## Abstract

The exceptional Lie group *E*_{8} plays a prominent role in both mathematics and theoretical physics. It is the largest symmetry group associated with the most general possible normed division algebra, namely, that of the non-associative real octonions, which—thanks to their non-associativity—form the only possible closed set of spinors (or rotors) that can parallelize the 7-sphere. By contrast, here we show how a similar 7-sphere also arises naturally from the algebraic interplay of the graded Euclidean primitives, such as points, lines, planes and volumes, which characterize the three-dimensional conformal geometry of the ambient physical space, set within its eight-dimensional Clifford-algebraic representation. Remarkably, the resulting algebra remains associative, and allows us to understand the origins and strengths of all quantum correlations locally, in terms of the geometry of the compactified physical space, namely, that of a quaternionic 3-sphere, *S*^{3}, with *S*^{7} being its algebraic representation space. Every quantum correlation can thus be understood as a correlation among a set of points of this *S*^{7}, computed using manifestly local spinors within *S*^{3}, thereby extending the stringent bounds of ±2 set by Bell inequalities to the bounds of *raison d’être* of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of *S*^{3} and *S*^{7}.

## 1. Introduction

The central source of intrinsic coherence, geometrical elegance and empirical success of Einstein’s theory of gravity is undoubtedly its strict adherence to local causality [1,2]. Indeed, despite the phenomenal empirical success of Newton’s theory of gravity for over two centuries [3], its founding on the unexplained ‘action at a distance’ was a reason enough for Einstein to search for its locally causal generalization. Today, we face a similar challenge in search of a theory that may unify quantum theory with Einstein’s theory of gravity. But in sharp contrast to Einstein’s theory, quantum theory seems to harbour a peculiar form of non-signalling non-locality, as noticed long ago by Einstein *et al.* [4]. They hoped, however, that quantum theory can be completed into a locally causal theory with addition of ‘hidden’ parameters or supplementary variables. Today such a hope of completing quantum theory into a realistic and local theory envisaged by Einstein is widely believed to have been dashed by Bell’s theorem [5,6], its variants [7] and the related experimental investigations [8–13]. Indeed, the claim of Bell’s theorem is remarkably comprehensive in scope: *no physical theory which is local and realistic as hoped for by Einstein can reproduce all of the strong correlations predicted by quantum mechanics* [14].

By contrast, our primary concern in this paper is not Bell’s theorem, but understanding the origins and strengths of all quantum correlations in terms of the algebraic, geometrical and topological properties of the physical space in which we are confined to perform our experiments. In our view, Bell’s theorem is a distraction that prevents us from understanding the true origins of quantum correlations, especially because it is neither a theorem in the strict mathematical sense, nor a result within quantum theory itself. Indeed, not a single concept from quantum theory is used in the derivation of the Bell-CHSH inequalities [5,6,15]. It is, in fact, an argument that depends on a number of physical assumptions about what is and what is not possible within any locally causal theory, and these assumptions can be and have been questioned before [16,17]. Consequently, by circumventing Bell’s argument, in this paper we set out to explain the origins and strengths of all quantum correlations within a locally causal framework of octonion-like spinors, which are constructed using a geometric algebra [18,19] of rudimentary Euclidean primitives, such as points, lines, planes and volumes. This is accomplished by recognizing and overcoming two neglected shortcomings of Bell’s argument [5,6,14,15,20]. The first of these shortcomings, which is discussed in greater detail in §4.2, amounts to averaging over measurement events in the derivation of the experimentally violated absolute bound of 2 on the CHSH string of expectation values that are *impossible* to occur in *any* possible world, classical or quantum, stemming from a mistaken application of the criterion of reality propounded by EPR [4,7]. The second shortcoming of Bell’s argument stems from the unjustified identification of the *image* {+1,−1} of the measurement functions, which represents the actual measurement results in Bell’s prescription [5,6], with the *co-domain* of these functions, which is neither specified by Bell explicitly nor observable directly in the so-called Bell-test experiments [8–14]. An explicit specification of the latter, however, is a prerequisite for the very definition of a mathematical function [16]. By contrast, in our prescription (3.1) of measurement results discussed in §3.1, the locally unobservable co-domain *S*^{7} of the measurement functions is explicitly specified with considerable detail, without compromising Bell’s pristine bivalued prescription, ±1, for the actually observed measurement results. It incorporates the Clifford algebraic properties of the physical space in which all such experiments are necessarily situated and performed [8–13,20].

As noted above, however, our primary focus in this paper is not on Bell’s theorem, but on understanding the origins and strengths of all quantum correlations as a consequence of the geometry and topology of the physical space (or more generally of space–time). As quantum correlations are necessarily observed within the confines of space–time, it is natural to view them as correlations among measurement events in space–time—i.e. among the ‘clicks’ of a set of detectors configured within space–time. On the other hand, what is actually recorded in the Bell-test experiments are coincidence counts among bivalued measurement results, observed simultaneously within space at a given time [8–13]. Therefore, without loss of generality, we will restrict our analysis to the physical space. With that in mind, in the next section we extensively review the algebraic properties of the compactified physical space, captured in definition (2.16), which is a quaternionic 3-sphere, and construct its algebraic representation space (2.60), which is an octonion-like 7-sphere. As such parallelizable 3- and 7-spheres play a vital role (see footnote 3) in our local-realistic framework, we have devoted a brief appendix (appendix A) to discuss their wider significance in physics and mathematics at a pedagogical level [16]. Our central theorem concerning the origins of all quantum correlations is then stated and proved in §3.2.

The proof presented in §3.2 includes a local-realistic derivation of the simplest yet emblematic quantum correlations—namely, those predicted by the rotationally invariant singlet or EPR-Bohm state—the strengths of which are well known to violate the theoretical bounds of ±2 set by the Bell-CHSH inequalities, in the Bell-test experiments [8–13]. Then, in §3.3, we derive the closely related Tsirel’son’s bounds of *all* quantum correlations within our framework. In the subsequent §3.4, we then explain the geometrical reasons for the fragility of quantum correlations as a counterpart of that of quantum entanglement. This brings us to §3.5 in which we derive the strong correlations predicted by the rotationally non-invariant 4-particle GHSZ state [7], together with the proof of Bell’s condition of factorizability within *S*^{7} in appendix B (a similar proof also goes through within *S*^{3}).

In §4.1, we then point out that the predictions of our local-realistic *S*^{7} framework is not in conflict with what is actually observed in the Bell-test experiments [8–13], as they simply reproduce the predictions of quantum mechanics [5–7]. In the subsequent §4.2, in the light of the widespread belief in Bell’s theorem, we reveal a serious oversight in Bell’s argument in some detail, independently of the constructive counterexamples provided by our *S*^{7} model for the strong correlations. This brings us to §4.3 in which we present an analytical disproof of the GHZ variant of Bell’s theorem, which does not involve Boole-type mathematical inequalities (see footnote 5) used by Bell in his original argument. In §5, we then present event-by-event numerical simulations of the 2-particle EPR-Bohm and 4-particle GHSZ correlations predicted by our local-realistic framework based on *S*^{7}. Finally, in §6, we summarize our findings.

## 2. Modern perspective on the Euclidean primitives

In physical experiments—which are usually confined to the three-dimensional physical space by necessity—we often measure relevant quantities by setting up a Cartesian coordinate system {*x*,*y*,*z*} in that space. Mathematically, this is equivalent to identifying the Euclidean space ^{3}. In practice, we sometimes even think of IR^{3} as *the* Euclidean space. Euclid himself, however, did not think of

It is, however, not always convenient to model the physical space in the spirit of Euclid. Therefore, in practice, we tend to identify ^{3} whenever possible. But there is no intrinsic way of identifying the two spaces in this manner without introducing an *unphysical* element of arbitrarily chosen coordinate system. This difficulty is relevant for understanding the origins of quantum correlations, for time and again we have learned that careless introduction of unphysical ideas in physics could lead to distorted views of the physical reality [16,21]. An intrinsic, coordinate-free representation of the Euclidean space is surely preferable, if what is at stake is the very nature of the physical reality (cf. §4.2).

Fortunately, precisely such a representation of *all treated on equal footing*. Given a set {**e**_{x},**e**_{y},**e**_{z}} of basis vectors representing lines in **e**_{i}} being a set of anti-commuting orthonormal vectors in IR^{3} such that **e**_{j}**e**_{i}=−**e**_{i}**e**_{j} for any *i*,*j*=*x*,*y*, or *z*. More generally, the unit vectors **e**_{i} satisfy the fundamental geometric or Clifford product in this (by definition) *associative* algebra,
**e**_{i}∧**e**_{j})^{2}=−1. Any vector

The normalized volume element *I*_{3} thus represents an element of the highest grade in the corresponding algebra, namely grade-3. It is also referred to as a pseudo-scalar, dual to the scalar, which in turn is the lowest possible grade in the algebra:
*I*_{3} implying (*I*_{3})^{2}=−1, and the duality relationship between the elements *Ω* of arbitrary grades is defined as
*Ω* and scalar part 〈 〉_{s} of the product of mixed-grade vectors **X** and **Y** of *n*-components defined as
**e**_{k} of grade-1 can be easily recovered from the unit bivectors **e**_{i}∧**e**_{j} of grade-2 using the above duality relation:
**e**_{i} of grade-1, three orthonormal bivectors **e**_{j}**e**_{k} of grade-2 and a trivector **e**_{i}**e**_{j}**e**_{k} of grade-3. Respectively, they represent points, lines, planes and volumes in ^{3}, there are 2^{3}=8 ways to combine the vectors **e**_{i} using the geometric product (2.2) such that no two products are linearly dependent, the resulting algebra, Cl_{3,0}, is a linear vector space of 2^{3}=8 dimensions, spanned by these graded bases:

### 2.1. One-point compactification of the three-dimensional Euclidean Space

The physical space represented by the above algebraic model is, however, not quite satisfactory. Stemming from an arbitrarily chosen origin, its points run off to infinity along every radial direction [19]. Moreover, there is no reason for these infinitely many infinities—which can be approached from infinitely many possible different directions—to be distinct from one another. It is therefore natural to assume that one and the same infinity is encountered along any radial direction, and identify it with a single point. One way to achieve this is by compactifying the space

Intuitively, this procedure is not difficult to understand with a two-dimensional analogue of *S*^{2} (cf. figure 3). If we surgically remove a single point from this surface and stretch the remainder out to infinity in every radial direction (like an infinite bed-sheet), then it provides an intuitive model for the two-dimensional Euclidean space, *S*^{2}-balloon from an

Similarly, figure 2 depicts an inverse stereographic projection of *S*^{3}, by the embedding map ^{4}. The crucial observation here is that, as an arbitrary vector ^{4}:
** ϕ**(

**x**) [or

**(**

*ψ***x**)] thus transforms the entire space

^{4}, thereby accomplishing a one-point compactification of

*S*

^{3}projects to the same angle between the projected curves on

*S*

^{3}projecting to an exact circle on

Now the tangent bundle of *S*^{3} happens to be trivial: *TS*^{3}=*S*^{3}×IR^{3}. This renders the tangent space at each point of *S*^{3} to be isomorphic to IR^{3}. Consequently, local experiences of the experimenters within *S*^{3} are no different from those of their counterparts within *S*^{3}, however, is clearly different from that of IR^{3} [16,17]. In particular, the triviality of the bundle *TS*^{3} means that *S*^{3} is parallelizable. As a result, a global *anholonomic* frame can be defined on *S*^{3} that fixes each of its points uniquely. Such a frame renders *S*^{3} diffeomorphic to the group SU(2)—i.e. to the set of all unit quaternions:
** ξ**(

**r**) is a bivector rotating about

**r**∈IR

^{3}with the rotation angle

*θ*in the range 0≤

*θ*<4

*π*. In terms of the even sub-algebra of (2.10), the bivector

**(**

*ξ***r**)∈

*S*

^{3}can be parametrized by the dual vector

**r**=

*r*

_{x}

**e**

_{x}+

*r*

_{y}

**e**

_{y}+

*r*

_{z}

**e**

_{z}∈IR

^{3}as

*ξ*^{2}(

**r**)=−1. Each configuration of any rotating rigid body can thus be represented by a quaternion

**q**(

*θ*,

**r**), which in turn can always be decomposed into a product of two bivectors, say

**(**

*ξ***u**) and

**(**

*ξ***v**), belonging to an

*S*

^{2}⊂

*S*

^{3},

*θ*being its rotation angle from

**q**(0,

**r**)=1. Note also that

**q**(

*θ*,

**r**) reduces to ±1 as

*κ*=0,1 or 2.

### 2.2. Conformal completion of the Euclidean primitives

Our interest now lies in the point *non-zero* vector of zero norm:
*null vector* in Conformal Geometric Algebra^{1} [18]. It is introduced to represent both finite points in space as well as points at infinity [19]. As points thus defined are null-dimensional or dimensionless, addition of

Equipped with **e**_{x}**e**_{y},**e**_{z}**e**_{x},**e**_{y}**e**_{z}} of bivectors as the orthonormal basis of the space **e**_{i}, the product of the basis bivectors works out to be
*I*_{3} for the Euclidean space (2.21):
^{3}. But we can now close it with the null vector

where we have used the subscript *c* on *I*_{c} to indicate that it is a volume element of the compact 3-sphere, *S*^{3}. As we noted earlier, in the Euclidean space the reverse of *I*_{3} is *I*_{c} is
*Ω* of any grade is given by
*I*_{3}, the dual of +1 in the conformal space also works out to be
^{4}, as depicted in figure 2. In this higher dimensional space, *unit* vector,
*even* sub-algebra of the 2^{4}=16-dimensional Clifford algebra Cl_{4,0}. Thus, a one-dimensional subspace—represented by the unit vector ^{4}—represents a *null*-dimensional space—i.e. the infinite point of *S*^{3}.

### 2.3. Orientation of representation space as a binary degree of freedom

Before we explore the properties of the above vector space, let us endow it with one more degree of freedom without which it is unjustifiably restrictive. To that end, we first recall what is meant by an orientation of a vector space [22]:

*Definition of orientation*: An orientation of a finite-dimensional vector space *b*_{1},…,*b*_{n}}, which determines the same orientation of *b*′_{1},…,*b*′_{n}} if *b*′_{i}=*ω*_{ij}*b*_{j} holds with det(*ω*_{ij})>0, and the opposite orientation of *b*′_{1},…,*b*′_{n}} if *b*′_{i}=*ω*_{ij}*b*_{j} holds with det(*ω*_{ij})<0.

Thus, each positive-dimensional real vector space has precisely two possible orientations, which we will denote as λ=+1 or λ=−1. More generally an oriented smooth manifold consists of that manifold together with a choice of orientation for each of its tangent spaces. It is worth noting that orientation is a *relative* concept. The orientation of a tangent space *b*_{1},…,*b*_{n}} is meaningful only with respect to that defined by the equivalence class of ordered basis {*b*′_{1},…,*b*′_{n}}, and vice versa.

Now in geometric algebra the choice of the sign of the unit pseudoscalar amounts to choosing an orientation of the space [18,19]. In our three-dimensional Euclidean space defined in (2.21) with an orthonormal set of unit bivector basis, *I*_{3}=**e**_{x}**e**_{y}**e**_{z} picks out the right-handed orientation for *ab initio*. But the algebra itself does not fix the handedness of the basis. In our presentation above, we could have equally well started out with a left-handed set of bivectors in (2.21) by letting −*I*_{3} instead of +*I*_{3} select the basis. Instead of the representation space (2.31), we would have then ended up with the space
^{7}<0. Consequently,

### 2.4. Representation space K λ remains closed under multiplication

As an eight-dimensional linear vector space, *closed* under multiplication. Suppose **X** and **Y** are two unit vectors in **X** and **Y** can be expanded in the basis of **Z**=**X****Y**:
**X** and **Y** in **X** and **Y** in **Z**=**X****Y** we have
*unit* vectors in *f*_{μνρ} is a totally anti-symmetric permutation tensor with only non-vanishing independent components being
*l*_{μνρ} is a totally *symmetric* permutation tensor with only non-vanishing independent components being
**X** and **Y** within

### 2.5. Representation space as a set of orthogonal pairs of quaternions

In his seminal works, Clifford introduced the concept of dual numbers, *z*, analogous to complex numbers, as follows:
*ε* is the dual operator, *r* is the real part and *d* is the dual part [19]. Similar to how the ‘imaginary’ operator *i* is introduced in the complex number theory to distinguish the ‘real’ and ‘imaginary’ parts of a complex number, Clifford introduced the dual operator *ε* to distinguish the ‘real’ and ‘dual’ parts of a dual number. The dual number theory can be extended to numbers of higher grades, including to numbers of composite grades, such as quaternions:
**q**_{r} and **q**_{d} are quaternions and *ϱ* and rewritten as
*ε* with the duality operator **q**_{d} at infinity. Note that we continue to write *ε* as if it were a scalar because it commutes with **q**_{d}. Comparing (2.50) and (2.52) with (2.35) we can now rewrite **q**_{r} be orthogonal to its dual **q**_{d}:
*ε*^{2}=+1, which gives
**q** in (2.16), it is not difficult to see that **q**_{r} must be orthogonal to **q**_{d}.

But there is more to the normalization condition *S*^{0}, *S*^{1}, *S*^{3} and *S*^{7} (cf. appendix A). To verify it, consider a product of two different members of the set *ϱ* of *S*^{3} to *S*^{7} will provide the conformal (see footnote 1) counterpart of the algebra Cl_{3,0} given in (2.10):
*associative* (but of course non-commutative) algebra.

Thus, to summarize this section, we started out with the observation that the correct model of the physical space is provided by the algebra of Euclidean primitives, such as points, lines, planes and volumes, as discovered by Grassmann and Clifford in the nineteenth century. We then recognized the need to ‘close’ the Euclidean space with a non-zero null vector *S*^{3}. The corresponding algebraic representation space of *S*^{7}. It is quite remarkable that *S*^{3} and *S*^{7}, which are the two spheres associated with the only two non-trivially possible normed division algebras, namely the quaternionic and octonionic algebras [16,23–29], emerge in this manner from the elementary algebraic properties of the Euclidean primitives (cf. appendix A). Unlike the non-associative octonionic algebra and the exceptional Lie groups such as *E*_{8} it gives rise to, however, the compact 7-sphere we have arrived at corresponds to an *associative* Clifford (or geometric) algebra [19,21], as noted above. And yet, as we shall soon see, it is sufficient to explain the origins of *all* quantum correlations. It remains to be seen what role, if any, the exceptional groups such as *G*_{2} and *E*_{8} may eventually play when the current framework is developed further.

## 3. Derivation of quantum correlations from Euclidean primitives

### 3.1. Constructing measurement functions in the manner of Bell

In order to derive quantum correlations predicted by arbitrary quantum states, our first task is to construct a set of measurement functions of the form:
*local* detections of binary measurement results, **n**. They are of the same realistic and deterministic form as that considered by Bell^{2} [5–7], except for their locally unobservable co-domain, which we have taken to be the algebraic representation space *S*^{7} constructed above, embedded in IR^{8}. For an explicit construction of the functions ^{8} analogous to (2.62):
**N**(**n**_{r},**n**_{d},−*n*_{7},λ) of the above *S*^{7}-vector,
^{2}=1. For our purposes, it will suffice to represent the detectors with the special case of this non-scalar part for which *n*_{7}≡0:

Next recall that, although global topology of *S*^{3} is different from that of IR^{3}, local experiences of experimenters within *S*^{3} are no different from those of their counterparts within IR^{3}, not the least because the tangent space at any point of *S*^{3} is isomorphic to IR^{3}. With this in mind, we identify the counterparts of measurement directions **n** within **n**_{r}+**n**_{d}*ε*_{+} within its algebraic representation space *S*^{7}. Then **n** relates to **D**(**n**_{r},**n**_{d},0) as
**D**(**n**_{r},**n**_{d},0) in (3.2) as a detector of the physical system represented by **N**(**n**_{r},**n**_{d},0,λ), originating in the initial state λ and producing the measurement results **n**↔**n**_{r}+**n**_{d}*ε*_{+} within IR^{3}. Indeed, using the definitions (3.4)–(3.14), it is easy to verify that
**a** and **b** the geometric product **N**(**a**_{r},**a**_{d},0,λ)**N**(**b**_{r},**b**_{d},0,λ) is highly non-trivial, as we saw in (2.43):
**a**⋅**b**=1 for the special case **a**=**b** and the normalization conditions for **a**_{r} and **b**_{d}, giving
*k*, we can now define the measurement functions (3.1) as maps of the form
**a** and **b**, as
*S*^{7} is a fair coin. Evidently, the functions ^{k} originating in the overlap of the backward lightcones of **a** [5,6]. And likewise, apart from the common cause λ^{k}, the event **b**. In particular, the function **b** or B, and the function **a** or A. This leads us to the following remarkable theorem.

### 3.2. Quantum correlations from the algebra of Euclidean primitives

### Theorem 3.1

*Every quantum mechanical correlation can be understood as a classical, local, realistic and deterministic correlation among a set of points of S*^{7} *constructed above, represented by maps of the form defined in (3.20) and (3.21).*

### Proof.

Recall that—as von Neumann recognized in his classic analysis [31]—regardless of the model of physics one is concerned with—whether it is the quantum mechanical model or a hidden variable model—it is sufficient to consider expectation values of the observables measured in possible states of the physical systems, since probabilities are but expectation values of the indicator random variables. Thus, probability *P*(*E*) of event *E* is expectation value

To that end, consider an arbitrary quantum state *Ψ*〉 or H. In particular, the state |*Ψ*〉 can be as entangled as one may wish [16]. Next, consider a self-adjoint operator **n**^{1}, **n**^{2},**n**^{3},**n**^{4},**n**^{5}, etc. The quantum mechanically expected value of this observable in the state |*Ψ*〉 is then defined by
**n**^{1}=**a**↔**a**_{r}+**a**_{d}*ε*_{+}, **n**^{2}=**b**↔**b**_{r}+**b**_{d}*ε*_{+}, etc., the corresponding local-realistic expectation value for the same system can be written as
*ρ*(λ),
**x** and **y**, depending in general on the measurement directions **a**, **b**, **c**, **d**, etc. Consequently, we have
^{k}, as in (3.13), is a fair coin. Here, equation (3.37) follows from equation (3.36) by using equation (3.13), which now takes the form
^{k}=±1. We can now identify the above local-realistic expectation with its quantum mechanical counterpart:

It is instructive to evaluate the sum in equation (3.35) somewhat differently to bring out the fundamental role played by the orientation λ^{k} in the derivation of the strong correlations (3.38). Instead of assuming λ^{k}=±1 to be an orientation of *S*^{7} as our starting point, we may view it as specifying the ordering relation between **N**(**x**_{r},**x**_{d},0,λ^{k}=±1) and **N**(**y**_{r},**y**_{d},0,λ^{k}=±1) and the corresponding detectors **D**(**x**_{r},**x**_{d},0) and **D**(**y**_{r},**y**_{d},0) with 50/50 chance of occurring, and only subsequently identify it with the orientation of *S*^{7}. Then, using relations (3.13) and (3.32), the sum in equation (3.35) can be evaluated directly by recognizing that in the right- and left-oriented *S*^{7} the following geometrical relations hold:
^{k} thus alternates the algebraic order of **N**(**x**_{r},**x**_{d},0,λ^{k}=±1) and **N**(**y**_{r},**y**_{d},0,λ^{k}=±1) *relative* to the algebraic order of the detectors **D**(**x**_{r},**x**_{d},0) and **D**(**y**_{r},**y**_{d},0). Consequently, the sum (3.35) reduces to
^{k} of *S*^{7} is a fair coin. Here

Evidently, the above method of calculating suggests that a given initial state λ of the physical system can indeed be viewed as specifying an ordering relation between **N**(**n**_{r},**n**_{d},0,λ) and the detectors **D**(**n**_{r},**n**_{d},0) that measure it:
**N**(**n**_{r},**n**_{d},0,λ) and **D**(**n**_{r},**n**_{d},0) are equivalent to the alternatively possible orientations of *S*^{7}.

#### 3.2.1. Special case of a two-level system entangled in the singlet state

Now, to complete the above proof of theorem 3.1, we must prove the step from equations (3.30) to (3.31). To that end, let us first consider observations of the spins of only two spin-**a** and **b**, which may be located at a space-like distance from one another [30]. As initially the emerging pair has zero net spin, its quantum mechanical state is described by the entangled singlet state
** σ**⋅

**z**|

**z**,±〉=±|

**z**,±〉 describing the eigenstates of the Pauli spin ‘vector’

**in which the particles have spin ‘up’ or ‘down’ along**

*σ***z**-axis, in the units of

**a**and

**b**. Here 𝟙 is the identity matrix. The corresponding locally causal description of this emblematic system within our framework thus involves only two contexts,

**n**

^{1}=

**a**↔

**a**

_{r}+

**a**

_{d}

*ε*

_{+}and

**n**

^{2}=

**b**↔

**b**

_{r}+

**b**

_{d}

*ε*

_{+}, with measurement results defined by the functions

**s**

_{1}↔

**s**

_{r1}+

**s**

_{d1}

*ε*

_{+}and

**s**

_{2}↔

**s**

_{r2}+

**s**

_{d2}

*ε*

_{+}represent the directions of the two spins emerging from the source.

Next, recalling that physically all bivectors ** ξ**(

**n**)∈

*S*

^{2}⊂

*S*

^{3}represent spins [16,17], we require that the total spin-zero angular momentum for the initial or ‘complete’ state associated with the above measurement functions is conserved,

**N**(

**s**

_{r},

**s**

_{d},0,λ

^{k}), this is equivalent to the condition

*S*

^{3}. Here, it leads to the following statistical equivalence, which can be viewed also as a geometrical identity:

*S*

^{3}; equation (3.61) follows from equation (3.60) by recalling that scalars λ

^{k}commute with the elements of all grades; equation (3.62) follows from equation (3.61) because λ

^{2}=+1, and by removing the superfluous limit operations; equation (3.63) follows from equation (3.62) by using the geometric product (3.19); equation (3.64) follows from equation (3.63) by using relations (3.13); and finally equation (3.65) follows from equation (3.64) by using equation (3.18) and because the scalar coefficient of

**D**vanishes in the

^{k}is a fair coin. This proves that singlet correlations (see footnote 3) are correlations among the scalar points of a quaternionic

*S*

^{3}.

As we did above for the general case, let us again evaluate the sum in equation (3.62) somewhat differently to bring out the crucial role played by λ^{k} in the derivation of the correlations (3.65). Using relations (3.13) and (3.19), sum (3.62) can be evaluated directly by recognizing that in the right- and left-oriented *S*^{7} the following geometrical relations hold:
^{k} thus alternate the *relative* order of **D**(**a**_{r},**a**_{d},0) **D**(**b**_{r},**b**_{d},0). As a result, sum (3.62) reduces to
^{k} of *S*^{7} is a fair coin. Here

The above method of calculating the correlations suggests that a given initial state λ of the physical system can be viewed also as specifying an ordering relation between **N**(**n**_{r},**n**_{d},0,λ) and the detectors **D**(**n**_{r},**n**_{d},0) that measure it:
**N**(**n**_{r},**n**_{d},0,λ) and **D**(**n**_{r},**n**_{d},0) are equivalent to the alternatively possible orientations of *S*^{7}.

#### 3.2.2. Conservation of the initial spin-0 from the twist in the Hopf bundle of *S*^{3}

Note that, apart from the initial state λ^{k}, the only other assumption used in the above derivation is that of the conservation of spin angular momentum (3.55). These two assumptions are necessary and sufficient to dictate the singlet correlations:
*S*^{3}≅*SU*(2). Recall that locally (in the topological sense) *S*^{3} can be written as a product *S*^{2}×*S*^{1}, but globally it has no cross-section [32,33]. It can be viewed also as a principal U(1) bundle over *S*^{2}, with the points of its base space *S*^{2} being the elements of the Lie algebra su(2), which are pure quaternions or bivectors [16,30,34]. The product of two such bivectors are, in general, non-pure quaternions of the form (2.18), and are elements of the group SU(2) itself. That is to say, they are points of the bundle space *S*^{3}, whose elements are the preimages of the points of the base space *S*^{2} [32,33]. These preimages are 1-spheres, *S*^{1}, called Hopf circles, or Clifford parallels [35]. As these 1-spheres are the fibres of the bundle, they do not share a single point in common. Each circle threads through every other circle in the bundle as shown in figure 6, making them linked together in a highly non-trivial configuration. This configuration can be quantified by the following relation among the fibres [34]:
*e*^{iψ−} and *e*^{iψ+}, respectively, are the U(1) fibre coordinates above the two hemispheres *H*_{−} and *H*_{+} of the base space *S*^{2}, with spherical coordinates (0≤*θ*<*π*,0≤*ϕ*<2*π*); *ϕ* is the angle parametrizing a thin strip *H*_{−}∩*H*_{+} around the equator of *S*^{2} [*θ*∼*π*/2] and *e*^{iϕ} is the transition function that glues the two sections *H*_{−} and *H*_{+} together, thus constituting the 3-sphere. It is evident from equation (3.73) that the fibres match perfectly at the angle *ϕ*=0 (modulo 2*π*), but differ from each other at all intermediate angles *ϕ*. For example, *e*^{iψ−} and *e*^{iψ+} differ by a minus sign at the angle *ϕ*=*π*. Now to derive the conservation of spin (3.55), we rewrite exponential relation (3.73) in our notation as
*η*_{arsr1} and *η*_{sr2br} between **a**_{r} and **s**_{r1} and **s**_{r2} and **b**_{r} with the fibres *ψ*_{−} and *ψ*_{+}, and the angle *η*_{arbr} between **a**_{r} and **b**_{r} with the generator of the transition function *e*^{iϕ} on the equator of *S*^{2}. Here, we have used the sign conventions to match the sign conventions in our definitions (3.50) and (3.51) and the correlations (3.65). The above representation of equation (3.73) is not as unusual as it may appear at first sight once we recall that geometric products of the bivectors appearing in it are all non-pure quaternions, which can be parametrized to take the exponential form
** ξ**(

**a**

_{r}) and noting that all unit bivectors square to −1, we obtain

**(**

*ξ***b**

_{r}) from the right and

**(**

*ξ***b**

_{r}) from the left then leads to the conservation of the spin angular momentum, just as we have specified in equation (3.52):

In fact, it is not difficult to see from the twist in the Hopf bundle of *S*^{3}, captured in equation (3.74), that if we set **a**_{r}=**b**_{r} (or equivalently *η*_{arbr}=0) for all fibres, then *S*^{3} reduces to the trivial bundle *S*^{2}×*S*^{1}, since then the fibre coordinates *η*_{arsr1} and *η*_{sr2br} would match up exactly on the equator of *S*^{2} [*θ*∼*π*/2]. In general, however, for **a**_{r}≠**b**_{r}, *S*^{3}≠*S*^{2}×*S*^{1}. For example, when **a**_{r}=−**b**_{r} (or equivalently when *η*_{arbr}=*π*) there will be a sign difference between the fibres at that point of the equator [32–34]. That in turn would produce a twist in the bundle analogous to the twist in a Möbius strip. It is this non-trivial twist in the *S*^{3} bundle that is responsible for the observed sign flips in the product **a**_{r}=**b**_{r} to **a**_{r}=−**b**_{r}, as evident from the correlations (3.65). In the appendix of the first chapter of [16] this is illustrated in a toy model of Alice and Bob in a Möbius world. But while the twist in a Möbius strip is in the *S*^{1} worth of parallel lines that make up the untwisted cylinder, the twist in *S*^{3} is in the arrangement of the *S*^{2} worth of circles that make up that 3-sphere (cf. figure 6) [34].

#### 3.2.3. The general case of arbitrarily entangled quantum state

We now proceed to generalize the above 2-particle case^{3} to the most general case of arbitrarily entangled quantum state considered in (3.26). To this end, let us consider any arbitrary number of measurement results corresponding to those in (3.26) and (3.27):
**n**^{3}=**c**↔**c**_{r}+**c**_{d}*ε*_{+} and **n**^{4}=**d**↔**d**_{r}+**d**_{d}*ε*_{+}:

It is important to recall here the elementary fact that any experiment of any kind in physics can always be reduced to a series of questions with ‘yes’/‘no’ answers, represented by binary measurement outcomes of the form (3.80)–(3.82). Therefore, the measurement framework we have developed here is completely general and applicable to any physical experiment.

Now, as in the EPR-Bohm-type experiment with a singlet state discussed above (cf. figure 5 and equations (3.54), (3.55) and (3.79)), for each pair of measurement outcomes such as (3.82) the twist in the Hopf bundle of *S*^{3} dictates the condition
^{k} is a fair coin. We can now identify this locally causal expectation with its quantum mechanical counterpart:

### 3.3. Derivation of Tsirel’son’s bounds on the correlation strength

Let us now investigate the bounds on the strengths of the local-realistic correlations (3.91) by deriving Tsirel’son’s bounds [17] for arbitrary quantum states [16]. To this end, instead of (3.80) consider an alternative set of measurement results such as
**n**′^{3}=**c**′↔**c**′_{r}+**c**′_{d}*ε*_{+} and **n**′^{4}=**d**′↔**d**′_{r}+**d**′_{d}*ε*_{+}. The correlation between these results can then be derived following steps analogous to those in the previous subsection:
**a**,**b**′,**c**′′,**d**′′′,**e**′′′′,…). Consequently, we may consider the following four relations corresponding to some alternative combinations of measurement contexts so that
**N**(**x**_{r},**x**_{d},0,λ^{k}) and **N**(**y**_{r},**y**_{d},0,λ^{k}) represent two independent equatorial points of an *S*^{6} within *S*^{7}, we take them to belong to two disconnected ‘sections’ of the bundle *S*^{5}×*S*^{1} (i.e. two disconnected *S*^{5}⊂*S*^{6}), satisfying
**x**_{r}×**y**_{d} exclusive to both **x**_{r} and **y**_{d}. If we now square the integrand of equation (3.106), use the above commutation relations, and use the fact that all **N**(**n**_{r},**n**_{d},0,λ^{k}) square to −1, then the absolute value of the above Bell-CHSH string (3.105) leads to the following variance inequality:
*S*^{7} [16,17]. Thus, it is the non-vanishing torsion T within *S*^{7}—the parallelizing torsion which makes the Riemann curvature of this representation space vanish—that is responsible for the stronger-than-linear correlations. We can see this from equation (3.108) by setting

Using the above expressions for the intrinsic torsions **u**⋅**v**:=**u**_{r}⋅**v**_{r}+**u**_{d}⋅**v**_{d} analogous to **a**⋅**b**:=**a**_{r}⋅**b**_{r}+**a**_{d}⋅**b**_{d} given in equation (3.18), we have the product
**u** and **v** are unnormalized vectors. Using the constraints analogous to those expressed in equation (3.18), we then have
**x**×**x**′)⋅(**y**′×**y**)≤+1, this inequality reduces to

### 3.4. Fragility of strong correlations increases with number of contexts

As we saw in equation (3.65), in the case of two contexts the scalar part of the geometric product **N**(**a**_{r},**a**_{d},0,λ^{k})**N**(**b**_{r},**b**_{d},0,λ^{k}) is
**N**(**a**_{r},**a**_{d},0,λ^{k})**N**(**b**_{r},**b**_{d},0,λ^{k})**N**(**c**_{r},**c**_{d},0,λ^{k}) works out to give
**a**_{r}±*Δ***a**_{r}, in only one of the four contexts **a**_{r}+**a**_{d}*ε*_{+} would lead to a dramatic change in the pattern of the corresponding correlation.

### 3.5. Reproducing the strong correlations exhibited by the GHSZ states

Now, as a second example of strong correlations, consider the four-particle Greenberger–Horne–Zeilinger state (or the GHSZ state [7]):
**z**-direction of the experimental set-up [7]. The **z**-direction thus represents the axis of anisotropy of the system. The quantum mechanical expectation value of the product of the four outcomes of the spin components in this state—namely, the products of finding the spin of particle 1 along **a**, the spin of particle 2 along **b**, etc.—is given by
*θ*_{a} and *ϕ*_{a} representing the polar and azimuthal angles, respectively, of the direction **a**, **b**, etc.—it works out to be

Our goal now is to reproduce this result within our locally causal framework described above (see also ch. 6 of [16]). To this end, we note that the state (3.124) represents, not a two-level, but a four-level quantum system (see footnote 3). Each of the two pairs of the spin-*γ*_{1}, *γ*_{2}, *γ*_{3} and *γ*_{4} are complex numbers satisfying |*γ*_{1}|^{2}+|*γ*_{2}|^{2}+|*γ*_{3}|^{2}+|*γ*_{4}|^{2}=1, which is equivalent to defining a unit 7-sphere, with |*γ*_{1}|^{2}, |*γ*_{2}|^{2}, |*γ*_{3}|^{2} and |*γ*_{4}|^{2} being the probabilities of actualizing the states |++〉, |+−〉, |−+〉 and |−−〉, respectively. Therefore, we may begin with four local maps of the form
**N**(**a**_{r},**a**_{d},0,λ^{k})**N**(**b**_{r},**b**_{d},0,λ^{k})**N**(**c**_{r},**c**_{d},0,λ^{k})**N**(**d**_{r},**d**_{d},0,λ^{k}), as spelled out in equation (3.121). Using a simple vector identity, this expectation value can be further simplified to take the form
**a**_{r}⋅**c**_{d}, etc., to zero, this expected value reduces to
**a**_{r}⋅**b**_{r} with **a**_{d}⋅**b**_{d}, etc., the expected value takes the form
**a**, **b**, **c** and **d**∈IR^{3}, chosen freely by the experimenters, with the directions **a**_{r}, **a**_{d}, etc., within our representation space **D** (instead of 2 as in equation (3.9)) arise because the product of four factors, **N**(**a**_{r},**a**_{d},0,λ^{k})**N**(**b**_{r},**b**_{d},0,λ^{k}) **N**(**c**_{r},**c**_{d},0,λ^{k}) **N**(**d**_{r},**d**_{d},0,λ^{k}), instead of two, **N**(**a**_{r},**a**_{d},0,λ^{k})**N**(**b**_{r},**b**_{d},0,λ^{k}), is involved in calculation (3.87) of the correlation, while maintaining the unity of the radius of *S*^{7}. Note also that components of only external vectors are involved in the definitions of the four detectors. And they do not mix with each other, so that Bell’s condition of local causality, or parameter independence [5,6], is strictly respected throughout. Substituting these coordinate values into the remaining vectors in the expected value (3.137) then reduces that value to
*θ*_{a} and *ϕ*_{a} representing, respectively, the polar and azimuthal angles of the direction **a**, etc., for all four measurement directions—this expression of the expected value can be further simplified to
*S*^{7}.

## 4. Bell’s theorem, its experimental tests and the GHSZ variant

### 4.1. Bell-test experiments: from inceptions to loophole-free advances

Contrary to what we have demonstrated above, it is widely believed that the so-called Bell-test experiments—from their initial conceptions summarized in the classic review paper by Clauser & Shimony [20] to their state-of-the-art ‘loophole-free’ variants [8–13]—undermine any prospects of a locally causal understanding of quantum correlations. It is important to appreciate, however, that all such experiments simply confirm the predictions of quantum mechanics. They neither contradict the quantum mechanical predictions nor go beyond them in any sense. Moreover, as in §i, we have reproduced all of the quantum mechanical predictions for the singlet state *exactly*, the Bell-test experiments [8–13] do not contradict the predictions of our model either. Rather, they simply corroborate them.

More precisely, in the analysis of all such experiments one averages over ‘coincidence counts’ to calculate expectation values in the form
*C*_{+−}(**a**,**b**), etc., represent the number of simultaneous occurrences of detections +1 along **a** and −1 along **b**, etc. In addition, they observe individual results

Finally, they observe that Bell-CHSH inequalities [15] with the absolute bound of 2 are exceeded by a factor of 2:
*S*^{7} model predicts precisely the relations (4.1)–(4.3) for the entangled state (3.47), as can be verified from our predictions (3.65), (3.50), (3.51) and (3.117).

Thus, the crucial difference between the predictions of our *S*^{7} model and those of quantum mechanics is *not* in the observational content, but in the interpretation of the latter in terms of non-locality.^{4} And this interpretation depends entirely on the argument put forward by Bell and his followers [5,6,20]. This argument, however, is fatally flawed, as we now demonstrate.

### 4.2. Surprising oversight in the derivation of the Bell-CHSH inequalities

From the outset let us stress that Bell’s so-called theorem is by no means a ‘theorem’ in the sense that word is used by mathematicians but rather a word-statement, which claims that *no physical theory which is realistic as well as local in the strict senses espoused by* Einstein [4] *and later formulated by* Bell (see footnote 2) [5,6] *can reproduce all of the statistical predictions of quantum theory* [14]. This word-statement is based on ‘violations’ of certain mathematical inequalities, which are derived by considering four *incompatible* EPR-Bohm-type experiments, and without using a single concept from quantum theory. While the bounds thus derived on the inequalities are exceeded by the predictions of quantum theory and ‘violated’ in actual experiments, their derivation happens to be marred by a serious conceptual oversight.

To appreciate this, consider the standard EPR type spin-**a** or **a**′ and Bob is free to choose a detector direction **b** or **b**′ to detect spins of the fermions they receive from a common source, at a space-like distance from each other. The objects of interest then are the bounds on the sum of possible averages put together in the manner of CHSH [15],

This should have been Bell’s final conclusion. However, by continuing, Bell overlooked something that is physically unjustifiable. He replaced the above sum of four separate averages of real numbers with the following single average:
*unobservable* and *unphysical* quantities. But it allows us to reduce the sum of four averages to
^{5} :

Let us now try to understand why the replacement in (4.7) is illegitimate.^{6} To begin with, Einstein’s (or even Bell’s own) notion of local realism does not, by itself, demand this replacement. As this notion is captured already in the very definition (see footnote 2) [5,6] of the functions *at the expense of what is physically possible in the actual experiments*. To be sure, mathematically there is nothing wrong with a replacement of four separate averages with a single average. Indeed, every school child knows that the sum of averages is equal to the average of the sum. But this rule of thumb is not valid in the above case, because (**a**,**b**), (**a**,**b**′), (**a**′,**b**) and (**a**′,**b**′) are *mutually exclusive pairs of measurement directions*, corresponding to four *incompatible* experiments. Each pair can be used by Alice and Bob for a given experiment, for all runs 1 to *n*, but no two of the four pairs can be used by them simultaneously. This is because Alice and Bob do not have the ability to make measurements along counterfactually possible pairs of directions such as (**a**,**b**) and (**a**,**b**′) simultaneously. Alice, for example, can make measurements along **a** or **a**′, but not along **a** *and* **a**′ at the same time.

But this fact is rather devastating for Bell’s argument, because it means that his replacement (4.7) is illegitimate. Consider, for example, a specific run of the EPR-Bohm-type experiment and the corresponding quantity being averaged in (4.7):
*k*=1 now represents a specific run of the experiment. But since Alice and Bob have only two particles at their disposal for each run, only one of the four terms of the above sum is physically meaningful. In other words, the above quantity is physically meaningless, because Alice, for example, cannot align her detector along **a** and **a**′ at the same time. And likewise, Bob cannot align his detector along **b** and **b**′ at the same time. What is more, this will be true for all possible runs of the experiment, or equivalently for all possible pairs of particles. Which implies that all of the quantities listed below, as they appear in average (4.9), are unobservable, and hence physically meaningless:

But as each of the quantities above is physically meaningless, their average appearing on the r.h.s. of (4.7), namely
^{7} [5,6,14,20]. That is to say, no physical experiment can ever be performed—*even in principle*—that can meaningfully allow to measure or evaluate the above average, as none of the above list of quantities could have experimentally observable values. Therefore the innocuous looking replacement (4.7) made by Bell is, in fact, illegal.

On the other hand, it is important to note that each of the averages appearing on the l.h.s. of replacement (4.7),

In summary, Bell and his followers derive the upper bound of 2 on the CHSH string of averages by an illegal move. In the middle of their derivation, they unjustifiably replace an observable, and hence physically meaningful quantity,

One may suspect that the above conclusion is perhaps an artefact of the discrete version (4.5) of the expectation values *Λ* is the space of all hidden variables λ, and *ρ*(λ) is the probability measure of λ [5–7]. Written in this form, it is now easy to see that the above CHSH sum of expectation values is both mathematically and physically identical to

To begin with, expression (4.19) involves an integration over fictitious quantities (see footnote 7) such as *not* closed under addition. Since each function

But there is also a much more serious physical problem with Bell’s version of reality. As noted above, the quantities *any* possible physical world, classical or quantum. That is because **b** and **b**′ are mutually exclusive directions. If

In conclusion, as the two integrands of (4.19) are physically meaningless, the stringent bounds of ±2 on expression (4.18) are also physically meaningless [30]. They are mathematical curiosities, without any relevance for the question of local realism.

### Corollary 4.1

*It is not possible to be in two places at once.*

It is instructive to consider the converse of the above argument. Consider the following hypothesis (see footnote 7): *It is possible—at least momentarily—to be in two places at once—for example, in New York and Miami—at exactly the same time.*

From this hypothesis, it follows that in a world in which it is possible to be in two places at once, it would be possible for Bob to detect a component of spin along two mutually exclusive directions, say **b** *and* **b**′, at exactly the same time as Alice detects a component of spin along the direction **a**, or **a**′. If we denote the measurement functions of Alice and Bob by **a** and **b** represent macroscopic directions) such a bizarre space–time event is never observed, because the measurement directions **a** and **b** freely chosen by Alice and Bob are mutually exclusive macroscopic measurement directions in physical space.

Likewise, nothing prevents Alice and Bob in such a bizarre world to simultaneously observe an event represented by

Consider now a large number of such initial states λ and corresponding simultaneous events like *Z*(**a**,**a**′,**b**,**b**′,λ). We can then calculate the expected value of such an event occurring in this bizarre world, by means of the integral