Complexifying the Spacetime Algebra by Means of an Extra Timelike Dimension: Pin, Spin and Algebraic Spinors

Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \ell}_{2,3}(\Bbb{R})$, it is possible to complexify spacetime Clifford algebra \ell}_{1,3}(\Bbb{R})$ by adding one additional timelike dimension Minkowski spacetime. In a recent work we showed how this treatment provide particular interpretation Dirac particles and antiparticles in terms new temporal dimension. article thoroughly study structure real \ell}_{2,3}(\Bbb{R})$ paying special attention embedding \ell}_{1,3}(\Bbb{R})\subseteq{C \ell}_{2,3}(\Bbb{R})$. On first half analyze Pin Spin groups construct an injective mapping $\operatorname{Pin}(1,3)\hookrightarrow\operatorname{Spin}(2,3)$, obtaining elements $\operatorname{Spin}(2,3)$ that represent parity time reversal. second paper spinor space prove usual complex spinors \ell}_{1,3}(\Bbb{C})$ reproduced conjugation inner product for