Origins of Quantum Correlations

 

In essence, quantum correlations are a measure of torsion within our physical space.

If you know about teleparallel gravity, then it will be easier for you to understand the above statement and the rest of my argument. Fortunately, it is not essential to know about teleparallel gravity to understand why quantum correlations are a manifestation of torsion in our physical space. It is, however, important to understand what is meant by torsion in a smooth Riemannian manifold. A simply-connected space such as {{S^3}} defined on the previous page is said to be absolutely parallelized (in the topological sense) if its curvature tensor {{{\cal R}^{\,\alpha}_{\;\;\,\beta\,\gamma\,\delta}}} (which, as a tensor, is defined locally, at every point p of S^{3\;}) vanishes identically with respect to the Weitzenböck connection

\displaystyle \stackrel{\bullet\;\;\;\;\;\,}{\Gamma_{\alpha\,\beta}^{\,\gamma}}\,\;:=\;\,\stackrel{\circ\;\;\;\;\;\,}{\Gamma_{\alpha\,\beta}^{\,\gamma}}\,+\; {\cal T}_{\,\alpha\,\beta}^{\,\gamma}\,, \ \ \ \ \ \ \ (1)

where {{\stackrel{\circ\;\;\;\;\;\,}{\Gamma_{\alpha\,\beta}^{\,\gamma}}}} is the symmetric Levi-Civita connection and {{{\cal T}_{\,\alpha\,\beta}^{\,\gamma}}} is the totally anti-symmetric torsion tensor. This vanishing of the curvature tensor renders the resulting parallelism on {{S^3}} absolute and guarantees the path-independence of any parallel transported vector within it. For a non-Euclidean space such as {{S^3}} this is possible if and only if the parallelizing torsion in the space is non-vanishing:

\displaystyle {\cal R}^{\,\alpha}_{\;\;\,\beta\,\gamma\,\delta}\,=\,0\;\;\;\;\;\;\;\;\text{but} \;\;\;\;\;\;\;\;{\cal T}_{\,\alpha\,\beta}^{\,\gamma}\,\not=\,0\,. \ \ \ \ \ \ \ (2)

Thus the non-Euclidean properties of the parallelized 3-sphere are encoded entirely within its torsion rather than a curvature. And as we saw on the previous page, it is this non-vanishing torsion within {{S^3}} that is responsible for the strong quantum correlations between measurement events {{{\,\cal A}=\pm\,1\,}} and {{{\,\cal B}=\pm\,1\,}} occurring within it. Mathematically the measurement functions thus take the form

\displaystyle \pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow S^3\hookrightarrow{\rm I\!R}^4\,, \ \ \ \ \ \ \ (3)

 

with their image points being

 

\displaystyle {\cal A}({\bf a},\,\lambda)\,=\,\;either\;\;+\!1\;\;\;\;or\;\;\;-\!1\,.\ \ \ \ \ \ \ \ \

It turns out that this local-realistic framework can be generalized to reproduce ALL quantum correlations, provided we take the codomain of the above functions to be an absolutely parallelized 7-sphere instead of a 3-sphere:

\displaystyle \pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow S^7\hookrightarrow{\rm I\!R}^8. \ \ \ \ \ \ \ (4)

In fact the choice of S^7 here is both necessary and sufficient to provide a complete local account of ALL quantum correlations. It may seem ad hoc, but it stems from the profound relationship between the normed division algebras and the parallelizability of the unit spheres. As is well known, the only spheres that can be parallelized absolutely are {{S^0}}, {{S^1}}, {{S^3}}, and {{S^7}}, and they correspond precisely to the four division algebras: {{\mathbb R}}, {{\mathbb C}}, {{\mathbb H}}, and {{\mathbb O}}. The 7-sphere is thus homeomorphic to the set of all unit octonions, and corresponds to the most general possible normed division algebra {{\mathbb O}}. It is therefore fitting that it also gives rise to the existence and strength of ALL quantum correlations. That is to say, quantum correlations exist and exhibit the strength they exhibit because the equality

\displaystyle (x_1^2 + x_2^2 + \dots + x_n^2)\,(y_1^2 + y_2^2 + \dots + y_n^2) \,=\, z_1^2 + z_2^2 + \dots + z_n^2 \ \ \ \ \ \ \ (5)

holds only for the integers {{n = 1,\;2,\;4}}, and {{8}}, for any generalized numbers x. Among the four parallelizable spheres, however, only {{S^3}} and {{S^7}} are non-trivially parallelized, and thus characterized by a non-vanishing torsion. Therefore, only with the choices {{S^3}} and {{S^7}} for the codomain of the measurement functions (4) can we reproduce the strong quantum correlations. Moreover, {{S^7}} can be viewed as a 4-sphere worth of 3-spheres in the language of Hopf fibration (cf. this page). Thus, in general, we only need to consider {{S^7}}, since {{S^3}} is already contained within it as one of its fibres.

This last observation suggestes that for general quantum correlaions the form of the measurement functions (4) has to be generalized to admit an interpediate stage (or map) by giving it the form

\displaystyle \pm\,1\,=\,{\cal A}({\bf a},\,\lambda):{\rm I\!R}^3\!\times\Lambda\longrightarrow {\rm I\!R}^7\!\times\Lambda\longrightarrow S^7\hookrightarrow{\rm I\!R}^8, \ \ \ \ \ \ \ (6)

where {\rm I\!R}^7 is a tangent space at a point of S^7, spanned by vectors of the form {\mathbf N}({\mathbf a}), with {\mathbf N}\in{\rm I\!R}^7 and {\mathbf a}\in{\rm I\!R}^3 (explicit constructions of how this map works in practice can be found in this paper). Thus, although the actual measurement results \pm\,1 are occurring in the familiar 3D tangent space {\rm I\!R}^3 of S^3, their observed values are dictated by the trivial tangent bundle structure {\rm T}S^7\!=S^7\times{\rm I\!R}^7 of the absolutely parallelized 7-sphere. In this sense, then, quantum correlations are the evidence of the fact that rotational symmetries respected by our physical space are those of a parallelized 7-sphere. It is important to note here, however, that, unlike the torsion within S^3, the non-associativity of the octonionic numbers leads to torsion within S^7 that is necessarily different at each point p of S^7:

S^3\ni{\cal T}_{\,{\bf a\,a'}}({\lambda})  \xrightarrow{\;S^3\longrightarrow\,S^7\;}{\cal T}_{\,{\bf N(a)\,N(a')}}(p,\,{\lambda})\in S^7. \ \ \ \ \ \ \ (7)

This variability of torsion gives rise to the rich variety of quantum correlations we observe in nature.

Suppose now we consider an arbitrary quantum state {|\Psi\rangle} and a corresponding self-adjoint operator {\cal\widehat O}({\bf a},\,{\bf b},\,{\bf c},\,{\bf d},\,\dots\,) in some Hilbert space {\cal H}, parameterized by an arbitrary number of local contexts {{{\bf a},\,{\bf b},\,{\bf c},\,{\bf d},}} etc. Note that I have imposed no restrictions on the state {|\Psi\rangle}, or on the size of the Hilbert space {\cal H}. In particular, {|\Psi\rangle} can be as entangled or unentangled as one may like, and {\cal H} can be as large or small as one may like. The quantum mechanical expectation value of the operator {\cal\widehat O}({\bf a},\,{\bf b},\,{\bf c},\,{\bf d},\,\dots\,) in the state {|\Psi\rangle} would then be

\displaystyle {\cal E}_{{\!}_{Q.M.}}({\bf a},\,{\bf b},\,{\bf c},\,{\bf d},\,\dots\,)\, =\,\text{Tr}\left\{{W}\,{\cal\widehat O}({\bf a},\,{\bf b},\,{\bf c},\,{\bf d},\,\dots\,)\right\}, \ \ \ \ \ \ \ (8)

where {{W}} is a statistical operator of unit trace representing the state. As noted, I have imposed no restrictions whatsoever on the state, the observable, or the number of local contexts. Nevertheless, it turns out that—because the octonionic division algebra remains closed under multiplication—the quantum correlation predicted by this expectation value can always be reproduced as local and realistic correlation among a set of points of a parallelized 7-sphere, by following a procedure very similar to the one discussed on the previous page. This leads us to the following awesome theorem:

Every quantum mechanical correlation can be understood as a deterministic, local-realistic correlation among a set of points of a parallelized 7-sphere, specified by maps of the form

\displaystyle \pm\,1\,=\,{\cal A} ({\bf a},\,\lambda): {\rm I\!R}^3\!\times\Lambda\longrightarrow {\rm I\!R}^7\!\times\Lambda\longrightarrow S^7 \hookrightarrow{\rm I\!R}^8. \ \ \ \ \ \ \ (9)

The proof of this theorem can be found in this paper as well as on the pages 13-17 of my book.

The Raison D’être of Quantum Correlations:

The above theorem demonstrates that the discipline of absolute parallelization in the manifold S^7 of all possible measurement results is responsible for the existence and strength of all {{\,}}quantum correlations. More precisely, it identifies quantum correlations as evidence that the physical space we live in respects the symmetries and topologies of a parallelized 7-sphere. There are profound mathematical and conceptual reasons why the topology of the 7-sphere plays such a significant role in the manifestation of quantum correlations. Essentially it is because 7-sphere happens to be homeomorphic to the most general possible division algebra. And it is the property of division that turns out to be responsible for maintaining strict local causality in the world we live in.

To understand this reasoning better, recall that, just as a parallelized 3-sphere is a 2-sphere worth of 1-spheres but with a twist in the manifold {{S^3\;(\not=S^2\times S^1)}}, a parallelized 7-sphere is a 4-sphere worth of 3-spheres but with a twist in the manifold {{S^7\;(\not=S^4\times S^3)}}. More precisely, just as {{S^3}} is a nontrivial fiber bundle over {{S^2}} with Clifford parallels {{S^1}} as its linked fibers, {{S^7}} is also a nontrivial fiber bundle, but over {{S^4}}, and with entire 3-dimensional spheres {{S^3}} as its linked fibers. Now it is the twist in the bundle {{S^3}} that forces one to forgo the commutativity of complex numbers (corresponding to the circles {{S^1}}) in favor of the non-commutativity of quaternions. In other words, a 3-sphere is not parallelizable by the commuting complex numbers but only by the non-commuting quaternions. And it is this non-commutativity that gives rise to the non-vanishing of the torsion in our physical space. In a similar vein, the twist in the bundle {{S^7\not=S^4\times S^3}} forces one to forgo the associativity of quaternions (corresponding to the fibers {{S^3}}) in favor of the non-associativity of octonions. In other words, a 7-sphere is not parallelizable by the associative quaternions but only by the non-associative octonions. And the reason why it can be parallelized at all is because its tangent bundle happens to be trivial:

\displaystyle {\rm T}S^7\,=\!\bigcup_{\,p\,\in\, S^7}\{p\}\times T_pS^7\,\equiv\,S^7\times{\rm I\!R}^7. \ \ \ \ \ \ \ (10)

Once parallelized by a set of unit octonions, both the 7-sphere and each of its 3-spherical fibers remain closed under multiplication. This, in turn, means that the factorizability or locality condition of Bell is automatically satisfied within a parallelized 7-sphere. The lack of associativity of octonions, however, entails that, unlike the unit 3-sphere [which is homeomorphic to the spinor group SU(2)], a 7-sphere is not a group manifold, but forms only a quasi-group. As a result, the torsion within the 7-sphere continuously varies from one point to another of the manifold. It is this variability of the parallelizing torsion within {{S^7}} that is ultimately responsible for the diversity and non-linearity of the quantum correlations we observe in nature:

\text{\small Parallelizing Torsion}\;\,{\cal T}_{\,\alpha\,\beta}^{\,\gamma}\not=0\;\;\;\Longleftrightarrow\;\;\;\text{\small Quantum Correlations.}

The upper bound on all possible quantum correlations is thus set by the maximum of possible torsion within the 7-sphere:

\text{\small Maximum of Torsion}\;\,{\cal T}_{\,\alpha\,\beta}^{\,\gamma}\not=0\;\;\;\Longrightarrow\;\;\,\text{\small The Upper Bound}\;\,2\sqrt{2}.

This reaffirms that quantum correlations are a measure of torsion within our physical space.

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