# On Dirac/ Clifford matrices

Let $$(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$$. The Dirac matrices $$\gamma^\mu$$, $$\mu=0,1,2,3$$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $$\{A,B\}=AB+BA$$ is the anti-commutator.

If we have another collection $$\tilde\gamma^\mu$$ of matrices of the same size satisfying the same commutation relations \eqref{1} then the Skolem–Noether theorem implies that there exists an invertible matrix $$C$$ such that $$\tilde\gamma^\mu=C\gamma^\mu C^{-1}\mbox{ for all } \mu.\tag{2}\label{2}$$

I am interested in the following situation. Let us require in addition to \eqref{1} that for Hermitian conjugates one has $$(\gamma^0)^*=\gamma^0, (\gamma^i)^*=-\gamma^i \text{ for }i=1,2,3.\tag{3}\label{3}$$ Is it possible to classify four tuples of matrices $$\gamma^\mu$$, $$\mu=0,\dotsc,3$$, satisfying \eqref{1} and \eqref{3}? Is there a generalization of such a classification to higher dimensions?

Remark 1. If one takes in \eqref{2} the matrix $$C$$ to be unitary then $$\tilde\gamma^\mu$$ also satisfy \eqref{3}.

Remark 2. My understanding is that the motivation to consider conditions \eqref{2} goes back to the original motivation of Dirac to construct an equation on the wave function $$\psi$$ of a free electron which would be not only relativistic and implied the Klein–Gordon equation, but also $$\lvert\psi\rvert^2$$ would be the time component of a conserved 4-current.

If you require that the matrix $$C$$ in \eqref{2} preserves the involutions \eqref{3}, then you get the consequence $$\tilde{\gamma}^\mu = C C^* \tilde{\gamma}^\mu (C C^*)^{-1}$$, which then implies that $$C C^* = u$$ is some unit in the center of the Clifford algebra such that $$u = u^*$$. Equivalently $$C^* = u C^{-1}$$. Rescaling $$C=v D$$, where $$v$$ is another scalar unit (which obviously cancels in the formula $$C \gamma C^{-1} = D \gamma D^{-1}$$), we get $$D^* = \frac{u}{v^* v} D^{-1}$$. So we can normalize the involution properties of $$C$$ by choosing $$u/(v^* v)$$ to be some canonical value.
The center of a complex Clifford algebra is either 1- or 2-dimensional, in a pattern that depends on the number of generators (which I'm always too lazy to look up). But supposing the center is 1-dimensional, that is consisting of scalar matrices $$\mathbb{C} I$$, for any $$u$$ we can choose $$v$$ such that $$u/(v^* v) = \pm 1$$. In that case, if you can find one tuple of $$\gamma$$-matrices satisfying \eqref{3}, the rest are classified by \eqref{2} where it is sufficient to take $$C^* = \pm C^{-1}$$. The case of the 2-dimensional center can be left as an exercise.
• Thank you. As far as I understand there are no matrices such that $C^*=-C^{-1}$. Indeed $0\leq tr(C^*C)=tr(-I)<0$.
• @asv Of course, you're right. I missed the positivity condition on $u$, in addition to its self-adjointness. Sep 23 at 15:31