Geometric Algebras of Light Cone Projective Graph Geometries

Abstract

A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by \({{\mathcal {N}}}\). The rules of addition and multiplication in \({{\mathcal {N}}}\) are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is positively or negatively correlated if their inner product is positive or negative, respectively. A basis of \(n+1\) null vectors, with pairwise inner products equal to plus or minus one half, defines the Clifford geometric algebras \({\mathbb {G}}_{1,n}\), or \({\mathbb {G}}_{n,1}\), respectively, and provides a foundation for a new Cayley–Grassman linear algebra, a theory of complete graphs, and other applications in pure and applied areas of science.