A fundamental difference between Bell’s model set within
$\mathrm{I\\!R}^{3}$ and the quaternionic 3-sphere model set within $S^{3}$ is
that, while in Bell’s model the spin angular momenta are incorrectly
represented by ordinary vectors such as ${\bf s}_{1}$ and ${\bf s}_{2}$ and
the conservation of initial zero spin is stipulated as ${\bf s}_{1}+{\bf
s}_{2}=0$ or as ${\bf s}_{1}=-{\bf s}_{2}$, in the $S^{3}$ model the spin
angular momenta are correctly represented by the bivectors
$-{\mathbf{L}}({\mathbf{s}}_{1})$ and ${\mathbf{L}}({\mathbf{s}}_{2})$ because
they are rotating in opposite senses. As a result, the conservation of net
zero spin is stipulated in equation (13) as

$\displaystyle-\,{\bf L}({\bf s}_{1})\,+\,{\bf L}({\bf s}_{2})=0\;$
$\displaystyle\Longleftrightarrow\;{\bf L}({\bf s}_{1})={\bf L}({\bf s}_{2})$
$\displaystyle\Longleftrightarrow\;{\bf s}_{1}=\,{\bf s}_{2}\,\equiv\,{\bf
s}.$ (140)

because $-{\mathbf{L}}({\mathbf{s}}_{1})=-I_{3}{\bf s}_{1}$ and
${\mathbf{L}}({\mathbf{s}}_{2})=I_{3}{\bf s}_{2}$, which is equivalent to the
following conservation condition stated in equation (14):

${\bf L}({\bf s}_{1})\,{\bf L}({\bf s}_{2})=\,I_{3}{\bf s}\,I_{3}{\bf
s}=(I_{3})^{2}\,{\bf s}\,{\bf s}=-1.$ (141)

Once these mathematical relations are strictly followed, no verbal description
with regard to what will happen in a given specific scenario such as “what if
one of the observers is rotated by 180 degrees to face the oncoming
particles,” etc. is necessary.

Question 9: Bell’s general argument and other impossibility arguments do not
assume that space has the structure of $\mathrm{I\\!R}^{3}$.

Answer 9: Although widely believed, this assertion is quite incorrect. Bell’s
theorem and other impossibility arguments do implicitly assume that physical
space has the structure of $\mathrm{I\\!R}^{3}$. It is unfortunate that
neither Bell nor other proponents of his theorem state this assumption
explicitly. The only exception is a brief remark by Bell, quoted in the
Introduction of the paper, which reads: “The space time structure has been
taken as given here. How then about gravitation?” By ignoring the qualitative
features of spacetime, Bell and his followers have neglected the geometrical
and topological properties of physical space in their theorems, such as the
non-commutativity of quaternions that constitute the 3-sphere. It is evident
from the model considered in the paper that when these properties are taken
into account, the observed strong singlet correlations can be easily derived.

Question 10: Bell’s general argument and other impossibility arguments do not
rely on the particular local model discussed in Section 3 of Bell’s 1964 paper
[4] and in Peres’s book [16].

Answer 10: This is not quite correct. The mathematical core of Bell’s theorem,
while not dependent on the specific local model discussed in Section 3 of his
paper, is nevertheless based on the correlations predicted by the singlet
state. As such, the local model in Section 3 of his paper is of considerable
interest. It is very much a part of the impossibility claim of the theorem.

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