matrices belong to a standard representation. The spin group, ${\mathrm{Spin }}(1,3)$, is defined by $${\mathrm{Spin }}(1,3) = \{ S \in \mathrm{SL}_4({\mathbb{C}}) \mid S\gamma_\alpha S^{-1} =
\gamma_\beta \Lambda^\beta_{\,\,\,\alpha} \textrm{ for some }\Lambda\in\mathcal{L} 
\} \ ,$$ and is known to be a two-to-one cover of $\mathfrak{L}^\uparrow_+$, i.e. the mapping $S\mapsto \Lambda(S)$ is a two-to-one covering homomorphism from the identity connected component ${\mathrm{Spin }}_0(1,3)$, of ${\mathrm{Spin }}(1,3)$, to $\mathfrak{L}^\uparrow_+$ with kernel $\{\mathbbm{1},-\mathbbm{1} \}$.

We now direct our attention to a curved spacetime setting. The frame bundle ${F{\mathcal{M}}}$ is the bundle of oriented and time-oriented tetrads $\{ e_\alpha^a
\}_{\alpha=0,1,2,3}$ over spacetime $({\mathcal{M}},g)$ with the convention that $e_0^a$ is a future pointing timelike vector; moreover, ${F{\mathcal{M}}}$ is a principal $\mathcal{L}^\uparrow_+$ bundle whose right action is given by $(R_\Lambda e)_\alpha = e_\beta \Lambda^\beta_{\,\,\,\alpha}$. A *spin structure* on $({\mathcal{M}},g)$ is a principal ${\mathrm{Spin }}_0(1,3)$ bundle, ${S{\mathcal{M}}}$, over $({\mathcal{M}},g)$ equipped with a fibre homomorphism $\varphi:{S{\mathcal{M}}}\mapsto {F{\mathcal{M}}}$ such that $\varphi\circ
R_S=R_{\Lambda(S)}\circ\varphi$, i.e. $\varphi$ intertwines the right action of the structure group on these bundles. Spin structures are not unique, however two such structures, ${S{\mathcal{M}}}$ and $\widetilde{{S{\mathcal{M}}}}$ equipped with $\varphi$ and $\widetilde\varphi$ respectively can be said to be equivalent if there is an isomorphism $\iota : {S{\mathcal{M}}}\mapsto \widetilde{{S{\mathcal{M}}}}$ such that 