TITLE: Clifford Algebra in Dirac Equation
QUESTION [9 upvotes]: 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!

REPLY [4 votes]: I am slightly confused by this question.  The fact that one can formulate the Dirac equation in either (3,1) or (1,3) signature, which have non-isomorphic Clifford algebras and hence Clifford modules of different type (real for (1,3) and quaternionic for (3,1) in my naming conventions), does not mean that one is complexifying the Clifford algebra in formulating the Dirac equation.
Unitarity of the time evolution -- a physical requirement independent of choices -- requires that $i D$, where $D$ is the Dirac operator, be hermitian, and in turn this forces a certain hermiticity condition on the "gamma" matrices, in essence choosing a real form of the complex Clifford algebra.  It is the spinor representation which can be taken to be complex since after all wave functions live in the tensor product of the Clifford module with a complex Hilbert space.  This is not the same thing as complexifying the Clifford algebra, though.
So to summarise, in the Dirac equation the Clifford algebra is real, but the pinor representation can be taken to be complex.