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

Abstract

Because of the isomorphism $C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})$, it is possible to complexify the spacetime Clifford algebra $C \ell _{1,3}({\mathbb {R}})$ 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 $C \ell _{2,3}({\mathbb {R}})$ paying special attention to the isomorphism $C \ell _{1,3}({\mathbb {C}})\cong C \ell _{2,3}({\mathbb {R}})$ and the embedding $C \ell _{1,3}({\mathbb {R}})\subseteq C \ell _{2,3}({\mathbb {R}})$. On the first half of this article we analyze the Pin and Spin groups and construct an injective mapping ${\text {Pin}}(1,3)\hookrightarrow {\text {Spin}}(2,3)$, obtaining in particular elements in ${\text {Spin}}(2,3)$ 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 $C \ell _{1,3}({\mathbb {C}})$ is reproduced by the Clifford conjugation inner product for real spinors in $C \ell _{2,3}({\mathbb {R}})$.

Appendix A. Code for computations on the \({C \ell }_{2,3}({\mathbb {R}})\) using the Clifford package for Maple

In this appendix we use the Clifford package for Maple [1], developed by Abłamowicz and Fauser to perform some calculations. We are going to use the following preamble, which contains information about the Clifford algebra and define the idempotent f, for all the calculations:

$$\begin{aligned}&{\texttt {with(Clifford):with(LinearAlgebra):}}\\&{\texttt {clibasis:=cbasis(5);}}\\&{\texttt {B:=linalg[diag](1,-1,-1,-1,1);}}\\&{\texttt {f:=cmul(1/2*(1+e2we3we4),1/2*(1-e1we4we5));}} \end{aligned}$$

1.1 A.1. Computation of the ${\mathbb {K}}$-basis for S

In order to compute the ${\mathbb {K}}$-basis for S we use the command spinorKbasis. This command takes four inputs:

• 1. a real basis for the space S,

• 2. the idempotent f,

• 3. a list of elements from the standard basis of ${C \ell }_{p,q}$ that generate ${\mathbb {K}}$ as a real vector space.

• 4. the string “left” or “right” indicating if S is the left minimal ideal, ${C \ell }_{p,q}{f}$, or the right one, $f{C \ell }_{p,q}$.

As a matter of fact spinorKbasis returns three elements, the ${\mathbb {K}}$-basis for S is the first one. Let’s find all this objects needed as inputs using the Clifford package. To generate a real basis of S we are going to use the command minimalideal in the following way:

$$\begin{aligned} {\texttt {realbasisS:=minimalideal(clibasis,f,'left');}} \end{aligned}$$

Lastly, we define the set of generators of ${\mathbb {K}}$, and compute the ${\mathbb {K}}$-basis for S:

$$\begin{aligned}&{\texttt {Kgenerators:=[Id,e1we2we3we4we5];}}\\&{\texttt {KbasisS:=spinorKbasis(realbasisS,f,Kgenerators,'left')[1];}} \end{aligned}$$

The basis obtained in this way, although rendering $\gamma _{0}=-ie_{4}e_{0}$ diagonal, it is not the matrix for the Dirac representation. However, by the following reordering:

$$\begin{aligned} {\texttt {DiracKbasisS:=[KbasisS[1],KbasisS[3],KbasisS[2],KbasisS[4]];}} \end{aligned}$$

we obtain the basis in Eq. (58).

1.2 A.2. Finding the element s for the reversion inner product

The following algorithm finds the element s in Eq. (51), as long as it belongs in the standard basis, which is the case for the idempotent f in Eq. (53).

$$\begin{aligned}&{\texttt {l:=nops(clibasis): s:=0:}} \\&{\texttt {for i from 1 to l do}} \\&\quad \quad \quad \,{\texttt {if cmul(clibasis[i],reversion(f))-cmul(f,clibasis[i])=0 and s=0}} \\&\quad \quad \,\,\,\,\,\qquad {\texttt {then s:=clibasis[i];}} \\&\quad \quad \quad \,{\texttt {end if;}} \\&{\texttt {end do;}} \\&{\texttt {print(s);}} \end{aligned}$$