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 2×2 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 4×4 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 4×4 real matrices. The challenge I am facing is in constructing a set of 4×4 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 4×4 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.
Best Answer
$ \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 $$ \gamma_0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},\quad \gamma_1\gamma_0 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},\quad i\gamma_2\gamma_0 = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix},\quad \gamma_3\gamma_0 = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. $$