## Introduction to Real Clifford Algebras: Cl(8) to E8 to Hyperfinite II1

**Authors:** Frank Dodd Tony Smith Jr

Real Clifford Algebras roughly represent the Geometry of Real Vector Spaces of signature (p,q) with the Euclidean Space (0,q) sometimes just being written (q) so that the Clifford algebra Cl(0,q) = Cl(q). A useful starting place for understanding how they work is to look at the most central example and then extend from it to others. This paper is only a rough introductory description to develop intuition and is NOT detailed or rigorous - for that see the references. Real Clifford Algebras have a tensor product periodicity property whereby Cl(q+8) = Cl(q) x Cl(8) so that if you understand Cl(8) you can understand larger Clifford Algebras such as Cl(16) = Cl(8) x Cl(8) and so on for as large as you want. So Cl(8) is taken to be the central example in this paper which has 4 parts: How Cl(8) works; What smaller Clifford Algebras inside Cl(8) look like; How the larger Clifford Algebra Cl(16) gives E8: How larger Clifford Algebras Cl(16N) = Cl(8(2N)) give in the large N limit a generalized Hyperfinite II1 von Neumann factor. V2 adds Creation and Annihilation Operators of AQFT as A7+h_92 Contraction of E8. V3 adds discussion about AQFT Possibility Space.

**Comments:** 16 Pages.

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### Submission history

[v1] 2013-04-15 16:46:45

[v2] 2013-05-22 11:56:06

[v3] 2015-05-30 11:09:51

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