spinor as being a quadruple of complex numbers at each point in spacetime does not generalise under the replacement $({\mathbb{R}}^4,\eta)\mapsto ({\mathcal{M}},g)$. A rigorous formulation of spinors on a manifold is given in terms of fibre bundles where the spin group is the structure group. (For a review of the necessary concepts related to fibre bundles, and in particular spinor bundles, the reader is directed to [@nakahara]). We shall review basic facts about the Dirac matrices and the Lorentz and spin groups in Minkowski spacetime, and use a local frame to generalise the results to a curved spacetime. What follows is based on [@dawson; @fewster] and benefits from the elaborations in [@dimock; @dirac; @fewster; @verch].

We begin by summarising several groups which appear in our discussion. The Lorentz group $O(1,3) = \{ \Lambda \in \mathrm{GL}_4({\mathbb{R}}) \mid \eta_{\alpha\beta}\Lambda^\alpha_{\,\,\,\gamma}\Lambda^\beta_{\,\,\,\delta}=\eta_{\gamma\delta} 
\}$ has the subgroup $\mathfrak{L}^\uparrow_+$, $$\mathfrak{L}^\uparrow_+ = \{ \Lambda \in O(1,3) \mid \det\Lambda=1 \,
\& \, \Lambda^0_{\,\,\,0}>0 \}$$ called the proper orthochronous Lorentz group. The Dirac gamma matrices $\gamma_\alpha$ satisfy the Clifford algebra relation $\{\gamma_\alpha , \gamma_\beta  \} =
2\eta_{\alpha\beta}\mathbbm{1}$ and are said to belong to a *standard representation* if $\gamma^\dagger_0=\gamma_0$ and $\gamma^\dagger_{\alpha}=-\gamma_{\alpha}$ for $\alpha=1,2,3$. From here on we shall assume that our Dirac 