For the moment, what I have is this:
Given the smooth oriented Lorentzian manifold $(M,g)$, we have the tangent bundle $TM\to M$.
We also have the $\mathrm{SO}(n,1)$-orthonormal frame bundle, $P\to M$. Now, $TM$ is the bundle associated to $FM$ via
$$ TM=P\times_{SO(n,1)}\mathbb R^{n+1} $$(see associated bundle).
Since $\mathrm{SO}(n,1)$ has the double cover $\mathrm{Spin}(n,1)$, we suppose the existence of a double cover bundle of $P$, $\mathrm{Spin}(n,1)\to Q\to M$. We then take a vector space $\Delta$ on which there is a representation of $\mathrm{Spin}(n,1)$, this is where the Clifford algebra comes in. The spinor bundle is finally $S=Q\times_{\mathrm{Spin}(n,1)}\Delta$. So the spinor bundle is a vector bundle associated to the double cover of the frame bundle.
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Author of the notes: Antonio J. Pan-Collantes
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