Spinors, Spacetime and Clifford algebra I'm looking to understand the intrinsic connection that Clifford algebra allows one to make between spin space and spacetime. For a while now I've trying to wrap my head around how the Clifford algebra fits into this story, with the members of my department consistently telling me "not to worry about it". However, I think there's something deep to be uncovered.
The gamma matrices present in the Dirac equation generate a Clifford algebra:
$\{\gamma^{\nu}, \gamma^{\mu}\} = 2\eta^{\nu \mu} I$. It is argued in the gamma matrices wikipedia page that this algebra is the complexification of the spacetime algebra: $Cl_{1,3}(\mathcal{C})$ is the complexification of $Cl_{1,3}(\mathcal{R})$. The answer given here (What is the role of the spacetime algebra?) would seem to suggest that this complex structure falls out naturally from the decomposition into degrees of $Cl_{1,3}(\mathcal{R})$. Is this the case?
Furthermore, is it the case that one can then use the gamma matrices that generate $Cl_{1,3}(\mathcal{C})$ to form the Lie Algebra of the Lorentz group, which to me gives the picture that these constructions in spin space can form spacetime transformations (as outlined here: Relation between the Dirac Algebra and the Lorentz group)?
Essentially (I think) the question I'm asking is does the Clifford algebra encapsulate some global space of which spacetime and the space of spinors belong - if so how then do the gamma matrices present in the Dirac  equation respect and link these two spaces? Were dealing with algebras, but does the isomorphism $SO(1,3)$~$SU(2)$ x $SU(2)$ come into play here?
I'm not a mathematician by trade, but I think technical answers will naturally come into play here - if people could try and hold onto some physical intuition it would be much appreciated.
Best wishes to all.
 A: *

*The reason we usually complexify the Clifford algebra is mostly convenience: The representation theory of complex algebras is simpler in general, and if we want to restrict to real representations for some reason later on we can always do that. In particular, Dirac spinors at least exist in all dimensions, while the "real" Majorana spinors are dependent on the number of dimensions and even on the signature (depending on what, exactly, you mean by "Majorana"), see also this Q&A of mine.

*The second degree of the Clifford algebra (complex or real doesn't matter here) is isomorphic as a Lie algebra to the Lorentz algebra (or, in the generalized version, the generalized Clifford algebra for a metric $\eta$ has the isometry algebra for that metric as its second degree). It is not the "gamma matrices" (= generators of the Clifford algebra, hence in particular first degree elements of it) that generate the Lorentz algebra, but their commutators $\sigma^{ij} = [\gamma^i, \gamma^j]$. (It may be that you are already aware of this, but this is a common point of confusion) 

*I'm not quite sure what your "does the Clifford algebra encapsulate some global space of which spacetime and the space of spinors belong" question is trying to ask, but let me point out that four dimensions - where one could identify the first degree of the Clifford algebra with both spacetime and the four-dimensional Dirac spinors - are an "accident". The Dirac spinor representation in $d$ dimensions is $2^{\lfloor d/2 \rfloor}$-dimensional, which you cannot identify with the $d$-dimensional first degree of the algebra in most other dimensions. Therefore, the Clifford algebra does not, in a general sense, "contain" spinors. 

*Lastly and most tangentially, there is no isomorphism $\mathrm{SO}(1,3)\cong \mathrm{SU}(2)\times \mathrm{SU}(2)$, regardless of how often you will read this lie in physics-oriented texts. See e.g. this answer by Qmechanic and the linked questions for details on the relation between the two groups and their algebras. The nutshell is that $\mathrm{su}(2)\oplus\mathrm{su}(2)$ is the compact real form of the complexification of $\mathrm{so}(1,3)$, hence the complex finite-dimensional representations of these algebras are equivalent, hence the projective representations of the group $\mathrm{SO}(1,3)$ are given equivalently by $\mathrm{su}(2)\oplus\mathrm{su}(2)$ representations. (For why projective representations matter, see this Q&A of mine)
A: Here is a simple path of research that I myself have been using to somewhat answer your question...
You can easily extend a Clifford algebra to a space with a non-flat metric tensor.  IF you assume all the elements of the Clifford objects are tensors, and thus the Clifford object itself is a scalar (from a tensor point of view), then all your equations will be generally covariant.
For example, you can do this using $Cl_{1,3}$ and using the electromagnetic field vector.  Maxwell's equations in curved space-time then reduce to:
$\partial F = \mu_0 J$
where
$F \equiv \partial A$
$\partial \equiv \gamma^{\mu} \frac{\partial}{\partial x^{\mu}}, A \equiv \gamma^{\mu} A_{\mu}$
Now to get to spinors, you use tetrads to rewrite your generators as linear combinations of the original flat generators (Dirac matrices for example), and you again treat every element of some object as tensors.  This then allows you to write a generally covariant version of Dirac's equation:
$\partial \Psi = \frac{E_0}{\hbar c} \Psi \gamma^{012}$
The trick in expanding this, is remembering to replace the derivatives of the "flat" Dirac matrices with covariant ones that use the spin-connection.  This is just a trick that compensates for your use of the tetrads and changing "frame".
I'm skipping a lot of detailed steps but this DOES give you a generally covariant equation for spinors.
