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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.

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It looks like there is a whole wiki page dedicated to this, unless this isn't what you are asking: Higher dimensional gamma matrices.

I believe the study of anti-commuting algebras is called Clifford Algebras. Also I think chirality is special to particular dimensions. I don't think you can find, for instance, a $\gamma^5$ matrix which anti-commutes with the pauli matrices in three dimensions... I think. I hope this helps.

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Thanks, this does partially answer the question. After a quick look at the article on gamma matrices in higher dimensions, it looks like they construct matrices for the Lorentzian case (-+...+). In that case, I suppose you can get a form of $\gamma$ such that $\bar{\psi}=\psi^{\dagger} \gamma^0$ still works... but if there are multiple timelike dimensions this approach breaks down. – sjasonw Dec 21 '12 at 6:10
Actually, the $\gamma$ matrices stem from irreducible representations of the Clifford algebra in the spinor space. $\gamma_0$ is a special case of an operator which maps the spinor space into its complex conjugate. In general, you can of course find irreducible representation of the Clifford algebra in higher dimensional spinor spaces, see for example P. Lounesto, "Clifford algebras and spinors". – AGP Apr 12 '13 at 18:08

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