Complexifying the spacetime algebra by means of an extra timelike dimension: Pin, Spin and algebraic spinors

Abstract

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