Christian Joy
Einstein Centre for Local-Realistic Physics, 15 Thackley End, Oxford OX2 6LB, UK.
R Soc Open Sci. 2018 May 30;5(5):180526. doi: 10.1098/rsos.180526. eCollection 2018 May.
The exceptional Lie group 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, , with being its algebraic representation space. Every quantum correlation can thus be understood as a correlation among a set of points of this , computed using manifestly local spinors within , thereby extending the stringent bounds of ±2 set by Bell inequalities to the bounds of 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 of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of and .
例外李群在数学和理论物理中都扮演着重要角色。它是与最一般的可能赋范可除代数相关联的最大对称群，即非结合实八元数的对称群，由于其非结合性，八元数构成了唯一可能的能使7维球面平行化的旋量（或转子）闭集。相比之下，在这里我们展示了一个类似的7维球面如何也自然地源于分级欧几里得基本元素（如点、线、面和体）的代数相互作用，这些元素刻画了周围物理空间的三维共形几何，该几何设定在其八维克利福德代数表示中。值得注意的是，所得代数仍然是结合的，并且使我们能够根据紧致化物理空间的几何结构，即四元数3维球面(S^3_{\mathbb{H}})（其中(\mathbb{H})是其代数表示空间），从局部理解所有量子关联的起源和强度。因此，每个量子关联都可以理解为(S^3_{\mathbb{H}})上一组点之间的关联，使用(\mathbb{H})内明显局部的旋量来计算，从而以与量子力学中预测的相同定量精确方式，将贝尔不等式设定的严格的(\pm2)界限扩展到所有可能强关联强度的(\sqrt{2})界限。由此产生的几何框架通过产生一个严格确定性和现实性的框架克服了贝尔定理，该框架允许对所有量子关联进行局部因果理解，而无需远程上下文相关性或反向因果关系。我们首先证明一个关于任意纠缠量子态预测的关联的几何起源的一般定理，然后重现EPR - 玻姆态和GHZ态预测的关联，以此来证明这一点。强关联的(\sqrt{2})界限结果是(S^3_{\mathbb{H}})和(S^7)的霍普夫丛中的类莫比乌斯扭转。
