I am wondering if there is any mathematical (or physical, besides the fact that classical quantum mechanics uses complex numbers) justification for why the complexified (1,3) Clifford algebra is used in Dirac's equation. A (the?) key point of special relativity is that spacetime is a real 4-d vector space with an inner product of signature (1,3). But by complexifying the signature becomes irrelevant-- all complex Clifford algebras in a given dimension are isomorphic where for real Clifford algebras, even signatures (p,q) and (q,p) are not isomorphic in general (as a side question: can there be a physical significance to this fact? or do only the spin group and the even subalgebra of the Clifford algebra, which are the same for (p,q) and (q,p), matter? I only hear about spinor bundles and spin structures, never pinor bundles or pin structures).


Thanks and I hope this isn't too physicsy of a question!



Clifford Algebra in Dirac Equation
This problem was investigated by Cecile DeWitt-Morette et al. This is a [review article](http://arxiv.org/PS_cache/math-ph/pdf/0012/0012006v1.pdf) describing the role of Pin groups in Physics. This article includes also a historical survey and a comprehensive list of references about Pin groups. 


Becides the fact that the Clifford algebras Cl(3,1) and Cl(1,3) are nonisomorphic, there is the question if the two types can be experimentally distinguished. 
The article gives a positive in the neutrinoless double beta decay experiment where it was observed that the neutrino has to be a Pin(1,3) particle. 


The article also describes solutions of the Dirac equation in topologically 
nontrivial spaces where the vacuum expectations of the Fermi currents are different in the two types of groups. (Thus one might conclude that this difference can be used to get information about the space-time topology).


The article also refers to Choquet-Bruhat, DeWitt-Morette and Dillard-Bleick's book about obstructions to the construction of Pin bundles.


