On Dirac/ Clifford matrices

Question

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.

Answer 1

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.