Metrics are Clifford algebra involutions

Abstract

Four-dimensional space-time, all relevant inner products, and some of the groups leaving these inner products invariant are manufactured from more basic algebraic ingredients, all inside the 8-dimensional Pauli algebra ℘: (1) Euclidean 3-spaceE ³, (2) Minkowski 4-spaceM ⁴, (3) complex 4-space ℂ⁴, and all three metrics and all three inner products. The groupsSO(3;ℝ) ⊂SO(3; 1;ℝ) ⊂SO (4;ℝ) are obtained as images of twofold covering maps of subgroups of ℘ or their direct product. A method of embedding ℘ in the Clifford algebraC(1;n−1) ofn-dimensional Minkowski space is given for anyn≥4. Furthermore, all three groups act not only on the relevant vector spaces, but on all ofC(1;n−1), leaving ℘ setwise invariant.