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Bell-Christian S7 Brown SL(8,R) E8 Bohm

Frank Dodd Tony Smith

Published May 1, 2015• 0 citations
OpenAlex

Abstract

Joy Christian has shown that Bell's Theorem should have been formulated in terms of the 7-sphere, and David Brown has shown that the resulting quantum structure indicates that physical reality uses the group SU(8). The non-compact version of SU(8) is SL(8,R) which is a fundamental part of the semi-direct product H92xSL(8,R) (where H92 is (8+28+56+1+56+28+8) = 185-dim and SL(8,R) is 63-dim) which is the Maximal Compactification of 248-dim E8 of E8 Physics described in viXra 1405.0030. Version 2 (v2) expands discussion, uses SL(8,R) in title, and states full viXra number.

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•May 2, 2026•0 citations
Joy Christian has shown that Bell's Theorem should have been formulated in terms of the 7-sphere, and David Brown has shown that the resulting quantum structure indicates that physical reality uses the group SU(8). The non-compact version of SU(8) is SL(8,R) which is a fundamental part of the semi-direct product H92xSL(8,R) (where H92 is (8+28+56+1+56+28+8) = 185-dim and SL(8,R) is 63-dim) which is the Maximal Compactification of 248-dim E8 of E8 Physics described in viXra 1405.0030. Version 2 (v2) expands discussion, uses SL(8,R) in title, and states full viXra number.
In 2007, and in a series of later papers, Joy Christian claimed to refute Bell's theorem, presenting an alleged local realistic model of the singlet correlations using techniques from Geometric Algebra (GA). Several authors published papers refuting his claims, and Christian's ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell's theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian's model used new devices to circumvent Bell's theorem or depended on a new way to misunderstand Bell's work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new "disproofs of Bell's theorem". Christian's fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation a.b = (ab + ba)/2. After that, ambiguous notation and technical complexity allow sign errors to be hidden from sight, and new mathematical errors can be introduced.
This work presents an elegant and conceptually clean geometric bridge between Pauli algebra and the topology of the Hopf fibration. It shows that the well-known Bell singlet correlation E(a, b) = −a·b is not an additional mystery of quantum theory, but is already fully encoded in the standard spinor geometry S³ → S² and in the non-commutative structure of SU(2). The main result can be understood as a representation statement: the Bell correlation admits a natural formulation as the real part of a U(1) holonomy on the Hopf bundle, namely E(a, b) = Re exp(i(π − θ)), where θ is the angle between measurement directions on the Bloch sphere. This holonomy arises from the Berry–Pancharatnam phase defined on the projective space CP¹ ≃ S², and depends only on the associated oriented solid angle. Importantly, this work does not introduce new physical laws and does not modify quantum mechanics. Instead, it demonstrates that the standard result E = −a·b already admits a rigorous geometric interpretation in terms of connections, curvature, and parallel transport. In this sense, the contribution is not a new theory, but a precise geometric reformulation of existing quantum structure. A key clarification concerns interpretation. The non-abelian SU(2) structure appearing here is not a hidden variable in the sense of Bell’s theorem. It describes the geometry of the quantum state space S³, not a set of pre-existing measurement outcomes. The reduction to U(1) reflects the fact that projective measurements access only equivalence classes on S², discarding the global phase. Accordingly, the violation of Bell inequalities arises from non-commutativity of observables, not from hidden parameters or signalling. The value of the work lies in structural unification. It brings together three standard but often separately presented elements: (i) the algebraic form of entangled correlations, (ii) the geometry of spinor space S³ and the Bloch sphere S², and (iii) the holonomy/phase nature of quantum quantities. In this formulation, Bell correlations appear not only as probabilistic objects, but as global geometric features of the underlying state space. This perspective is particularly useful for geometric quantum mechanics, quantum information theory, and pedagogy, where it provides a clear and rigorous insight into the origin of the cosine correlation law. At the same time, the scope is deliberately modest: the construction is fully consistent with standard quantum theory and does not attempt to extend it. The main risk lies in interpretational overreach. While the geometric picture is compelling, it should not be mistaken for an alternative to quantum mechanics or for a hidden-variable completion. Rather, it reveals an internal geometric layer that is already present in the theory. In summary, this work represents a high-quality incremental advance in quantum foundations: not a paradigm shift, but a compact and illuminating reconstruction of a fundamental result in the language of topology and geometry.
In 1991 with Abner Shimony as thesis advisor, J. Christian obtained his PhD in the foundations of quantum theory. As of 2012, he is affiliated with both Wolfson College of the University of Oxford and the Perimeter Institute in Waterloo, Ontario. From 2007 to 2012 Christian published some highly controversial work related to Bell’s theorem. Christian has proposed a parallelized 7-sphere model for a local, realistic theory that replaces quantum theory. This communication considers how his model might be relevant to a physical interpretation of M-theory. There is overwhelming empirical evidence in favor of the Ranada-Milgrom effect, at least as an approximation. Christian’s theory of local realism might be empirically valid if and only if the Ranada-Milgrom effect is empirically valid.
This paper attempts to derive the SU(2) algebraic structure of quantum spin fromthe bipolar n = 2 torsion recursion of the Cohesion UFT, and thereby derive the Bellcorrelation E(a, b) = − cos θ without importing quantum mechanical postulates. Thederivation proceeds as follows. The bipolar torsion axis is a unit vector n ∈ S2. The2-sphere carries a natural symplectic structure — the area form ω = sin θ dθ ∧ dϕ —that is a geometric property of S2, not a quantum mechanical input. The Poissonbrackets induced by this symplectic structure are {ni, nj} = εijknk, which is the classicalsu(2) Poisson algebra. The slip operator of the bipolar torsion cycle defines ℏ as theminimum torsion action. Replacing Poisson brackets with commutators via ℏ gives[Jˆi, Jˆj ] = iℏεijkJˆk, which is SU(2). The Bell correlation E(a, b) = − cos θ then followsfrom the standard SU(2) result for anti-phase bipolar pairs in the spin- 12representation.Two steps in this chain are identified rather than formally derived: that the area formof S2is the correct symplectic structure for the torsion dynamics (plausible from thegeometry but not yet proven from the Lagrangian), and that n = 2 specifically yieldsthe spin- 12representation rather than higher spin. These gaps are precisely located.The derivation is substantially complete; the remaining steps are formal rather thanconceptual.