I am currently delving into the study of Clifford algebras, particularly $ \text{Cl}(3,1) $, in the context of theoretical physics and am seeking clarity on a specific representation issue.
As I understand, the Clifford algebra $ \text{Cl}(1,3) $ is associated with 2x2 quaternion matrices, which is reminiscent of the matrices used in the Dirac equation. However, my focus is on $ \text{Cl}(3,1) $, which is said to correspond to the algebra of 4x4 real matrices.
I am aware that the Dirac matrices typically generate the Clifford algebra $ \text{Cl}(1,3) $ and involve complex numbers, which would not satisfy the requirement for $ \text{Cl}(3,1) $ to be represented by 4x4 real matrices. The challenge I am facing is in constructing a set of 4x4 real matrices that can act as generators for $ \text{Cl}(3,1) $, satisfying the necessary anticommutation relations with the Minkowski metric of signature (3,1).
Could someone provide a representation or guide me towards a set of generators for $ \text{Cl}(3,1) $ that are exclusively real 4x4 matrices? Any insights into the nuances of these representations, particularly in the context of their application in theoretical physics, would be greatly appreciated.
Thank you for your assistance.
$ \newcommand\Cl{\mathrm{Cl}} $The following was inspired by Proposition 15.20 of Clifford Algebras and the Classical Groups by Ian Porteous.
Recall that the gamma matrices are $$ \gamma_0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},\quad \gamma_1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},\quad \gamma_2 = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix},\quad \gamma_3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} $$ and they generate the real algebra $\Cl(1,3)$. We easily get generators with all real entries for $\Cl(2,2)$ by replacing $\gamma_2$ with $i\gamma_2$. Now we employ an isomorphism $\Cl(p+1,q) \cong \Cl(q+1,p)$; this stems from the isomorphism of even subalgebras $\Cl^+(p,q) \cong \Cl^+(q,p)$, but we can implement it directly by noting that $$ \gamma_0,\quad \gamma_1\gamma_0,\quad i\gamma_2\gamma_0,\quad \gamma_3\gamma_0 $$ generates the same algebra as $\gamma_0,\gamma_1,i\gamma_2,\gamma_3$ but with the commutation relations of $\Cl(3,1)$ generators. Explicitly these matrices are $$ \gamma_0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},\quad \gamma_1\gamma_0 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},\quad i\gamma_2\gamma_0 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix},\quad \gamma_3\gamma_0 = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. $$