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

TL;DR

This paper explores how adding an extra timelike dimension to spacetime algebra enriches the mathematical structure, providing new insights into Dirac particles, spin groups, and spinor spaces within an extended Clifford algebra framework.

Contribution

It introduces a detailed study of the real Clifford algebra with an extra timelike dimension, establishing isomorphisms and embeddings that relate to Dirac particles and spinor structures.

Findings

Constructed an injective map from Pin(1,3) to Spin(2,3)

Represented parity and time reversal within Spin(2,3)

Reproduced complex spinor structures using real Clifford algebra

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 ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$.