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arXiv:1201.0775

On the Origins of Quantum Correlations

Joy Christian

Published Dec 28, 2011• 5 citations
arXivHTMLPDF

Abstract

It is well known that quantum correlations are not only more disciplined (and hence stronger) compared to classical correlations, but they are more disciplined in a mathematically very precise sense. This raises an important physical question: What is responsible for making quantum correlations so much more disciplined? Here we explain the observed discipline of quantum correlations by identifying the symmetries of our physical space with those of a parallelized 7-sphere. We substantiate this identification by proving that any quantum correlation can be understood as a classical, local-realistic correlation among a set of points of a parallelized 7-sphere.

DOI
10.48550/arxiv.1201.0775

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It is well known that quantum correlations are not only more disciplined (and hence stronger) compared to classical correlations, but they are more disciplined in a mathematically very precise sense. This raises an important physical question: What is responsible for making quantum correlations so much more disciplined? Here we explain the observed discipline of quantum correlations by identifying the symmetries of our physical space with those of a parallelized 7-sphere. We substantiate this identification by proving that any quantum correlation can be understood as a classical, local-realistic correlation among a set of points of a parallelized 7-sphere.
In recent decades it was established that the measurements of physical quantities in space-time points divided by large space-like intervals can be correlated.The correlation is created by the measurements while their mutual influence requires the corresponding supraliminal velocities.This situation is commonly known as the violation of the Bell inequality in the Bohm version of Einstein-Podolsky-Rosen paradox.Nearly 40 years the question remains disputable with a variety of contradicting views.The most frequently the violation is explained by the existence of some momentary action at a distance which obliges the quantum mechanics (as well as all nature) to be nonlocal.We find the "common cause in the past" as the origin of the observed quantum correlation which so far was unnoticed either by the partisans of quantum non-locality or their opponents.The novelty of our approach is the accent on the expressions for the observed physical quantities rather than on the "entangled" two-particle wave functions.We show the crucial role of phases in the correlation process that is hard to notice using the traditional approach.
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 ± 2 2 on the strengths of all possible strong correlations, in the same quantitatively precise manner as that predicted within quantum mechanics. The resulting geometrical framework thus overcomes Bell’s theorem by producing a strictly deterministic and realistic framework that allows a locally causal understanding of all quantum correlations, without requiring either remote contextuality or backward causation. We demonstrate this by first proving a general theorem concerning the geometrical origins of the correlations predicted by arbitrarily entangled quantum states, and then reproducing the correlations predicted by the EPR-Bohm and the GHZ states. The raison d’être of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of S 3 and S 7 .
Nowadays, no one should doubt that quantum entanglement is (at least, one of) the \ncharacteristic traits of quantum mechanics. Furthermore, quantum information science \nis strongly influenced by the advantages provided by quantum entanglement, as the \nlatter can be regarded as a genuine resource for an efficient (better-than-classical) \nprocessing of information in certain computational tasks. The very basic notion of \ncorrelation is fundamental for the conceptualization of the physical communicational \nproblem and, as a consequence, for the notion of information itself. Thus, the question \nabout the nature of quantum information is close (and unavoidably linked) to the \nquestion about the nature of quantum correlations.
The discipline of parallelization in the manifold of all possible measurement results is shown to be responsible for the existence of all quantum correlations, with the upper bound on their strength stemming from the maximum of possible torsion within all norm-composing parallelizable manifolds. A profound interplay is thus uncovered between the existence and strength of quantum correlations and the parallelizability of the spheres S^0, S^1, S^3, and S^7 necessitated by the four real division algebras. In particular, parallelization within a unit 3-sphere is shown to be responsible for the existence of EPR and Hardy type correlations, whereas that within a unit 7-sphere is shown to be responsible for the existence of all GHZ type correlations. Moreover, parallelizability in general is shown to be equivalent to the completeness criterion of EPR, in addition to necessitating the locality condition of Bell. It is therefore shown to predetermine both the local outcomes as well as the quantum correlations among the remote outcomes, dictated by the infinite factorizability of points within the spheres S^3 and S^7. The twin illusions of quantum entanglement and non-locality are thus shown to stem from the topologically incomplete accountings of the measurement results.
In relativity there is space-time out there. In quantum mechanics there is entanglement. Entanglement manifests itself by producing correlations between classical events (e.g. the firing of some detectors) at any two space-time locations. If the locations are time-like separated, i.e. one is in the future of the other, then there is no specific difficulty to understand the correlations. But if the two locations are space-like separated, the problem is different. How can the two space-time locations out there know about what happens in each other without any sort of communication? If space-time really exists, the locations must do something like communicating. Or it was all set up at the Beginning. But the correlations depend also on the free choice of the experimentalists, one in each space-time location. This allowed John Bell to derive his inequality and the experimentalists to violate it, thus refuting the assumption that it was all set up at the beginning: the Correlations can't be explained by common causes.