I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional Minkowski space (signature $-+++$) and the 4 (complex) dimensional Dirac representation of $\text{spin}(3,1)$ via $\gamma^a= \left( \begin{array}{ccc} 0 & \sigma^a \\ \bar{\sigma}^a & 0 \\ \end{array} \right)$. Then we would define the Dirac adjoint as $\bar{\psi}=\psi^\dagger \gamma^0$. Let us denote the representation by $D:\text{spin}(3,1) \rightarrow \text{Lin}(\bf{C^4})$.
The fundamental property of the Dirac adjoint of $\psi$ is that under a Lorentz transformation, $\bar{\psi}$ transforms to $\bar{\psi} D^{-1}$ which ensures, for instance, that $\bar{\psi} \psi$ is a Lorentz scalar.
So now to my concern. It seems to me that this construction of $\bar{\psi}$ is super specific to 1+3 dimensions and maybe also to the chiral form of $\gamma$. What if we consider a Diracy representation of $\text{spin}(n,m)$? That is, we take a collection of $d$ dimensional matrices $\left\{ \gamma^a \mid 1\leq a \leq n+m \right\}$ such that $\left\{ \gamma^a,\gamma^b \right\} = 2 \eta^{ab}$ ($\eta$ is the diagonal matrix with $n$ 1's and $m$ $-1$'s) and use these to construct a $D$ dimensional representation. Then, if $\psi \in \bf{C}^D$, I'd like to find a (hopefully real) function $\psi \mapsto \bar{\psi} \psi$ that is invariant under the action we just (kind of) constructed.
